Some invariance principles for mized rank statistics and induced order statistics and some application

The structural affinity of mixed rank statistics and linear combinations of functions of concomitants of order statistics (or induced order statistics) is examined here. Some weal as well as strong invariance principles for these statistics are studied. A variety of models (depend on the nature of stochastic dependence of the two variates) is considered and the regularity conditions are tailored for these diverse situations. Some possible applications of these results in some problems of sequential (statistical) inference are also considered.     

Outliers and order statistics

Outliers are to be found among the extremes of a data set. Extremes are examples of order statistics. It is thus relevant to ask to what extent the statistical methods (and probabilistic properties) of outliers and of order statistics coincide and depend on each other. Whilst clear overlap is identifiable, aims and procedures are often quite distinct and each topic plays its own important role in the panoply of statistical principles and methodology.     

More bounds for moments of some rank statistics

Let X₁,…,X₇ be i.i.d. random variables with a common continuous distribution F, Two parameters, μ(F) = P(X₁ < X₅ and X₁+X₄ < X₂+X₃) and λ(F) = P(X₁+X₄ < X₂+X₃ and X₁+X₇ < X₅+X₆), which appear in the moments of some rank statistics have been studied by several authors. It is shown that the existing lower bound, 3/10 ≤ μ(F) can be improved to 3/10 < μ(F) and that no further improvement is possible. It is also shown that the existing upper bounds μ(F) ≤ (2^1/2+6)/24 ≈ 0.30893 and λ(F) ≤ 7/24 ≈ 0.29167 can be improved to [14+(2/3)^1/2]/48 ≈ 0.30868 and {7 ? [1 ? (2/3)^1/2]²/4}/24 ≈ 0.29132.     

Higher order moments of order statistics from INID symmetric random variables

In this paper we present analogues of Balakrishnan's (1989) relations that relate the triple and quadruple moments of order statistics from independent and nonidentically distributed (I.NI.D.) random variables from a symmetric distribution to those of the folded distribution. We then apply these results, along with the corresponding recurrence relations for the exponential distribution derived recently by Childs (2003), to study the robustness of the Winsorized variance.     

Improved bounds for moments of some rank statistics

Improved lower hounds are obtained for two parameters μ(F) and λ(F) which occur in the moments of linear rank statistics studied by Doksum, Hollander, Lehmann and Hsu. IT is shown that if F is a continuous distribution, then μ(F) > 3/10 and λ(F) >= 89/315 - 0.28254. These results decrease the range of possible values of μ(F) and λ(F) by more than 84% and 34%, respectively, over those previously reported.     

Best linear unbiased interpolation of order statistics

We derive the best linear unbiased interpolation for the missing order statistics of a random sample using the well-known projection theorem. The proposed interpolation method only needs the first two moments on both sides of a missing order statistic. A simulation study is performed to compare the proposed method with a few interpolation methods for exponential and Lévy distributions.     

Multivariate signed rank statistics for shift alternatives

Let X ₁,X ₂,…,X _N be successive independent random P-vectors drawn from some continuous diagonally symmetric distribution. The problem of detecting a shift in level of the sequence at an unknown time point M, ≦M ≦ N-1, is studied. Test statistics based on multivariate analogues of the rank statistics derived by BHATTACHARYYA and JOHNSON (1888) are proposed, and their asymptotic properties are investigated.     

Distribution of concomitants of order statistics and their order statistics

For a random sample of size n$n$ from an absolutely continuous random vector (X,Y)$(X,Y)$, let _Yi:n$Y_{i:n}$ be i$i$th Y$Y$-order statistic and _Y[j:n]$Y_{[j:n]}$ be the Y$Y$-concomitant of _Xj:n$X_{j:n}$. We determine the joint pdf of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$ for all i,j=1$i,j=1$ to n$n$, and establish some symmetry properties of the joint distribution for symmetric populations. We discuss the uses of the joint distribution in the computation of moments and probabilities of various ranks for _Y[j:n]$Y_{[j:n]}$. We also show how our results can be used to determine the expected cost of mismatch in broken bivariate samples and approximate the first two moments of the ratios of linear functions of _Yi:n$Y_{i:n}$ and _Y[j:n]$Y_{[j:n]}$. For the bivariate normal case, we compute the expectations of the product of _Yi:n$Y_{i:n}$ and _Y[i:n]$Y_{[i:n]}$ for n=2$n=2$ to 8 for selected values of the correlation coefficient and illustrate their uses.     

On distributions Of generalized order statistics

In a wide subclass of generalized order statistics, representations of marginal density and distribution functions are developed. The results are applied to obtain several relations, such as recurrence relations, and explicit expressions for the moments of generalized order statistics from Pareto, power function and Weibull distributions Moreover, characterizations of exponential distributions are shown by means of a distributional identity as well as by* an identity of expectations involving a subrange and a corresponding generalized order statistic.     

An alternative proof of order statistics moment problem

A necessary and sufficient condition that two distributions having finite means are identical is that for any fixed integer r > 0, the expected values of their rth (n ? r) order statistics are equal [or the expected values of their (n-r)th (n > r ? 0) order statistics are equal] for all n where n is the sample size.     

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.     

Some identities among moments of order statistics

Some new identities among the m oments of order statistics are derived. These are more general in nature and are applicable when moments of Some extreme order statistics do not exist.     

Fisher information in generalized order statistics

A representation of the Fisher information in generalized order statistics in terms of the hazard rate of the underlying distribution function is derived under mild regularity conditions. This expression supplements results for complete, Type-II censored, and progressively Type-II censored data. As a byproduct, we find a hazard rate based representation for samples of k-records which apparently has not been known so far. Moreover, sufficient conditions for the validity of this representation in location and scale family settings are given. The result is illustrated by considering generalized order statistics based on logistic, Laplace, and extreme value distributions.     

Random translation of discrete order statistics

We investigate the existence and uniqueness of a discrete parent distribution supported on the integers whose order statistics are related by a random translation. We also provide some examples using the constructive method that we propose.     

Regression via order statistics and their concomitants

We consider a five-dimensional normal distribution and derive the exact joint distribution one variable, linear combinations of order statistics from two other variables, and linear combinations of the corresponding concomitants of these order statistics. We show that this joint distribution is a mixture of trivariate unified skew-normal distributions. This mixture representation enables us to predict one variable based on linear combinations of order statistics from two other variables and linear combinations of the corresponding concomitants. We finally illustrate the usefulness of these results by using a real data.     

Some distributional problems connected with order statistics

Two characterizations of distributions symmetric about zero are given. These are based on the distributional properties of the squates of the order statistics from a random sample from these distributions. A result explering the relation between the distribution funcitons of two unordered (not necessarily independent) variables and those of their order statistics is presented. This has some interesting applications.     

A note on hazard rates of order statistics

Let F_k:m be the cumulative disribution function of the kth order statistic in a sample of size n from a distribution

F(x) with density function f(x).The primary objective of this paper is to show that F_k+1mis IHR(increasing hazard rate) if F_km(x)is IHH and that F_k-1:n(x)is DHR.(decreasing hazard rate) if F_km(x) is DHR.