Dempster's combination rule and sufficient statistics

In his articles (1966-1968) concerning statistical inference based on lower and upper probabilities, Dempster refers to the connection between Fisher's fiducial argument and his own ideas of statistical inference. Dempster's main concern however focuses on the “Bayesian” aspects of his theory and not on an elaboration of the relation between Fisher's and his ideas. This article attempts to work out the connection between those two approaches and focuses primarily on the question, whether Dempster's combination rule, his upper and lower probabilty based on sufficient statistics and inference based on sufficient statistics in Fisher's sense are consistent. To be adequate to Fisher's reasoning, we deal with absolutely continuous, one parametric families of distributions.This is certainly not the usual assumption in context with Dempster's theory and implies a normative but straightforward definition concerning the underlying conditional distribution; this definition however is done in Dempster's spirit as can be seen from his articles, (1966, 1968,a,b). Under those assumptions it can be shown that - similar to Lindley's results concerning consistency in fiducial reasoning (1958) - the combination rule, Dempster's procedure based on sufficient statistics and fiducial inference by sufficient statistics agree iff the parametric family under consideration can be transformed to location parameter form.     

Applications of schur theory to reliability inference for k of n systems

The results of Hoeffding (1956), Pledger and Proschan (1971), Gleser (1975) and Boland and Proschan (1983) are used to obtain Buehler (1957) 1-α lower confidence limits for the reliability of k of n systems of independent components when the subsystem data have equal sample sizes and the observed failures satisfy certain conditions. To the best of our knowledge, for k ≠ 1 or n, this is the first time the exact optimal lower confidence limits for system reliability have been given. The observed failure vectors are a generalization of key test results for k of n systems, k ≠ n (Soms (1984) and Winterbottom (1974)). Two examples applying the above theory are also given.     

Testing a main effect in a three factor mixed model

In the present paper, the author has considered and compared the power of three test procedures, based on two preliminary tests of significance (PTS), for testing a main effect in a three factor factorial experiment. Davenport and Webster (1973) type test statistics are used in the final test. Recommendation for a suitable size of PTS has been made so that the power of a test procedure is adequately high.     

Quadratic perturbation expansions of certain functions of eigenvalues and eigenvectors and their application to sensitivity analysis in multivariate methods

Tanaka (1988) lias derived the influence functions, which are equivalent to the perturbation expansions up to linear terms, of two functions of eigenvalues and eigenvectors of a real symmetric matrix, and applied them to principal component analysis. The present paper deals with the perturbation expansions up to quadratic terms of the same functions and discusses their application to sensitivity analysis in multivariate methods, in particular, principal component analysis and principal factor analysis. Numerical examples are given to show how the approximation improves with the quadratic terms.     

The weighted jackknife for ratio estimation

Using the idea of impirical influence function, Hinkley (1977), the weighted jackknife technique is extended to ratio estimation. A weighted jackknife variance estimator for the ratio estimator is developed. Using the prediction theory approach, the properties of the weighted jackknifed variance estimator are examined. The implications of the failures of regression model on the behaviour of the weighted jackknifed variance estimator, for ratio estimation, are also studied.     

Non-parametric estimation of the conditional mode

The problem of estimating the mode of a conditional probability density function is considered. It is shown that under some regularity conditions the estimate of the conditional mode obtained by maximizing a kernel estimate of the conditional probability density function is strongly consistent and asymptotically normally distributed.     

Moment expressions and summary statistics for the complete and truncated weibull distribution

Traditionally, the moments of the Weibull distribution have been calculated using the standard Weibull (Johnson and Kotz, 1970) . This article will expand on that idea and cover the truncated cases for the standard Weibull distributions. Also, the same techniques used for the standard form will be used to derive the moment expressions for the three-parameter complete and truncated Weibull distributions. The summary statistics are then calculated from the moment expressions. Weibull moments involve the gamma and incomplete gamma functions.     

Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi‐Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse‐Gaussian based saddlepoint approximation of Wood, Booth & Butler (1993) is shown to accurately deal with the nonstationarity whereas the normal‐based Lugannani & Rice (1980) approximation cannot, Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.     

Copula-based regression models: A survey

In this review paper we collect several results about copula-based models, especially concerning regression models, by focusing on some insurance applications.