Smallest Order Statistics Research Articles

Detection of frailty in weibull lifetime data using outlier tests

Heterogeneity in lifetime data may be modelled by multiplying an individual's hazard by an unobserved frailty. We test for the presence of frailty of this kind in univariate and bivariate data with Weibull distributed lifetimes, using statistics based on the ordered Cox–Snell residuals from the null model of no frailty. The form of the statistics is suggested by outlier testing in the gamma distribution. We find through simulation that the sum of the k largest or k smallest order statistics, for suitably chosen k, provides a powerful test when the frailty distribution is assumed to be gamma or positive stable, respectively. We provide recommended values of k for sample sizes up to 100 and simple formulae for estimated critical values for tests at the 5% level.

Estimation After Subset Selection from Exponential Populations: Location Parameter Case

Synoptic Independent random samples of size n each are drawn from k (≥ 2) exponential populations Π1,…, Πk, having unknown location parameters θ1,…,θk and common known scale parameter σ. Let X1,…,Xk denote the smallest order statistics based on samples from k populations. Without loss of generality we take = 1. For selecting a non-empty subset S of {1,…,k}, containing the index of the “best” population (one associated with max(θ1,…, θk)), we consider the decision rule which selects the index i in S if and only if Xi ≥ max(X1,…,Xk) – d, i = 1,…,k, where d ≥ 0 is chosen so that the probability of including the index of the best population in the selected subset is at least P* (1/k ≤ P* < 1), a pre-assigned level. The problem of estimating W, the mean of all θis such that i ε S, is investigated. The uniformly minimum variance unbiased estimator (which is also the uniformly minimum risk unbiased estimator) is obtained and the maximum likelihood estimator is shown to be inadmissible. For k = 2, the UMVUE is compared with a natural estimator (analog of the minimum risk estimators of θ1,…, θk), numerically. A numerical example is considered to demonstrate the computations of various estimates of W.

A note on the mean of the order statistics based on independent exponential random variables

Upper and lower bounds are obtained on the mean of the r-th smallest order statistics based on n independent exponential random variables under a certain condition on r.

Characterizations of distributions via two expected values of order statistics

LetXk,n be thek-th smallest order statistic of a random sample of sizen from a distributionF(x) withE|X|<∞. Then in general two expected values of order statistics, sayEXk,n andEXl,m, are not enough to characterizeF in the class of all distributions on the real lineR ≡ (−∞, ∞). However, in the following class of distributions associated with any fixed non-degenerate distributionF0 through the location and scale parameters, $$F_{F_0 } = \{ F_{\mu ,\sigma } :F_{\mu ,\sigma } (x) = F_0 ((x - \mu )/\sigma ) on R,\mu \in R,\sigma > 0\} ,$$ F is characterized by any pairEXk,n andEXl,m with indices satisfying the condition: $$(l - k,m - n) \notin \{ (a,b):a = b = 0 or 0< a< b or b< a< 0\} .$$ Three special cases concerning the uniform, exponential and logistic distributions are investigated completely.

Two characterizations of the exponential distribution

A sequence Xn ≥ 1 of independent and identically distrinbuted random variable with absoiutely continuous cumulative distribution function F ( x ) is considered. X j is a record value of this sequence if .Let X L(n) n≤o with L(0)=1 be the sequence of such record values and Z n,m =X L(n)-X L(m) suppose is the with smallest order statistic in a random sample of size n from F and . Two characterizations of the exponential distribution are given based on the distributional properties of Zn,m and Tk,n.

On characterization of the exponential distribution by spacings

Let X be a non-negative random variable with cumulative probability distribution function F. Suppse X1, X2, ..., Xn be a random sample of size n from F and Xi,n is the i-th smallest order statistics. We define the standardized spacings Dr,n=(n-r) (Xr+1,n-Xr,n), 1≤r≤n, with DO,n=nX1,n and Dn,n=0. Characterizations of the exponential distribution are given by considering the expectation and hazard rates of Dr,n.

SOME LAW OF THE ITERATED LOGARITHM TYPE RESULTS FOR THE EMPIRICAL PROCESS

SummaryWe consider Z±n= sup0&lt; t ≤ 1/22 U±n (t)/(t(1‐ t))1/2, where + and ‐denote the positive and negative parts respectively of the sample paths of the empirical process Un. U±n and Un are seen to behave rather differently, which is tied to the asymmetry of the binomial distribution, or to the asymmetry of the distribution of small order statistics. Csáki (1975) showed that log Z±n/log2n is the appropriate normalization for a law of the iterated logarithm (LIL) for Z±n we show that Z‐n/(2 log2n)1/2 is the appropriate normalization for Z‐n. Csörgö &amp; Révész (1975) posed the question: if we replace the sup over (0,1/2) above, by ‐the sup over [an, 1‐an] where an→0, how fast can an→0 and still have |Zn|/(2 log2n)1/2 maintain a finite lim sup a.s.? This question is answered herein. The techniques developed are then used in Section 4 to give an interesting new proof of the upper class half of a result of Chung (1949) for |Un(t)|. The proofs draw heavily on James (1975); two basic inequalities of that paper are strengthened to their potential, and are felt to be of independent interest.

On the moments of gamma order statistics

AbstractA recurrence relation between the moments of order statistics from the gamma distribution having an integer parameter r is obtained. It is shown that if the negative moments of orders ‐(r−1), …, −1 of the smallest order statistic in random samples of size n are known, then one can obtain all the moments. Tables of negative moments for r = 2 (1) 5 are also given.

A Characteristic Property of the Exponential Distribution

Let $X$ be a nonnegative random variable with probability distribution function $F$. Suppose $X_{i,n} (i = 1,\cdots, n)$ is the $i$th smallest order statistics in a random sample of size $n$ from $F$. A necessary and sufficient condition for $F$ to be exponential is given which involves the identical distribution of the random variables $X$ and $(n - i) (X_{i+1,n} - X_{i,n})$ for some $i$ and $n$, $(1 \leqq i < n)$.

Moment relations for order statistics of the standardized gamma distribution and the inverse multinomial distribution

In inverse multinomial sampling, observations are selected one at a time from a multinomial distribution with k cells and associated probabilities il1, . . ., 11k, where Z HII = 1. The order statistics represent the numbers of multinomial observations required to obtain a predetermined number of observations, A, say, from any one of the k cells, from each of any two of the cells, and so on. The smallest order statistic is of particular interest since it is used as a sampling termination criterion in a selection procedure proposed by Cacoullos & Sobel (1966), for determining the most probable category in a multinomial distribution. The maximum expected amount of sampling occurs for the configuration Hi = k-1 (i 1 l,k), which we refer to as the symmetrical case. In this note, we establish a very simple relation between moments of the distributions of the order statistics of the symmetrical inverse multinomial distribution and the order statistics of independent standardized gamma variables with integer parameter A. Gupta (1960) considered the order statistics of the gamma distribution with integer parameter and presented tables of the first four moments of their distributions for A = 1 (1) 5 and k = 1 (1) 10. We also present and compare some asymptotic expressions for the moments.

Sample Sizes for Approximate Independence of Largest and Smallest Order Statistics

Abstract Let Xn and X 1 be the largest and smallest order statistics, respectively, of a random sample of size n. Quite generally, Xn and X 1 are approximately independent for n sufficiently large. Minimum n for attaining at least specified levels of independence are developed. Level of independence is measured by the maximum difference between the true values of P(X 1 ≤x 1, Xn ≤xn ) and the corresponding values assuming independence of Xn and X 1. The results apply to all possible distributions for the population sampled. The value of minimum n is the smallest allowable n for the continuous case but can be too large otherwise. Minimum n is finite for all nonzero differences.

Sample Sizes for Approximate Independence of Largest and Smallest Order Statistics

Abstract Let Xn and X 1 be the largest and smallest order statistics, respectively, of a random sample of size n. Quite generally, Xn and X 1 are approximately independent for n sufficiently large. Minimum n for attaining at least specified levels of independence are developed. Level of independence is measured by the maximum difference between the true values of P(X 1 ≤x 1, Xn ≤xn ) and the corresponding values assuming independence of Xn and X 1. The results apply to all possible distributions for the population sampled. The value of minimum n is the smallest allowable n for the continuous case but can be too large otherwise. Minimum n is finite for all nonzero differences.

Asymptotic independence between largest and smallest of a set of independent observations

Abstract : Let X sub n and X sub l be the largest and smallest order statistics, respectively, of a set of n independent univariate observations. Under rather general conditions, with respect to the distributions of the individual observations, X sub n and X sub l are asymptotically independent. However, asymptotic independence does not occur for all cases.