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## An identity for the product moments of order statistics

### Metrika (1996-12-01) 44: 95-100 , December 01, 1996

A general identity for the product moments of successive order statistics is given, which is valid in a class of probability distributions including Weibull, Pareto, exponential and Burr distributions.

## A note on the mean past and the mean residual life of a (n − k + 1)-out-of-n system under multi monitoring

### Statistical Papers (2010-06-01) 51: 409-419 , June 01, 2010

In the study of the reliability of technical systems, *k*-out-of-*n* systems play an important role. In the present paper, we consider a (*n* − *k* + 1)-*out*-of-*n* system consisting of *n* identical components such that the lifetimes of components are independent and have a common distribution function *F*. It is assumed that the number of monitoring is *l* and the total number of failures of the components at time *t*_{i} is *m*_{i}, *i* = 1, . . . , *l* − 1. Also at time *t*_{l}(*t*_{1} < . . . < *t*_{l}) the system have failed or the system is still working. Under these conditions, the mean past lifetime, the mean residual lifetime of system and their properties are investigated.

## Runs in an ordered sequence of random variables

### Metrika (2008-04-01) 67: 299-313 , April 01, 2008

Let
$$X_{1},\ldots,X_{n}$$
be independent and identically distributed random variables with continuous distribution function. Denote by
$$X_{1:n} \leq \cdots \leq X_{n:n}$$
the corresponding order statistics. In the present paper, the concept of
$$\varepsilon$$
-*neighbourhood* runs, which is an extension of the usual run concept to the continuous case, is developed for the sequence of ordered random variables
$$X_{1:n}\leq\cdots\leq X_{n:n}.$$

## Approximate maximum likelihood predictors of future failure times of shifted exponential distributions under multiple type II censoring

### Statistical Methods and Applications (2004-04-01) 13: 43-54 , April 01, 2004

### Abstract.

Suppose we consider a general multiple type II censored sample (some middle observations being censored) from a shifted exponential distribution. The maximum likelihood prediction method does not admit explicit solutions. We introduce a simple approximation to one of prediction likelihood equations and derive approximate predictors of missing failure times. We compute their mean square prediction errors by simulation and compare them with the best linear predictors. Further, we present two real examples to illustrate this method of prediction.

## Rao-Blackwell Modifications

### Adaptive Sampling Designs (2013-01-01): 27-36 , January 01, 2013

This chapter summarizes some foundational theory for adaptive sampling methods. The Rao-Blackwell theorem can be applied to unbiased estimators to provide more efficient estimators. Closed form expressions for these and related estimators are discussed. The theory is also applied to selecting networks without replacement, and the question of ignoring information from labels is considered.

## Sintesis numerica de un batch unidimensional

### Trabajos de Estadistica (1986-03-01) 1: 97-111 , March 01, 1986

### Resumen

El análisis exploratorio de datos, según Tukey (1977), propone la síntesis numérica como el más importante resumen de un batch.^{*} En este trabajo estudiamos la relación entre los cuasicuantiles^{**} y los cuantiles de un batch, hallando cotas a la diferencia entre sus valores. Asimismo, se analiza el resumen numérico como herramienta exploratoria que contiene un alto nivel de información.

## Asymptotic efficiency of new exponentiality tests based on a characterization

### Metrika (2016-02-01) 79: 221-236 , February 01, 2016

Two new tests for exponentiality, of integral- and Kolmogorov-type, are proposed. They are based on a recent characterization and formed using appropriate V-statistics. Their asymptotic properties are examined and their local Bahadur efficiencies against some common alternatives are found. A class of locally optimal alternatives for each test is obtained. The powers of these tests, for some small sample sizes, are compared with different exponentiality tests.

## A Simple Method for Testing Global and Individual Hypotheses Involving a Limited Number of Possibly Correlated Outcomes

### Applied Statistics in Biomedicine and Clinical Trials Design (2015-01-01): 519-546 , January 01, 2015

Tests for the presence of a global treatment effect expressed by possibly correlated “primary” outcome variables taken together frequently use Bonferroni-type adjustments. These procedures accommodate an arbitrary number of comparisons, but can be conservative if the outcome variables are highly correlated. This conservatism can be ameliorated by a simple rule requiring essentially no calculation (and therefore convenient to apply when exact calculation is impractical) that is relatively robust to the correlation structure of the responses when the number of comparisons is not large (16 or less for 5 % level tests). The recommended global testing rule is: For a type 1 error rate of *α* and up to *K*(*α*) “primary” response variables, reject the global null hypothesis if (a) the smallest marginal *p* value is slightly less than *α*_{1} = *α*/*K*, (b) the second smallest marginal *p* value is ≤ 2*α*_{1}, or (c) the third smallest marginal *p* value is ≤ *α*. Analytic expressions that do not assume independence or any particular distribution for the responses are provided for the probability of rejecting the global null hypothesis. The type 1 error rates and power generally are preserved regardless of the correlation structure. Individual comparisons can be tested if the global null hypothesis is rejected, with reasonable preservation of comparison-wise type 1 error rates and of the false discovery rates (FDRs).

## On optimal choice of order statistics in large samples for the construction of confidence regions for the location and scale

### Metrika (2013-05-01) 76: 577-593 , May 01, 2013

Given a large sample from a location-scale population we estimate the unknown parameters by means of confidence regions constructed on the basis of two order statistics. The problem of the best choice of those statistics to obtain good estimates, as $$n\rightarrow \infty ,$$ is considered.

## Normex, a new method for evaluating the distribution of aggregated heavy tailed risks

### Extremes (2014-12-01) 17: 661-691 , December 01, 2014

We develop theoretically as well as numerically a new method, Normex, for the sum of independent heavy tailed distributed random variables, to obtain the most accurate evaluation of its entire distribution. Normex provides sharp results, whatever the number of summands and the tail index are. It is particularly suited when the Central Limit Theorem (CLT) applies but with slow convergence of the mean and with a poor approximation for the tail. Hence, it is filling up a gap in the literature by giving an appropriate limit distribution in this case, in general better than with most standard methods. An application is developed to evaluate the Value-at-Risk of the yearly log returns of financial assets.