An introduction to relationship algebras

Since the introduction and definition of relationship algebras by James (1957), rolatively little use has been made of this theory in experimental design and analysis problems This may be due to the underlying theory of relationship algebras being previously accessible only in books on mathematical structure theory, e.g. van der Waerden (1950) This paper provides a development of the theory of relationship algebras beginning at an elementray level and including only those parts of mathematical structure theory which are essential to this development. An example of its application in derving an analysis due to alling (1967) is presented.     

A note on the estimation of pr{y<x} for truncation parameter distributions

In this note a relationship in the treatment of the lower and upper truncations considered in Beg (1980) is pointed out and the minimum variance unbiased estimator of P = Pr{Y<X) for the (upper) truncated exponential distribution is obtained.     

A multivariate solution of the multivariate ranking and selection problem

The problem of selection of the best multivariate population is given a new formulation which does not involve reducing the populations to univariate quantities. This formulation's solution is developed for known, and (using the Heteroscedastic Method) also for unknown, variance-covariance matrices. Preference reversals and arbitrary nonlinear preference functions are explicitly allowed in this new theory     

A note on the central limit theorem for the bootstrap mean

Following Yang (1988), a simple, more self-contained derivation of the asymptotic normality of the bootstrap mean is presented, as well as other asymptotic results. The derivations are appropriate for beginning graduate students in statistics, relying only on fundamental notions of probability theory and analysis,     

Discrimination between the complementary log-log and logistic model for ordinal data

An assessment is made of the number of observations on ordinal data necessary for reasonable power in a significance test of the null hypothesis of a logit model versus an alternative of a complementary log-log and vice versa. The conclusion is that quite modest numbers of observations, e.g. 50-100, are adequate provided that the boundaries of the ordinal scale are suitably defined.     

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.     

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.