Note on the Cramér-Rao inequality in the nonregular case: the family of uniform distributions

We consider the family of uniform distributions with range of unit length. The main result of this note asserts that the average variance of any unbiased estimator of the midpoint of the range is not less than (2(n+1))(n+2))^-1 and this lower bound is sharp. The proof is based upon a nonregular version of the Cramér-Rao inequality.     

Conditional inference for subject-specific and marginal agreement: Two families of agreement measures

A two-stage procedure is described for assessing subject-specific and marginal agreement for data from a test-retest reliability study of a binary classification procedure. Subject-specific agreement is parametrized through the log odds ratio, while marginal agreement is reflected by the log ratio of the off-diagonal Poisson means. A family of agreement measures in the interval [-1, 1] is presented for both types of agreement. The conditioning argument described facilitates exact inference. The proposed methodology is demonstrated by way of an example involving hypothetical data chosen for illustrative purposes, and data from a National Health Survey Study (Rogot and Goldberg 1966).     

A note on the asymptotic optimality of a two population sequential probability ratio test

It is shown that a two population sequential probability ratio test studied by a number of recent authors in the context of sequential medical trials is asymptotically optimal.     

Complete designs with blocks of maximal multiplicity

The set of distinct blocks of a block design is known as its support. We construct complete designs with parameters v(?7), k=3, λ=v ? 2 which contain a block of maximal multiplicity and with support size $\text{b}{}^1\text{= (}{}^{\mathrm{v}}{}_3\text{) ? 4(v ? 2)}$. Any complete design which contains such a block, and has parameters v, k, λ as above, must be supported on at most (^v₃) ? 4(v ? 2) blocks. Attention is given to complete designs because of their direct relationship to simple random sampling.     

Robustness of the linear discriminant function to nonnormality: Johnson's system

The asymptotic distribution of the Errors of Misclassification in using the Linear Discriminant Function is investigated here. The purpose is to study the effects of nonnormality on these errors. The class of distributions considered is the Johnson's system. Each of the three random variables can be transformed to normality. In one particular case numerical evaluations are made, based on which it is possible to recommend whether or not it is necessary to make the transformation prior to classification. In a parallel study, we present similar results for the Edgeworth Series distribution, where the random variables cannot be transformed to normality.     

A note on the universal optimality criterion for full rank models

The aim of this note is to suggest a revised formulation of the universal optimality criterion for full rank models as stated in Kiefer (1975). We have presented the relevant results with indications of some possible applications.     

A class of tests for testing ‘new is better than used’

A class of tests is proposed for testing H₀ F?(x) = e^?λx, λ > 0, x≥0 vs. H₁ F?(x + y) ≤ F?(x)F?(y), x, y≥0, with strict inequality for some x, y ≥ 0 (F = new is better than used). Efficiency comparisons of some tests within the class are made and a new test is proposed on the basis of these comparisons. Consistency and the asymptotic normality of the class of tests is proved under fairly broad conditions on the underlying entities.     

On regressing regression coefficients

This paper considers a regression model in which coefficients obtained from a previous regression are themselves the object of analysis. It is shown that the parameters of interest can be obtained in two ways: pooling across observations and subsamples, or a two-stage process of first estimating the coefficients within each subsample, and then using these coefficients as dependent variables in a second stage regression. The relative properties of these estimators are analyzed, and the conditions under which the two estimators are equivalent are derived.     

Recursive techniques for the construction of experimental designs

The construction of experimental designs by recursive techniques is studied in this paper. Formulae for the recursive addition or deletion of data from a design are derived for a typical sub-hypothesis situation of a general experimental design. These results are used to consider the recursive construction of experimental designs with respect to different optimality criteria. This approach to the construction of designs is quite different to that of the well-established theory of optimal design.     

Minimax subset selection for the multinomial distribution

Let (X₁,…,X_k) be a multinomial vector with unknown cell probabilities (p₁,?,p_k). A subset of the cells is to be selected in a way so that the cell associated with the smallest cell probability is included in the selected subset with a preassigned probability, P¹. Suppose the loss is measured by the size of the selected subset, S. Using linear programming techniques, selection rules can be constructed which are minimax with respect to S in the class of rules which satisfy the P¹-condition. In some situations, the rule constructed by this method is the rule proposed by Nagel (1970). Similar techniques also work for selection in terms of the largest cell probability.