Functions Of Order Statistics Research Articles

Linear Order Statistic Estimation for the Two-Parameter Weibull and Extreme-Value Distributions from Type II Progressively Censored Samples

Point estimation for the scale and location parameters of the extreme-value (Type I) distribution by linear functions of order statistics from Type II progressively censored samples is investigated. Four types of linear estimators are considered: the best linear unbiased (BLU), an approximation to the BLU, unweighted regression, and a linearized maximum likelihood. Linear transformations of the estimators are also considered for reducing mean square errors. Exact bias, variance, and mean square error comparisons of the estimators are made for several censoring patterns. Since the natural logarithms of Weibull variates have extreme-value distributions, the investigation is applicable to estimation for Weibull distributions.

Functions of Order Statistics

Two theorems on the asymptotic normality of linear combinations of functions of order statistics are given. Theorem 1 requires a smooth scoring function but the underlying df need not be continuous even and can also depend on the sample size. Theorem 2 allows general scoring functions but places additional restrictions on the df. Applications included.

Reverse Submartingale and Some Functions of Order Statistics

Reverse Submartingale and Some Functions of Order Statistics

Asymptotic Normality of Linear Combinations of Functions of Order Statistics

FN'(t) = inf {x:FN(x) > t}; and we write g o FN-l for the composed function g (FN-1). Note that TN = aO g o FN-1 dVN when the signed measure VN puts mass CNi/N at i/N for i = 1, * * , N and puts 0 mass elsewhere. Let v denote a signed measure on (0, 1). (The signed measures VN will not be used, but v is in some sense their limit.) For technical reasons to be displayed below, we bound ourselves away from 0 and 1 by an amount #N where NiN -3 0 as N - oo at a rate to be specified later. Let (1.2) 1N - fANd (1.2) Y~~~~~~.N = fAN g dv

Linear Functions of Order Statistics

Linear Functions of Order Statistics

The Use of Order Statistics in Estimation

A survey is given of recent advances, including a number of original contributions by the author, in the use of order statistics to obtain point and interval estimates of the parameters of various statistical populations from complete and from censored samples. In a few cases estimators based on order statistics are the efficient estimators, but more often they are substitute estimators that sacrifice some efficiency in the interest of computational simplicity and/or robustness in the presence of outliers. In life testing, they are often used to obtain estimates of the parameters of the life distribution before all of the items placed on test have failed. Point estimators based on order statistics may be either best linear unbiased estimators, based on all available observations from complete or censored samples or on one or more observations chosen in some optimal manner, or maximum-likelihood estimators, based on censored samples, of a single parameter or of two or more parameters jointly. Interval estimators may be based on percentage points of order statistics or of functions of order statistics, or they may be based on asymptotic variances and covariances of maximum-likelihood estimators. A summary is given of available results, together with a list of references and examples of applications to such problems as estimating the scatter of bullets aimed at a target and the reliability of an electronic device.

An Elementary Proof of Asymptotic Normality of Linear Functions of Order Statistics

Let $X_1, \cdots, X_n$ be independent random variables, identically distributed with continuous distribution function $F$. Let $X_{(1)}, \cdots, X_{(n)}$ denote the corresponding order statistics and $F_n$ the empiric distribution function, which we assume to be right continuous. Jung [4] found the asymptotic mean and variance of linear functions of the form $T_n = n^{-1} \sum^n_{i=1} J(i/n)X_{(i)} = \int^\infty_{-\infty} xJ(F_n(x)) dF_n(x)$ when the function $J$ has four bounded derivatives. More recently, it has been shown ([1], [2]) that under suitable restrictions \begin{equation*}\tag{1.1}\mathscr{L}\{n^{\frac{1}{2}}\lbrack T_n - \int^\infty_{-\infty} xJ(F(x)) dF(x)\rbrack\} \rightarrow N(0, \sigma^2)\end{equation*} where \begin{equation*}\tag{1.2}\sigma^2 = 2 \int\int_{s < t} J(F(s))J(F(t))F(s)\lbrack 1 - F(t)\rbrack ds dt.\end{equation*} Here $N(0, \sigma^2)$ denotes the normal distribution with mean zero and variance $\sigma^2$, and (1.1) uses a standard notation for convergence in distribution. The purpose of this note is to give a self-contained proof of (1.1) under the assumptions that the $X_i$ have finite mean and that $J'$ exists and is continuous and of bounded variation except at finitely many jumps of $J$. This theorem can be subsumed in a corrected version of Govindarajulu [2], where appeal is made to the results of [3]. The present proof is both more elementary and shorter. Chernoff, Gastwirth and Johns [1] do not require boundedness of $J$, but invoke compensating assumptions on $F$. Their methods of proof are quite different from those of [2] and of the present paper.

Convergence of Distributions Generated by Stationary Stochastic Processes

Convergence of Distributions Generated by Stationary Stochastic Processes

Miscellanea: Gini's mean difference rediscovered

A linear function of order statistics, recently proposed by Downton (1966) and Barnett, Mullen & Saw (1967) as an estimator of the standard deviation of a normal distribution, is essentially Gini's mean difference, g. This relation helps, for example, in a study of the behaviour of the estimator under non-normality. Some other occurrences of g are cited.

Sample Size Comparisons Based on Functions of Order Statistics

It is well known that the factor of economy of costs is an important consideration in the design and analysis of experiments. Mosteller [2] proposed some statistical techniques economically analysing large masses of applicable, in particular, to the estimation of parameters when the underlying random variables are assumed to be normally distributed. The methods were designed primarily for situations where the collection of data is inexpensive compared to the cost of analysis by means of standard procedures.

Asymptotic Distribution of Linear Combinations of Functions of Order Statistics with Applications to Estimation

Asymptotic Distribution of Linear Combinations of Functions of Order Statistics with Applications to Estimation

A Note on Recurrence Relations between Expected Values of Functions of Order Statistics

In this note, some recurrence relations are derived between the expected values of functions of order statistics from any arbitrary distribution with continuous cumulative distribution function (cdf). These recurrence relations are closely related to some results obtained by Srikantan [2].

On the distribution of linear functions and ratios of linear functions of ordered correlated normal random variables with emphasis on range

SUMMARY This paper deals with the distribution of linear functions and ratios of linear functions of order statistics from an equally correlated set of normal random variables. Some special cases of this family of linear functions are considered. The case of range is studied in great detail both for the case of equal correlations as well as in the general case. For the general case, formulae for the distribution of range are obtained for sample sizes of two, three and four. In the equally correlated case, expressions are obtained for the probability integral, percentage points and moments of the linear functions in terms of the corresponding expressions for the uncorrelated cases. It is also shown that ratios of certain linear functions of the order statistics have a distribution independent of this common correlation.

On Uncorrelated Linear Functions of Order Statistics

Abstract In this paper the correlation between the mean and certain linear functions of Order Statistics are obtained for samples from any population and the results for the symmetrical population are derived therefrom. The results obtained in this paper answer in part a question raised by Ostle and Steck in their 1959 paper [2].

On Uncorrelated Linear Functions of Order Statistics

Abstract In this paper the correlation between the mean and certain linear functions of Order Statistics are obtained for samples from any population and the results for the symmetrical population are derived therefrom. The results obtained in this paper answer in part a question raised by Ostle and Steck in their 1959 paper [2].

Recurrence Relations Between the PDF's of Order Statistics, and Some Applications

In this paper we obtain a recurrence relation between the pdf's (probability density functions) of order statistics. This relation is then extended to their cdf's (cumulative distribution functions) and to the expected values of a given function of each order statistic and a function of pairs of them. The interrelation between these results and other known formulas is shown. The application of the results to the preparation of tables of cdf's, expected values, variances, etc., of order statistics is indicated, and the numerical errors propagated in such computations analysed. Finally, an illustration is given. It is hoped that the unified approach presented here will provide greater insight and flexibility to procedures of numerical evaluation of expected values, etc., of order statistics.

A Note on the Stochastic Independence of Functions of Order Statistics

A Note on the Stochastic Independence of Functions of Order Statistics

On Minimum Variance Among Certain Linear Functions of Order Statistics

Suppose there are $n$ normal populations $N(\mu_i, 1), i = 1, \cdots, n$ and that one random observation from each of these $n$ populations is given. Let $x_1 \leqq x_2 \leqq \cdots \leqq x_n$ be the observations when arranged in order of magnitude and let the corresponding $n$ random variables be denoted by $X_i, i = 1, \cdots, n.$ The following theorem is proved: THEOREM. \begin{equation*}\operatorname{Var}\big(\sum^n_{i = 1} c_i X_i\big), \text{where}\end{equation*}\begin{equation*}\tag{1}\sum^n_{i = 1} c_i = 1,\end{equation*} is minimum when $c_i = 1/n, i = 1, \cdots, n.$ The above theorem may be applied to provide a direct proof of the result that $\sum^n_{i = 1}X_i$ is the best unbiased linear function of order statistics for estimating the sum $\sum^n_{i = 1}\mu_i.$

On Some Useful "Inefficient" Statistics

Several statistical techniques are proposed for economically analyzing large masses of data by means of punched-card equipment; most of these techniques require only a counting sorter. The methods proposed are designed especially for situations where data are inexpensive compared to the cost of analysis by means of statistically "efficient" or "most powerful" procedures. The principal technique is the use of functions of order statistics, which we call systematic statistics. It is demonstrated that certain order statistics are asymptotically jointly distributed according to the normal multivariate law. For large samples drawn from normally distributed variables we describe and give the efficiencies of rapid methods: i) for estimating the mean by using $1, 2, \cdots, 10$ suitably chosen order statistics; (cf. p. 386) ii) for estimating the standard deviation by using 2, 4, or 8 suitably chosen order statistics; (cf. p. 389) iii) for estimating the correlation coefficient whether other parameters of the normal bivariate distribution are known or not (three sorting and three counting operations are involved) (cf. p. 394). The efficiencies of procedures ii) and iii) are compared with the efficiencies of other estimates which do not involve sums of squares or products.