On order statistics from non-identical right-truncated exponential random variables and some applications

By considering order statistics arising from n independent non-identically distributed right-truncated exponential random variables, we derive in this paper several recurrence relations for the single and the product moments of order statistics. These recurrence relations are simple in nature and could be used systematically in order to compute all the single and the product moments of order statistics for all sample sizes in a simple recursive manner. The results for order statistics from a multiple-outlier model (with a slippage of p observations) from a right-truncated exponential population are deduced as special cases. These results will be useful in assessing robustness properties of any linear estimator of the unknown parameter of the right-truncated exponential distribution, in the presence of one or more outliers in the sample. These results generalize those for the order statistics arising from an i.i.d. sample from a right-truncated exponential population established by Joshi (1978, 1982).     

Rényi entropy of m-generalized order statistics

Abstract

Characterizing relations via Rényi entropy of m-generalized order statistics are considered along with examples and related stochastic orderings. Previous results for common order statistics are included.     

Extended exponential distribution based on order statistics

The extended exponential distribution due to Nadarajah and Haghighi (2011 Nadarajah, S., Haghighi, F. (2011). An extension of the exponential distribution. Statistics 45:543–558.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) is an alternative to and always provides better fits than the gamma, Weibull, and the exponentiated exponential distributions whenever the data contain zero values. We establish recurrence relations for the single and product moments of order statistics from the extended exponential distribution. These recurrence relations enable computation of the means, variances, and covariances of all order statistics for all sample sizes in a simple and efficient manner. By using these relations, we tabulate the means, variances, and covariances of order statistics and derive best linear unbiased estimates of the extended exponential distribution. Finally, a data application is provided.     

Csiszar’s ϕ-divergences for testing the order in a Markov chain

x ₁, ..., x _n+r can be treated as the sample values of a Markov chain of order r or less (chain in which the dependence extends over r+1 consecutive variables only), and consider the problem of testing the hypothesis H ₀ that a chain of order r− 1 will be sufficient on the basis of the tools given by the Statistical Information Theory: _ϕ-Divergences. More precisely, if p _{a 1 ....., a r}: _{a r +1} denotes the transition probability for a r ^th order Markov chain, the hypothesis to be tested is H ₀:p _{a 1 ....., a r}: _{a r +1} = p _{a 2 ....., a r}: _{a r +1}, a _i ∈{1, ..., s}, i = 1, ..., r + 1 The tests given in this paper, for the first time, will have as a particular case the likelihood ratio test and the test based on the chi-squared statistic. Received: August 3, 1998; revised version: November 25, 1999     

Order statistics from non-identical right-truncated Lomax random variables with applications

In this paper, we derive some recurrence relations for the single and the product moments of order statistics from n independent and non-identically distributed Lomax and right-truncated Lomax random variables. These recurrence relations are simple in nature and could be used systematically in order to compute all the single and product moments of all order statistics in a simple recursive manner. The results for order statistics from the multiple-outlier model (with a slippage of p observations) are deduced as special cases. We then apply these results by examining the robustness of censored BLUE's to the presence of multiple outliers. Received: November 30, 1998; revised version: March 8, 2000     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from a generalized half-logistic distribution with application to inference

In this paper, we establish several recurrence relations for the single and product moments of progressively Type-II right-censored order statistics from a generalized half-logistic distribution. The use of these relations in a systematic recursive manner enables the computation of all the means, variances, and covariances of progressively Type-II right-censored order statistics from the generalized half-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁, …, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Sandhu [Recurrence relations for single and product moments of order statistics from a generalized half-logistic distribution with applications to inference, J. Stat. Comput. Simul. 52 (1995), pp. 385–398.]. The moments so determined are then utilized to derive the best linear unbiased estimators of the scale and location–scale parameters of the generalized half-logistic distribution. The best linear unbiased predictors of censored failure times are discussed briefly. Finally, a numerical example is presented to illustrate the inferential method developed here.     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from generalized logistic distribution with applications to inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a generalized logistic distribution. The use of these relations in a systematic manner allow us to compute all the means, variances, and covariances of progressively Type-II right censored order statistics from the generalized logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁, …, R_m). These moments are then utilized to derive best linear unbiased estimators of the scale and location-scale parameters of the generalized logistic distribution. A comparison of these estimators with the maximum likelihood estimates is then made through Monte Carlo simulations. Finally, the best linear unbiased predictors of censored failure times is discussed briefly.     

Asymptotic distributions of order statistics and record values arising from the class of beta-generated distributions

In this paper, we show that both the class of beta-generated distributions G_F and its base distribution F belong to the same domain of maximal (or minimal or upper record value or lower record value) attraction. Moreover, it is shown that the weak convergence of any non-extreme order statistic (central or intermediate order statistic), based on a base distribution F, to a non-degenerate limit type implies the weak convergence of G_F to a non-degenerate limit type. The relations between the two limit types are deduced.     

On an application of concomitants of order statistics

ABSTRACT

Let (X_i, Y_i), i = 1, …, n be a pair where the first coordinate X_i represents the lifetime of a component, and the second coordinate Y_i denotes the utility of the component during its lifetime. Then the random variable Y_{[r: n]} which is known to be the concomitant of the rth order statistic defines the utility of the component which has the rth smallest lifetime. In this paper, we present a dynamic analysis for an n component system under the above-mentioned concomitant setup.     

Simulation of Balakrishnan skew-normal order statistics

ABSTRACT

The novel Balakrishnan skew-normal distribution introduced in 2008 has received considerable interest. Here, we derive stochastic representations for simulating order statistics of the novel Balakrishnan skew-normal distribution. The resulting algorithms are more efficient than the ordinary sorting algorithm.     

Recurrence relations for the moments of order statistics from a beta distribution

Certain recurrence relations for the single and product moments of the order statistics of a random sample of sizen arising from a beta distribution are derived. The usefulness of these relations in evaluating the single and product moments of beta order statistics is also discussed.     

Estimation and test of several multivariate normal means under an order restriction when the dimension is larger than two

Suppose that an order restriction is imposed among several p-variate normal mean vectors. We are interested in the problems of estimating these mean vectors and testing their homogeneity under this restriction. These problems are multivariate extensions of Bartholomew's (1959) ones. For the bivariate case, these problems have been studied by Sasabuchi et al. (1983) and (1998) and some others. In the present paper we examine the convergence of an iterative algorithm for computing the maximum likelihood estimator when p is larger than two. We also study some test procedures for testing homogeneity when p is larger than two.     

Maximum variance of order statistics from symmetric populations revisited

We consider i.i.d. samples of size n with symmetric non-degenerate parent distributions and finite variances. Papadatos [A note on maximum variance of order statistics from symmetric populations, Ann. Inst. Statist. Math. 48 (1997), pp. 117–121] proved that the maximal variance of each non-extreme order statistic, expressed in the population variance units, is attained in a one-parametric family of symmetric two- and three-point distributions. The parameters of the extreme variance distributions coincide with the arguments maximizing some polynomials of degree 2n?1 over a finite interval. The bounds for variances are equal to the maximal values of the polynomials. We present a more precise solution to the problem by applying the variation diminishing property of Bernstein polynomials.     

A pseudo restricted MLE under multivariate order restrictions and its algorithm

Abstract

It is shown in this paper that a quasi order for the vectors in R^p is a cone induced if and only if the order is preservable under limits and under linear combinations with non-negative coefficients. For the mean vectors in MANOVA subject to the restriction of simple ordering, a pseudo restricted MLE is proposed. This estimator is a matrix projection onto a closed convex set inside the restricted domain. An algorithm for the pseudo restricted MLE is developed, that computes the matrix projections using only vector projections.     

Non-parametric predictive inference for future order statistics

This article presents non-parametric predictive inference for future order statistics. Given the data consisting of n real-valued observations, m future observations are considered and predictive probabilities are presented for the rth-ordered future observation. In addition, joint and conditional probabilities for events involving multiple future order statistics are presented. The article further presents the use of such predictive probabilities for order statistics in statistical inference, in particular considering pairwise and multiple comparisons based on two or more independent groups of data.     

Tests for the order of integration against higher order integration

Locally best invariant tests for the null hypothesis of I(p) against the alternative hypothesis of I(q), < q, are developed for models with independent normal errors. The tests are semiparametrically extended for models with autocorrelated errors. The method is illustrated by two real data sets in terms of double unit roots. The proposed tests can be used for determining integration orders of nonstationary time series.     

Partial order relations for classification comparisons

The Bayes classification rule offers the optimal classifier, minimizing the classification error rate, whereas the Neyman–Pearson lemma offers the optimal family of classifiers to maximize the detection rate for any given false alarm rate. These motivate studies on comparing classifiers based on similarities between the classifiers and the optimal. In this article, we define partial order relations on classifiers and families of classifiers, based on rankings of rate function values and rankings of test function values, respectively. Each partial order relation provides a sufficient condition, which yields better classification error rates or better performance on the receiver operating characteristic analysis. Various examples and applications of the partial order theorems are discussed to provide comparisons of classifiers and families of classifiers, including the comparison of cross-validation methods, training data that contains outliers, and labelling errors in training data. The Canadian Journal of Statistics 48: 152–166; 2020 © 2019 Statistical Society of Canada     

Slope-rotatable designs with equal maximum directional variance for second order response surface models

The concept of sloperotaiability with equal maximum directional vari ance for second order response surface models is introduced as a new design property. This requires that the maximum variance of the estimated slope over all possible directions be only a function of p, which is the distance from the design originif is shown that a rotatable design satisfies this property Also, minimization of tiie maximum variance of the estimated slope over all possible directions is proposed as a new design optirnality criterion, and op¬timal designs are called slope-directional minirnax designs. For the class of cquiradial designs, the slope-directional minirnax designs are compared with D— optimal designs.     

A zero-one law for large order statistics

Fix r ≥ 1, and let {M_nr} be the rth largest of {X₁,X₂,…X_n}, where X₁,X₂,… is a sequence of i.i.d. random variables with distribution function F. It is proved that P[M_nr ≤ u_n i.o.] = 0 or 1 according as the series Σ^∞_n=3Fⁿ(u_n)(log log n)^r/n converges or diverges, for any real sequence {u_n} such that n{1 -F(u_n)} is nondecreasing and divergent. This generalizes a result of Bamdorff-Nielsen (1961) in the case r = 1.