Tests for Symmetric Error Distribution in Linear and Nonparametric Regression Models

In linear and nonparametric regression models, the problem of testing for symmetry of the distribution of errors is considered. We propose a test statistic which utilizes the empirical characteristic function of the corresponding residuals. The asymptotic null distribution of the test statistic as well as its behavior under alternatives is investigated. A simulation study compares bootstrap versions of the proposed test to other more standard procedures.     

Bayesian estimation of parameters of mixed geometric failure models from Type I group censored sample

The estimation problem of the parameters of a mixed geometric lifetime model, using Bayesian approach and Type I group censored sample, will be investigated in the case of two subpopulations. The Bayes estimates are derived for squared error, minimum expected, general entropy and linex loss functions under informative and diffuse priors. A practical example given by Nelson (W.B. Nelson, Hazard plotting methods for analysis of the life data with different failure models, J. Qual. Technol. 2 (1970), pp. 126–149) is considered. A simulation study is carried out along with risk.     

Statistical estimation for a failure model with damage accumulation in a case of small samples

A failure model with damage accumulation is considered. Damages occur according to a Poisson process and they degenerate into failures in a random time. The rate of the Poisson process and the degeneration time distribution are unknown. Two sample populations are available: a sample of intervals between damages and a sample of degeneration times. The case of small samples is considered. The purpose is to estimate the expectation and the distribution of the number of damages and failures at time t. We consider the plug-in and resampling estimators of the above mentioned characteristics. The expectations and variances of the suggested estimators are investigated. The numerical examples show that the resampling estimator has some advantages.     

Preliminary phi-divergence test estimators for linear restrictions in a logistic regression model

The problem of estimation of the parameters in a logistic regression model is considered under multicollinearity situation when it is suspected that the parameter of the logistic regression model may be restricted to a subspace. We study the properties of the preliminary test based on the minimum ϕ -divergence estimator as well as in the ϕ -divergence test statistic. The minimum ϕ -divergence estimator is a natural extension of the maximum likelihood estimator and the ϕ -divergence test statistics is a family of the test statistics for testing the hypothesis that the regression coefficients may be restricted to a subspace.     

Weibull inference using trimmed samples and prior information

Trimmed samples are commonly used in several branches of statistical methodology, especially when the presence of contaminated data is suspected. Assuming that certain proportions of the smallest and largest observations from a Weibull sample are unknown or have been eliminated, a Bayesian approach to point and interval estimation of the scale parameter, as well as hypothesis testing and prediction, is presented. In many cases, the use of substantial prior information can significantly increase the quality of the inferences and reduce the amount of testing required. Some Bayes estimators and predictors are derived in closed-forms. Highest posterior density estimators and credibility intervals can be computed using iterative methods. Bayes rules for testing one- and two-sided hypotheses are also provided. An illustrative numerical example is included.     

VARIANCE ESTIMATION IN TWO-PHASE SAMPLING

Two‐phase sampling is often used for estimating a population total or mean when the cost per unit of collecting auxiliary variables, x, is much smaller than the cost per unit of measuring a characteristic of interest, y. In the first phase, a large sample s₁ is drawn according to a specific sampling design p(s₁) , and auxiliary data x are observed for the units i∈s₁ . Given the first‐phase sample s₁ , a second‐phase sample s₂ is selected from s₁ according to a specified sampling design {p(s₂∣s₁) } , and (y, x) is observed for the units i∈s₂ . In some cases, the population totals of some components of x may also be known. Two‐phase sampling is used for stratification at the second phase or both phases and for regression estimation. Horvitz–Thompson‐type variance estimators are used for variance estimation. However, the Horvitz–Thompson ( Horvitz & Thompson, J. Amer. Statist. Assoc. 1952 ) variance estimator in uni‐phase sampling is known to be highly unstable and may take negative values when the units are selected with unequal probabilities. On the other hand, the Sen–Yates–Grundy variance estimator is relatively stable and non‐negative for several unequal probability sampling designs with fixed sample sizes. In this paper, we extend the Sen–Yates–Grundy ( Sen , J. Ind. Soc. Agric. Statist. 1953; Yates & Grundy , J. Roy. Statist. Soc. Ser. B 1953) variance estimator to two‐phase sampling, assuming fixed first‐phase sample size and fixed second‐phase sample size given the first‐phase sample. We apply the new variance estimators to two‐phase sampling designs with stratification at the second phase or both phases. We also develop Sen–Yates–Grundy‐type variance estimators of the two‐phase regression estimators that make use of the first‐phase auxiliary data and known population totals of some of the auxiliary variables.     

Relative efficiency of first difference estimator in panel data regression with serially correlated error components

We consider the pooled cross-sectional and time series regression model when the disturbances follow a serially correlated one-way error components. In this context we discovered that the first difference estimator for the regression coefficients is equivalent to the generalized least squares estimator irrespective of the particular form of the regressor matrix when the disturbances are generated by a first order autoregressive process where the autocorrelation is close to unity.     

Convergence rates of change-point estimators and tail probabilities of the first-passage-time process

In the classical setting of the change-point problem, the maximum-likelihood estimator and the traditional confidence region for the change-point parameter are considered. It is shown that the probability of the correct decision, the coverage probability and the expected size of the confidence set converge exponentially fast as the sample size increases to infinity. For this purpose, the tail probabilities of the first passage times are studied. General inequalities are established, and exact asymptotics are obtained for the case of Bernoulli distributions. A closed asymptotic form for the expected size of the confidence set is derived for this case via the conditional distribution of the first passage times.     

Multivariate boundary kernels and a continuous least squares principle

Whereas there are many references on univariate boundary kernels, the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail. The use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean-squared error or sup-norm deviation for functions with compact support. We develop a class of boundary kernels which work for any support, regardless of the complexity of its boundary. Our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated. These boundary kernels provide a natural continuation of non-negative kernels used in the interior onto the boundary. They are obtained as solutions of the same kernel-generating variational problem which also produces the kernel function used in the interior as its solution. We discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares. Along the way we establish a continuous least squares principle and a continuous analogue of the Gauss–Markov theorem.     

Sur la convergence des tests de Schlee et de Yatchew

L'auteur adapte l'approche de Schlee (1980) et de Yatchew (1992) à la construction de deux tests de convexité pour une fonction de régression dans un modèle non paramétrique. Il établit de nouveaux résultats de convergence pour ces tests et étudie leur comportement dans de petits échantillons à l'aide de simulations.