Some Properties of Weighted Distributions in the Context of Repairable Systems

ABSTRACT

In this article we introduce some structural relationships between weighted and original variables in the context of maintainability function and reversed repair rate. Furthermore, we prove some characterization theorems for specific models such as power, exponential, Pareto II, beta, and Pearson system of distributions using the relationships between the original and weighted random variables.     

Interval Partitioning Arising from Size-Biased Picking

We consider the following fragmentation model of the unit interval 𝕀. We start fragmenting 𝕀 into two pieces with uniform random sizes. One of these two subintervals is then chosen at random according to a β-size-biased picking procedure, β ∈ ?. This particular tagged fragment is next broken into two random pieces, one of which is chosen accordingly at random for further splitting; the process is then iterated independently. This model constitutes a recursive β-partitioning procedure where splitting occurs only at one of the two fragments formed in the previous step of fragmentation. We investigate some statistical features of this model.     

A Note on Influence Assessment in Score Tests

Recently, [1] Ebrahimi, N. 1996. How to measure uncertainty about residual life time. Sankhya Ser. A, 58: 48–57.  [Google Scholar] proposed a dynamic measure based on differential entropy applied to the residual lifetime. This measure has been used for the classification and ordering of survival functions. More recently, [2] Ebrahimi, N. 1997. Testing whether lifetime distribution is decreasing uncertainty. Journal of Statistical Planning and Inference, 64: 9–19. [Crossref], [Web of Science ®] , [Google Scholar] has considered the problem of testing the monotonicity of this measure. We propose and study several kernel type estimators of the entropy of residual life through the estimation of f(x) log f(x). These estimators can be applied to the classification and comparison of lifetime distribution.     

Presmoothed Estimation with Left-Truncated and Right-Censored Data

We propose a new method to estimate the cumulative hazard function and the corresponding distribution function of survival times under randomly left-truncated and right-censored observations (LTRC). The new estimators are based on presmoothing ideas, the estimation of the conditional expectation m of the censoring indicator. An almost sure representation for both estimators is established, from which a strong consistency rate and asymptotic normality are derived. It is shown that the presmoothed modification leads to a gain in terms of asymptotic mean squared error. This efficiency with respect to the classical estimators is also shown in a simulation study. Finally, an application to a real data set is provided.     

The 3 F 2 with Complex Parameters as Generating Function of Discrete Distribution

A new discrete family of probability distributions that are generated by the ₃ F ₂ function with complex parameters is presented. Some of the properties of this new family are studied as well as methods of estimation for its parameters. It affords considerable flexibility of shape which turns the distribution into an appropriate candidate for modeling data that cannot be adequately fitted by classical families with fewer parameters. Finally, three examples in the fields of Agriculture and Education are included in order to show the versatility and utility of this distribution.     

Clustering Discrete Data Through the Multinomial Mixture Model

In this work, the multinomial mixture model is studied, through a maximum likelihood approach. The convergence of the maximum likelihood estimator to a set with characteristics of interest is shown. A method to select the number of mixture components is developed based on the form of the maximum likelihood estimator. A simulation study is then carried out to verify its behavior. Finally, two applications on real data of multinomial mixtures are presented.     

Optimum spacings for the joint estimation and tests of hypothesis of location and scale parameters of the cauchy distribution

A simultaneous test for the location and scale parameters of the Cauchy distribution is considered based on selected order statistics.. It is shown that optimum spacings that maximise the Pitman ARE of the test coincide with that of the optimum spacings for the estimation problem.     

Large sample bias of maximum likeihood estimates from a series of sparse ixj tables

For a large series of IxJ tables, each containing two observations, the bias of the maximum likelihood estimates of log linear partial association parameters is shown to be equal to the parameters, regardless of the size of I and J. The partial association considered is that between row and column variables; the three way interactions are assumed to be O. This is a generalization of Andersen's results (1973a, 1973b) for a series of 2x2 tables.     

Conditions for fourth order rotatability in K dimensions

In this paper, conditions for a set of points to form a rotatable design of order four have been worked out. These condi- tions form the basis for obtaining a set of rotatable designs of order four.     

A new rank test for the K-Sample problem

The treatment sum of squares in the one-way analysis of variance can be expressed in two different ways: as a sum of comparisons between each treatment and the remaining treatments combined, or as a sum of comparisons between the treatments two at a time. When comparisons between treatments are made with the Wilcoxon rank sum statistic, these two expressions lead to two different tests; namely, that of Kruskal and Wallis and one which is essentially the same as that proposed by Crouse (1961,1966). The latter statistic is known to be asymptotically distributed as a chi-squared variable when the numbers of replicates are large. Here it is shown to be asymptotically normal when the replicates are few but the number of treatments is large. For all combinations of numbers of replicates and treatments its empirical distribution is well approximated by a beta distribution