Order Statistics In Samples Research Articles

Bounds on expectations of order statistics for dependent samples

Arnold and Groeneveld (1979) presented distribution free bounds for E( X g:n ) the expected value of the gth order statistic when sampling n possibly dependent observations from populations each with finite expectations μ i and finite variances σ 2 i . Here, the problem of incorporating known dependencies among the distributions of observations in a sample to produce bounds for the order statistics of that sample is considered. Limits are obtained for expectations of the gth order statistic and the range. These limits hold even when the population variances are not finite. An application of the upper bound for E( X n:n ) is given.

On a test statistic for testing normality

This paper presents a statistic for testing a complete sample for normality. The test statistic is defined to be a linear combination of the standardized sample order statistics. It is a location and scale invariant statistic and also could be generalized to test the distributional assumptions for the other location and scale family of distributions. Empirical percentage points are provided for samples of size n = 3(1), 40(10), 150. A Monte Carlo study was conducted to compare the power of the proposed test with that of W test for samples of sizes n = 10, 15, 20, 35, 50 and 36 non-normal alternatives. It is concluded that for the skewed alternatives both statistics have similar power.

Relationships among moments of order statistics in samples from two related outlier models and some applications

Balakrishnan (1987a) has recently shown that the moments of order statistics in samples drawn from a continuous population with pdf f(x) symmetric about zero comprising a single outlier with pdf g(x) also symmetric about zero can be expressed in terms of the moments of order statistics in samples drawn from the population obtained by folding the pdf f(x) at zero and the moments of order statistics in samples drawn from the population obtained by folding the pdf f(x) at zero comprising a single outlier with pdf obtained by folding g(x) at zero. The cumulative round off error involved in the numerical evaluation of the moments of order statistics from the symmetric outlier model, using a table of the moments of order statistics from the folded population and the moments of order statistics from the folded outlier model, has also been studied by Balakrishnan (1987a) and shown to be not serious. Making use of these results we study here the robustness of some estimators of th location and scale parameters of ...

Recurrence Relations Among Moments of Order Statistics from Two Related Outlier Models

AbstractIt is shown that the moments of order statistics in samples drawn from a continuous population with pdf f(x) symmetric about zero comprising a single outlier with pdf g(x) symmetric about zero can be expressed in terms of the moments of order statistics in samples drawn from the population obtained by folding the pdf f(x) at zero and the moments of order statistics in samples drawn from the population obtained by folding the pdf f(x) at zero comprising a single outlier with pdf obtained by folding g(x) at zero. The cumulative round off error involved in numerical evaluation of the moments of order statistics from the symmetric‐outlier model, using a table of the moments of order statistics from the folded population and the moments of order statistics from the folded‐outlier model, is not serious.

Characterizations and o-statistic representations of lu-statistics

The LU-statistics, that is, the class of all L-statistics that are also U-statistics, are characterised both via conditions on their coefficients and as linear combinations of the 0-statistics, which generalize the usual sample order statistics. The explicit computational formulas provided can be used to decide whether a given L-statistic is an LU-statistic and, if so, to find its kernel and its simplest 0-statistic representation.

O-statistics and their applications

O-statistic distributional properties and nonparametric applications including numerical and graphical data summary, quantile point and interval estimation, and tests of uniformity are presented and discussed in a survey of both published and new results. Initially termed generalized sample quantiles and proposed as smoothed alternatives to the conventional sample quantile, the O-statistics generalize the usual sample order statistics and are obtained by averaging subsample quantiles over all subsamples of a fixed size. The resulting statistics form natural bases for the LU-statistics.

Numerical and Graphical Data Summary UsingO-Statistics

Abstract Numerical and graphical techniques for descriptive data analysis and presentation are developed using O-statistics, which are smoothed generalizations of the usual sample order statistics. Following the derivation of O-statistics and their elementary properties, the statistical relevance and descriptive capabilities of O-statistics are demonstrated through a series of examples.

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.

Best Linear Unbiased Estimators for Normal Distribution Quantiles for Sample Sizes up to 20

This paper provides tables for estimating the quantiles of the normal distribution by using order statistics in small samples. Linear unbiased estimators with minimum variance and based on ordered observations are constructed for sample sizes from 2 through 20.

Best linear unbiased quantile estimators of the logistic distribution using order statistics

This paper deals with the problem of estimating the Quantiles Q(ξ), with ξ=0.01, 0.05(0.05)0.95, 0.99 of the logistic distribution by using order statistics in small samples. Linear unbiased estimators with minimum variance BLUE based on ordered observations are constructed for sample size n = 2(1)10, 15 and 20.

A study of shifting models in life tests via bayesian approach using semi-or-used priors (SOUPS)

This paper analyses the shift in parameter of a life test model. This analysis depends on the prediction of order statistics in future samples based on order statistics in a series of earlier samples in life tests having a general exponential model. While a series ofk samples are being drawn, model itself undergoes a change. Firstly, a single shift is considered and the effect of this shift on the variance is discussed. Generalisation withs shifts (s≦k) ink samples in also taken up and the semi-or-used priors (SOUPS) have been used to get predictive distributions. Finally, shift afteri (i≦k) stages, from exponential to gamma model is considered and for this case effect of the shift on the variance as well as on the Bayesian prediction region (BPR) is analysed along with set of tables.

Bounds for the mean of second largest order statistic in large samples

Some bounds and approximations for the mean of second largest order sta¬tistic from an arbitrary continuous distribution symmetric about zero are obtained. For the standard normal case, FOURIER coefficients involved in the approximation procedure are tabulated and the bounds obtained are compared with other similar methods

Quantile Estimates in Complete and Censored Samples From Extreme-Value and Weibull Distributions

This paper deals with the problem of estimating the quantiles (ξ) = 0.01, 0.05(0.05)0.95, 0.99 of the largest extreme-value distribution by using order statistics in small samples. Linear unbiased estimators, with minimum variance, based on ordered observations are constructed for sample sizes 5(5)20 for both complete and right-censored samples.

A note on optimum grouping and the relative discriminating power of qualitative to continuous normal variates

Ogawa (1951) considered the efficiency of estimation of the population mean from suitably chosen order statistics in large samples. Cox (1957) has considered the relative amount of information retained by grouping the normal curve. Cochran and Hopkins (1961) determined the discriminating power retained after partitioning normally distributed variates into qualitative ones in multivariate classification problems. And Connor (1972) discussed the asymptotic efficiencies of the test for the trend using m groups formed from a continuous variable. The same expression appears in all these investigations. This note throws some more light on the occurrence of the same expression in these seemingly unrelated problems.

Independent scale-free spacings for the exponential and uniform distributions

It is shown that, by suitable random normalisation, the spacings (differences between order statistics) of exponential and uniform samples are independent. These normalised spacings can be used for consecutive discordancy testing when scale (and possibly also location) parameters are unknown.

Bounds for the moments of extreme order statistics for large samples

The object of this paper is to obtain some bounds and approximations to the moments of extreme order statistics in large, samples. These are of same form as those of SUGIURA [11]. The proposed method is applicable whenever a few moments of extreme order statistics for large sample sizes are available. Fourier coefficients useful for approximating the first two moments of normal order statistics are tabulated. Numerical calculations for normal distribution show that this method gives a considerable improvement over the method of SUGIURA.

Expected cost bounds for the selection and ordering procedures based on binary-type questions

The problems considered in this paper are similar to those studied by Arrow, Pesotchinsky and Sobel (1981). We search for the optimal methods of ordering of a sample or of identifying sample order statistics by asking binary type questions and employing the criterion which defines the cost of a question as the number of responses. The asymptotic results show that our procedures with simple algorithms are comparable (in terms of cost) with the best known methods for ordering by using pairwise comparisons.

Rank order statistics for unequal samples

Rank order statistics for unequal samples

A note on the mixed moments of order statistics from exponential and truncated exponential distributions

Abstract Some recurrence relations for mixed moments of order statistics in random samples of size n from an exponential and right truncated exponential distributions are derived. It is shown that one can calculate all of these moments without evaluating any single or double integral.

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.