An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. In recent work Bubenik introduced the persistence landscape, a stable representation of persistence diagrams amenable to statistical analysis and machine learning tools. In this paper we generalise the persistence landscape to multiparameter persistence modules providing a stable representation of the rank invariant. We show that multiparameter landscapes are stable with respect to the interleaving distance and persistence weighted Wasserstein distance, and that the collection of multiparameter landscapes faithfully represents the rank invariant. Finally we provide example calculations and statistical tests to demonstrate a range of potential applications and how one can interpret the landscapes associated to a multiparameter module.

We develop an encompassing framework for matching, covariate balancing, and doubly-robust methods for causal inference from observational data called generalized optimal matching (GOM). The framework is given by generalizing a new functional-analytical formulation of optimal matching, giving rise to the class of GOM methods, for which we provide a single unified theory to analyze tractability and consistency. Many commonly used existing methods are included in GOM and, using their GOM interpretation, can be extended to optimally and automatically trade off balance for variance and outperform their standard counterparts. As a subclass, GOM gives rise to kernel optimal matching (KOM), which, as supported by new theoretical and empirical results, is notable for combining many of the positive properties of other methods in one. KOM, which is solved as a linearly-constrained convex-quadratic optimization problem, inherits both the interpretability and model-free consistency of matching but can also achieve the $sqrt{n}$-consistency of well-specified regression and the bias reduction and robustness of doubly robust methods. In settings of limited overlap, KOM enables a very transparent method for interval estimation for partial identification and robust coverage. We demonstrate this in examples with both synthetic and real data.

Derivatives play an important role in bandwidth selection methods (e.g., plug-ins), data analysis and bias-corrected confidence intervals. Therefore, obtaining accurate derivative information is crucial. Although many derivative estimation methods exist, the majority require a fixed design assumption. In this paper, we propose an effective and fully data-driven framework to estimate the first and second order derivative in random design. We establish the asymptotic properties of the proposed derivative estimator, and also propose a fast selection method for the tuning parameters. The performance and flexibility of the method is illustrated via an extensive simulation study.

Anomaly detection is not an easy problem since distribution of anomalous samples is unknown a priori. We explore a novel method that gives a trade-off possibility between one-class and two-class approaches, and leads to a better performance on anomaly detection problems with small or non-representative anomalous samples. The method is evaluated using several data sets and compared to a set of conventional one-class and two-class approaches.

Jun Zhang, Yujie Gai, Xia Cui, Gaorong Li.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 370--393.

Abstract:
This paper studies the measure of symmetry or asymmetry of a continuous variable under the multiplicative distortion measurement errors setting. The unobservable variable is distorted in a multiplicative fashion by an observed confounding variable. First, two direct plug-in estimation procedures are proposed, and the empirical likelihood based confidence intervals are constructed to measure the symmetry or asymmetry of the unobserved variable. Next, we propose four test statistics for testing whether the unobserved variable is symmetric or not. The asymptotic properties of the proposed estimators and test statistics are examined. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test statistics. These methods are applied to analyze a real dataset for an illustration.

Danilo Covaes Nogarotto, Caio Lucidius Naberezny Azevedo, Jorge Luis Bazán.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 304--322.

Abstract:
The interest on the analysis of the zero–one augmented beta regression (ZOABR) model has been increasing over the last few years. In this work, we developed a Bayesian inference for the ZOABR model, providing some contributions, namely: we explored the use of Jeffreys-rule and independence Jeffreys prior for some of the parameters, performing a sensitivity study of prior choice, comparing the Bayesian estimates with the maximum likelihood ones and measuring the accuracy of the estimates under several scenarios of interest. The results indicate, in a general way, that: the Bayesian approach, under the Jeffreys-rule prior, was as accurate as the ML one. Also, different from other approaches, we use the predictive distribution of the response to implement Bayesian residuals. To further illustrate the advantages of our approach, we conduct an analysis of a real psychometric data set including a Bayesian residual analysis, where it is shown that misleading inference can be obtained when the data is transformed. That is, when the zeros and ones are transformed to suitable values and the usual beta regression model is considered, instead of the ZOABR model. Finally, future developments are discussed.

Mohsen Maleki, Arezo Hajrajabi, Reinaldo B. Arellano-Valle.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 273--290.

Abstract:
In this paper, we study the finite mixtures of autoregressive processes assuming that the distribution of innovations (errors) belongs to the class of scale mixture of skew-normal (SMSN) distributions. The SMSN distributions allow a simultaneous modeling of the existence of outliers, heavy tails and asymmetries in the distribution of innovations. Therefore, a statistical methodology based on the SMSN family allows us to use a robust modeling on some non-linear time series with great flexibility, to accommodate skewness, heavy tails and heterogeneity simultaneously. The existence of convenient hierarchical representations of the SMSN distributions facilitates also the implementation of an ECME-type of algorithm to perform the likelihood inference in the considered model. Simulation studies and the application to a real data set are finally presented to illustrate the usefulness of the proposed model.

Zhengwei Liu, Qi Li, Fukang Zhu.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 251--272.

Abstract:
To predict time series of counts with small values and remarkable fluctuations, an available model is the $r$ states random environment process based on the negative binomial thinning operator and the geometric marginal. However, we argue that the aforementioned model may suffer from the following two drawbacks. First, under the condition of no prior information, the overdispersed property of the geometric distribution may cause the predictions fluctuate greatly. Second, because of the constraints on the model parameters, some estimated parameters are close to zero in real-data examples, which may not objectively reveal the correlation relationship. For the first drawback, an $r$ states random environment process based on the binomial thinning operator and the Poisson marginal is introduced. For the second drawback, we propose a generalized $r$ states random environment integer-valued autoregressive model based on the binomial thinning operator to model fluctuations of data. Yule–Walker and conditional maximum likelihood estimates are considered and their performances are assessed via simulation studies. Two real-data sets are conducted to illustrate the better performances of the proposed models compared with some existing models.

Mohd Arshad, Omer Abdalghani.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 167--182.

Abstract:
Suppose that $pi_{1},pi_{2},ldots ,pi_{k}$ be $k(geq2)$ independent exponential populations having unknown location parameters $mu_{1},mu_{2},ldots,mu_{k}$ and known scale parameters $sigma_{1},ldots,sigma_{k}$. Let $mu_{[k]}=max {mu_{1},ldots,mu_{k}}$. For selecting the population associated with $mu_{[k]}$, a class of selection rules (proposed by Arshad and Misra [ Statistical Papers 57 (2016) 605–621]) is considered. We consider the problem of estimating the location parameter $mu_{S}$ of the selected population under the criterion of the LINEX loss function. We consider three natural estimators $delta_{N,1},delta_{N,2}$ and $delta_{N,3}$ of $mu_{S}$, based on the maximum likelihood estimators, uniformly minimum variance unbiased estimator (UMVUE) and minimum risk equivariant estimator (MREE) of $mu_{i}$’s, respectively. The uniformly minimum risk unbiased estimator (UMRUE) and the generalized Bayes estimator of $mu_{S}$ are derived. Under the LINEX loss function, a general result for improving a location-equivariant estimator of $mu_{S}$ is derived. Using this result, estimator better than the natural estimator $delta_{N,1}$ is obtained. We also shown that the estimator $delta_{N,1}$ is dominated by the natural estimator $delta_{N,3}$. Finally, we perform a simulation study to evaluate and compare risk functions among various competing estimators of $mu_{S}$.

Andreas Anastasiou, Robert E. Gaunt.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 136--149.

Abstract:
We use the delta method and Stein’s method to derive, under regularity conditions, explicit upper bounds for the distributional distance between the distribution of the maximum likelihood estimator (MLE) of a $d$-dimensional parameter and its asymptotic multivariate normal distribution. Our bounds apply in situations in which the MLE can be written as a function of a sum of i.i.d. $t$-dimensional random vectors. We apply our general bound to establish a bound for the multivariate normal approximation of the MLE of the normal distribution with unknown mean and variance.

Ahmad Younso.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 112--126.

Abstract:
We consider a new nonparametric rule of classification, inspired from the classical moving window rule, that allows for the classification of spatially dependent functional data containing some completely missing curves. We investigate the consistency of this classifier under mild conditions. The practical use of the classifier will be illustrated through simulation studies.

Durba Bhattacharya, Sourabh Bhattacharya.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 71--89.

Abstract:
Present day bio-medical research is pointing towards the fact that cognizance of gene–environment interactions along with genetic interactions may help prevent or detain the onset of many complex diseases like cardiovascular disease, cancer, type2 diabetes, autism or asthma by adjustments to lifestyle. In this regard, we propose a Bayesian semiparametric model to detect not only the roles of genes and their interactions, but also the possible influence of environmental variables on the genes in case-control studies. Our model also accounts for the unknown number of genetic sub-populations via finite mixtures composed of Dirichlet processes. An effective parallel computing methodology, developed by us harnesses the power of parallel processing technology to increase the efficiencies of our conditionally independent Gibbs sampling and Transformation based MCMC (TMCMC) methods. Applications of our model and methods to simulation studies with biologically realistic genotype datasets and a real, case-control based genotype dataset on early onset of myocardial infarction (MI) have yielded quite interesting results beside providing some insights into the differential effect of gender on MI.

Xiufang Liu, Dehui Wang.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 638--673.

Abstract:
This paper discusses a $p$th-order dependence-driven random coefficient integer-valued autoregressive time series model ($operatorname{DDRCINAR}(p)$). Stationarity and ergodicity properties are proved. Conditional least squares, weighted least squares and maximum quasi-likelihood are used to estimate the model parameters. Asymptotic properties of the estimators are presented. The performances of these estimators are investigated and compared via simulations. In certain regions of the parameter space, simulative analysis shows that maximum quasi-likelihood estimators perform better than the estimators of conditional least squares and weighted least squares in terms of the proportion of within-$Omega$ estimates. At last, the model is applied to two real data sets.

Caio L. N. Azevedo, Dalton F. Andrade

Source: Braz. J. Probab. Stat., Volume 24, Number 3, 415--433.

Abstract:
The nominal response model (NRM) was proposed by Bock [ Psychometrika 37 (1972) 29–51] in order to improve the latent trait (ability) estimation in multiple choice tests with nominal items. When the item parameters are known, expectation a posteriori or maximum a posteriori methods are commonly employed to estimate the latent traits, considering a standard symmetric normal distribution as the latent traits prior density. However, when this item set is presented to a new group of examinees, it is not only necessary to estimate their latent traits but also the population parameters of this group. This article has two main purposes: first, to develop a Monte Carlo Markov Chain algorithm to estimate both latent traits and population parameters concurrently. This algorithm comprises the Metropolis–Hastings within Gibbs sampling algorithm (MHWGS) proposed by Patz and Junker [ Journal of Educational and Behavioral Statistics 24 (1999b) 346–366]. Second, to compare, in the latent trait recovering, the performance of this method with three other methods: maximum likelihood, expectation a posteriori and maximum a posteriori. The comparisons were performed by varying the total number of items (NI), the number of categories and the values of the mean and the variance of the latent trait distribution. The results showed that MHWGS outperforms the other methods concerning the latent traits estimation as well as it recoveries properly the population parameters. Furthermore, we found that NI accounts for the highest percentage of the variability in the accuracy of latent trait estimation.

Umberto Amato, Anestis Antoniadis, Italia De Feis.

Source: Statistics Surveys, Volume 14, 32--70.

Abstract:
We present and compare some nonparametric estimation methods (wavelet and/or spline-based) designed to recover a one-dimensional piecewise-smooth regression function in both a fixed equidistant or not equidistant design regression model and a random design model. Wavelet methods are known to be very competitive in terms of denoising and compression, due to the simultaneous localization property of a function in time and frequency. However, boundary assumptions, such as periodicity or symmetry, generate bias and artificial wiggles which degrade overall accuracy. Simple methods have been proposed in the literature for reducing the bias at the boundaries. We introduce new ones based on adaptive combinations of two estimators. The underlying idea is to combine a highly accurate method for non-regular functions, e.g., wavelets, with one well behaved at boundaries, e.g., Splines or Local Polynomial. We provide some asymptotic optimal results supporting our approach. All the methods can handle data with a random design. We also sketch some generalization to the multidimensional setting. To study the performance of the proposed approaches we have conducted an extensive set of simulations on synthetic data. An interesting regression analysis of two real data applications using these procedures unambiguously demonstrates their effectiveness.

Norbert Hirschauer, Sven Grüner, Oliver Mußhoff, Claudia Becker.

Source: Statistics Surveys, Volume 12, 136--172.

Abstract:
Data on how many scientific findings are reproducible are generally bleak and a wealth of papers have warned against misuses of the $p$-value and resulting false findings in recent years. This paper discusses the question of what we can(not) learn from the $p$-value, which is still widely considered as the gold standard of statistical validity. We aim to provide a non-technical and easily accessible resource for statistical practitioners who wish to spot and avoid misinterpretations and misuses of statistical significance tests. For this purpose, we first classify and describe the most widely discussed (“classical”) pitfalls of significance testing, and review published work on these misuses with a focus on regression-based “confirmatory” study. This includes a description of the single-study bias and a simulation-based illustration of how proper meta-analysis compares to misleading significance counts (“vote counting”). Going beyond the classical pitfalls, we also use simulation to provide intuition that relying on the statistical estimate “$p$-value” as a measure of evidence without considering its sample-to-sample variability falls short of the mark even within an otherwise appropriate interpretation. We conclude with a discussion of the exigencies of informed approaches to statistical inference and corresponding institutional reforms.

George Karabatsos, Fabrizio Leisen.

Source: Statistics Surveys, Volume 12, 66--104.

Abstract:
We are living in the big data era, as current technologies and networks allow for the easy and routine collection of data sets in different disciplines. Bayesian Statistics offers a flexible modeling approach which is attractive for describing the complexity of these datasets. These models often exhibit a likelihood function which is intractable due to the large sample size, high number of parameters, or functional complexity. Approximate Bayesian Computational (ABC) methods provides likelihood-free methods for performing statistical inferences with Bayesian models defined by intractable likelihood functions. The vastity of the literature on ABC methods created a need to review and relate all ABC approaches so that scientists can more readily understand and apply them for their own work. This article provides a unifying review, general representation, and classification of all ABC methods from the view of approximate likelihood theory. This clarifies how ABC methods can be characterized, related, combined, improved, and applied for future research. Possible future research in ABC is then outlined.

Michael Fop, Thomas Brendan Murphy.

Source: Statistics Surveys, Volume 12, 18--65.

Abstract:
Model-based clustering is a popular approach for clustering multivariate data which has seen applications in numerous fields. Nowadays, high-dimensional data are more and more common and the model-based clustering approach has adapted to deal with the increasing dimensionality. In particular, the development of variable selection techniques has received a lot of attention and research effort in recent years. Even for small size problems, variable selection has been advocated to facilitate the interpretation of the clustering results. This review provides a summary of the methods developed for variable selection in model-based clustering. Existing R packages implementing the different methods are indicated and illustrated in application to two data analysis examples.

Zeinab Mashreghi, David Haziza, Christian Léger.

Source: Statistics Surveys, Volume 10, 1--52.

Abstract:
We review bootstrap methods in the context of survey data where the effect of the sampling design on the variability of estimators has to be taken into account. We present the methods in a unified way by classifying them in three classes: pseudo-population, direct, and survey weights methods. We cover variance estimation and the construction of confidence intervals for stratified simple random sampling as well as some unequal probability sampling designs. We also address the problem of variance estimation in presence of imputation to compensate for item non-response.

José A. Ferreira.

Source: Statistics Surveys, Volume 9, 106--208.

Abstract:
This article provides a concise and essentially self-contained exposition of some of the most important models and non-parametric methods for the analysis of observational data, and a substantial number of illustrations of their application. Although for the most part our presentation follows P. Rosenbaum’s book, “Observational Studies”, and naturally draws on related literature, it contains original elements and simplifies and generalizes some basic results. The illustrations, based on simulated data, show the methods at work in some detail, highlighting pitfalls and emphasizing certain subjective aspects of the statistical analyses.

Didier Chauveau, David R. Hunter, Michael Levine.

Source: Statistics Surveys, Volume 9, 1--31.

Abstract:
The conditional independence assumption for nonparametric multivariate finite mixture models, a weaker form of the well-known conditional independence assumption for random effects models for longitudinal data, is the subject of an increasing number of theoretical and algorithmic developments in the statistical literature. After presenting a survey of this literature, including an in-depth discussion of the all-important identifiability results, this article describes and extends an algorithm for estimation of the parameters in these models. The algorithm works for any number of components in three or more dimensions. It possesses a descent property and can be easily adapted to situations where the data are grouped in blocks of conditionally independent variables. We discuss how to adapt this algorithm to various location-scale models that link component densities, and we even adapt it to a particular class of univariate mixture problems in which the components are assumed symmetric. We give a bandwidth selection procedure for our algorithm. Finally, we demonstrate the effectiveness of our algorithm using a simulation study and two psychometric datasets.

Aki Vehtari, Janne Ojanen.

Source: Statistics Surveys, Volume 8, , 1--1.

Abstract:
Errata for “A survey of Bayesian predictive methods for model assessment, selection and comparison” by A. Vehtari and J. Ojanen, Statistics Surveys , 6 (2012), 142–228. doi:10.1214/12-SS102.

Aki Vehtari, Janne Ojanen

Source: Statist. Surv., Volume 6, 142--228.

Abstract:
To date, several methods exist in the statistical literature for model assessment, which purport themselves specifically as Bayesian predictive methods. The decision theoretic assumptions on which these methods are based are not always clearly stated in the original articles, however. The aim of this survey is to provide a unified review of Bayesian predictive model assessment and selection methods, and of methods closely related to them. We review the various assumptions that are made in this context and discuss the connections between different approaches, with an emphasis on how each method approximates the expected utility of using a Bayesian model for the purpose of predicting future data.

Nancy Heckman

Source: Statist. Surv., Volume 6, 113--141.

Abstract:
The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $sum_{j}(Y_{j}-mu(t_{j}))^{2}+lambda int_{a}^{b}[mu''(t)]^{2},dt$, where the data are $t_{j},Y_{j}$, $j=1,ldots,n$. The minimization is taken over an infinite-dimensional function space, the space of all functions with square integrable second derivatives. But the calculations can be carried out in a finite-dimensional space. The reduction from minimizing over an infinite dimensional space to minimizing over a finite dimensional space occurs for more general objective functions: the data may be related to the function $mu$ in another way, the sum of squares may be replaced by a more suitable expression, or the penalty, $int_{a}^{b}[mu''(t)]^{2},dt$, might take a different form. This paper reviews the Reproducing Kernel Hilbert Space structure that provides a finite-dimensional solution for a general minimization problem. Particular attention is paid to the construction and study of the Reproducing Kernel Hilbert Space corresponding to a penalty based on a linear differential operator. In this case, one can often calculate the minimizer explicitly, using Green’s functions.

Gery Geenens

Source: Statist. Surv., Volume 5, 30--43.

Abstract:
Recently, some nonparametric regression ideas have been extended to the case of functional regression. Within that framework, the main concern arises from the infinite dimensional nature of the explanatory objects. Specifically, in the classical multivariate regression context, it is well-known that any nonparametric method is affected by the so-called “curse of dimensionality”, caused by the sparsity of data in high-dimensional spaces, resulting in a decrease in fastest achievable rates of convergence of regression function estimators toward their target curve as the dimension of the regressor vector increases. Therefore, it is not surprising to find dramatically bad theoretical properties for the nonparametric functional regression estimators, leading many authors to condemn the methodology. Nevertheless, a closer look at the meaning of the functional data under study and on the conclusions that the statistician would like to draw from it allows to consider the problem from another point-of-view, and to justify the use of slightly modified estimators. In most cases, it can be entirely legitimate to measure the proximity between two elements of the infinite dimensional functional space via a semi-metric, which could prevent those estimators suffering from what we will call the “curse of infinite dimensionality”.

References:
[1] Ait-Saïdi, A., Ferraty, F., Kassa, K. and Vieu, P. (2008). Cross-validated estimations in the single-functional index model, Statistics, 42, 475–494.

[2] Aneiros-Perez, G. and Vieu, P. (2008). Nonparametric time series prediction: A semi-functional partial linear modeling, J. Multivariate Anal., 99, 834–857.

[3] Baillo, A. and Grané, A. (2009). Local linear regression for functional predictor and scalar response, J. Multivariate Anal., 100, 102–111.

[4] Burba, F., Ferraty, F. and Vieu, P. (2009). k-Nearest Neighbour method in functional nonparametric regression, J. Nonparam. Stat., 21, 453–469.

[5] Cardot, H., Ferraty, F. and Sarda, P. (1999). Functional linear model, Stat. Probabil. Lett., 45, 11–22.

[6] Crambes, C., Kneip, A. and Sarda, P. (2009). Smoothing splines estimators for functional linear regression, Ann. Statist., 37, 35–72.

[7] Delsol, L. (2009). Advances on asymptotic normality in nonparametric functional time series analysis, Statistics, 43, 13–33.

[8] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications, Chapman and Hall, London.

[9] Fan, J. and Zhang, J.-T. (2000). Two-step estimation of functional linear models with application to longitudinal data, J. Roy. Stat. Soc. B, 62, 303–322.

[10] Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis, Springer-Verlag, New York.

[11] Ferraty, F., Laksaci, A. and Vieu, P. (2006). Estimating Some Characteristics of the Conditional Distribution in Nonparametric Functional Models, Statist. Inf. Stoch. Proc., 9, 47–76.

[12] Ferraty, F., Mas, A. and Vieu, P. (2007). Nonparametric regression on functional data: inference and practical aspects, Aust. NZ. J. Stat., 49, 267–286.

[13] Ferraty, F., Van Keilegom, I. and Vieu, P. (2010). On the validity of the bootstrap in nonparametric functional regression, Scand. J. Stat., 37, 286–306.

[14] Ferraty, F., Laksaci, A., Tadj, A. and Vieu, P. (2010). Rate of uniform consistency for nonparametric estimates with functional variables, J. Stat. Plan. Inf., 140, 335–352.

[15] Ferraty, F. and Romain, Y. (2011). Oxford handbook on functional data analysis (Eds), Oxford University Press.

[16] Gasser, T., Hall, P. and Presnell, B. (1998). Nonparametric estimation of the mode of a distribution of random curves, J. Roy. Stat. Soc. B, 60, 681–691.

[17] Geenens, G. (2011). A nonparametric functional method for signature recognition, Manuscript.

[18] Härdle, W., Müller, M., Sperlich, S. and Werwatz, A. (2004). Nonparametric and semiparametric models, Springer-Verlag, Berlin.

[19] James, G.M. (2002). Generalized linear models with functional predictors, J. Roy. Stat. Soc. B, 64, 411–432.

[20] Masry, E. (2005). Nonparametric regression estimation for dependent functional data: asymptotic normality, Stochastic Process. Appl., 115, 155–177.

[21] Nadaraya, E.A. (1964). On estimating regression, Theory Probab. Applic., 9, 141–142.

[22] Quintela-Del-Rio, A. (2008). Hazard function given a functional variable: nonparametric estimation under strong mixing conditions, J. Nonparam. Stat., 20, 413–430.

[23] Rachdi, M. and Vieu, P. (2007). Nonparametric regression for functional data: automatic smoothing parameter selection, J. Stat. Plan. Inf., 137, 2784–2801.

[24] Ramsay, J. and Silverman, B.W. (1997). Functional Data Analysis, Springer-Verlag, New York.

[25] Ramsay, J. and Silverman, B.W. (2002). Applied functional data analysis; methods and case study, Springer-Verlag, New York.

[26] Ramsay, J. and Silverman, B.W. (2005). Functional Data Analysis, 2nd Edition, Springer-Verlag, New York.

[27] Stone, C.J. (1982). Optimal global rates of convergence for nonparametric regression, Ann. Stat., 10, 1040–1053.

[28] Watson, G.S. (1964). Smooth regression analysis, Sankhya A, 26, 359–372.

[29] Yeung, D.T., Chang, H., Xiong, Y., George, S., Kashi, R., Matsumoto, T. and Rigoll, G. (2004). SVC2004: First International Signature Verification Competition, Proceedings of the International Conference on Biometric Authentication (ICBA), Hong Kong, July 2004.

Gregory J. Matthews, Ofer Harel

Source: Statist. Surv., Volume 5, 1--29.

Abstract:
There is an ever increasing demand from researchers for access to useful microdata files. However, there are also growing concerns regarding the privacy of the individuals contained in the microdata. Ideally, microdata could be released in such a way that a balance between usefulness of the data and privacy is struck. This paper presents a review of proposed methods of statistical disclosure control and techniques for assessing the privacy of such methods under different definitions of disclosure.

References:
Abowd, J., Woodcock, S., 2001. Disclosure limitation in longitudinal linked data. Confidentiality, Disclosure, and Data Access: Theory and Practical Applications for Statistical Agencies, 215–277.

Adam, N.R., Worthmann, J.C., 1989. Security-control methods for statistical databases: a comparative study. ACM Comput. Surv. 21 (4), 515–556.

Armstrong, M., Rushton, G., Zimmerman, D.L., 1999. Geographically masking health data to preserve confidentiality. Statistics in Medicine 18 (5), 497–525.

Bethlehem, J.G., Keller, W., Pannekoek, J., 1990. Disclosure control of microdata. Jorunal of the American Statistical Association 85, 38–45.

Blum, A., Dwork, C., McSherry, F., Nissam, K., 2005. Practical privacy: The sulq framework. In: Proceedings of the 24th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems. pp. 128–138.

Bowden, R.J., Sim, A.B., 1992. The privacy bootstrap. Journal of Business and Economic Statistics 10 (3), 337–345.

Carlson, M., Salabasis, M., 2002. A data-swapping technique for generating synthetic samples; a method for disclosure control. Res. Official Statist. (5), 35–64.

Cox, L.H., 1980. Suppression methodology and statistical disclosure control. Journal of the American Statistical Association 75, 377–385.

Cox, L.H., 1984. Disclosure control methods for frequency count data. Tech. rep., U.S. Bureau of the Census.

Cox, L.H., 1987. A constructive procedure for unbiased controlled rounding. Journal of the American Statistical Association 82, 520–524.

Cox, L.H., 1994. Matrix masking methods for disclosure limitation in microdata. Survey Methodology 6, 165–169.

Cox, L.H., Fagan, J.T., Greenberg, B., Hemmig, R., 1987. Disclosure avoidance techniques for tabular data. Tech. rep., U.S. Bureau of the Census.

Dalenius, T., 1977. Towards a methodology for statistical disclosure control. Statistik Tidskrift 15, 429–444.

Dalenius, T., 1986. Finding a needle in a haystack - or identifying anonymous census record. Journal of Official Statistics 2 (3), 329–336.

Dalenius, T., Denning, D., 1982. A hybrid scheme for release of statistics. Statistisk Tidskrift.

Dalenius, T., Reiss, S.P., 1982. Data-swapping: A technique for disclosure control. Journal of Statistical Planning and Inference 6, 73–85.

De Waal, A., Hundepool, A., Willenborg, L., 1995. Argus: Software for statistical disclosure control of microdata. U.S. Census Bureau.

DeGroot, M.H., 1962. Uncertainty, information, and sequential experiments. Annals of Mathematical Statistics 33, 404–419.

DeGroot, M.H., 1970. Optimal Statistical Decisions. Mansell, London.

Dinur, I., Nissam, K., 2003. Revealing information while preserving privacy. In: Proceedings of the 22nd ACM SIGMOD-SIGACT-SIGART Symposium on Principlesof Database Systems. pp. 202–210.

Domingo-Ferrer, J., Torra, V., 2001a. A Quantitative Comparison of Disclosure Control Methods for Microdata. In: Doyle, P., Lane, J., Theeuwes, J., Zayatz, L. (Eds.), Confidentiality, Disclosure and Data Access - Theory and Practical Applications for Statistical Agencies. North-Holland, Amsterdam, Ch. 6, pp. 113–135.

Domingo-Ferrer, J., Torra, V., 2001b. Disclosure control methods and information loss for microdata. In: Doyle, P., Lane, J., Theeuwes, J., Zayatz, L. (Eds.), Confidentiality, Disclosure and Data Access - Theory and Practical Applications for Statistical Agencies. North-Holland, Amsterdam, Ch. 5, pp. 93–112.

Duncan, G., Lambert, D., 1986. Disclosure-limited data dissemination. Journal of the American Statistical Association 81, 10–28.

Duncan, G., Lambert, D., 1989. The risk of disclosure for microdata. Journal of Business & Economic Statistics 7, 207–217.

Duncan, G., Pearson, R., 1991. Enhancing access to microdata while protecting confidentiality: prospects for the future (with discussion). Statistical Science 6, 219–232.

Dwork, C., 2006. Differential privacy. In: ICALP. Springer, pp. 1–12.

Dwork, C., 2008. An ad omnia approach to defining and achieving private data analysis. In: Lecture Notes in Computer Science. Springer, p. 10.

Dwork, C., Lei, J., 2009. Differential privacy and robust statistics. In: Proceedings of the 41th Annual ACM Symposium on Theory of Computing (STOC). pp. 371–380.

Dwork, C., Mcsherry, F., Nissim, K., Smith, A., 2006. Calibrating noise to sensitivity in private data analysis. In: Proceedings of the 3rd Theory of Cryptography Conference. Springer, pp. 265–284.

Dwork, C., Nissam, K., 2004. Privacy-preserving datamining on vertically partitioned databases. In: Advances in Cryptology: Proceedings of Crypto. pp. 528–544.

Elliot, M., 2000. DIS: a new approach to the measurement of statistical disclosure risk. International Journal of Risk Assessment and Management 2, 39–48.

Federal Committee on Statistical Methodology (FCSM), 2005. Statistical policy working group 22 - report on statistical disclosure limitation methodology. U.S. Census Bureau.

Fellegi, I.P., 1972. On the question of statistical confidentiality. Journal of the American Statistical Association 67 (337), 7–18.

Fienberg, S.E., McIntyre, J., 2004. Data swapping: Variations on a theme by Dalenius and Reiss. In: Domingo-Ferrer, J., Torra, V. (Eds.), Privacy in Statistical Databases. Vol. 3050 of Lecture Notes in Computer Science. Springer Berlin/Heidelberg, pp. 519, http://dx.doi.org/10.1007/ 978-3-540-25955-8_2

Fuller, W., 1993. Masking procedurse for microdata disclosure limitation. Journal of Official Statistics 9, 383–406.

General Assembly of the United Nations, 1948. Universal declaration of human rights.

Gouweleeuw, J., P. Kooiman, L.W., de Wolf, P.-P., 1998. Post randomisation for statistical disclosure control: Theory and implementation. Journal of Official Statistics 14 (4), 463–478.

Greenberg, B., 1987. Rank swapping for masking ordinal microdata. Tech. rep., U.S. Bureau of the Census (unpublished manuscript), Suitland, Maryland, USA.

Greenberg, B.G., Abul-Ela, A.-L.A., Simmons, W.R., Horvitz, D.G., 1969. The unrelated question randomized response model: Theoretical framework. Journal of the American Statistical Association 64 (326), 520–539.

Harel, O., Zhou, X.-H., 2007. Multiple imputation: Review and theory, implementation and software. Statistics in Medicine 26, 3057–3077.

Hundepool, A., Domingo-ferrer, J., Franconi, L., Giessing, S., Lenz, R., Longhurst, J., Nordholt, E.S., Seri, G., paul De Wolf, P., 2006. A CENtre of EXcellence for Statistical Disclosure Control Handbook on Statistical Disclosure Control Version 1.01.

Hundepool, A., Wetering, A. v.d., Ramaswamy, R., Wolf, P.d., Giessing, S., Fischetti, M., Salazar, J., Castro, J., Lowthian, P., Feb. 2005. τ-argus 3.1 user manual. Statistics Netherlands, Voorburg NL.

Hundepool, A., Willenborg, L., 1996. μ- and τ-argus: Software for statistical disclosure control. Third International Seminar on Statistical Confidentiality, Bled.

Karr, A., Kohnen, C.N., Oganian, A., Reiter, J.P., Sanil, A.P., 2006. A framework for evaluating the utility of data altered to protect confidentiality. American Statistician 60 (3), 224–232.

Kaufman, S., Seastrom, M., Roey, S., 2005. Do disclosure controls to protect confidentiality degrade the quality of the data? In: American Statistical Association, Proceedings of the Section on Survey Research.

Kennickell, A.B., 1997. Multiple imputation and disclosure protection: the case of the 1995 survey of consumer finances. Record Linkage Techniques, 248–267.

Kim, J., 1986. Limiting disclosure in microdata based on random noise and transformation. Bureau of the Census.

Krumm, J., 2007. Inference attacks on location tracks. Proceedings of Fifth International Conference on Pervasive Computingy, 127–143.

Li, N., Li, T., Venkatasubramanian, S., 2007. t-closeness: Privacy beyond k-anonymity and l-diversity. In: Data Engineering, 2007. ICDE 2007. IEEE 23rd International Conference on. pp. 106–115.

Liew, C.K., Choi, U.J., Liew, C.J., 1985. A data distortion by probability distribution. ACM Trans. Database Syst. 10 (3), 395–411.

Little, R.J.A., 1993. Statistical analysis of masked data. Journal of Official Statistics 9, 407–426.

Little, R.J.A., Rubin, D.B., 1987. Statistical Analysis with Missing Data. John Wiley & Sons.

Liu, F., Little, R.J.A., 2002. Selective multiple mputation of keys for statistical disclosure control in microdata. In: Proceedings Joint Statistical Meet. pp. 2133–2138.

Machanavajjhala, A., Kifer, D., Abowd, J., Gehrke, J., Vilhuber, L., April 2008. Privacy: Theory meets practice on the map. In: International Conference on Data Engineering. Cornell University Comuputer Science Department, Cornell, USA, p. 10.

Machanavajjhala, A., Kifer, D., Gehrke, J., Venkitasubramaniam, M., 2007. L-diversity: Privacy beyond k-anonymity. ACM Trans. Knowl. Discov. Data 1 (1), 3.

Manning, A.M., Haglin, D.J., Keane, J.A., 2008. A recursive search algorithm for statistical disclosure assessment. Data Min. Knowl. Discov. 16 (2), 165–196.

Marsh, C., Skinner, C., Arber, S., Penhale, B., Openshaw, S., Hobcraft, J., Lievesley, D., Walford, N., 1991. The case for samples of anonymized records from the 1991 census. Journal of the Royal Statistical Society 154 (2), 305–340.

Matthews, G.J., Harel, O., Aseltine, R.H., 2010a. Assessing database privacy using the area under the receiver-operator characteristic curve. Health Services and Outcomes Research Methodology 10 (1), 1–15.

Matthews, G.J., Harel, O., Aseltine, R.H., 2010b. Examining the robustness of fully synthetic data techniques for data with binary variables. Journal of Statistical Computation and Simulation 80 (6), 609–624.

Moore, Jr., R., 1996. Controlled data-swapping techniques for masking public use microdata. Census Tech Report.

Mugge, R., 1983. Issues in protecting confidentiality in national health statistics. Proceedings of the Section on Survey Research Methods.

Nissim, K., Raskhodnikova, S., Smith, A., 2007. Smooth sensitivity and sampling in private data analysis. In: STOC ’07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing. pp. 75–84.

Paass, G., 1988. Disclosure risk and disclosure avoidance for microdata. Journal of Business and Economic Statistics 6 (4), 487–500.

Palley, M., Simonoff, J., 1987. The use of regression methodology for the compromise of confidential information in statistical databases. ACM Trans. Database Systems 12 (4), 593–608.

Raghunathan, T.E., Reiter, J.P., Rubin, D.B., 2003. Multiple imputation for statistical disclosure limitation. Journal of Official Statistics 19 (1), 1–16.

Rajasekaran, S., Harel, O., Zuba, M., Matthews, G.J., Aseltine, Jr., R., 2009. Responsible data releases. In: Proceedings 9th Industrial Conference on Data Mining (ICDM). Springer LNCS, pp. 388–400.

Reiss, S.P., 1984. Practical data-swapping: The first steps. CM Transactions on Database Systems 9, 20–37.

Reiter, J.P., 2002. Satisfying disclosure restriction with synthetic data sets. Journal of Official Statistics 18 (4), 531–543.

Reiter, J.P., 2003. Inference for partially synthetic, public use microdata sets. Survey Methodology 29 (2), 181–188.

Reiter, J.P., 2004a. New approaches to data dissemination: A glimpse into the future (?). Chance 17 (3), 11–15.

Reiter, J.P., 2004b. Simultaneous use of multiple imputation for missing data and disclosure limitation. Survey Methodology 30 (2), 235–242.

Reiter, J.P., 2005a. Estimating risks of identification disclosure in microdata. Journal of the American Statistical Association 100, 1103–1112.

Reiter, J.P., 2005b. Releasing multiply imputed, synthetic public use microdata: An illustration and empirical study. Journal of the Royal Statistical Society, Series A: Statistics in Society 168 (1), 185–205.

Reiter, J.P., 2005c. Using CART to generate partially synthetic public use microdata. Journal of Official Statistics 21 (3), 441–462.

Rubin, D.B., 1987. Multiple Imputation for Nonresponse in Surveys. John Wiley & Sons.

Rubin, D.B., 1993. Comment on “Statistical disclosure limitation”. Journal of Official Statistics 9, 461–468.

Rubner, Y., Tomasi, C., Guibas, L.J., 1998. A metric for distributions with applications to image databases. Computer Vision, IEEE International Conference on 0, 59.

Sarathy, R., Muralidhar, K., 2002a. The security of confidential numerical data in databases. Information Systems Research 13 (4), 389–403.

Sarathy, R., Muralidhar, K., 2002b. The security of confidential numerical data in databases. Info. Sys. Research 13 (4), 389–403.

Schafer, J.L., Graham, J.W., 2002. Missing data: Our view of state of the art. Psychological Methods 7 (2), 147–177.

Singh, A., Yu, F., Dunteman, G., 2003. MASSC: A new data mask for limiting statistical information loss and disclosure. In: Proceedings of the Joint UNECE/EUROSTAT Work Session on Statistical Data Confidentiality. pp. 373–394.

Skinner, C., 2009. Statistical disclosure control for survey data. In: Pfeffermann, D and Rao, C.R. eds. Handbook of Statistics Vol. 29A: Sample Surveys: Design, Methods and Applications. pp. 381–396.

Skinner, C., Marsh, C., Openshaw, S., Wymer, C., 1994. Disclosure control for census microdata. Journal of Official Statistics 10, 31–51.

Skinner, C., Shlomo, N., 2008. Assessing identification risk in survey microdata using log-linear models. Journal of the American Statistical Association 103, 989–1001.

Skinner, C.J., Elliot, M.J., 2002. A measure of disclosure risk for microdata. Journal of the Royal Statistical Society. Series B (Statistical Methodology) 64 (4), 855–867.

Smith, A., 2008. Efficient, dfferentially private point estimators. arXiv:0809.4794v1 [cs.CR].

Spruill, N.L., 1982. Measures of confidentiality. Statistics of Income and Related Administrative Record Research, 131–136.

Spruill, N.L., 1983. The confidentiality and analytic usefulness of masked business microdata. In: Proceedings of the Section on Survey Reserach Microdata. American Statistical Association, pp. 602–607.

Sweeney, L., 1996. Replacing personally-identifying information in medical records, the scrub system. In: American Medical Informatics Association. Hanley and Belfus, Inc., pp. 333–337.

Sweeney, L., 1997. Guaranteeing anonymity when sharing medical data, the datafly system. Journal of the American Medical Informatics Association 4, 51–55.

Sweeney, L., 2002a. Achieving k-anonymity privacy protection using generalization and suppression. International Journal of Uncertainty, Fuzziness and Knowledge Based Systems 10 (5), 571–588.

Sweeney, L., 2002b. k-anonymity: A model for protecting privacy. International Journal of Uncertainty, Fuzziness and Knowledge Based Systems 10 (5), 557–570.

Tendick, P., 1991. Optimal noise addition for preserving confidentiality in multivariate data. Journal of Statistical Planning and Inference 27 (2), 341–353.

United Nations Economic Comission for Europe (UNECE), 2007. Manging statistical cinfidentiality and microdata access: Principles and guidlinesof good practice.

Warner, S.L., 1965. Randomized response: A survey technique for eliminating evasive answer bias. Journal of the American Statistical Association 60 (309), 63–69.

Wasserman, L., Zhou, S., 2010. A statistical framework for differential privacy. Journal of the American Statistical Association 105 (489), 375–389.

Willenborg, L., de Waal, T., 2001. Elements of Statistical Disclosure Control. Springer-Verlag.

Woodward, B., 1995. The computer-based patient record and confidentiality. The New England Journal of Medicine, 1419–1422.

Methods for generating synthetic data have become of increasing importance to build large datasets required for Convolution Neural Networks (CNN) based deep learning techniques for a wide range of computer vision applications. In this work, we extend existing methodologies to show how 2D thermal facial data can be mapped to provide 3D facial models. For the proposed research work we have used tufts datasets for generating 3D varying face poses by using a single frontal face pose. The system works by refining the existing image quality by performing fusion based image preprocessing operations. The refined outputs have better contrast adjustments, decreased noise level and higher exposedness of the dark regions. It makes the facial landmarks and temperature patterns on the human face more discernible and visible when compared to original raw data. Different image quality metrics are used to compare the refined version of images with original images. In the next phase of the proposed study, the refined version of images is used to create 3D facial geometry structures by using Convolution Neural Networks (CNN). The generated outputs are then imported in blender software to finally extract the 3D thermal facial outputs of both males and females. The same technique is also used on our thermal face data acquired using prototype thermal camera (developed under Heliaus EU project) in an indoor lab environment which is then used for generating synthetic 3D face data along with varying yaw face angles and lastly facial depth map is generated.

Data-space inversion (DSI) and related procedures represent a family of methods applicable for data assimilation in subsurface flow settings. These methods differ from model-based techniques in that they provide only posterior predictions for quantities (time series) of interest, not posterior models with calibrated parameters. DSI methods require a large number of flow simulations to first be performed on prior geological realizations. Given observed data, posterior predictions can then be generated directly. DSI operates in a Bayesian setting and provides posterior samples of the data vector. In this work we develop and evaluate a new approach for data parameterization in DSI. Parameterization reduces the number of variables to determine in the inversion, and it maintains the physical character of the data variables. The new parameterization uses a recurrent autoencoder (RAE) for dimension reduction, and a long-short-term memory (LSTM) network to represent flow-rate time series. The RAE-based parameterization is combined with an ensemble smoother with multiple data assimilation (ESMDA) for posterior generation. Results are presented for two- and three-phase flow in a 2D channelized system and a 3D multi-Gaussian model. The RAE procedure, along with existing DSI treatments, are assessed through comparison to reference rejection sampling (RS) results. The new DSI methodology is shown to consistently outperform existing approaches, in terms of statistical agreement with RS results. The method is also shown to accurately capture derived quantities, which are computed from variables considered directly in DSI. This requires correlation and covariance between variables to be properly captured, and accuracy in these relationships is demonstrated. The RAE-based parameterization developed here is clearly useful in DSI, and it may also find application in other subsurface flow problems.

We study the problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as $dX_t=(mu+ heta X_t)dt+dG_t, tgeq0$ with unknown parameters $ heta>0$ and $muinR$, where $G$ is a Gaussian process. We provide least square-type estimators $widetilde{ heta}_T$ and $widetilde{mu}_T$ respectively for the drift parameters $ heta$ and $mu$ based on continuous-time observations ${X_t, tin[0,T]}$ as $T ightarrowinfty$.

Our aim is to derive some sufficient conditions on the driving Gaussian process $G$ in order to ensure that $widetilde{ heta}_T$ and $widetilde{mu}_T$ are strongly consistent, the limit distribution of $widetilde{ heta}_T$ is a Cauchy-type distribution and $widetilde{mu}_T$ is asymptotically normal. We apply our result to fractional Vasicek, subfractional Vasicek and bifractional Vasicek processes. In addition, this work extends the result of cite{EEO} studied in the case where $mu=0$.

When the response mechanism is believed to be not missing at random (NMAR), a valid analysis requires stronger assumptions on the response mechanism than standard statistical methods would otherwise require. Semiparametric estimators have been developed under the model assumptions on the response mechanism. In this paper, a new statistical test is proposed to guarantee model identifiability without using any instrumental variable. Furthermore, we develop optimal semiparametric estimation for parameters such as the population mean. Specifically, we propose two semiparametric optimal estimators that do not require any model assumptions other than the response mechanism. Asymptotic properties of the proposed estimators are discussed. An extensive simulation study is presented to compare with some existing methods. We present an application of our method using Korean Labor and Income Panel Survey data.

This paper considers the problem of estimation of the Fisher information for location from a random sample of size $n$. First, an estimator proposed by Bhattacharya is revisited and improved convergence rates are derived. Second, a new estimator, termed a clipped estimator, is proposed. Superior upper bounds on the rates of convergence can be shown for the new estimator compared to the Bhattacharya estimator, albeit with different regularity conditions. Third, both of the estimators are evaluated for the practically relevant case of a random variable contaminated by Gaussian noise. Moreover, using Brown's identity, which relates the Fisher information and the minimum mean squared error (MMSE) in Gaussian noise, two corresponding consistent estimators for the MMSE are proposed. Simulation examples for the Bhattacharya estimator and the clipped estimator as well as the MMSE estimators are presented. The examples demonstrate that the clipped estimator can significantly reduce the required sample size to guarantee a specific confidence interval compared to the Bhattacharya estimator.

Text summarization refers to the process that generates a shorter form of text from the source document preserving salient information. Recently, many models for text summarization have been proposed. Most of those models were evaluated using recall-oriented understudy for gisting evaluation (ROUGE) scores. However, as ROUGE scores are computed based on n-gram overlap, they do not reflect semantic meaning correspondences between generated and reference summaries. Because Korean is an agglutinative language that combines various morphemes into a word that express several meanings, ROUGE is not suitable for Korean summarization. In this paper, we propose evaluation metrics that reflect semantic meanings of a reference summary and the original document, Reference and Document Aware Semantic Score (RDASS). We then propose a method for improving the correlation of the metrics with human judgment. Evaluation results show that the correlation with human judgment is significantly higher for our evaluation metrics than for ROUGE scores.

Uplift modeling is a predictive modeling technique that estimates the user-level incremental effect of a treatment using machine learning models. It is often used for targeting promotions and advertisements, as well as for the personalization of product offerings. In these applications, there are often hundreds of features available to build such models. Keeping all the features in a model can be costly and inefficient. Feature selection is an essential step in the modeling process for multiple reasons: improving the estimation accuracy by eliminating irrelevant features, accelerating model training and prediction speed, reducing the monitoring and maintenance workload for feature data pipeline, and providing better model interpretation and diagnostics capability. However, feature selection methods for uplift modeling have been rarely discussed in the literature. Although there are various feature selection methods for standard machine learning models, we will demonstrate that those methods are sub-optimal for solving the feature selection problem for uplift modeling. To address this problem, we introduce a set of feature selection methods designed specifically for uplift modeling, including both filter methods and embedded methods. To evaluate the effectiveness of the proposed feature selection methods, we use different uplift models and measure the accuracy of each model with a different number of selected features. We use both synthetic and real data to conduct these experiments. We also implemented the proposed filter methods in an open source Python package (CausalML).

Ridge regression (RR) is a regularization technique that penalizes the L2-norm of the coefficients in linear regression. One of the challenges of using RR is the need to set a hyperparameter ($alpha$) that controls the amount of regularization. Cross-validation is typically used to select the best $alpha$ from a set of candidates. However, efficient and appropriate selection of $alpha$ can be challenging, particularly where large amounts of data are analyzed. Because the selected $alpha$ depends on the scale of the data and predictors, it is not straightforwardly interpretable. Here, we propose to reparameterize RR in terms of the ratio $gamma$ between the L2-norms of the regularized and unregularized coefficients. This approach, called fractional RR (FRR), has several benefits: the solutions obtained for different $gamma$ are guaranteed to vary, guarding against wasted calculations, and automatically span the relevant range of regularization, avoiding the need for arduous manual exploration. We provide an algorithm to solve FRR, as well as open-source software implementations in Python and MATLAB (https://github.com/nrdg/fracridge). We show that the proposed method is fast and scalable for large-scale data problems, and delivers results that are straightforward to interpret and compare across models and datasets.

We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. Numerically we demonstrate the effectiveness of the method on a class of parametric elliptic PDE problems, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare our method with existing algorithms from the literature.

The random-effects or normal-normal hierarchical model is commonly utilized in a wide range of meta-analysis applications. A Bayesian approach to inference is very attractive in this context, especially when a meta-analysis is based only on few studies. The bayesmeta R package provides readily accessible tools to perform Bayesian meta-analyses and generate plots and summaries, without having to worry about computational details. It allows for flexible prior specification and instant access to the resulting posterior distributions, including prediction and shrinkage estimation, and facilitating for example quick sensitivity checks. The present paper introduces the underlying theory and showcases its usage.

This paper is devoted to the R package JSM which performs joint statistical modeling of survival and longitudinal data. In biomedical studies it has been increasingly common to collect both baseline and longitudinal covariates along with a possibly censored survival time. Instead of analyzing the survival and longitudinal outcomes separately, joint modeling approaches have attracted substantive attention in the recent literature and have been shown to correct biases from separate modeling approaches and enhance information. Most existing approaches adopt a linear mixed effects model for the longitudinal component and the Cox proportional hazards model for the survival component. We extend the Cox model to a more general class of transformation models for the survival process, where the baseline hazard function is completely unspecified leading to semiparametric survival models. We also offer a non-parametric multiplicative random effects model for the longitudinal process in JSM in addition to the linear mixed effects model. In this paper, we present the joint modeling framework that is implemented in JSM, as well as the standard error estimation methods, and illustrate the package with two real data examples: a liver cirrhosis data and a Mayo Clinic primary biliary cirrhosis data.

Eric D. Kolaczyk, Lizhen Lin, Steven Rosenberg, Jackson Walters, Jie Xu.

Source: The Annals of Statistics, Volume 48, Number 1, 514--538.

Abstract:
It is becoming increasingly common to see large collections of network data objects, that is, data sets in which a network is viewed as a fundamental unit of observation. As a result, there is a pressing need to develop network-based analogues of even many of the most basic tools already standard for scalar and vector data. In this paper, our focus is on averages of unlabeled, undirected networks with edge weights. Specifically, we (i) characterize a certain notion of the space of all such networks, (ii) describe key topological and geometric properties of this space relevant to doing probability and statistics thereupon, and (iii) use these properties to establish the asymptotic behavior of a generalized notion of an empirical mean under sampling from a distribution supported on this space. Our results rely on a combination of tools from geometry, probability theory and statistical shape analysis. In particular, the lack of vertex labeling necessitates working with a quotient space modding out permutations of labels. This results in a nontrivial geometry for the space of unlabeled networks, which in turn is found to have important implications on the types of probabilistic and statistical results that may be obtained and the techniques needed to obtain them.

Hock Peng Chan.

Source: The Annals of Statistics, Volume 48, Number 1, 346--373.

Abstract:
Lai and Robbins ( Adv. in Appl. Math. 6 (1985) 4–22) and Lai ( Ann. Statist. 15 (1987) 1091–1114) provided efficient parametric solutions to the multi-armed bandit problem, showing that arm allocation via upper confidence bounds (UCB) achieves minimum regret. These bounds are constructed from the Kullback–Leibler information of the reward distributions, estimated from specified parametric families. In recent years, there has been renewed interest in the multi-armed bandit problem due to new applications in machine learning algorithms and data analytics. Nonparametric arm allocation procedures like $epsilon $-greedy, Boltzmann exploration and BESA were studied, and modified versions of the UCB procedure were also analyzed under nonparametric settings. However, unlike UCB these nonparametric procedures are not efficient under general parametric settings. In this paper, we propose efficient nonparametric procedures.

Xi Chen, Jason D. Lee, Xin T. Tong, Yichen Zhang.

Source: The Annals of Statistics, Volume 48, Number 1, 251--273.

Abstract:
The stochastic gradient descent (SGD) algorithm has been widely used in statistical estimation for large-scale data due to its computational and memory efficiency. While most existing works focus on the convergence of the objective function or the error of the obtained solution, we investigate the problem of statistical inference of true model parameters based on SGD when the population loss function is strongly convex and satisfies certain smoothness conditions. Our main contributions are twofold. First, in the fixed dimension setup, we propose two consistent estimators of the asymptotic covariance of the average iterate from SGD: (1) a plug-in estimator, and (2) a batch-means estimator, which is computationally more efficient and only uses the iterates from SGD. Both proposed estimators allow us to construct asymptotically exact confidence intervals and hypothesis tests. Second, for high-dimensional linear regression, using a variant of the SGD algorithm, we construct a debiased estimator of each regression coefficient that is asymptotically normal. This gives a one-pass algorithm for computing both the sparse regression coefficients and confidence intervals, which is computationally attractive and applicable to online data.