Logspline Density Estimation under Censoring and Truncation

ABSTRACT. In this paper we consider logspline density estimation for data that may be left-truncated or right-censored. For randomly left-truncated and right-censored data the product-limit estimator is known to be a consistent estimator of the survivor function, having a faster rate of convergence than many density estimators. The product-limit estimator and B-splines are used to construct the logspline density estimate for possibly censored or truncated data. Rates of convergence are established when the log-density function is assumed to be in a Besov space. An algorithm involving a procedure similar to maximum likelihood, stepwise knot addition, and stepwise knot deletion is proposed for the estimation of the density function based upon sample data. Numerical examples are used to show the finite-sample performance of inference based on the logspline density estimation.     

On the asymptotic normality of the L1- and L2-errors in histogram density estimation

The L₁ and L₂-errors of the histogram estimate of a density f from a sample X₁,X₂,…,Xn using a cubic partition are shown to be asymptotically normal without any unnecessary conditions imposed on the density f. The asymptotic variances are shown to depend on f only through the corresponding norm of f. From this follows the asymptotic null distribution of a goodness-of-fit test based on the total variation distance, introduced by Györfi and van der Meulen (1991). This note uses the idea of partial inversion for obtaining characteristic functions of conditional distributions, which goes back at least to Bartlett (1938).     

Logspline Deconvolution in Besov Space

ABSTRACT. In this paper we consider logspline density estimation for random variables which are contaminated with random noise. In the logspline density estimation for data without noise, the logarithm of an unknown density function is estimated by a polynomial spline, the unknown parameters of which are given by maximum likelihood. When noise is present, B-splines and the Fourier inversion formula are used to construct the logspline density estimator of the unknown density function. Rates of convergence are established when the log-density function is assumed to be in a Besov space. It is shown that convergence rates depend on the smoothness of the density function and the decay rate of the characteristic function of the noise. Simulated data are used to show the finite-sample performance of inference based on the logspline density estimation.     

EMPIRICAL LIKELIHOOD-BASED KERNEL DENSITY ESTIMATION

This paper considers the estimation of a probability density function when extra distributional information is available (e.g. the mean of the distribution is known or the variance is a known function of the mean). The standard kernel method cannot exploit such extra information systematically as it uses an equal probability weight n^-1 at each data point. The paper suggests using empirical likelihood to choose the probability weights under constraints formulated from the extra distributional information. An empirical likelihood-based kernel density estimator is given by replacing n^-1 by the empirical likelihood weights, and has these advantages: it makes systematic use of the extra information, it is able to reflect the extra characteristics of the density function, and its variance is smaller than that of the standard kernel density estimator.     

An extended Gompertz-Makeham distribution with application to lifetime data

In this paper, we propose an extension of the Gompertz-Makeham distribution. This distribution is called the transmuted Gompertz-Makeham (TGM). The new model which can handle bathtub-shaped, increasing, increasing-constant and constant hazard rate functions. This property makes TGM is useful in survival analysis. Various statistical and reliability measures of the model are obtained, including hazard rate function, moments, moment generating function (mgf), quantile function, random number generating, skewness, kurtosis, conditional moments, mean deviations, Bonferroni curve, Lorenz curve, Gini index, mean inactivity time, mean residual lifetime and stochastic ordering; we also obtain the density of the ith order statistic. Estimation of the model parameters is justified by the method of maximum likelihood. An application to real data demonstrates that the TGM distribution can provides a better fit than some other very well known distributions.     

Adaptive Warped Kernel Estimators

In this work, we develop a method of adaptive non‐parametric estimation, based on ‘warped’ kernels. The aim is to estimate a real‐valued function s from a sample of random couples (X,Y). We deal with transformed data (Φ(X),Y), with Φ a one‐to‐one function, to build a collection of kernel estimators. The data‐driven bandwidth selection is performed with a method inspired by Goldenshluger and Lepski (Ann. Statist., 39, 2011, 1608). The method permits to handle various problems such as additive and multiplicative regression, conditional density estimation, hazard rate estimation based on randomly right‐censored data, and cumulative distribution function estimation from current‐status data. The interest is threefold. First, the squared‐bias/variance trade‐off is automatically realized. Next, non‐asymptotic risk bounds are derived. Lastly, the estimator is easily computed, thanks to its simple expression: a short simulation study is presented.     

Estimation of scale functions to model heteroscedasticity by regularised kernel-based quantile methods

A main goal of regression is to derive statistical conclusions on the conditional distribution of the output variable Y given the input values x. Two of the most important characteristics of a single distribution are location and scale. Regularised kernel methods (RKMs) – also called support vector machines in a wide sense – are well established to estimate location functions like the conditional median or the conditional mean. We investigate the estimation of scale functions by RKMs when the conditional median is unknown, too. Estimation of scale functions is important, e.g. to estimate the volatility in finance. We consider the median absolute deviation (MAD) and the interquantile range as measures of scale. Our main result shows the consistency of MAD-type RKMs.     

On maximum likelihood estimation of the binomial parameter n

This paper gives a necessary and sufficient condition for the existence of a finite conditional maximum-likelihood estimate for the binomial parameter n. Upper bounds for the conditional and beta-binomial maximum-likelihood estimators are derived. An example is given to show that the conditional likelihood function and the beta-binomial likelihood function may not be unimodal.     

A Logspline Estimation for a Linear Regression Model with an Interval-Censored Continuous Covariate

The study of a linear regression model with an interval-censored covariate, which was motivated by an acquired immunodeficiency syndrome (AIDS) clinical trial, was first proposed by Gómez et al. They developed a likelihood approach, together with a two-step conditional algorithm, to estimate the regression coefficients in the model. However, their method is inapplicable when the interval-censored covariate is continuous. In this article, we propose a novel and fast method to treat the continuous interval-censored covariate. By using logspline density estimation, we impute the interval-censored covariate with a conditional expectation. Then, the ordinary least-squares method is applied to the linear regression model with the imputed covariate. To assess the performance of the proposed method, we compare our imputation with the midpoint imputation and the semiparametric hierarchical method via simulations. Furthermore, an application to the AIDS clinical trial is presented.     

Non-parametric estimation of the conditional mode

The problem of estimating the mode of a conditional probability density function is considered. It is shown that under some regularity conditions the estimate of the conditional mode obtained by maximizing a kernel estimate of the conditional probability density function is strongly consistent and asymptotically normally distributed.     

Linear. empirical bayes estimation in the case of the wishart distribution

We consider independent pairs (X₁,∑₁), (X₂,∑₂),…,(X_n∑_n), where each Si is distributed according to some unknown density function g(∑) and, given ∑_i = ∑, X has a conditional density function g(x|∑) of the Wishart type. In each pair, the first component is observable but the second is not. After the (n + l)-th observation X_n+i is obtained, the objective is to estimate ∑ _n+i corresponding to X_n+i. This estimator is called an empirical Bayes (EB) estimator of ∑. We construct a linear EB estimator of ∑ and examine its precision.     

Optimal bandwidth matrices in functional principal component analysis of density functions

In order to explore and compare a finite number T of data sets by applying functional principal component analysis (FPCA) to the T associated probability density functions, we estimate these density functions by using the multivariate kernel method. The data set sizes being fixed, we study the behaviour of this FPCA under the assumption that all the bandwidth matrices used in the estimation of densities are proportional to a common parameter h and proportional to either the variance matrices or the identity matrix. In this context, we propose a selection criterion of the parameter h which depends only on the data and the FPCA method. Then, on simulated examples, we compare the quality of approximation of the FPCA when the bandwidth matrices are selected using either the previous criterion or two other classical bandwidth selection methods, that is, a plug-in or a cross-validation method.     

On the strong Kotz approximation of Dirichlet random vectors

Let (X ₁, X ₂) be a bivariate L _p -norm generalized symmetrized Dirichlet (LpGSD) random vector with parameters α₁,α₂. If p=α₁=α₂=2, then (X ₁, X ₂) is a spherical random vector. The estimation of the conditional distribution of Z _u *:=X ₂ | X ₁>u for u large is of some interest in statistical applications. When (X ₁, X ₂) is a spherical random vector with associated random radius in the Gumbel max-domain of attraction, the distribution of Z _u * can be approximated by a Gaussian distribution. Surprisingly, the same Gaussian approximation holds also for Z _u :=X ₂| X ₁=u. In this paper, we are interested in conditional limit results in terms of convergence of the density functions considering a d-dimensional LpGSD random vector. Stating our results for the bivariate setup, we show that the density function of Z _u * and Z _u can be approximated by the density function of a Kotz type I LpGSD distribution, provided that the associated random radius has distribution function in the Gumbel max-domain of attraction. Further, we present two applications concerning the asymptotic behaviour of concomitants of order statistics of bivariate Dirichlet samples and the estimation of the conditional quantile function.     

Selection of linear mixed-effects models for clustered data

We consider model selection for linear mixed-effects models with clustered structure, where conditional Kullback–Leibler (CKL) loss is applied to measure the efficiency of the selection. We estimate the CKL loss by substituting the empirical best linear unbiased predictors (EBLUPs) into random effects with model parameters estimated by maximum likelihood. Although the BLUP approach is commonly used in predicting random effects and future observations, selecting random effects to achieve asymptotic loss efficiency concerning CKL loss is challenging and has not been well studied. In this paper, we propose addressing this difficulty using a conditional generalized information criterion (CGIC) with two tuning parameters. We further consider a challenging but practically relevant situation where the number, $m$, of clusters does not go to infinity with the sample size. Hence the random-effects variances are not consistently estimable. We show that via a novel decomposition of the CKL risk, the CGIC achieves consistency and asymptotic loss efficiency, whether $m$ is fixed or increases to infinity with the sample size. We also conduct numerical experiments to illustrate the theoretical findings.     

A New Regression Model: Modal Linear Regression

The mode of a distribution provides an important summary of data and is often estimated on the basis of some non‐parametric kernel density estimator. This article develops a new data analysis tool called modal linear regression in order to explore high‐dimensional data. Modal linear regression models the conditional mode of a response Y given a set of predictors x as a linear function of x . Modal linear regression differs from standard linear regression in that standard linear regression models the conditional mean (as opposed to mode) of Y as a linear function of x . We propose an expectation–maximization algorithm in order to estimate the regression coefficients of modal linear regression. We also provide asymptotic properties for the proposed estimator without the symmetric assumption of the error density. Our empirical studies with simulated data and real data demonstrate that the proposed modal regression gives shorter predictive intervals than mean linear regression, median linear regression and MM‐estimators.     

Weak Convergence of the Regularization Path in Penalized M‐Estimation

Abstract. We consider a function defined as the pointwise minimization of a doubly index random process. We are interested in the weak convergence of the minimizer in the space of bounded functions. Such convergence results can be applied in the context of penalized M‐estimation, that is, when the random process to minimize is expressed as a goodness‐of‐fit term plus a penalty term multiplied by a penalty weight. This weight is called the regularization parameter and the minimizing function the regularization path. The regularization path can be seen as a collection of estimators indexed by the regularization parameter. We obtain a consistency result and a central limit theorem for the regularization path in a functional sense. Various examples are provided, including the ?¹‐regularization path for general linear models, the ?¹‐ or ?²‐regularization path of the least absolute deviation regression and the Akaike information criterion.     

Specification testing in nonparametric AR‐ARCH models

In this paper, an autoregressive time series model with conditional heteroscedasticity is considered, where both conditional mean and conditional variance function are modeled nonparametrically. Tests for the model assumption of independence of innovations from past time series values are suggested. Tests based on weighted L²‐distances of empirical characteristic functions are considered as well as a Cramér–von Mises‐type test. The asymptotic distributions under the null hypothesis of independence are derived, and the consistency against fixed alternatives is shown. A smooth autoregressive residual bootstrap procedure is suggested, and its performance is shown in a simulation study.     

Interpreting the Principal Component Analysis of Multivariate Density Functions

Functional principal component analysis (FPCA) as a reduction data technique of a finite number T of functions can be used to identify the dominant modes of variation of numeric three-way data.

We carry out the FPCA on multidimensional probability density functions, relate this method to other standard methods and define its centered or standardized versions. Grounded on the relationship between FPCA of densities, FPCA of their corresponding characteristic functions, PCA of the MacLaurin expansions of these characteristic functions and dual STATIS method applied to their variance matrices, we propose a method for interpreting the results of the FPCA of densities. This method is based on the investigations of the relationships between the scores of the FPCA and the moments associated to the densities.

The method is illustrated using known Gaussian densities. In practice, FPCA of densities deals with observations of multidimensional variables on T occasions. These observations can be used to estimate the T associated densities (i) by estimating the parameters of these densities, assuming that they are Gaussian, or (ii) by using the Gaussian kernel method and choosing the matrix bandwidth by the normal reference rule. Thereafter, FPCA estimate is derived from these estimates and the interpretation method is carried out to explore the dominant modes of variation of the types of three-way data encountered in sensory analysis and archaeology.     

Lessons From Quantile Panel Estimation of the Environmental Kuznets Curve

We employ quantile regression fixed effects models to estimate the income-pollution relationship on NO _x (nitrogen oxide) and SO ₂ (sulfur dioxide) using U.S. data. Conditional median results suggest that conditional mean methods provide too optimistic estimates about emissions reduction for NO _x , while the opposite is found for SO ₂. Deleting outlier states reverses the absence of a turning point for SO ₂ in the conditional mean model, while the conditional median model is robust to them. We also document the relationship's sensitivity to including additional covariates for NO _x , and undertake simulations to shed light on some estimation issues of the methods employed.     

Adaptive Deconvolution on the Non‐negative Real Line

In this paper, we consider the problem of adaptive density or survival function estimation in an additive model defined by Z=X+Y with X independent of Y, when both random variables are non‐negative. This model is relevant, for instance, in reliability fields where we are interested in the failure time of a certain material that cannot be isolated from the system it belongs. Our goal is to recover the distribution of X (density or survival function) through n observations of Z, assuming that the distribution of Y is known. This issue can be seen as the classical statistical problem of deconvolution that has been tackled in many cases using Fourier‐type approaches. Nonetheless, in the present case, the random variables have the particularity to be supported. Knowing that, we propose a new angle of attack by building a projection estimator with an appropriate Laguerre basis. We present upper bounds on the mean squared integrated risk of our density and survival function estimators. We then describe a non‐parametric data‐driven strategy for selecting a relevant projection space. The procedures are illustrated with simulated data and compared with the performances of a more classical deconvolution setting using a Fourier approach. Our procedure achieves faster convergence rates than Fourier methods for estimating these functions.