Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

Classical estimation of hazard rate and mean residual life functions of Pareto distribution

Abstract

In this paper we find the maximum likelihood estimates (MLEs) of hazard rate and mean residual life functions (MRLF) of Pareto distribution, their asymptotic non degenerate distribution, exact distribution and moments. We also discuss the uniformly minimum variance unbiased estimate (UMVUE) of hazard rate function and MRLF. Finally, two numerical examples with simulated data and real data set, are presented to illustrate the proposed estimates.     

A Test for Constant Hazard Against a Change-Point Alternative

In this article, we consider the change-point hazard rate model which arises quite commonly in mechanical or biological systems, which experience a high hazard rate early in their lifetime due to infant mortality and then a constant or steady hazard rate after the threshold time. We first derive the corresponding mean residual life function (MRLF) and observe that the MRLF is initially increasing and then constant. Here, we derive a test statistic for exponentiality against Increasing Initially then Constant Mean Residual Life (ICMRL). We also derive the asymptotic distribution of the test statistic and compare the power of the test with other existing tests such as likelihood ratio, Weibull, and Log gamma tests considered in the literature. The test performs quite well as compared to other alternatives studied.     

On the Nonparametric Mean Residual Life Estimator in Length-biased Sampling

In this article, we discuss nonparametric estimation of a mean residual life function from length-biased data. Precisely, we prove strong uniform consistency and weak converge of the nonparametric mean residual life estimator in length-biased setting.     

On Upshifted Reversed Mean Residual Life Order

Recently, the concept of reversed mean residual life order based on the mean of the random variable X _t  = (t ? X | X ≤ t), t > 0, called the reversed residual life, defined for the nonnegative random variable X, has been introduced in the literature. In this paper, a stochastic order based on the shifted version of the reversed mean residual life is proposed, based on the reversed mean residual life function for a random variable X with support (l _X , ∞), where l _X may be negative infinity, and its properties are studied. Closure under the Poisson shock model and properties for spare allocation are also discussed.     

A generalized scheffé method of multiple comparisons

Standard resulrs on the extrema of quotients of quadratic forms are extended to the non-negative definite case. The maximum and the set over which it is achieved are characterized explicitly both in terms of generalized inverse matrices and generalized eigenvalues. These results become the basis of Scheffe type multiple comparisons in the usual way. To demonstrate their application to statistics with singular covariance matrices, the method is detailed for Mantel-Haenszel, Breslow, and Cox statistics. An example is presented illustrating a situation where the proposed Scheffe type comparisons may be better than the pairwise method.     

Progressively censored tests for clinical experiments and life testing problems based on weighted empirical distributions

In the context of time-sequential studies, progressively censored tests for a simple regression model based on weighted empirical distributions are considered for ungrouped as well as grouped data situations. Early decision rules based on such tests are formulated. The asymptotic theory of the proposed tests rests on a construction of suitable empirical processes and their convergence (in distribution) to appropriate Gaussian functions. Critical values of the proposed test statistics are obtained by simulation, For a hypothetical example (of practical interest), a comparative study is made for the empirical powers and stopping times for some rival tests.     

A time series approach to life table construction

This paper deals with the construction of the life table. A discussion of basic facts about the life table is followed by the proposal of a nonstationary, autoregressive model for the life table. The moment structure of the nonstationary, autoregressive model is developed. Some estimation procedures are proposed followed by several examples.     

Time Series Modeling of Degradation Due to Outdoor Weathering

Environmental variables have an important effect on the reliability of many products such as coatings and polymeric composites. Long-term prediction of the performance or service life of such products must take into account the probabilistic/stochastic nature of the outdoor weather. In this article, we propose a time series modeling procedure to model the time series data of daily accumulated degradation. Daily accumulated degradation is the total amount of degradation accrued within one day and can be obtained by using a degradation rate model for the product and the weather data. The fitted model of the time series can then be used to estimate the future distribution of cumulative degradation over a period of time, and to compute reliability measures such as the probability of failure. The modeling technique and estimation method are illustrated using the degradation of a solar reflector material. We also provide a method to construct approximate confidence intervals for the probability of failure.     

Statistical analysis of a two-unit dependent repairable system

Maximum likelihood estimators are investigated a for lr-out-of 2:G repairable system when failures and repairs are statistically dependent. Under two censoring schemes Accelerated Life Testing Procedure (ALTP) is used to obtain the estimates. Information matrices are supplied and a special case when the failure and repair phenomenon are independent of each other is considered.