A consistent nonparametric density estimator for the deconvolution problem

The problem of nonparametric estimation of a probability density function when the sample observations are contaminated with random noise is studied. A particular estimator f?_n(x) is proposed which uses kernel-density and deconvolution techniques. The estimator f?_n(x) is shown to be uniformly consistent, and its appearance and properties are affected by constants M_n and h_n which the user may choose. The optimal choices of M_n and h_n depend on the sample size n, the noise distribution, and the true distribution which is being estimated. Particular selections for M_n and h_n which minimize upper-bound functions of the mean squared error for f?_n(x) are recommended.     

Bivariate uniform deconvolution

We construct a density estimator in the bivariate uniform deconvolution model. For this model, we derive four inversion formulas to express the bivariate density that we want to estimate in terms of the bivariate density of the observations. By substituting a kernel density estimator of the density of the observations, we then obtain four different estimators. Next we construct an asymptotically optimal convex combination of these four estimators. Expansions for the bias, variance, as well as asymptotic normality are derived. Some simulated examples are presented.     

Bayesian deconvolution of oil well test data using Gaussian processes

We use Bayesian methods to infer an unobserved function that is convolved with a known kernel. Our method is based on the assumption that the function of interest is a Gaussian process and, assuming a particular correlation structure, the resulting convolution is also a Gaussian process. This fact is used to obtain inferences regarding the unobserved process, effectively providing a deconvolution method. We apply the methodology to the problem of estimating the parameters of an oil reservoir from well-test pressure data. Here, the unknown process describes the structure of the well. Applications to data from Mexican oil wells show very accurate results.     

Penalized contrast estimator for adaptive density deconvolution

The authors consider the problem of estimating the density g of independent and identically distributed variables X_I, from a sample Z₁,… Z_n such that Z_I = X_I + σ? for i = 1,…, n, and E is noise independent of X, with σ? having a known distribution. They present a model selection procedure allowing one to construct an adaptive estimator of g and to find nonasymptotic risk bounds. The estimator achieves the minimax rate of convergence, in most cases where lower bounds are available. A simulation study gives an illustration of the good practical performance of the method.     

Asymptotic properties of maximum quasi-likelihood estimators in generalized linear models with adaptive designs

In this paper, we present the asymptotic properties of maximum quasi-likelihood estimators (MQLEs) in generalized linear models with adaptive designs under some mild regular conditions. The existence of MQLEs in quasi-likelihood equation is discussed. The rate of convergence and asymptotic normality of MQLEs are also established. The results are illustrated by Monte-Carlo simulations.     

Partial deconvolution estimation in nonparametric regression

In this article, we propose a class of partial deconvolution kernel estimators for the nonparametric regression function when some covariates are measured with error and some are not. The estimation procedure combines the classical kernel methodology and the deconvolution kernel technique. According to whether the measurement error is ordinarily smooth or supersmooth, we establish the optimal local and global convergence rates for these proposed estimators, and the optimal bandwidths are also identified. Furthermore, lower bounds for the convergence rates of all possible estimators for the nonparametric regression functions are developed. It is shown that, in both the super and ordinarily smooth cases, the convergence rates of the proposed partial deconvolution kernel estimators attain the lower bound. The Canadian Journal of Statistics 48: 535–560; 2020 © 2020 Statistical Society of Canada     

On adaptive estimation

Point estimates that are weighted averages of other estimates are considered. They are adaptive because the weights are also functions of the sample observations.In particular, the weights are functions of new measures of peakedness and skewness. Five adaptive estimators are compared (in a Monte Carlo study using the swindle) to some of the usual estimators, including those robust ones of Huber and Tukey. In addition, the swindle constant is considered in some detail. All of the adaptive estimators do extremely well with an adaptive biweight statistic being the best one in this study. Suggestions are made about future directions in this area.     

An identity for the Fisher information and Mahalanobis distance

Consider a mixture problem consisting of k classes. Suppose we observe an s-dimensional random vector X   whose distribution is specified by the relations P(X∈A|Y=i)=_Pi(A)$P(X\in A|Y=i)=P_i(A)$, where Y   is an unobserved class identifier defined on {1,…,k}$\{1,\dots ,k\}$, having distribution P(Y=i)=_pi$P(Y=i)=p_i$. Assuming the distributions _Pi$P_i$ having a common covariance matrix, elegant identities are presented that connect the matrix of Fisher information in Y   on the parameters _p1,…,_pk$p_1,\dots ,p_k$, the matrix of linear information in X, and the Mahalanobis distances between the pairs of P  's. Since the parameters are not free, the information matrices are singular and the technique of generalized inverses is used. A matrix extension of the Mahalanobis distance and its invariant forms are introduced that are of interest in their own right. In terms of parameter estimation, the results provide an independent of the parameter upper bound for the loss of accuracy by esimating _p1,…,_pk$p_1,\dots ,p_k$ from a sample of X^′$X^{\prime }$s, as compared with the ideal estimator based on a random sample of Y^′$Y^{\prime }$s.     

Integrated Square Error Asymptotics for Supersmooth Deconvolution

Abstract.  We derive the asymptotic distribution of the integrated square error of a deconvolution kernel density estimator in supersmooth deconvolution problems. Surprisingly, in contrast to direct density estimation as well as ordinary smooth deconvolution density estimation, the asymptotic distribution is no longer a normal distribution but is given by a normalized chi-squared distribution with 2 d.f. A simulation study shows that the speed of convergence to the asymptotic law is reasonably fast.     

On estimating the mean of the selected normal population in two-stage adaptive designs

Adaptive design is widely used in clinical trials. In this paper, we consider the problem of estimating the mean of the selected normal population in two-stage adaptive designs. Under the LINEX and L₂ loss functions, admissibility and minimax results are derived for some location invariant estimators of the selected normal mean. The naive sample mean estimator is shown to be inadmissible under the LINEX loss function and to be not minimax under both loss functions.     

Statistical inference for the extreme value distribution under adaptive Type-II progressive censoring schemes

Adaptive Type-II progressive censoring schemes have been shown to be useful in striking a balance between statistical estimation efficiency and the time spent on a life-testing experiment. In this article, some general statistical properties of an adaptive Type-II progressive censoring scheme are first investigated. A bias correction procedure is proposed to reduce the bias of the maximum likelihood estimators (MLEs). We then focus on the extreme value distributed lifetimes and derive the Fisher information matrix for the MLEs based on these properties. Four different approaches are proposed to construct confidence intervals for the parameters of the extreme value distribution. Performance of these methods is compared through an extensive Monte Carlo simulation.     

A comparison of some robust, adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.     

A comparison of some robust,adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.

     

Profile Hellinger distance estimation

The successful application of the Hellinger distance approach to fully parametric models is well known. The corresponding optimal estimators, known as minimum Hellinger distance (MHD) estimators, are efficient and have excellent robustness properties [Beran R. Minimum Hellinger distance estimators for parametric models. Ann Statist. 1977;5:445–463]. This combination of efficiency and robustness makes MHD estimators appealing in practice. However, their application to semiparametric statistical models, which have a nuisance parameter (typically of infinite dimension), has not been fully studied. In this paper, we investigate a methodology to extend the MHD approach to general semiparametric models. We introduce the profile Hellinger distance and use it to construct a minimum profile Hellinger distance estimator of the finite-dimensional parameter of interest. This approach is analogous in some sense to the profile likelihood approach. We investigate the asymptotic properties such as the asymptotic normality, efficiency, and adaptivity of the proposed estimator. We also investigate its robustness properties. We present its small-sample properties using a Monte Carlo study.     

Infill asymptotics for adaptive kernel estimators of spatial intensity

We apply the Abramson principle to define adaptive kernel estimators for the intensity function of a spatial point process. We derive asymptotic expansions for the bias and variance under the regime that n independent copies of a simple point process in Euclidean space are superposed. The method is illustrated by means of a simple example and applied to tornado data.     

Optimal Change-point Estimation in Inverse Problems

We develop a method of estimating a change-point of an otherwise smooth function in the case of indirect noisy observations. As two paradigms we consider deconvolution and non-parametric errors-in-variables regression. In a similar manner to well-established methods for estimating change-points in non-parametric regression, we look essentially at the difference of one-sided kernel estimators. Because of the indirect nature of the observations we employ deconvoluting kernels. We obtain an estimate of the change-point by the extremal point of the differences between these two-sided kernel estimators. We derive rates of convergence for this estimator. They depend on the degree of ill-posedness of the problem, which derives from the smoothness of the error density. Analysing the Hellinger modulus of continuity of the problem we show that these rates are minimax     

AN EXTENSION OF THE EM ALGORITHM FOR OPTIMIZATION OF CONSTRAINED LIKELIHOOD: AN APPLICATION IN TOXICOLOGY

Since its introduction in the mid 1970's, the EM algorithm has found a widespread popularity for solving the likelihood equations. Several investigators have used the algorithm in a variety of problems with incomplete information to obtain maximum likelihood estimates in numerous applications. The algorithm, however, becomes inappropriate when the underlying equations are subject to some constraints. Although an extension has been proposed to derive solutions when the parameters are subject to a set of linear constraints, the evaluation of likelihood equations from incomplete data when the equations are subject to a nonlinear constraint is still an open problem. Here, we consider a mixture model, a classical example of incomplete data, and discuss the problem of maximum likelihood estimation of the model parameters when the parameters have to satisfy a nonlinear constraint. An extension of the EM algorithm based on the celebrated Lagrange multiplier will be proposed to solve the equations. An application of the methodology in animal bioassay experiments for risk assessment of toxic substances will be described and data from a toxicological experiment will be used to illustrate the results.