Biased estimation and hypothesis testing in linear regression

We consider a test of the simple hypothesis =₀ based on some biased estimator. Under a certain condition the corresponding test statistic coincides with the usualF-statistic based on the least squares estimator. Surprisingly, this condition is met by several well-known biased estimators.     

Asymptotics of Bayesian median loss estimation

We establish the consistency, asymptotic normality, and efficiency for estimators derived by minimizing the median of a loss function in a Bayesian context. We contrast this procedure with the behavior of two Frequentist procedures, the least median of squares (LMS) and the least trimmed squares (LTS) estimators, in regression problems. The LMS estimator is the Frequentist version of our estimator, and the LTS estimator approaches a median-based estimator as the trimming approaches 50% on each side. We argue that the Bayesian median-based method is a good tradeoff between the two Frequentist estimators.     

Statistical estimation of a mixture of Gaussian distributions

The paper is devoted to the problem of statistical estimation of a multivariate distribution density, which is a discrete mixture of Gaussian distributions. A heuristic approach is considered, based on the use of the EM algorithm and nonparametric density estimation with a sequential increase in the number of components of the mixture. Criteria for testing of model adequacy are discussed.     

Parametric tail copula estimation and model testing

Parametric models for tail copulas are being used for modeling tail dependence and maximum likelihood estimation is employed to estimate unknown parameters. However, two important questions seem unanswered in the literature: (1) What is the asymptotic distribution of the MLE and (2) how does one test the parametric model? In this paper, we answer these two questions in the case of a single parameter for ease of illustration. A simulation study is provided to investigate the finite sample performance of the proposed estimator and test.     

Robust Bayesian prediction and estimation under a squared log error loss function

Robust Bayesian analysis is concerned with the problem of making decisions about some future observation or an unknown parameter, when the prior distribution belongs to a class Γ instead of being specified exactly. In this paper, the problem of robust Bayesian prediction and estimation under a squared log error loss function is considered. We find the posterior regret Γ-minimax predictor and estimator in a general class of distributions. Furthermore, we construct the conditional Γ-minimax, most stable and least sensitive prediction and estimation in a gamma model. A prequential analysis is carried out by using a simulation study to compare these predictors.     

Hypothesis testing using affine linear estimators

The general linear hypothesis is usually tested by means of anF-statistic dependent on the least squares estimator. In this paper, a class of linear estimators is identified which can also serve as a basis for such anF-statistic. Conditions are derived under which thisF-statistic coincides with the usual one. This opens the possibility of constructing minimax-estimators which dominate LS with respect to risk, yielding the same test results.Support by Deutsche Forschungsgemeinschaft, Grant No. Tr 253/1-2 is gratefully acknowledged.     

Admissibility of linear estimators with respect to inequality constraints under matrix loss function

In this paper we investigate the admissibility of linear estimators in the multivariate linear model with respect to inequality constraints under matrix loss function. The necessary and sufficient conditions for a linear estimator to be admissible in the class of homogeneous linear estimators and the class of inhomogeneous linear estimators are obtained, respectively.     

An estimation method for the Neyman chi-square divergence with application to test of hypotheses

We propose a new definition of the Neyman chi-square divergence between distributions. Based on convexity properties and duality, this version of the χ² is well suited both for the classical applications of the χ² for the analysis of contingency tables and for the statistical tests in parametric models, for which it is advocated to be robust against outliers.We present two applications in testing. In the first one, we deal with goodness-of-fit tests for finite and infinite numbers of linear constraints; in the second one, we apply χ²-methodology to parametric testing against contamination.     

Dual divergence estimators and tests: Robustness results

The class of dual ?-divergence estimators (introduced in Broniatowski and Keziou (2009) [5]) is explored with respect to robustness through the influence function approach. For scale and location models, this class is investigated in terms of robustness and asymptotic relative efficiency. Some hypothesis tests based on dual divergence criteria are proposed and their robustness properties are studied. The empirical performances of these estimators and tests are illustrated by Monte Carlo simulation for both non-contaminated and contaminated data.     

Likelihood ratio tests for continuous monotone hazards with an unknown change point

We consider the likelihood ratio tests (LRT) for two continuous monotone hazards with an unknown change point. We establish the convergence in distribution and weak convergence of LRT. Simulation studies show that the proposed tests compare favorably to other existing tests.     

Parametric estimation of a bivariate stable Lévy process

We propose a parametric model for a bivariate stable Lévy process based on a Lévy copula as a dependence model. We estimate the parameters of the full bivariate model by maximum likelihood estimation. As an observation scheme we assume that we observe all jumps larger than some ε>0 and base our statistical analysis on the resulting compound Poisson process. We derive the Fisher information matrix and prove asymptotic normality of all estimates when the truncation point ε→0. A simulation study investigates the loss of efficiency because of the truncation.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

Item response theory for longitudinal data: population parameter estimation

In this work we propose IRT models to estimate ability distribution parameters of a population of individuals submitted to different tests along the time, having or not common items. The item parameters are considered known and several covariance structures are proposed to accommodate the possible dependence among the abilities of the same individual, measured at different instants. Maximum likelihood equations and some simulation results are presented.     

Optimum two level fractional factorial plans for model identification and discrimination

Model identification and discrimination are two major statistical challenges. In this paper we consider a set of models M_k for factorial experiments with the parameters representing the general mean, main effects, and only k out of all two-factor interactions. We consider the class D of all fractional factorial plans with the same number of runs having the ability to identify all the models in M_k, i.e., the full estimation capacity.The fractional factorial plans in D with the full estimation capacity for k?2 are able to discriminate between models in M_u for u?k^*, where k^*=(k/2) when k is even, k^*=((k-1)/2) when k is odd. We obtain fractional factorial plans in D satisfying the six optimality criterion functions AD, AT, AMCR, GD, GT, and GMCR for 2^m factorial experiments when m=4 and 5. Both single stage and multi-stage (hierarchical) designs are given. Some results on estimation capacity of a fractional factorial plan for identifying models in M_k are also given. Our designs D4.1 and D10 stand out in their performances relative to the designs given in Li and Nachtsheim [Model-robust factorial designs, Technometrics 42(4) (2000) 345-352.] for m=4 and 5 with respect to the criterion functions AD, AT, AMCR, GD, GT, and GMCR. Our design D4.2 stands out in its performance relative the Li-Nachtsheim design for m=4 with respect to the four criterion functions AT, AMCR, GT, and GMCR. However, the Li-Nachtsheim design for m=4 stands out in its performance relative to our design D4.2 with respect to the criterion functions AD and GD. Our design D14 does have the full estimation capacity for k=5 but the twelve run Li-Nachtsheim design does not have the full estimation capacity for k=5.     

Asymptotics for testing hypothesis in some multivariate variance components model under non-normality

We consider the problem of deriving the asymptotic distribution of the three commonly used multivariate test statistics, namely likelihood ratio, Lawley-Hotelling and Bartlett-Nanda-Pillai statistics, for testing hypotheses on the various effects (main, nested or interaction) in multivariate mixed models. We derive the distributions of these statistics, both in the null as well as non-null cases, as the number of levels of one of the main effects (random or fixed) goes to infinity. The robustness of these statistics against departure from normality will be assessed.Essentially, in the asymptotic spirit of this paper, both the hypothesis and error degrees of freedom tend to infinity at a fixed rate. It is intuitively appealing to consider asymptotics of this type because, for example, in random or mixed effects models, the levels of the main random factors are assumed to be a random sample from a large population of levels.For the asymptotic results of this paper to hold, we do not require any distributional assumption on the errors. That means the results can be used in real-life applications where normality assumption is not tenable.As it happens, the asymptotic distributions of the three statistics are normal. The statistics have been found to be asymptotically null robust against the departure from normality in the balanced designs. The expressions for the asymptotic means and variances are fairly simple. That makes the results an attractive alternative to the standard asymptotic results. These statements are favorably supported by the numerical results.     

Parameter estimation under ambiguity and contamination with the spurious model

Recently, we proposed variants as a statistical model for treating ambiguity. If data are extracted from an object with a machine then it might not be able to give a unique safe answer due to ambiguity about the correct interpretation of the object. On the other hand, the machine is often able to produce a finite number of alternative feature sets (of the same object) that contain the desired one. We call these feature sets variants of the object. Data sets that contain variants may be analyzed by means of statistical methods and all chapters of multivariate analysis can be seen in the light of variants. In this communication, we focus on point estimation in the presence of variants and outliers. Besides robust parameter estimation, this task requires also selecting the regular objects and their valid feature sets (regular variants). We determine the mixed MAP-ML estimator for a model with spurious variants and outliers as well as estimators based on the integrated likelihood. We also prove asymptotic results which show that the estimators are nearly consistent.The problem of variant selection turns out to be computationally hard; therefore, we also design algorithms for efficient approximation. We finally demonstrate their efficacy with a simulated data set and a real data set from genetics.     

Second-order optimality of randomized estimation and test procedures

Let P_{(Θ, τ)} 6 $\text{A}$, θ ∈ Θ ? $\text{R}$, τ ∈ T ? $\text{R}$^p denote a family of probability measures, where τ denotes the vector of nuisance parameters. Starting from randomized asymptotic maximum likelihood (as. m. l.) estimators for (θ, τ) we construct randomized estimators which are asymptotically median unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ resp. test procedures which are as. similar of level $\text{α + o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ (for testing θ = θ₀, τ ∈ T against one sided alternatives). The estimation procedures are second-order efficient in the class of estimators which are median unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ and the test procedures are second-order efficient in the class of tests which are as. of level $\text{α + o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$. These results hold without any continuity condition on the family of probability measures.     

Asymptotics for functionals of self-normalized residuals of discretely observed stochastic processes

The purpose of this paper is to derive the stochastic expansion of self-normalized-residual functionals stemming from a class of diffusion type processes observed at high frequency, where total observing period may or may not tend to infinity. The result enables us to construct some explicit statistics for goodness of fit tests, consistent against “presence of a jump component” and “diffusion-coefficient misspecification”; then, the acceptance of the null hypothesis may serve as a collateral evidence for using the correctly specified diffusion type model. Especially, our asymptotic result clarifies how to remove the bias caused by plugging in a diffusion-coefficient estimator.     

Nonparametric likelihood ratio goodness-of-fit tests for survival data

Berk and Jones (Z. Wahrsch. Verw. Gebiete 47 (1979) 47) described a nonparametric likelihood test of uniformity that is more efficient, in Bahadur's sense, than any weighted Kolmogorov-Smirnov test at any alternative. This article shows how to obtain a nonparametric likelihood test of a general parametric family for incomplete survival data. A nonparametric likelihood ratio test process is employed to measure the discrepancy between a parametric family and the observed data. Large sample properties of the likelihood ratio test process are studied under both the null and alternative hypotheses. A Monte Carlo simulation method is proposed to estimate its null distribution. We show how to produce a likelihood ratio graphical check as well as a formal test of a parametric family based on the developed theory. Our method is developed for the right-censorship model, but can be easily extended to some other survival models. Illustrations are given using both real and simulated data.