On the asymptotic independence of numbers of observations near order statistics

In this paper, we consider the numbers of observations in two-sided neighbourhoods of the kth and (n?r)th order statistics from a sample of size n and show that they are asymptotically independent as n→∞. We also establish a result that generalizes all the existing results regarding the asymptotic independence of numbers of observations in the left and right neighbourhoods of order statistics. Finally, we consider the limiting joint behaviour of numbers of observations in the neighbourhoods of s central order statistics and establish that they are asymptotically independent.     

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.     

Parameter Estimation for Bivariate Shock Models with Singular Distribution for Censored Data with Concomitant Order Statistics

When two‐component parallel systems are tested, the data consist of Type‐II censored data X_(i), i= 1, n, from one component, and their concomitants Y _[i] randomly censored at X_(r), the stopping time of the experiment. Marshall & Olkin's (1967) bivariate exponential distribution is used to illustrate statistical inference procedures developed for this data type. Although this data type is motivated practically, the likelihood is complicated, and maximum likelihood estimation is difficult, especially in the case where the parameter space is a non‐open set. An iterative algorithm is proposed for finding maximum likelihood estimates. This article derives several properties of the maximum likelihood estimator (MLE) including existence, uniqueness, strong consistency and asymptotic distribution. It also develops an alternative estimation method with closed‐form expressions based on marginal distributions, and derives its asymptotic properties. Compared with variances of the MLEs in the finite and large sample situations, the alternative estimator performs very well, especially when the correlation between X and Y is small.     

Bounds for Expectations of Differences of Generalized Order Statistics Based on General and Life Distributions

Let X _? ^(r), r ≥ 1, denote generalized order statistics based on an arbitrary distribution function F with finite pth absolute moment for some 1 ≤ p ≤ ∞. We present sharp upper bounds on E(X _? ^(s) ? X _? ^(r)), 1 ≤ r < s, for F being either general or life distribution. The bounds are expressed in various scale units generated by pth central absolute or raw moments of F, respectively. The distributions achieving the bounds are specified.     

DETERMINATION OF DOMAINS OF ATTRACTION BASED ON A SEQUENCE OF MAXIMA

Suppose that the maximum of a random sample from a distribution F(x) may be obtained in each of k equally spaced observation periods. This paper proposes a test to determine the domain of attraction of F(x), and investigates the properties when the sample size is very large and perhaps unknown and k is fixed and small. The test statistic is a function of the spacings between the order statistics based on the sequence of maxima and is suggested by reference to one studied previously when inference was based on the largest k observations of a random sample. A Monte Carlo study shows that the proposed test is more powerful than its main competitor. The test is illustrated by two examples.     

OPTIMAL SEMIPARAMETRIC INFERENCE FOR THE TAIL INDEX BASED ON RATIOS OF THE LARGEST EXTREMES

The k largest order statistics in a random sample from a common heavy‐tailed parent distribution with a regularly varying tail can be characterized as Fréchet extremes. This paper establishes that consecutive ratios of such Fréchet extremes are mutually independent and distributed as functions of beta random variables. The maximum likelihood estimator of the tail index based on these ratios is derived, and the exact distribution of the maximum likelihood estimator is determined for fixed k, and the asymptotic distribution as k →∞ . Inferential procedures based upon the maximum likelihood estimator are shown to be optimal. The Fréchet extremes are not directly observable, but a feasible version of the maximum likelihood estimator is equivalent to Hill's statistic. A simple diagnostic is presented that can be used to decide on the largest value of k for which an assumption of Fréchet extremes is sustainable. The results are illustrated using data on commercial insurance claims arising from fires and explosions, and from hurricanes.     

Some Related Minima Stability and Minima Infinite Divisibility of the General Multivariate Pareto Distributions

This article studies the minima stable property of the general multivariate Pareto distributions MP^(k)(I), MP^(k)(II), MP^(k)(III), MP^(k)(IV) which can be applied to characterize the MP^(k) distribution via its weighted ordered coordinates minima and marginal distribution. Also, the multivariate semi-Pareto distribution (denoted by MSP) is discerned in the class of geometric minima infinite divisible and geometric minima stable distributions. If the exponent measure is satisfied by some functional equation, then the geometric minima stable property can be used to characterize the MSP distribution. Finally, the finite sample minima infinite divisible property of the MP^(k)(I), (II), and (IV) distributions is also discussed.     

Prediction Intervals for Future Order Statistics from a Generalized Modified Weibull

Viewing the future order statistics as latent variables at each Gibbs sampling iteration, several Bayesian approaches to predict future order statistics based on type-II censored order statistics, X₍₁₎, X₍₂₎, …, X_(r), of a size n( > r) random sample from a four-parameter generalized modified Weibull (GMW) distribution, are studied. Four parameters of the GMW distribution are first estimated via simulation study. Then various Bayesian approaches, which include the plug-in method, the Monte Carlo method, the Gibbs sampling scheme, and the MCMC procedure, are proposed to develop the prediction intervals of unobserved order statistics. Finally, four type-II censored samples are utilized to investigate the predictions.     

Data depth‐based nonparametric scale tests

Liu and Singh (1993, 2006) introduced a depth‐based d‐variate extension of the nonparametric two sample scale test of Siegel and Tukey (1960). Liu and Singh (2006) generalized this depth‐based test for scale homogeneity of k ≥ 2 multivariate populations. Motivated by the work of Gastwirth (1965), we propose k sample percentile modifications of Liu and Singh's proposals. The test statistic is shown to be asymptotically normal when k = 2, and compares favorably with Liu and Singh (2006) if the underlying distributions are either symmetric with light tails or asymmetric. In the case of skewed distributions considered in this paper the power of the proposed tests can attain twice the power of the Liu‐Singh test for d ≥ 1. Finally, in the k‐sample case, it is shown that the asymptotic distribution of the proposed percentile modified Kruskal‐Wallis type test is χ² with k ? 1 degrees of freedom. Power properties of this k‐sample test are similar to those for the proposed two sample one. The Canadian Journal of Statistics 39: 356–369; 2011 © 2011 Statistical Society of Canada     

Sign test based on k-tuple general ranked set samples

ABSTRACT

The sign test based on the k-tuple ranked set samples is discussed here. We first derive the distribution of the k-tuple ranked set sample sign test statistic, and then the asymptotic distribution is also obtained. We then compare its performance with its counterparts based on simple random sample and classical ranked set sample. The asymptotic relative efficiency and the power are then derived. Finally, the effect of imperfect ranking on the procedure is assessed.     

SELECTION PROCEDURES FOR SCALE PARAMETERS USING TWO-SAMPLE U-STATISTICS

Let be k independent populations having the same known quantile of order p (0 p 1) and let F(x)=F(x/_i) be the absolutely continuous cumulative distribution function of the ith population indexed by the scale parameter ₁, i = 1,…, k. We propose subset selection procedures based on two-sample U-statistics for selecting a subset of k populations containing the one associated with the smallest scale parameter. These procedures are compared with the subset selection procedures based on two-sample linear rank statistics given by Gill & Mehta (1989) in the sense of Pitman asymptotic relative efficiency, with interesting results.     

Sample moments of the autocorrelations of moving average processes and a modification to bartlett'sasymptotic variance formula

In this paper we express the sample autocorrelations for a moving average process of order q as a function of its own theoretical autocorrelations and the sample autocorrelations for the generating white noise series. Approximate analytic expressions are then obtained forthe moments of the sample autocorrelations of the moving average process.

Using these expressions, together with numerical evidence, we show that Bartlett's asymptotic formula for the variance of the sample autocorrelations of moving average processes, which is used widely in identifying these processes, is a large overestimate when considering finitesample sizes.

Our approach is for motivational purposes and so is purely formal, the amount of mathematics presented being kept to a minimum.     

A comparative study on a permutation statistic

We studied asymptotic distribution and finite sample properties of a randomly weighted permutation statistic. The asymptotic normality and the finite sample simulations derived from our studies provided theoretical and numerical justifications for distributional assumption of many useful test statistics used in identifying spatial autocorrelations of mapped data. We compared a new method in computing the mean and the approximated variance of the randomly weighted D statistic, a special permutation statistic, with the Walter’s conditional method. In the numerical illustration of the method, we calculated the standardized values of the D statistic by subtracting the mean from the D statistic and dividing the difference by the standard deviation for the standardized mortality ratios (SMRs) and the life expectancies among the 48 states of the continental USA. Spatial autocorrelations of the SMRs and the life expectancies were found to be statistically significant.     

A Fast Wavelet Algorithm for Analyzing One-Dimensional Signal Processing and Asymptotic Distribution of Wavelet Coefficients With Numerical Example and Simulation

In this article, we consider a sample point (t _j , s _j ) including a value s _j  = f(t _j ) at height s _j and abscissa (time or location) t _j . We apply wavelet decomposition by using shifts and dilations of the basic Häar transform and obtain an algorithm to analyze a signal or function f. We use this algorithm in practical to approximating function by numerical example. Some relationships between wavelets coefficients and asymptotic distribution of wavelet coefficients are investigated. At the end, we illustrate the results on simulated data by using MATLAB and R software.     

Inference on P(Y < X) in Bivariate Rayleigh Distribution

This paper deals with the estimation of reliability R = P(Y < X) when X is a random strength of a component subjected to a random stress Y, and (X, Y) follows a bivariate Rayleigh distribution. The maximum likelihood estimator of R and its asymptotic distribution are obtained. An asymptotic confidence interval of R is constructed using the asymptotic distribution. Also, two confidence intervals are proposed based on Bootstrap method and a computational approach. Testing of the reliability based on asymptotic distribution of R is discussed. Simulation study to investigate performance of the confidence intervals and tests has been carried out. Also, a numerical example is given to illustrate the proposed approaches.     

An (s,k, S) fluid inventory model with exponential leadtimes and order cancellations

We consider a stochastic fluid inventory model based on a (s, k, S) policy. The content level W = {W(t): t ≥ 0} increases or decreases according to a fluid-flow rate modulated by an n-state continuous time Markov chain (CTMC). W starts at W(0) = S; whenever W(t) drops to level s, an order is placed to take the inventory back to level S, which the supplier will carry out after an exponential leadtime. However, if during the leadtime the content level reaches k, the order is suppressed. We obtain explicit formulas for the expected discounted costs. The derivations are based on the optional sampling theorem (OST) to the multidimensional martingale and on fluid flow techniques.     

Test for the equality of several correlation coefficients

We derive two C(α) statistics and the likelihood-ratio statistic for testing the equality of several correlation coefficients, from k ≥ 2 independent random samples from bivariate normal populations. The asymptotic relationship of the C(α) tests, the likelihood-ratio test, and a statistic based on the normality assumption of Fisher's Z-transform of the sample correlation coefficient is established. A comparative performance study, in terms of size and power, is then conducted by Monte Carlo simulations. The likelihood-ratio statistic is often too liberal, and the statistic based on Fisher's Z-transform is conservative. The performance of the two C(α) statistics is identical. They maintain significance level well and have almost the same power as the other statistics when empirically calculated critical values of the same size are used. The C(α) statistic based on a noniterative estimate of the common correlation coefficient (based on Fisher's Z-transform) is recommended.     

Information Properties for Concomitants of Order Statistics in Farlie–Gumbel–Morgenstern (FGM) Family

Let (X _i , Y _i ), i = 1, 2,…, n be independent and identically distributed random variables from some continuous bivariate distribution. If X _(r) denotes the rth-order statistic, then the Y's associated with X _(r) denoted by Y _[r] is called the concomitant of the rth-order statistic. In this article, we derive an analytical expression of Shannon entropy for concomitants of order statistics in FGM family. Applying this expression for some well-known distributions of this family, we obtain the exact form of Shannon entropy, some of the information properties, and entropy bounds for concomitants of order statistics. Some comparisons are also made between the entropy of order statistics X _(r) and the entropy of its concomitants Y _[r]. In this family, we show that the mutual information between X _(r) and Y _[r], and Kullback–Leibler distance among the concomitants of order statistics are all distribution-free. Also, we compare the Pearson correlation coefficient between X _(r) and Y _[r] with the mutual information of (X _(r), Y _[r]) for the copula model of FGM family.     

Distributions of patterns of two successes separated by a string of <Emphasis Type="Italic">k</Emphasis>-2 failures

Let Z ₁, Z ₂, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z _t  = 1) and q = Pr(Z _t  = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns E₁{\mathcal{E}_{1}}: two successes are separated by at most k−2 failures, E₂{\mathcal{E}_{2}}: two successes are separated by exactly k −2 failures, and E₃{\mathcal{E}_{3}} : two successes are separated by at least k − 2 failures. Denote by N_n,k⁽ⁱ⁾{ N_{n,k}^{(i)}} (respectively M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}) the number of occurrences of the pattern E_i{\mathcal{E}_{i}} , i = 1, 2, 3, in Z ₁, Z ₂, . . . , Z _n when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} (resp. W_r,k⁽ⁱ⁾){W_{r,k}^{(i)})} be the waiting time for the r − th occurrence of the pattern E_i{\mathcal{E}_{i}}, i = 1, 2, 3, in Z ₁, Z ₂, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of N_n,k⁽ⁱ⁾{N_{n,k}^{(i)}}, M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}, T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} and W_r,k⁽ⁱ⁾{W_{r,k}^{(i)}} (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.