ABSTRACT In a progressively censored life testing experiment, the censored units can be used for some other life testing experiments. In a step-stress stage life testing experiment, we run parallel experiments while placing the progressively censored units at the higher stress level along with the main experiment. In this work, multiple step-stress stage life testing experiments are performed under the assumption that the experimental units at different stress levels follow Weibull distributions with the same shape parameter, but different scale parameters. The main objectives of this paper are to provide an order-restricted classical inference of the model parameters and also to find out the optimal life testing scheme. We provide the order-restricted maximum likelihood estimators of the unknown parameters and the associated confidence intervals based on the observed Fisher information matrix. An extensive simulation study and a data analysis are provided for illustrative purposes. In search of the optimal life testing plan under this set ups, a meta-heuristic approach based on a variable neighbourhood search algorithm is employed using D-optimality criterion. Finally, we have indicated how the method can be extended when the shape parameter of the Weibull distributions are also different at different stress levels.

ABSTRACT The frequency polygon estimation, which is based on histogram technique, has a similar convergence rate as those of modern non-negative kernel estimators and the advantage of computational simplicity. This paper will study the uniformly complete consistency and the rates of uniformly complete consistency of frequency polygon estimation for widely orthant dependent samples under some general conditions, which improve and extend the corresponding ones in the literature. As an application, the uniformly complete consistency and the corresponding rates of hazard rate function estimator are also obtained.

ABSTRACT In this paper, we study the asymptotic behaviour, including Cramér-type moderate deviations and optimal Berry-Esseen bounds for maximum likelihood estimator of drift parameter in α-Wiener bridge process. For statistical practice, a self-normalized version asymptotic result for the maximum likelihood estimator is also established. Numerical simulations show that our new established maximum likelihood estimator-based test statistic outperforms than the moment estimator-based test statistic and the likelihood ratio test statistic in literature.

In this paper, we propose robust statistical inference and variable selection method for generalized linear models that accommodate the outliers, nonignorable dropouts and within-subject correlations. The purpose of our study is threefold. First, we construct the robust and bias-corrected generalized estimating equations (GEEs) by combining the Mallows-type weights, Huber's score function and inverse probability weighting approaches to against the influence of outliers and account for nonignorable dropouts. Subsequently, the generalized method of moments is utilized to estimate the parameters in the nonignorable dropout propensity based on sufficient instrumental estimating equations. Second, in order to incorporate the within-subject correlations under an informative working correlation structure, we borrow the idea of quadratic inference function and hybrid-GEE to obtain the improved empirical likelihood procedures. The asymptotic properties of the proposed estimators and their confidence regions are derived. Third, the robust variable selection and algorithm are investigated. We evaluate the performance of proposed estimators through simulation and illustrate our method in an application to HIV-CD4 data.

Truncated survival data are observed retrospectively, if the death event falls into the study period. We find that estimating the survival distribution requires a model for the birth event. With births generated by a parametric Poisson process, we analyse the likelihood of parametric survival, stochastically independent of birth. Conditioning on the number of observed units reduces the number of parameters by one, and to two in our application. The compact support of an observation simplifies the proof of consistency. Furthermore, that we only need to show identification separately for the birth and the death distribution is helpful for demonstrating asymptotic normality. For identification of the survival parameters, a stronger criterion is needed than for a simple random sample, but is fulfilled in our application. In a simulation study, we find that the variance inflation by truncation can be substantial, and apparently is indeed so in our application. From 55,000 German companies that went insolvent between 2014 and 2016, we infer an average time to insolvency of six years and a negative linear trend of corporate foundations after the German reunification in 1990. Companies in Northern Germany survive longer than in the south.

Length-biased data arise when the probability of observing a subject in a sample is proportional to its corresponding value. The statistical tools that should be applied to analyse such sets of data differ from those used for random samples. Ignoring this issue in an investigation results in a biased inference. In this paper, we propose an empirical likelihood-based method to draw inference on covariate effects in an accelerated failure time model while the observations are subject to length-bias. The asymptotic distribution of the empirical log-likelihood ratio statistic is derived to be a weighted sum of independent chi-square distributions. Hence, we then extend the results by an exploration into the adjusted empirical likelihood. We derive an asymptotic standard chi-square distribution for the adjusted log-likelihood statistic. The limiting distributions are applied to obtain confidence regions for the regression parameters. A simulation study is carried out to evaluate and compare the performance of the proposed procedures with an existing method based on the normal approximation approach. Finally, the procedures are illustrated by modelling the regression parameter and estimating confidence intervals for a set of real data on widths of shrubs.

We study the non-parametric estimation of an unknown survival function S with support on R + based on a sample with multiplicative measurement errors. The proposed fully data-driven procedure is based on the estimation of the Mellin transform of the survival function and a regularization of the inverse of the Mellin transform by a spectral cut-off. The upcoming bias-variance trade-off is handled by a data-driven choice of the cut-off parameter. In order to discuss the bias term, we consider the Mellin–Sobolev spaces which characterize the regularity of the unknown survival function S through the decay of its Mellin transform. For the analysis of the variance term, we consider the independent and identically distributed case and incorporate dependent observations in form of Bernoulli shift processes and β-mixing sequences. Additionally, we show minimax optimality over Mellin–Sobolev spaces of the spectral cut-off estimator.

In this paper, we introduce two new families of distributions containing unimodal as well as bimodal and trimodal distributions. The models extend the normal model to trimodal symmetric and asymmetric situations, and typically involve fewer parameters to be estimated than the mixtures of normal distributions. Statistical properties of these new families are studied in detail. The problem of estimating parameters is addressed by considering the maximum likelihood method and Fisher information matrices are derived. A small Monte Carlo simulation study is conducted to examine the performance of the obtained estimators. The methodology developed is illustrated with three real data applications.

Recently, a new class of uncertainty measure, namely the dynamic cumulative residual Renyi's entropy, has been introduced and studied in the literature. In parallel to that here, we have defined cumulative residual Renyi's entropy for a doubly truncated random variable called interval cumulative residual Renyi's entropy (ICRRE). Several properties, including the effect of monotone transformation, and bounds of ICRRE are discussed. Furthermore, it is shown that the proposed measure uniquely determines the distribution function and obtained characterizations of some well-known life distributions. Finally, we have proposed four nonparametric estimators of ICRRE and compared their performance by using the Monte-Carlo simulation study. A real-life data set is also illustrated based on the best-proposed estimator.

In this paper, we study a class of empirical processes associated with U-statistics in the context of right-censored data when the censoring is independent, and the censored data is reweighted using the inverse of survival function of the censoring random variable. Asymptotic properties of these U-empirical processes are obtained. Theoretical results are applied on two classes of goodness-of-fit tests.