Rth Order Statistic Research Articles

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.

Estimation of the scale parameter of a log-logistic distribution

In this work the ranked set sampling technique has been applied to estimate the scale parameter \(\alpha \) of a log-logistic distribution under a situation where the units in a sample can be ordered by judgement method without any error. We have evaluated the Fisher information contained in the order statistics arising from this distribution and observed that median of a random sample contains the maximum information about the parameter \(\alpha \). Accordingly we have used median ranked set sampling to estimate \(\alpha \). We have further carried out the multistage median ranked set sampling to estimate \(\alpha \) with improved precision. Suppose it is not possible to rank the units in a sample according to judgement method without error but the units can be ordered based on an auxiliary variable \(Z\) such that \((X, Z)\) has a Morgenstern type bivariate log-logistic distribution (MTBLLD). In such a situation we have derived the Fisher information contained in the concomitant of rth order statistic of a random sample of size \(n\) from MTBLLD and identified those concomitants among others which possess largest amount of Fisher information and defined an unbalanced ranked set sampling utilizing those units in the sample and thereby proposed an estimator of \(\alpha \) using the measurements made on those units in this ranked set sample.

Estimation and goodness-of-fit procedures for Farlie–Gumbel–Morgenstern bivariate copula of order statistics

In this study, we provide the Farlie–Gumbel–Morgenstern bivariate copula of rth and sth order statistics. The main emphasis in this study is on the inference procedure which is based on the maximum pseudo-likelihood estimate for the copula parameter. As for the methodology, goodness-of-fit test statistic for copulas which is based on a Cramér–von Mises functional of the empirical copula process is applied for selecting an appropriate model by bootstrapping. An application of the methodology to simulated data set is also presented.

Concomitants of order statistics from multivariate elliptical distributions

In this paper, by considering a 2 n-dimensional elliptically contoured random vector ( X T , Y T ) T = ( X 1 , … , X n , Y 1 , … , Y n ) T , we derive the exact joint distribution of linear combinations of concomitants of order statistics arising from X . Specifically, we establish a mixture representation for the distribution of the rth concomitant order statistic, and also for the joint distribution of the rth order statistic and its concomitant. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The two most important special cases of multivariate normal and multivariate t distributions are then discussed in detail. Finally, an application of the established results in an inferential problem is outlined.

Estimating variances of strata in ranked set sampling

In RSS, the variance of observations in each ranked set plays an important role in finding an optimal design for unbalanced RSS and in inferring the population mean. The empirical estimator (i.e., the sample variance in a given ranked set) is most commonly used for estimating the variance in the literature. However, the empirical estimator does not use the information in the entire data over different ranked sets. Further, it is highly variable when the sample size is not large enough, as is typical in RSS applications. In this paper, we propose a plug-in estimator for the variance of each set, which is more efficient than the empirical one. The estimator uses a result in order statistics which characterizes the cumulative distribution function (CDF) of the rth order statistics as a function of the population CDF. We analytically prove the asymptotic normality of the proposed estimator. We further apply it to estimate the standard error of the RSS mean estimator. Both our simulation and empirical study show that our estimators consistently outperform existing methods.

Statistical inference for induced L-statistics: a random perturbation approach

Suppose that X and Y are two numerical characteristics defined for each individual in a population. In a random sample of (X, Y) with sample size n, denote the rth ordered X variate by X r:n and the associated Y variate, the induced rth order statistics, by Y [r:n], respectively. Induced order statistics arise naturally in the context of selection where individuals ought to be selected by their ranks in a related X value due to difficulty or high costs of obtaining Y at the time of selection. The induced L-statistics, which take the form of , are very useful in regression analysis, especially when the observations are subject to a type-II censoring scheme with respect to the dependent variable, or when the regression function at a given quantile of the predictor variable is of interest. The limiting variance of the induced L-statistics involve the underlying regression function and inferences based on nonparametric estimation are often unstable. In this paper, we consider the distributional approximation of the induced L-statistics by the random perturbation method. Large sample properties of the randomly perturbed induced L-statistics are established. Numerical studies are also conducted to illustrate the method and to assess its finite-sample performance.

Error performance of general order selection in correlated Nakagami fading channels

A procedure for determining the probability distribution of the rth order statistic, Gr:L, r=1, 2, , L, among a set of L correlated Nakagami diversity branch gains G1, G2, , GL has been described in David and Nagaraja (2003) and Elkashlan et al. (2008). The results are used to evaluate the bit error rate (BER) of general order selection (GOS), a diversity method in which the rth order branch is selected for transmission, over correlated Nakagami fading branches. GOS can be used to improve system throughput and provide various levels of services, both of which are highly desirable in high-speed communication systems. Numerical and simulation results are presented and used to illustrate the effects of fading correlation on the BER associated with the rth order gain branch.

Smooth non‐parametric estimation of the distribution function from balanced ranked set samples

AbstractIn certain sampling schemes, such as ranked set sampling, the observed data set can consist entirely of independent order statistics. A balanced ranked set sample consists of n independent cycles, in each of which the experimenter selects k random samples, each of size k. The recorded observation in sample r of cycle i is the rth order statistic, X(r). Stokes and Sager (1988) have shown that the empirical distribution function based on such samples is an unbiased estimate of the underlying distribution function and is more precise than the empirical distribution function calculated from a simple random sample of the same size. In this article, the properties of the empirical distribution function developed by Stokes and Sager (1988) are further investigated. In addition, a kernel‐type estimator of the distribution function is developed. For large n, the smooth estimator is shown to be consistent and asymptotically normal. Finally, small sample properties of the two estimators are investigated and compared via computer simulations. Copyright © 2004 John Wiley & Sons, Ltd.

Stochastic Comparisons and Dependence among Concomitants of Order Statistics

Let (Xi, Yi) i=1, 2, …, n be n independent and identically distributed random variables from some continuous bivariate distribution. If X(r) denotes the rth ordered X-variate then the Y-variate, Y[r], paired with X(r) is called the concomitant of the rth order statistic. In this paper we obtain new general results on stochastic comparisons and dependence among concomitants of order statistics under different types of dependence between the parent random variables X and Y. The results obtained apply to any distribution with monotone dependence between X and Y. In particular, when X and Y are likelihood ratio dependent, it is shown that the successive concomitants of order statistics are increasing according to likelihood ratio ordering and they are TP2 dependent in pairs. If we assume that the conditional hazard rate of Y given X=x is decreasing in x, then the concomitants are increasing according to hazard rate ordering and are dependent according to the right corner set increasing property. Finally, it is proved that if Y is stochastically increasing in X, then the concomitants of order statistics are stochastically increasing and are associated. Analogous results are obtained when the variables X and Y are negatively dependent. We also prove that if the hazard rate of the conditional distribution of Y given X=x is decreasing in x and y, then the concomitants have DFR (decreasing failure rate) distributions and are ordered according to dispersive ordering.

A Note on Extremes of Concomitants of Order Statistics

Let $$\{ (X_n ,Y_n )\} _{n \geqslant 1} $$ be a sequence of identically distributed variables. We study the asymptotic distribution of $$M_{k:n} = \max (Y_{[n - k + 1:n]} , \ldots ,Y_{[n:n]} )$$ , where Y [r:n] denotes the concomitant of the rth order statistic X r:n , corresponding to $$X_1 , \ldots ,X_n $$ , and $$k \geqslant 1$$ is held fixed while $$n \to \infty $$ . Conditions are given for the $$Y_{[r:n]} $$ and $$M_{k:n} $$ to have the same asymptotic behavior as that we would apply if $$\{ (X_n ,Y_n )\} _{n \geqslant 1} $$ were i.i.d. The result is illustrated with a simple linear regression model $$Y_n = \beta X_n + \varepsilon _n ,n \geqslant 1$$ , where $$\{ (X_n \} _{n \geqslant 1} $$ is a stationary sequence with extremal index $$\theta < 1$$ .

On characterizations of independent identically distributed random variables via order structures

Let X 1,X 2,…,X n be compatible independent random variables of continuous type, X (r) be the rth order statistic and I ( r) : I ( r) = i if X ( r) = X i . It is shown that I (r) and X (r) are independent if and only if identically distributed.

Covariance between variables and their order statistics for multivariate normal variables

Siegel (1993, J. Amer. Statist. Assoc. 88, 77–80) showed that when ( X 1, …, X n ) have a multivariate normal distribution then Cov( X 1, X (1)) = Σ n i = 1 Cov( X 1, X i ) P( X i = X (1)), where X (1) is the minimum of ( X 1, …, X n ). We show that a similar result holds for any order statistic. Thus X (1) can be replaced by X ( r) , the rth order statistic, everywhere in the above formula. Normality is essentially also necessary for this result to hold.

Characterizations of identically distributed independent random variables using order statistics

Suppose X 1,… X n are independent random variables of continuous type with proportional hazard (or failure) rates. Let X ( r) denote the rth order statistic and let I ( r) = i if X i = X ( r) . It is proved that X i 's are identically distributed if and only if X ( r) and I ( r) are independent for some r ∈ {2,…, n}. The second characterization is in terms of order statistics from subsamples.

Asymptotic ( r− 1)-dependent representation for rth order statistic from a stationary sequence

Let X 1, X 2,… be a stationary sequence of random variables. Denote by M ( k) n the kth largest value of X 1, X 2, …, X n . We find necessary and sufficient conditions for the existence of an ( r− 1)-dependent stationary sequence X̃ 1, X̃ 2, …(determined by a distribution function G and numbers β 1, β 2,…, β r ⩾ 0, Σ r q=1 β q =1), such that for each k, 1⩽k⩽r, sup χ∈ R 1|P (M (k) n⩽x)minus;P( M ̃ (k) n⩽x)|→0 as n→+∞, where M ̃ (1) n, M ̃ (2) n, …, M ̃ (r) n are order statistics of X ̃ 1, X ̃ 2, …, X ̃ n . If such asymptotic ( G, β 1, β 2, …, β r )-representation exists, then for each k, 1⩽ k⩽ r, there are numbers 0⩽ γ k, j ⩽1, j=1, 2, …, k−1, satisfying sup χ∈ R 1|P(M (k) n⩽x)− P(M (l) n⩽x)( 1 + Σ k−1 j=1 ((-log P(M (1) n⩽ x)) j j! )·γ k,j)|→0 as n→∞. This corresponds to limit theorems for M ( q) n obtained by Hsing (1988). Convergence of all order statistics is also discussed.

Bounds on distribution functions of order statistics for dependent variates

Upper and lower bounds are given for F X r:n , the distribution function of the rth order statistic from n possibly dependent random variables. We show that these bounds may be reached when the random variables have a common distribution function. For any distribution function F we may construct a set of n exchangeable variates (with c.d.f. F), whose dependency structure is such that the bounds are attained.

On a recurrence relation for order statistics

For n random variables a recurrence relation for the joint distribution function of the rth and sth order statistics is derived without any assumption about the joint distribution function of the n random variables.

Moments of order statistics from one truncation parameter densities

Analogs of some classical recurrence relations for moments of order statistics for a one truncation parameter densities are derived. These relate the kthorder moment of the rthorder statistic Xr:nto the kthorder moment of Xl:ior Xr:rvia an integral operator. Similar results are obtained for product moments. An application is also given.

A class of distributions connected to order statistics with nonintegral sample size

Newton's binomial series expansion is used to develop a (class of) distribution function(s) Fr:∝ which may be interpreted as the distribution of the rthorder statistic with nonintegral sample size∝. It is shown that Fr:∝ has properties similar to Fr:n. Multivariate extension is considered and an elementary proof of the integral representation for the joint distribution of a subset of order statistics is given. An application is included.

Unbiased plotting positions for the general extreme value distribution

Probability plots maintain their popularity for flood frequency analysis, and are used to fit distributions, to identify outliers and to assess goodness-of-fit. If the objective of a plot is to determine quantiles or distribution parameters then one must employ unbiased plotting positions, defined as the mean of the rth order statistic in samples from the reduced variate population. These plotting positions differ from one distribution to another. The general extreme value (GEV) distribution is commonly used for flood frequency analysis, and contains the Gumbel distribution as a special case. Whilst plotting position formulae for this special case are well established, results are not available for the general case. By careful sequencing of the individual terms it is possible to integrate numerically the basic expression for expected order statistics to compute plotting positions for all practical sample sizes. However this integration consumes considerable computer resources, and for sample sizes of 35 and less exact unbiased plotting positions can be more readily evaluated using expressions derived using probability weighted moment theory. Numerical problems force an alternative approach for larger samples, and an asymptotic approximation has been developed. This has been found to be very accurate even for small samples, although an exact formula should be used for the largest flood in a sample. An even simpler level of calculation may be appropriate in some circumstances, and plotting position formulae have been developed based on expressions for the non-exceedance probability of the form F r = (r − α) (n − β) . The parameters α and β are allowed to vary with sample size and the shape parameter of the general extreme value distribution.

Estimating the Scale Parameter of an Exponential Distribution from a Sample of Time-Censoredrth-Order Statistics

Abstract The problem of estimating the mean of an exponential distribution is studied, when the data available are a random sample of time-censored rth-order statistics. Examples for such empirical situations are cited. Maximum likelihood estimators (MLE's) and moment-equation estimators (MEE's) are studied. Theoretical derivations are provided for the large sample variances and distributions of these estimators. The efficiency of the MEE compared to the MLE is studied.