Smallest Order Statistics Research Articles

Orderings of extremes from dependent Gaussian variables with Archimedean copula under simple tree order restrictions

Suppose the random variables follow normal distributions , , and their dependence is modelled by Archimedean copula with generator ϕ. Then, the following results are established for the largest order statistics: (1) Suppose and are increasing in t>0, then, if and for we have ; (2) Suppose is decreasing and is increasing in t>0, then, if and for we have ; (3) Suppose and are increasing in t>0, then, if and for we have ; (4) Suppose is decreasing and is increasing in t>0, then, if and for we have . Analogous results are then established for smallest order statistics as well. In addition, we present some numerical examples to illustrate all the results established here. Finally, some concluding remarks are made.

Stochastic comparisons of extreme order statistic from dependent and heterogeneous lower-truncated Weibull variables under Archimedean copula

<abstract><p>This article studies the stochastic comparisons of order statistics with dependent and heterogeneous lower-truncated Weibull samples under Archimedean copula. To begin, we obtain the usual stochastic and hazard rate orders of the largest and smallest order statistics from heterogeneous and dependent lower-truncated Weibull samples under Archimedean copula. Second, under Archimedean copula, we get the convex transform and the dispersive orders of the largest and smallest order statistics from dependent and heterogeneous lower-truncated Weibull samples. Finally, several numerical examples are given to demonstrate the theoretical conclusions.</p></abstract>

Ordering results of the smallest order statistics from independent heterogeneous new modified generalized linear failure rate random variables

This article deals with stochastic comparisons of series systems comprising a new modified generalized linear failure rate family of distributions. The results are established in three directions. First, we establish based on vector majorization the usual stochastic order and the hazard rate order of series systems. Next, multiple-outlier models are considered. When the number of components is equal, the results are obtained in terms of the likelihood ratio order. For an unequal number of components, we develop comparisons according to the usual stochastic order, and hazard rate order. Then, based on weak majorization and p-larger orders between the vector of parameters, some characterization results for comparing the lifetimes of series systems are also established. Finally, we discuss based on multivariate chain majorization the usual stochastic order of series systems. Numerical examples are considered to illustrate the results.

Generalized Modified Inverse Weibull Distribution: Its Properties and Applications

In this paper, we introduce a new useful continuous distribution called generalized modified inverse Weibull distribution. This distribution is a four-parameter extension of the modified inverse Weibull which generalizes some well-known distributions. Various statistical and probabilistic properties are derived such as rth moment, moment generating function, Renyi and Shannon entropies and hazard rate function. We also discuss estimation of the parameters by maximum likelihood and provide the information matrix. The likelihood ratio order (which implies the hazard rate and usual stochastic orders) between smallest order statistics from two independent heterogeneous samples of this new family are discussed. Finally, a real numerical example is also considered for illustrative purposes.

Comparisons between largest and smallest order statistics from Pareto distributions

Comparisons between largest and smallest order statistics from Pareto distributions

STOCHASTIC MONOTONICITY OF CONDITIONAL ORDER STATISTICS IN MULTIPLE-OUTLIER SCALE POPULATION

This paper discusses the stochastic monotonicity property of the conditional order statistics from independent multiple-outlier scale variables in terms of the likelihood ratio order. Let X1, …, Xn be a set of non-negative independent random variables with Xi, i=1, …, p, having common distribution function F(λ1x), and Xj, j=p+1, …, n, having common distribution function F(λ2x), where F(·) denotes the baseline distribution. Let Xi:n(p, q) be the ith smallest order statistics from this sample. Denote by $X_{i,n}^{s}(p,q)\doteq [X_{i:n}(p,q)|X_{i-1:n}(p,q)=s]$. Under the assumptions that xf′(x)/f(x) is decreasing in x∈ℛ+, λ1≤λ2 and s1≤s2, it is shown that $X_{i:n}^{s_{1}}(p+k,q-k)$ is larger than $X_{i:n}^{s_{2}}(p,q)$ according to the likelihood ratio order for any 2≤i≤n and k=1, 2, …, q. Some parametric families of distributions are also provided to illustrate the theoretical results.

On extreme order statistics from heterogeneous Weibull variables

ABSTRACTIn this paper, we focus on stochastic comparisons of extreme order statistics from heterogeneous independent/interdependent Weibull samples. Specifically, we study extreme order statistics from Weibull distributions with (i) common shape parameter but different scale parameters, and (ii) common scale parameter but different shape parameters. Several new comparison results in terms of the likelihood ratio order, reversed hazard rate order and usual stochastic order are studied in those scenarios. The results established here strengthen and generalize some of the results known in the literature including Khaledi and Kochar [Weibull distribution: some stochastic comparisons. J Statist Plann Inference. 2006;136:3121–3129], Fang and Zhang [Stochastic comparisons of series systems with heterogeneous Weibull components. Statist Probab Lett. 2013;83:1649–1653], Torrado [Comparisons of smallest order statistics from Weibull distributions with different scale and shape parameters. J Korean Statist Soc. 2015;44:68–76] and Torrado and Kochar [Stochastic order relations among parallel systems from Weibull distributions. J Appl Probab. 2015;52:102–116]. Some numerical examples are also provided for illustration.

An Erd\xf6s\u2013R\xe9v\xe9sz Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Let {X(t):tin mathbb R_+} be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, mathbb E X(t) = 0, mathbb E X^2(t) = 1 and correlation function satisfying (i) r(t) = 1 - C|t|^{alpha } + o(|t|^{alpha }) as trightarrow 0 for some 0le alpha le 2 and C>0; (ii) sup _{tge s}|r(t)|<1 for each s>0 and (iii) r(t) = O(t^{-lambda }) as trightarrow infty for some lambda >0. For any nge 1, consider n mutually independent copies of X and denote by {X_{r:n}(t):tge 0} the rth smallest order statistics process, 1le rle n. We provide a tractable criterion for assessing whether, for any positive, non-decreasing function f, mathbb P(mathscr {E}_f)=mathbb P(X_{r:n}(t) > f(t), text { i.o.}) equals 0 or 1. Using this criterion we find, for a family of functions f_p(t) such that z_p(t)=mathbb P(sup _{sin [0,1]}X_{r:n}(s)>f_p(t))=O((tlog ^{1-p} t)^{-1}), that mathbb P(mathscr {E}_{f_p})= 1_{{pge 0}}. Consequently, with xi _p (t) = sup {s:0le sle t, X_{r:n}(s)ge f_p(s)}, for pge 0 we have lim _{trightarrow infty }xi _p(t)=infty and limsup _{trightarrow infty }(xi _p(t)-t)=0 a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on xi _p(t), namely, that liminf _{trightarrow infty }(xi _p(t)-t)/h_p(t) = -1 a.s. for p>1 and liminf _{trightarrow infty }log (xi _p(t)/t)/(h_p(t)/t) = -1 a.s. for pin (0,1], where h_p(t)=(1/z_p(t))plog log t.

Role of concomitants of order statistics in determining parent bivariate distributions

ABSTRACTIn this paper, we establish the role of concomitants of order statistics in the unique identification of the parent bivariate distribution. From the results developed, we have illustrated by examples the process of determination of the parent bivariate distribution using a marginal pdf and the pdf of either of the concomitant of largest or smallest order statistic on the other variable. An application of the results derived in modeling of a bivariate distribution for data sets drawn from a population as well is discussed.

Ordering results for the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables

ABSTRACTIn this paper, we discuss stochastic comparisons of the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables. Let be independent random variables with Further, let be another set of independent random variables with First, when and a matrix with different parameters changes to another matrix in the sense of multivariate chain majorization and row majorization, we investigate the usual stochastic order of the largest order statistics. Next, when and , we establish the usual stochastic order of the largest and smallest order statistics. Finally, we provide sufficient conditions for the hazard rate order of the smallest order statistics.

Stochastic comparisons on sample extremes of dependent and heterogenous observations

Stochastic comparison on order statistics from heterogeneous-dependent observations has been paid lots of attention recently. This paper devotes to investigating the ordering properties of order statistics from dependent observations. We derive the usual stochastic order for sample minimums and the second smallest order statistic, the dispersive order and the star order for minimums of samples having proportional hazards and Archimedean survival copulas. Similar ordering results are also obtained for maximums and the second largest order statistic of samples having proportional reversed hazards and Archimedean copulas. Several examples illustrating the main results are presented as well.

Stochastic somparisons of order statistics from scaled and interdependent random variables

This paper studies order statistics from random variables following the scale model. In the presence of the Archimedean copula or survival copula for the random variables, we obtain the usual stochastic order of the sample extremes and the second smallest order statistic, the dispersive order and the star order of the sample extremes.

Size-biased permutation of a finite sequence with independent and identically distributed terms

This paper focuses on the size-biased permutation of $n$ independent and identically distributed (i.i.d.) positive random variables. This is a finite dimensional analogue of the size-biased permutation of ranked jumps of a subordinator studied in Perman-Pitman-Yor (PPY) [Probab. Theory Related Fields 92 (1992) 21-39], as well as a special form of induced order statistics [Bull. Inst. Internat. Statist. 45 (1973) 295-300; Ann. Statist. 2 (1974) 1034-1039]. This intersection grants us different tools for deriving distributional properties. Their comparisons lead to new results, as well as simpler proofs of existing ones. Our main contribution, Theorem 25 in Section 6, describes the asymptotic distribution of the last few terms in a finite i.i.d. size-biased permutation via a Poisson coupling with its few smallest order statistics.

Non-positive upper bounds on expectations of small order statistics from DDA and DFRA populations

Rychlik [Appl Math (Warsaw) 29:15–32, 2002] presented positive sharp upper bounds on the expectations of order statistics with sufficiently large ranks, based on i.i.d. samples from the decreasing density and failure rate populations (DDA and DFRA, for short). They were expressed in terms of the population mean and standard deviation. Here we provide respective non-positive upper tight evaluations for expected small order statistics centered about the population mean, measured in various scale units.

Some Aspects of Probability of Correct Selections

In an experiment of treatment selections, random samples are drawn from k populations with ordered means. The probability that a sample statistic from the population with the highest mean turns out to be ranked the highest is referred to as the probability of correct selection (PCS). An inequality was proved previously that shows the monotonicity of PCS with respect to change in variance of the samples. In this article, we first present a more general form of the probability inequality to be used to investigate PCS. An extension of the monotonicity of PCS to order statistics is considered. We show that the PCS of the smallest order statistic preserves the monotonicity. Additionally, a normal approximation method is used to further generalize the theory. The general order statistics will not enjoy the same properties, as we reveal the obstacles, and a numerical counter example.

Moments of Order Statistics from Nonidentically Distributed Three Parameters Beta typeI and Erlang Truncated Exponential Variables

Problem statement: Moments of order statistics of independent non-identically distributed (INID) random variables is not an easy subject to deal with for continuous distributions. One is forced to use messy algebraic calculations (whether one uses permanents or not). This was the motivation behind this study. In this study the moments of order statistics arising from independent nonidentically distributed three parameters Beta type I distribution and Erlang Truncated Exponential distribution were derived. Approach: We employed an easier technique established by Barakat & Abdelkader will be referred to as (BAT). Results: The mean, the second moment and the variance of the median and the smallest order statistics for the first distribution were given for different values of the shape parameter and different sample sizes. Conclusion: The results can be used to make some inferences and used the BAT technique to derive moments of order statistics arising from independent nonidentically distributed for any other continuous distribution with distribution function (cdf) in the form: F(x) = 1-λ (x).

Tight evaluations for expectations of small order statistics from symmetric and symmetric unimodal populations

We present sharp non-positive upper bounds on the deviations of the expectations of small order statistics from the population mean, measured in the scale units generated by the population absolute central moments, when the parent distribution is either symmetric or symmetric unimodal.

On Dependence Properties of Random Minima and Maxima

Let X (n) and X (1) be the largest and smallest order statistics, respectively, of a random sample of fixed size n. Quite generally, X (1) and X (n) are approximately independent for n sufficiently large. In this article, we study the dependence properties of random extremes in terms of their copula, when the sample size has a left-truncated binomial distribution and show that they tend to be more dependent in this case. We also give closed-form formulas for the measures of association Kendall's τ and Spearman's ρ to measure the amount of dependence between two extremes.

Bounds on expectations of small order statistics from decreasing density populations

For i > (n + 1)/2, Danielak (Statistics 37:305–324, 2003) established an optimal positive upper mean-variance bound on the expectation of ith order statistic based on the i.i.d. sample of size n from the decreasing density population. We show that the best bounds on the expected deviation of the ith order statistics from the population mean, i ≤ (n + 1)/2, expressed in more general scale units generated by pth absolute central moments with p > 1 amount to zero. We also determine the respective strictly negative bounds in the mean absolute deviation units.

Fisher information and optimal schemes in progressive Type-II censored samples

This article deals with two results for the computation of the Fisher information (FI) in progressive Type-II censored data; the results simplifies the calculation of the FI in any progressive censored data. The FI is presented as a sum of the FI in the smallest order statistics. The results are used to obtain the optimal censoring schemes in the progressive Type-II censored data for two optimality criteria that are based on the inverse of the exact FI. The methodology is illustrated with the normal and the Weibull distributions, which are popular models in the lifetime analysis.