A class of weighted normal distributions and its variants useful for inequality constrained analysis

This article develops a class of the weighted normal distributions for which the probability density function has the form of a product of a normal density and a weight function. The class constitutes marginal distributions obtained from various kinds of doubly truncated bivariate normal distributions. This class of distributions strictly includes the normal, skew–normal and two-piece skew–normal and is useful for selection modelling and inequality constrained normal mean analysis. Some distributional properties and Bayesian perspectives of the class are given. Probabilistic representation of the distributions is also given. The representation is shown to be straightforward to specify distribution and to implement computation, with output readily adapted for required analysis. Necessary theories and illustrative examples are provided.     

Characterizations of geometric distribution through progressively Type-II right-censored order statistics

In this paper, we consider characterizations of geometric distribution based on some properties of progressively Type-II right-censored order statistics. Specifically, we establish characterizations through conditional expectation, identical distribution, and independence of functions of progressively Type-II right-censored order statistics. Moreover, extensions of these results to generalized order statistics are also sketched. These generalize the corresponding results known for the case of ordinary order statistics.     

Bivariate exponential-type distributions with linear regression

Let Xbe a bivariate exponential-type random vector (BIDLIKAR, PATIL (1968)), than it is proved:

1. If P(X ≥0) = 1 is valid, then Xhas linear regression to both directions if and only if Xpossesses a symmetric Γ-distribution.

2. Xpossesses linear regression to both directions with constant regression coefficients (independent of the parameter vector ? of the exponential-type distribution (BIDLIKAR, PATIL (1968)) if and only if Xis normal distributed.     

Optimal designs for exponential regression

This paper is concerned with the optimal design problem for the particular case of non-linear parametrisation:the parameters to be estimated are included in exponents.Some properties of locally optimal designs as functions of estimated parameters are investigated and a table of such designs in given.We consider also designs to be optimal in the sense of minimax approach.     

Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring

Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored samples. The marginal distributions are derived and exact confidence bounds for the parameters are obtained. The results are also used to derive the exact distribution of the MLE of the pth quantile, as well as the corresponding confidence bounds. These exact confidence intervals are then compared with parametric bootstrap confidence intervals in terms of coverage probabilities. Finally, we present some numerical examples to illustrate the methods of inference developed here.     

A note on Stein-type shrinkage estimator in partial linear models

In this paper, an exact sufficient condition for the dominance of the Stein-type shrinkage estimator over the usual unbiased estimator in a partial linear model is exhibited. Comparison result is then done under the balanced loss function. It is assumed that the vector of disturbances is typically distributed according to the law belonging to the sub-class of elliptically contoured models. It is also shown that the dominance condition is robust. Furthermore, a nonparametric estimation after estimation of the linear part is added for detecting the efficiency of the obtained results.     

On estimation and diagnostics analysis in log-generalized gamma regression model for interval-censored data

The interval-censored survival data appear very frequently, where the event of interest is not observed exactly but it is only known to occur within some time interval. In this paper, we propose a location-scale regression model based on the log-generalized gamma distribution for modelling interval-censored data. We shall be concerned only with parametric forms. The proposed model for interval-censored data represents a parametric family of models that has, as special submodels, other regression models which are broadly used in lifetime data analysis. Assuming interval-censored data, we consider a frequentist analysis, a Jackknife estimator and a non-parametric bootstrap for the model parameters. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some techniques to perform global influence.     

Optimally robust estimators in generalized Pareto models

In this paper, we study the robustness properties of several procedures for the joint estimation of shape and scale in a generalized Pareto model. The estimators that we primarily focus upon, most bias robust estimator (MBRE) and optimal MSE-robust estimator (OMSE), are one-step estimators distinguished as optimally robust in the shrinking neighbourhood setting; that is, they minimize the maximal bias, respectively, on such a specific neighbourhood, the maximal mean squared error (MSE). For their initialization, we propose a particular location–dispersion estimator, MedkMAD, which matches the population median and kMAD (an asymmetric variant of the median of absolute deviations) against the empirical counterparts. These optimally robust estimators are compared to the maximum-likelihood, skipped maximum-likelihood, Cramér–von-Mises minimum distance, method-of-medians, and Pickands estimators. To quantify their deviation from robust optimality, for each of these suboptimal estimators, we determine the finite-sample breakdown point and the influence function, as well as the statistical accuracy measured by asymptotic bias, variance, and MSE – all evaluated uniformly on shrinking neighbourhoods. These asymptotic findings are complemented by an extensive simulation study to assess the finite-sample behaviour of the considered procedures. The applicability of the procedures and their stability against outliers are illustrated for the Danish fire insurance data set from the package evir.