A federal investigation of Chicago's failures to respond to sexual violence in schools raises troubling questions for school districts nationwide.

British Columbia Premier John Horgan has offered the NHL a place to play if the league can find a way to resume the season. Speaking at a COVID-19 media briefing Wednesday, Horgan said he has written a letter to both NHL Commissioner Gary Bettman and NHL Players’ Association head Donald Fehr to let them know B.C. is “a place to potentially restart the NHL assuming the games would be played without audiences, but instead played for television.” The NHL suspended its season March 12 with 189 regular-season games left.

Wednesday's collection of links.

How to tune in to Tom Wilson's best games on Sunday.

Alex Ovechkin can lay claim as the best Capitals player to wear No. 8, but what about the rest of the jersey numbers? Who are the best Caps for every number?

In a video posted along with the schedule, the Seahawks gave season tickets to 12 workers who had been nominated for consideration. The 12 were told of receiving season tickets in video calls with Seahawks players Shaquill and Shaquem Griffin, DK Metcalf, Tyler Lockett, Will Dissly and coach Pete Carroll. The Royal St. John’s Regatta in Canada has been canceled because of the COVID-19 pandemic.

Alex Ovechkin can lay claim as the best Capitals player to wear No. 8, but what about the rest of the jersey numbers? Who are the best Caps for every number?

A 24-team Stanley Cup Playoffs? Here's a look at the latest NHL buzz and a hypothetical Flyers matchup. By Jordan Hall

Capitals forward Evgeny Kuznetsov doesn't consider his series-winning goal against the Penguins in 2018 is the biggest goal of his life.

One of many possible plans if the league can resume play.

The century-old Reno Rodeo has canceled the 10-day event in June because of the coronavirus pandemic. Reno Rodeo President Craig Downie said in a letter to the rodeo’s board of directors that canceling the event scheduled for June 18-27 was necessary to ensure the safety of participants, fans, vendors, sponsors and volunteers. General Manager George Combs said it was a difficult decision, made in consultation with health experts as well as the Professional Rodeo Cowboys Association.

Troy Brouwer scored the game-winning goal as the Capitals beat the Blackhawks at Nationals Park on Jan. 1, 2015. Relive the excitement with Monday night's replay.

Kerfoot discussing his first Maple Leafs season spurred a larger discussion.

Châtillon, France : Association Hommefleur, [date of publication not identified]

2017

Stuttgart : Franz Steiner Verlag, 2019.

Rockville, Maryland : National Institute on Drug Abuse, 1986.

Rockville, Maryland : National Institute on Drug Abuse, 1978.

Rockville, Maryland : National Institute on Drug Abuse, 1979.

New York : Experimental Manpower Laboratory at Mobilization for Youth, Inc., [1971]

[Dorchester] : Dorset Community NHS Trust, 1997.

São Paulo : Governo do Estado de Sao Paulo, Secretaria da Segurança Pública, 1975.

London : Reproductive Health Matters, 1993-2018.

London : Reproductive Health Matters, 1993-2018.

Oregon State basketball player Destiny Slocum has opted to enter the transfer portal for her final season of eligibility. Slocum, a 5-foot-7 guard, averaged a team-best 14.9 points and had 4.7 assists a game this past season with the Beavers, who finished the season ranked No. 14 with a 23-9 record. In a statement released by the university on Thursday, Slocum thanked everyone who supported her in the decision.

COLUMBIA, S.C. (AP) -- South Carolina freshman Aliyah Boston has won the Lisa Leslie Award given to the top center in women’s college basketball.

Eunji Lim.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 2053--2097.

Abstract:
We study the asymptotic behavior of the least squares estimators when the model is possibly misspecified. We consider the setting where we wish to estimate an unknown function $f_{*}:(0,1)^{d} ightarrow mathbb{R}$ from observations $(X,Y),(X_{1},Y_{1}),cdots ,(X_{n},Y_{n})$; our estimator $hat{g}_{n}$ is the minimizer of $sum _{i=1}^{n}(Y_{i}-g(X_{i}))^{2}/n$ over $gin mathcal{G}$ for some set of functions $mathcal{G}$. We provide sufficient conditions on the metric entropy of $mathcal{G}$, under which $hat{g}_{n}$ converges to $g_{*}$ as $n ightarrow infty $, where $g_{*}$ is the minimizer of $|g-f_{*}| riangleq mathbb{E}(g(X)-f_{*}(X))^{2}$ over $gin mathcal{G}$. As corollaries of our theorem, we establish $|hat{g}_{n}-g_{*}| ightarrow 0$ as $n ightarrow infty $ when $mathcal{G}$ is the set of monotone functions or the set of convex functions. We also make a connection between the convergence rate of $|hat{g}_{n}-g_{*}|$ and the metric entropy of $mathcal{G}$. As special cases of our finding, we compute the convergence rate of $|hat{g}_{n}-g_{*}|^{2}$ when $mathcal{G}$ is the set of bounded monotone functions or the set of bounded convex functions.

Emanuel Ben-David, Bala Rajaratnam.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 580--604.

Abstract:
The Wishart distribution defined on the open cone of positive-definite matrices plays a central role in multivariate analysis and multivariate distribution theory. Its domain of parameters is often referred to as the Gindikin set. In recent years, varieties of useful extensions of the Wishart distribution have been proposed in the literature for the purposes of studying Markov random fields and graphical models. In particular, generalizations of the Wishart distribution, referred to as Type I and Type II (graphical) Wishart distributions introduced by Letac and Massam in Annals of Statistics (2007) play important roles in both frequentist and Bayesian inference for Gaussian graphical models. These distributions have been especially useful in high-dimensional settings due to the flexibility offered by their multiple-shape parameters. Concerning Type I and Type II Wishart distributions, a conjecture of Letac and Massam concerns the domain of multiple-shape parameters of these distributions. The conjecture also has implications for the existence of Bayes estimators corresponding to these high dimensional priors. The conjecture, which was first posed in the Annals of Statistics, has now been an open problem for about 10 years. In this paper, we give a necessary condition for the Letac and Massam conjecture to hold. More precisely, we prove that if the Letac and Massam conjecture holds on a decomposable graph, then no two separators of the graph can be nested within each other. For this, we analyze Type I and Type II Wishart distributions on appropriate Markov equivalent perfect DAG models and succeed in deriving the aforementioned necessary condition. This condition in particular identifies a class of counterexamples to the conjecture.

Gregor Pasemann, Wilhelm Stannat.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 547--579.

Abstract:
A parameter estimation problem for a class of semilinear stochastic evolution equations is considered. Conditions for consistency and asymptotic normality are given in terms of growth and continuity properties of the nonlinear part. Emphasis is put on the case of stochastic reaction-diffusion systems. Robustness results for statistical inference under model uncertainty are provided.

Anatoli Juditsky, Arkadi Nemirovski.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 458--502.

Abstract:
We consider the problem of recovering linear image of unknown signal belonging to a given convex compact signal set from noisy observation of another linear image of the signal. We develop a simple generic efficiently computable non linear in observations “polyhedral” estimate along with computation-friendly techniques for its design and risk analysis. We demonstrate that under favorable circumstances the resulting estimate is provably near-optimal in the minimax sense, the “favorable circumstances” being less restrictive than the weakest known so far assumptions ensuring near-optimality of estimates which are linear in observations.

Jean-Marc Bardet, Kare Kamila, William Kengne.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 2009--2052.

Abstract:
This paper studies the model selection problem in a large class of causal time series models, which includes both the ARMA or AR($infty $) processes, as well as the GARCH or ARCH($infty $), APARCH, ARMA-GARCH and many others processes. To tackle this issue, we consider a penalized contrast based on the quasi-likelihood of the model. We provide sufficient conditions for the penalty term to ensure the consistency of the proposed procedure as well as the consistency and the asymptotic normality of the quasi-maximum likelihood estimator of the chosen model. We also propose a tool for diagnosing the goodness-of-fit of the chosen model based on a Portmanteau test. Monte-Carlo experiments and numerical applications on illustrative examples are performed to highlight the obtained asymptotic results. Moreover, using a data-driven choice of the penalty, they show the practical efficiency of this new model selection procedure and Portemanteau test.

François Bachoc, José Betancourt, Reinhard Furrer, Thierry Klein.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1962--2008.

Abstract:
The asymptotic analysis of covariance parameter estimation of Gaussian processes has been subject to intensive investigation. However, this asymptotic analysis is very scarce for non-Gaussian processes. In this paper, we study a class of non-Gaussian processes obtained by regular non-linear transformations of Gaussian processes. We provide the increasing-domain asymptotic properties of the (Gaussian) maximum likelihood and cross validation estimators of the covariance parameters of a non-Gaussian process of this class. We show that these estimators are consistent and asymptotically normal, although they are defined as if the process was Gaussian. They do not need to model or estimate the non-linear transformation. Our results can thus be interpreted as a robustness of (Gaussian) maximum likelihood and cross validation towards non-Gaussianity. Our proofs rely on two technical results that are of independent interest for the increasing-domain asymptotic literature of spatial processes. First, we show that, under mild assumptions, coefficients of inverses of large covariance matrices decay at an inverse polynomial rate as a function of the corresponding observation location distances. Second, we provide a general central limit theorem for quadratic forms obtained from transformed Gaussian processes. Finally, our asymptotic results are illustrated by numerical simulations.

Wanli Qiao.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 302--344.

Abstract:
Bandwidth selection is crucial in the kernel estimation of density level sets. A risk based on the symmetric difference between the estimated and true level sets is usually used to measure their proximity. In this paper we provide an asymptotic $L^{p}$ approximation to this risk, where $p$ is characterized by the weight function in the risk. In particular the excess risk corresponds to an $L^{2}$ type of risk, and is adopted to derive an optimal bandwidth for nonparametric level set estimation of $d$-dimensional density functions ($dgeq 1$). A direct plug-in bandwidth selector is developed for kernel density level set estimation and its efficacy is verified in numerical studies.

Gianluca Finocchio, Johannes Schmidt-Hieber.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 239--271.

Abstract:
Consider the Gaussian sequence model under the additional assumption that a fixed fraction of the means is known. We study the problem of variance estimation from a frequentist Bayesian perspective. The maximum likelihood estimator (MLE) for $sigma^{2}$ is biased and inconsistent. This raises the question whether the posterior is able to correct the MLE in this case. By developing a new proving strategy that uses refined properties of the posterior distribution, we find that the marginal posterior is inconsistent for any i.i.d. prior on the mean parameters. In particular, no assumption on the decay of the prior needs to be imposed. Surprisingly, we also find that consistency can be retained for a hierarchical prior based on Gaussian mixtures. In this case we also establish a limiting shape result and determine the limit distribution. In contrast to the classical Bernstein-von Mises theorem, the limit is non-Gaussian. We show that the Bayesian analysis leads to new statistical estimators outperforming the correctly calibrated MLE in a numerical simulation study.

Patrick L. Combettes, Christian L. Müller.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 207--238.

Abstract:
We introduce a flexible optimization model for maximum likelihood-type estimation (M-estimation) that encompasses and generalizes a large class of existing statistical models, including Huber’s concomitant M-estimator, Owen’s Huber/Berhu concomitant estimator, the scaled lasso, support vector machine regression, and penalized estimation with structured sparsity. The model, termed perspective M-estimation, leverages the observation that convex M-estimators with concomitant scale as well as various regularizers are instances of perspective functions, a construction that extends a convex function to a jointly convex one in terms of an additional scale variable. These nonsmooth functions are shown to be amenable to proximal analysis, which leads to principled and provably convergent optimization algorithms via proximal splitting. We derive novel proximity operators for several perspective functions of interest via a geometrical approach based on duality. We then devise a new proximal splitting algorithm to solve the proposed M-estimation problem and establish the convergence of both the scale and regression iterates it produces to a solution. Numerical experiments on synthetic and real-world data illustrate the broad applicability of the proposed framework.

David Azriel, Armin Schwartzman.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 174--206.

Abstract:
In this work we study estimation of signals when the number of parameters is much larger than the number of observations. A large body of literature assumes for these kind of problems a sparse structure where most of the parameters are zero or close to zero. When this assumption does not hold, one can focus on low-dimensional functions of the parameter vector. In this work we study one-dimensional linear projections. Specifically, in the context of high-dimensional linear regression, the parameter of interest is ${oldsymbol{eta}}$ and we study estimation of $mathbf{a}^{T}{oldsymbol{eta}}$. We show that $mathbf{a}^{T}hat{oldsymbol{eta}}$, where $hat{oldsymbol{eta}}$ is the least squares estimator, using pseudo-inverse when $p>n$, is minimax and admissible. Thus, for linear projections no regularization or shrinkage is needed. This estimator is easy to analyze and confidence intervals can be constructed. We study a high-dimensional dataset from brain imaging where it is shown that the signal is weak, non-sparse and significantly different from zero.

Heiko Werner, Hajo Holzmann, Pierre Vandekerkhove.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1816--1871.

Abstract:
We investigate a flexible two-component semiparametric mixture of regressions model, in which one of the conditional component distributions of the response given the covariate is unknown but assumed symmetric about a location parameter, while the other is specified up to a scale parameter. The location and scale parameters together with the proportion are allowed to depend nonparametrically on covariates. After settling identifiability, we provide local M-estimators for these parameters which converge in the sup-norm at the optimal rates over Hölder-smoothness classes. We also introduce an adaptive version of the estimators based on the Lepski-method. Sup-norm bounds show that the local M-estimator properly estimates the functions globally, and are the first step in the construction of useful inferential tools such as confidence bands. In our analysis we develop general results about rates of convergence in the sup-norm as well as adaptive estimation of local M-estimators which might be of some independent interest, and which can also be applied in various other settings. We investigate the finite-sample behaviour of our method in a simulation study, and give an illustration to a real data set from bioinformatics.

Tuo Chen, Zhihua Su, Yi Yang, Shanshan Ding.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 143--173.

Abstract:
As a generalization of the classical linear regression, expectile regression (ER) explores the relationship between the conditional expectile of a response variable and a set of predictor variables. ER with respect to different expectile levels can provide a comprehensive picture of the conditional distribution of the response variable given the predictors. We adopt an efficient estimation method called the envelope model ([8]) in ER, and construct a novel envelope expectile regression (EER) model. Estimation of the EER parameters can be performed using the generalized method of moments (GMM). We establish the consistency and derive the asymptotic distribution of the EER estimators. In addition, we show that the EER estimators are asymptotically more efficient than the ER estimators. Numerical experiments and real data examples are provided to demonstrate the efficiency gains attained by EER compared to ER, and the efficiency gains can further lead to improvements in prediction.

Gwenaëlle Castellan, Anthony Cousien, Viet Chi Tran.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 50--81.

Abstract:
Global sensitivity analysis is a set of methods aiming at quantifying the contribution of an uncertain input parameter of the model (or combination of parameters) on the variability of the response. We consider here the estimation of the Sobol indices of order 1 which are commonly-used indicators based on a decomposition of the output’s variance. In a deterministic framework, when the same inputs always give the same outputs, these indices are usually estimated by replicated simulations of the model. In a stochastic framework, when the response given a set of input parameters is not unique due to randomness in the model, metamodels are often used to approximate the mean and dispersion of the response by deterministic functions. We propose a new non-parametric estimator without the need of defining a metamodel to estimate the Sobol indices of order 1. The estimator is based on warped wavelets and is adaptive in the regularity of the model. The convergence of the mean square error to zero, when the number of simulations of the model tend to infinity, is computed and an elbow effect is shown, depending on the regularity of the model. Applications in Epidemiology are carried to illustrate the use of non-parametric estimators.

Aurélie Fischer, Dominique Picard.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1648--1689.

Abstract:
In this paper, we consider the estimation of a change-point for possibly high-dimensional data in a Gaussian model, using a maximum likelihood method. We are interested in how dimension reduction can affect the performance of the method. We provide an estimator of the change-point that has a minimax rate of convergence, up to a logarithmic factor. The minimax rate is in fact composed of a fast rate —dimension-invariant— and a slow rate —increasing with the dimension. Moreover, it is proved that considering the case of sparse data, with a Sobolev regularity, there is a bound on the separation of the regimes above which there exists an optimal choice of dimension reduction, leading to the fast rate of estimation. We propose an adaptive dimension reduction procedure based on Lepski’s method and show that the resulting estimator attains the fast rate of convergence. Our results are then illustrated by a simulation study. In particular, practical strategies are suggested to perform dimension reduction.

Kentaro Minami.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1508--1576.

Abstract:
We study the problem of estimating piecewise monotone vectors. This problem can be seen as a generalization of the isotonic regression that allows a small number of order-violating changepoints. We focus mainly on the performance of the nearly-isotonic regression proposed by Tibshirani et al. (2011). We derive risk bounds for the nearly-isotonic regression estimators that are adaptive to piecewise monotone signals. The estimator achieves a near minimax convergence rate over certain classes of piecewise monotone signals under a weak assumption. Furthermore, we present an algorithm that can be applied to the nearly-isotonic type estimators on general weighted graphs. The simulation results suggest that the nearly-isotonic regression performs as well as the ideal estimator that knows the true positions of changepoints.

Claudia Wehrhahn, Samuel Leonard, Abel Rodriguez, Tatiana Xifara.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1449--1478.

Abstract:
Identifying disease clusters (areas with an unusually high incidence of a particular disease) is a common problem in epidemiology and public health. We describe a Bayesian nonparametric mixture model for disease clustering that constrains clusters to be made of adjacent areal units. This is achieved by modifying the exchangeable partition probability function associated with the Ewen’s sampling distribution. We call the resulting prior the Restricted Chinese Restaurant Process, as the associated full conditional distributions resemble those associated with the standard Chinese Restaurant Process. The model is illustrated using synthetic data sets and in an application to oral cancer mortality in Germany.

Xuening Zhu.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1413--1448.

Abstract:
High dimensional time series receive considerable attention recently, whose temporal and cross-sectional dependency could be captured by the vector autoregression (VAR) model. To tackle with the high dimensionality, penalization methods are widely employed. However, theoretically, the existing studies of the penalization methods mainly focus on $i.i.d$ data, therefore cannot quantify the effect of the dependence level on the convergence rate. In this work, we use the spectral properties of the time series to quantify the dependence and derive a nonasymptotic upper bound for the estimation errors. By focusing on the nonconcave penalization methods, we manage to establish the oracle properties of the penalized VAR model estimation by considering the effects of temporal and cross-sectional dependence. Extensive numerical studies are conducted to compare the finite sample performance using different penalization functions. Lastly, an air pollution data of mainland China is analyzed for illustration purpose.

Rahul Mazumder, Haolei Weng.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1348--1385.

Abstract:
Estimating a low rank matrix from its linear measurements is a problem of central importance in contemporary statistical analysis. The choice of tuning parameters for estimators remains an important challenge from a theoretical and practical perspective. To this end, Stein’s Unbiased Risk Estimate (SURE) framework provides a well-grounded statistical framework for degrees of freedom estimation. In this paper, we use the SURE framework to obtain degrees of freedom estimates for a general class of spectral regularized matrix estimators—our results generalize beyond the class of estimators that have been studied thus far. To this end, we use a result due to Shapiro (2002) pertaining to the differentiability of symmetric matrix valued functions, developed in the context of semidefinite optimization algorithms. We rigorously verify the applicability of Stein’s Lemma towards the derivation of degrees of freedom estimates; and also present new techniques based on Gaussian convolution to estimate the degrees of freedom of a class of spectral estimators, for which Stein’s Lemma does not directly apply.

Vincent Brault, Christine Keribin, Mahendra Mariadassou.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1234--1268.

Abstract:
The Latent Block Model (LBM) is a model-based method to cluster simultaneously the $d$ columns and $n$ rows of a data matrix. Parameter estimation in LBM is a difficult and multifaceted problem. Although various estimation strategies have been proposed and are now well understood empirically, theoretical guarantees about their asymptotic behavior is rather sparse and most results are limited to the binary setting. We prove here theoretical guarantees in the valued settings. We show that under some mild conditions on the parameter space, and in an asymptotic regime where $log (d)/n$ and $log (n)/d$ tend to $0$ when $n$ and $d$ tend to infinity, (1) the maximum-likelihood estimate of the complete model (with known labels) is consistent and (2) the log-likelihood ratios are equivalent under the complete and observed (with unknown labels) models. This equivalence allows us to transfer the asymptotic consistency, and under mild conditions, asymptotic normality, to the maximum likelihood estimate under the observed model. Moreover, the variational estimator is also consistent and, under the same conditions, asymptotically normal.

Tomáš Rubín, Victor M. Panaretos.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1137--1210.

Abstract:
Functional time series analysis, whether based on time or frequency domain methodology, has traditionally been carried out under the assumption of complete observation of the constituent series of curves, assumed stationary. Nevertheless, as is often the case with independent functional data, it may well happen that the data available to the analyst are not the actual sequence of curves, but relatively few and noisy measurements per curve, potentially at different locations in each curve’s domain. Under this sparse sampling regime, neither the established estimators of the time series’ dynamics nor their corresponding theoretical analysis will apply. The subject of this paper is to tackle the problem of estimating the dynamics and of recovering the latent process of smooth curves in the sparse regime. Assuming smoothness of the latent curves, we construct a consistent nonparametric estimator of the series’ spectral density operator and use it to develop a frequency-domain recovery approach, that predicts the latent curve at a given time by borrowing strength from the (estimated) dynamic correlations in the series across time. This new methodology is seen to comprehensively outperform a naive recovery approach that would ignore temporal dependence and use only methodology employed in the i.i.d. setting and hinging on the lag zero covariance. Further to predicting the latent curves from their noisy point samples, the method fills in gaps in the sequence (curves nowhere sampled), denoises the data, and serves as a basis for forecasting. Means of providing corresponding confidence bands are also investigated. A simulation study interestingly suggests that sparse observation for a longer time period may provide better performance than dense observation for a shorter period, in the presence of smoothness. The methodology is further illustrated by application to an environmental data set on fair-weather atmospheric electricity, which naturally leads to a sparse functional time series.

Fabien Panloup, Samy Tindel, Maylis Varvenne.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1075--1136.

Abstract:
In this paper we consider the drift estimation problem for a general differential equation driven by an additive multidimensional fractional Brownian motion, under ergodic assumptions on the drift coefficient. Our estimation procedure is based on the identification of the invariant measure, and we provide consistency results as well as some information about the convergence rate. We also give some examples of coefficients for which the identifiability assumption for the invariant measure is satisfied.

Christophe A. N. Biscio, Nicolas Chenavier, Christian Hirsch, Anne Marie Svane.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 1024--1074.

Abstract:
We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. We analyze the power of tests derived from this statistic on simulated point patterns and compare its performance with global envelope tests. Finally, we apply the tests to a point pattern from an application context in neuroscience. As the main methodological contribution, we derive sufficient conditions for a functional central limit theorem on bounded persistent Betti numbers of point processes with exponential decay of correlations.

Xianzheng Huang, Haiming Zhou.

Source: Electronic Journal of Statistics, Volume 14, Number 1, 970--1023.

Abstract:
We consider estimating the density of a response conditioning on an error-prone covariate. Motivated by two existing kernel density estimators in the absence of covariate measurement error, we propose a method to correct the existing estimators for measurement error. Asymptotic properties of the resultant estimators under different types of measurement error distributions are derived. Moreover, we adjust bandwidths readily available from existing bandwidth selection methods developed for error-free data to obtain bandwidths for the new estimators. Extensive simulation studies are carried out to compare the proposed estimators with naive estimators that ignore measurement error, which also provide empirical evidence for the effectiveness of the proposed bandwidth selection methods. A real-life data example is used to illustrate implementation of these methods under practical scenarios. An R package, lpme, is developed for implementing all considered methods, which we demonstrate via an R code example in Appendix B.2.