Akio Fujiwara, Koichi Yamagata.

Source: Bernoulli, Volume 26, Number 3, 2105--2142.

Abstract:
We herein develop a theory of contiguity in the quantum domain based upon a novel quantum analogue of the Lebesgue decomposition. The theory thus formulated is pertinent to the weak quantum local asymptotic normality introduced in the previous paper [Yamagata, Fujiwara, and Gill, Ann. Statist. 41 (2013) 2197–2217], yielding substantial enlargement of the scope of quantum statistics.

Marie Ernst, Gesine Reinert, Yvik Swan.

Source: Bernoulli, Volume 26, Number 3, 2051--2081.

Abstract:
We propose probabilistic representations for inverse Stein operators (i.e., solutions to Stein equations) under general conditions; in particular, we deduce new simple expressions for the Stein kernel. These representations allow to deduce uniform and nonuniform Stein factors (i.e., bounds on solutions to Stein equations) and lead to new covariance identities expressing the covariance between arbitrary functionals of an arbitrary univariate target in terms of a weighted covariance of the derivatives of the functionals. Our weights are explicit, easily computable in most cases and expressed in terms of objects familiar within the context of Stein’s method. Applications of the Cauchy–Schwarz inequality to these weighted covariance identities lead to sharp upper and lower covariance bounds and, in particular, weighted Poincaré inequalities. Many examples are given and, in particular, classical variance bounds due to Klaassen, Brascamp and Lieb or Otto and Menz are corollaries. Connections with more recent literature are also detailed.

Adriana Coutinho, Rodrigo Lambert, Jérôme Rousseau.

Source: Bernoulli, Volume 26, Number 3, 2021--2050.

Abstract:
We investigate the length of the longest common substring for encoded sequences and its asymptotic behaviour. The main result is a strong law of large numbers for a re-scaled version of this quantity, which presents an explicit relation with the Rényi entropy of the source. We apply this result to the zero-inflated contamination model and the stochastic scrabble. In the case of dynamical systems, this problem is equivalent to the shortest distance between two observed orbits and its limiting relationship with the correlation dimension of the pushforward measure. An extension to the shortest distance between orbits for random dynamical systems is also provided.

Arnak S. Dalalyan, Lionel Riou-Durand.

Source: Bernoulli, Volume 26, Number 3, 1956--1988.

Abstract:
Langevin diffusion processes and their discretizations are often used for sampling from a target density. The most convenient framework for assessing the quality of such a sampling scheme corresponds to smooth and strongly log-concave densities defined on $mathbb{R}^{p}$. The present work focuses on this framework and studies the behavior of the Monte Carlo algorithm based on discretizations of the kinetic Langevin diffusion. We first prove the geometric mixing property of the kinetic Langevin diffusion with a mixing rate that is optimal in terms of its dependence on the condition number. We then use this result for obtaining improved guarantees of sampling using the kinetic Langevin Monte Carlo method, when the quality of sampling is measured by the Wasserstein distance. We also consider the situation where the Hessian of the log-density of the target distribution is Lipschitz-continuous. In this case, we introduce a new discretization of the kinetic Langevin diffusion and prove that this leads to a substantial improvement of the upper bound on the sampling error measured in Wasserstein distance.

Galatia Cleanthous, Athanasios G. Georgiadis, Gerard Kerkyacharian, Pencho Petrushev, Dominique Picard.

Source: Bernoulli, Volume 26, Number 3, 1832--1862.

Abstract:
We consider the problem of estimating the density of observations taking values in classical or nonclassical spaces such as manifolds and more general metric spaces. Our setting is quite general but also sufficiently rich in allowing the development of smooth functional calculus with well localized spectral kernels, Besov regularity spaces, and wavelet type systems. Kernel and both linear and nonlinear wavelet density estimators are introduced and studied. Convergence rates for these estimators are established and discussed.

Xin Bing, Florentina Bunea, Marten Wegkamp.

Source: Bernoulli, Volume 26, Number 3, 1765--1796.

Abstract:
Topic models have become popular for the analysis of data that consists in a collection of n independent multinomial observations, with parameters $N_{i}inmathbb{N}$ and $Pi_{i}in[0,1]^{p}$ for $i=1,ldots,n$. The model links all cell probabilities, collected in a $p imes n$ matrix $Pi$, via the assumption that $Pi$ can be factorized as the product of two nonnegative matrices $Ain[0,1]^{p imes K}$ and $Win[0,1]^{K imes n}$. Topic models have been originally developed in text mining, when one browses through $n$ documents, based on a dictionary of $p$ words, and covering $K$ topics. In this terminology, the matrix $A$ is called the word-topic matrix, and is the main target of estimation. It can be viewed as a matrix of conditional probabilities, and it is uniquely defined, under appropriate separability assumptions, discussed in detail in this work. Notably, the unique $A$ is required to satisfy what is commonly known as the anchor word assumption, under which $A$ has an unknown number of rows respectively proportional to the canonical basis vectors in $mathbb{R}^{K}$. The indices of such rows are referred to as anchor words. Recent computationally feasible algorithms, with theoretical guarantees, utilize constructively this assumption by linking the estimation of the set of anchor words with that of estimating the $K$ vertices of a simplex. This crucial step in the estimation of $A$ requires $K$ to be known, and cannot be easily extended to the more realistic set-up when $K$ is unknown. This work takes a different view on anchor word estimation, and on the estimation of $A$. We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates $K$ from the observed data. We derive new finite sample minimax lower bounds for the estimation of $A$, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any $n,N_{i},p$ and $K$, and both $p$ and $K$ are allowed to increase with $n$, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics $K$, while we provide the competing methods with the correct value in our simulations.

Cristina Butucea, Amandine Dubois, Martin Kroll, Adrien Saumard.

Source: Bernoulli, Volume 26, Number 3, 1727--1764.

Abstract:
We address the problem of non-parametric density estimation under the additional constraint that only privatised data are allowed to be published and available for inference. For this purpose, we adopt a recent generalisation of classical minimax theory to the framework of local $alpha$-differential privacy and provide a lower bound on the rate of convergence over Besov spaces $mathcal{B}^{s}_{pq}$ under mean integrated $mathbb{L}^{r}$-risk. This lower bound is deteriorated compared to the standard setup without privacy, and reveals a twofold elbow effect. In order to fulfill the privacy requirement, we suggest adding suitably scaled Laplace noise to empirical wavelet coefficients. Upper bounds within (at most) a logarithmic factor are derived under the assumption that $alpha$ stays bounded as $n$ increases: A linear but non-adaptive wavelet estimator is shown to attain the lower bound whenever $pgeq r$ but provides a slower rate of convergence otherwise. An adaptive non-linear wavelet estimator with appropriately chosen smoothing parameters and thresholding is shown to attain the lower bound within a logarithmic factor for all cases.

Arun K. Kuchibhotla, Rohit K. Patra.

Source: Bernoulli, Volume 26, Number 2, 1587--1618.

Abstract:
We consider estimation and inference in a single index regression model with an unknown but smooth link function. In contrast to the standard approach of using kernels or regression splines, we use smoothing splines to estimate the smooth link function. We develop a method to compute the penalized least squares estimators (PLSEs) of the parametric and the nonparametric components given independent and identically distributed (i.i.d.) data. We prove the consistency and find the rates of convergence of the estimators. We establish asymptotic normality under mild assumption and prove asymptotic efficiency of the parametric component under homoscedastic errors. A finite sample simulation corroborates our asymptotic theory. We also analyze a car mileage data set and a Ozone concentration data set. The identifiability and existence of the PLSEs are also investigated.

Amir Najafi, Seyed Abolfazl Motahari, Hamid R. Rabiee.

Source: Bernoulli, Volume 26, Number 2, 1535--1559.

Abstract:
A Bernoulli Mixture Model (BMM) is a finite mixture of random binary vectors with independent dimensions. The problem of clustering BMM data arises in a variety of real-world applications, ranging from population genetics to activity analysis in social networks. In this paper, we analyze the clusterability of BMMs from a theoretical perspective, when the number of clusters is unknown. In particular, we stipulate a set of conditions on the sample complexity and dimension of the model in order to guarantee the Probably Approximately Correct (PAC)-clusterability of a dataset. To the best of our knowledge, these findings are the first non-asymptotic bounds on the sample complexity of learning or clustering BMMs.

Denis Talay, Milica Tomašević.

Source: Bernoulli, Volume 26, Number 2, 1323--1353.

Abstract:
In this paper, we analyze a stochastic interpretation of the one-dimensional parabolic–parabolic Keller–Segel system without cut-off. It involves an original type of McKean–Vlasov interaction kernel. At the particle level, each particle interacts with all the past of each other particle by means of a time integrated functional involving a singular kernel. At the mean-field level studied here, the McKean–Vlasov limit process interacts with all the past time marginals of its probability distribution in a similarly singular way. We prove that the parabolic–parabolic Keller–Segel system in the whole Euclidean space and the corresponding McKean–Vlasov stochastic differential equation are well-posed for any values of the parameters of the model.

Emanuele Dolera, Stefano Favaro.

Source: Bernoulli, Volume 26, Number 2, 1294--1322.

Abstract:
Given a sequence ${X_{n}}_{ngeq 1}$ of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that $frac{1}{n}sum_{i=1}^{n}X_{i}stackrel{a.s.}{longrightarrow }Y$ for a suitable random variable $Y:Omega ightarrow [0,1]$ satisfying $mathsf{P}[X_{1}=x_{1},dots ,X_{n}=x_{n}|Y]=Y^{sum_{i=1}^{n}x_{i}}(1-Y)^{n-sum_{i=1}^{n}x_{i}}$. In this paper, we study the rate of convergence in law of $frac{1}{n}sum_{i=1}^{n}X_{i}$ to $Y$ under the Kolmogorov distance. After showing that a rate of the type of $1/n^{alpha }$ can be obtained for any index $alpha in (0,1]$, we find a sufficient condition on the distribution of $Y$ for the achievement of the optimal rate of convergence, that is $1/n$. Besides extending and strengthening recent results under the weaker Wasserstein distance, our main result weakens the regularity hypotheses on $Y$ in the context of the Hausdorff moment problem.

Nina Gantert, Markus Heydenreich, Timo Hirscher.

Source: Bernoulli, Volume 26, Number 2, 1269--1293.

Abstract:
We investigate a model for opinion dynamics, where individuals (modeled by vertices of a graph) hold certain abstract opinions. As time progresses, neighboring individuals interact with each other, and this interaction results in a realignment of opinions closer towards each other. This mechanism triggers formation of consensus among the individuals. Our main focus is on strong consensus (i.e., global agreement of all individuals) versus weak consensus (i.e., local agreement among neighbors). By extending a known model to a more general opinion space, which lacks a “central” opinion acting as a contraction point, we provide an example of an opinion formation process on the one-dimensional lattice $mathbb{Z}$ with weak consensus but no strong consensus.

Michael Schweinberger.

Source: Bernoulli, Volume 26, Number 2, 1205--1233.

Abstract:
We consider the challenging problem of statistical inference for exponential-family random graph models based on a single observation of a random graph with complex dependence. To facilitate statistical inference, we consider random graphs with additional structure in the form of block structure. We have shown elsewhere that when the block structure is known, it facilitates consistency results for $M$-estimators of canonical and curved exponential-family random graph models with complex dependence, such as transitivity. In practice, the block structure is known in some applications (e.g., multilevel networks), but is unknown in others. When the block structure is unknown, the first and foremost question is whether it can be recovered with high probability based on a single observation of a random graph with complex dependence. The main consistency results of the paper show that it is possible to do so under weak dependence and smoothness conditions. These results confirm that exponential-family random graph models with block structure constitute a promising direction of statistical network analysis.

Chao Gao.

Source: Bernoulli, Volume 26, Number 2, 1139--1170.

Abstract:
This paper studies robust regression in the settings of Huber’s $epsilon$-contamination models. We consider estimators that are maximizers of multivariate regression depth functions. These estimators are shown to achieve minimax rates in the settings of $epsilon$-contamination models for various regression problems including nonparametric regression, sparse linear regression, reduced rank regression, etc. We also discuss a general notion of depth function for linear operators that has potential applications in robust functional linear regression.

Ester Mariucci, Kolyan Ray, Botond Szabó.

Source: Bernoulli, Volume 26, Number 2, 1070--1097.

Abstract:
The estimation of a log-concave density on $mathbb{R}$ is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We further present computationally more feasible approximations and both an empirical and hierarchical Bayes approach. All priors are illustrated numerically via simulations.

Sumit Mukherjee.

Source: Bernoulli, Volume 26, Number 2, 1016--1043.

Abstract:
A sufficient criterion for “non-degeneracy” is given for Exponential Random Graph Models on sparse graphs with sufficient statistics which are functions of the degree sequence. This criterion explains why statistics such as alternating $k$-star are non-degenerate, whereas subgraph counts are degenerate. It is further shown that this criterion is “almost” tight. Existence of consistent estimates is then proved for non-degenerate Exponential Random Graph Models.

Leif Döring, Philip Weissmann.

Source: Bernoulli, Volume 26, Number 2, 980--1015.

Abstract:
Conditioning stable Lévy processes on zero probability events recently became a tractable subject since several explicit formulas emerged from a deep analysis using the Lamperti transformations for self-similar Markov processes. In this article, we derive new harmonic functions and use them to explain how to condition stable processes to hit continuously a compact interval from the outside.

Gilles Bonnet, Nicolas Chenavier.

Source: Bernoulli, Volume 26, Number 2, 948--979.

Abstract:
We investigate the maximal degree in a Poisson–Delaunay graph in $mathbf{R}^{d}$, $dgeq 2$, over all nodes in the window $mathbf{W}_{ ho }:= ho^{1/d}[0,1]^{d}$ as $ ho $ goes to infinity. The exact order of this maximum is provided in any dimension. In the particular setting $d=2$, we show that this quantity is concentrated on two consecutive integers with high probability. A weaker version of this result is discussed when $dgeq 3$.

Christian Hirsch, Christian Mönch.

Source: Bernoulli, Volume 26, Number 2, 927--947.

Abstract:
This paper considers two asymptotic properties of a spatial preferential-attachment model introduced by E. Jacob and P. Mörters (In Algorithms and Models for the Web Graph (2013) 14–25 Springer). First, in a regime of strong linear reinforcement, we show that typical distances are at most of doubly-logarithmic order. Second, we derive a large deviation principle for the empirical neighbourhood structure and express the rate function as solution to an entropy minimisation problem in the space of stationary marked point processes.

Nhat Ho, XuanLong Nguyen, Ya’acov Ritov.

Source: Bernoulli, Volume 26, Number 2, 828--857.

Abstract:
In finite mixture models, apart from underlying mixing measure, true kernel density function of each subpopulation in the data is, in many scenarios, unknown. Perhaps the most popular approach is to choose some kernel functions that we empirically believe our data are generated from and use these kernels to fit our models. Nevertheless, as long as the chosen kernel and the true kernel are different, statistical inference of mixing measure under this setting will be highly unstable. To overcome this challenge, we propose flexible and efficient robust estimators of the mixing measure in these models, which are inspired by the idea of minimum Hellinger distance estimator, model selection criteria, and superefficiency phenomenon. We demonstrate that our estimators consistently recover the true number of components and achieve the optimal convergence rates of parameter estimation under both the well- and misspecified kernel settings for any fixed bandwidth. These desirable asymptotic properties are illustrated via careful simulation studies with both synthetic and real data.

Richard A. Davis, Mikkel Slot Nielsen, Victor Rohde.

Source: Bernoulli, Volume 26, Number 2, 799--827.

Abstract:
In this paper, we introduce a model, the stochastic fractional delay differential equation (SFDDE), which is based on the linear stochastic delay differential equation and produces stationary processes with hyperbolically decaying autocovariance functions. The model departs from the usual way of incorporating this type of long-range dependence into a short-memory model as it is obtained by applying a fractional filter to the drift term rather than to the noise term. The advantages of this approach are that the corresponding long-range dependent solutions are semimartingales and the local behavior of the sample paths is unaffected by the degree of long memory. We prove existence and uniqueness of solutions to the SFDDEs and study their spectral densities and autocovariance functions. Moreover, we define a subclass of SFDDEs which we study in detail and relate to the well-known fractionally integrated CARMA processes. Finally, we consider the task of simulating from the defining SFDDEs.

Jing Lei.

Source: Bernoulli, Volume 26, Number 1, 767--798.

Abstract:
We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under Bernstein-type or log Sobolev-type conditions.

Elena Issoglio, Francesco Russo.

Source: Bernoulli, Volume 26, Number 1, 728--766.

Abstract:
In this paper, we investigate BSDEs where the driver contains a distributional term (in the sense of generalised functions) and derive general Feynman–Kac formulae related to these BSDEs. We introduce an integral operator to give sense to the equation and then we show the existence of a strong solution employing results on a related PDE. Due to the irregularity of the driver, the $Y$-component of a couple $(Y,Z)$ solving the BSDE is not necessarily a semimartingale but a weak Dirichlet process.

Mingjie Liang, René L. Schilling, Jian Wang.

Source: Bernoulli, Volume 26, Number 1, 664--693.

Abstract:
We present a general method to construct couplings of stochastic differential equations driven by Lévy noise in terms of coupling operators. This approach covers both coupling by reflection and refined basic coupling which are often discussed in the literature. As applications, we prove regularity results for the transition semigroups and obtain successful couplings for the solutions to stochastic differential equations driven by additive Lévy noise.

Keisuke Yano, Kengo Kato.

Source: Bernoulli, Volume 26, Number 1, 616--641.

Abstract:
In this paper, we study frequentist coverage errors of Bayesian credible sets for an approximately linear regression model with (moderately) high dimensional regressors, where the dimension of the regressors may increase with but is smaller than the sample size. Specifically, we consider quasi-Bayesian inference on the slope vector under the quasi-likelihood with Gaussian error distribution. Under this setup, we derive finite sample bounds on frequentist coverage errors of Bayesian credible rectangles. Derivation of those bounds builds on a novel Berry–Esseen type bound on quasi-posterior distributions and recent results on high-dimensional CLT on hyperrectangles. We use this general result to quantify coverage errors of Castillo–Nickl and $L^{infty}$-credible bands for Gaussian white noise models, linear inverse problems, and (possibly non-Gaussian) nonparametric regression models. In particular, we show that Bayesian credible bands for those nonparametric models have coverage errors decaying polynomially fast in the sample size, implying advantages of Bayesian credible bands over confidence bands based on extreme value theory.

Eric Beutner, Laurent Bordes, Laurent Doyen.

Source: Bernoulli, Volume 26, Number 1, 557--586.

Abstract:
Virtual age models are very useful to analyse recurrent events. Among the strengths of these models is their ability to account for treatment (or intervention) effects after an event occurrence. Despite their flexibility for modeling recurrent events, the number of applications is limited. This seems to be a result of the fact that in the semiparametric setting all the existing results assume the virtual age function that describes the treatment (or intervention) effects to be known. This shortcoming can be overcome by considering semiparametric virtual age models with parametrically specified virtual age functions. Yet, fitting such a model is a difficult task. Indeed, it has recently been shown that for these models the standard profile likelihood method fails to lead to consistent estimators. Here we show that consistent estimators can be constructed by smoothing the profile log-likelihood function appropriately. We show that our general result can be applied to most of the relevant virtual age models of the literature. Our approach shows that empirical process techniques may be a worthwhile alternative to martingale methods for studying asymptotic properties of these inference methods. A simulation study is provided to illustrate our consistency results together with an application to real data.

Xiucai Ding.

Source: Bernoulli, Volume 26, Number 1, 387--417.

Abstract:
We consider the recovery of a low rank $M imes N$ matrix $S$ from its noisy observation $ ilde{S}$ in the high dimensional framework when $M$ is comparable to $N$. We propose two efficient estimators for $S$ under two different regimes. Our analysis relies on the local asymptotics of the eigenstructure of large dimensional rectangular matrices with finite rank perturbation. We derive the convergent limits and rates for the singular values and vectors for such matrices.

Luca M. Giordano, Maria Jolis, Lluís Quer-Sardanyons.

Source: Bernoulli, Volume 26, Number 1, 352--386.

Abstract:
In this article, we consider the quasi-linear stochastic wave and heat equations on the real line and with an additive Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $Hin (0,1)$. The drift term is assumed to be globally Lipschitz. We prove that the solution of each of the above equations is continuous in terms of the index $H$, with respect to the convergence in law in the space of continuous functions.

Tobias Zwingmann, Hajo Holzmann.

Source: Bernoulli, Volume 26, Number 1, 323--351.

Abstract:
We show weak convergence of quantile and expectile processes to Gaussian limit processes in the space of bounded functions endowed with an appropriate semimetric which is based on the concepts of epi- and hypo- convergence as introduced in A. Bücher, J. Segers and S. Volgushev (2014), ‘ When Uniform Weak Convergence Fails: Empirical Processes for Dependence Functions and Residuals via Epi- and Hypographs ’, Annals of Statistics 42 . We impose assumptions for which it is known that weak convergence with respect to the supremum norm generally fails to hold. For quantiles, we consider stationary observations, where the marginal distribution function is assumed to be strictly increasing and continuous except for finitely many points and to admit strictly positive – possibly infinite – left- and right-sided derivatives. For expectiles, we focus on independent and identically distributed (i.i.d.) observations. Only a finite second moment and continuity at the boundary points but no further smoothness properties of the distribution function are required. We also show consistency of the bootstrap for this mode of convergence in the i.i.d. case for quantiles and expectiles.

Eduard Belitser, Nurzhan Nurushev.

Source: Bernoulli, Volume 26, Number 1, 191--225.

Abstract:
In the general signal$+$noise (allowing non-normal, non-independent observations) model, we construct an empirical Bayes posterior which we then use for uncertainty quantification for the unknown, possibly sparse, signal. We introduce a novel excessive bias restriction (EBR) condition, which gives rise to a new slicing of the entire space that is suitable for uncertainty quantification. Under EBR and some mild exchangeable exponential moment condition on the noise, we establish the local (oracle) optimality of the proposed confidence ball. Without EBR, we propose another confidence ball of full coverage, but its radius contains an additional $sigma n^{1/4}$-term. In passing, we also get the local optimal results for estimation , posterior contraction problems, and the problem of weak recovery of sparsity structure . Adaptive minimax results (also for the estimation and posterior contraction problems) over various sparsity classes follow from our local results.

Michael V. Boutsikas, Eutichia Vaggelatou

Source: Bernoulli, Volume 16, Number 2, 301--330.

Abstract:
Let X 1 , X 2 , …, X n be a sequence of independent or locally dependent random variables taking values in ℤ + . In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum ∑ i =1 n X i and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This “smoothness factor” is of order O( σ −2 ), according to a heuristic argument, where σ 2 denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration.

Braddock, William, 1798-1869 -- Correspondence.

Hewish Henry -- Family.

South Australia -- Genealogy.

Klemm (Family)

King's Orphan Schools (New Town, Tas.)

Cheesman, John -- Family.

Traeger, Ernst Wilhelm, 1805-1874.

Huppatz (Family).

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