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.     

Sequential procedures based on M-estimators with discontinuous score functions

Bounded-width sequential confidence intervals and sequential tests for regression parameter based on M-estimators are extended to the case where the score-functions generating the M-estimators have jump-discontinuities. In the context of the asymptotic normality of the stopping variable, for the confidence interval problem, it is observed that the jump-discontinuities induce a slower rate of convergence. The proofs of the main theorems rest on the weak convergence of some related processes and this is also studied.     

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.     

A formula for P(bi≤Ri≤ai, 1≤i≤m) under a general class of alternatives

Let X_i≤?≤X_m and Y_i≤?≤Y_n be two sets of independent order statistics from continous distributions with distribution functions F and G respectively. Let R_i denote the rank of X_i in the combined order sample. Steck (1980) has found an expression for P(b_i≤R_i≤a_i, all i) when F = h(G), h being the incomplete beta function with parameters (α,β?α+1). An alternative expression for the same probability is obtained which is computationally a substantial improvement on Steck's result.     

Information in experiments and sufficiency

We define measures of information contained in an experiment which are by-products of the parametric measures of Fisher, Vajda, Mathai and Boekee and the non-parametric measures of Bhattacharyya, Rényi, Matusita, Kagan and Csiszár. We use these measures to compare sufficient experiments according to Blackwell's definition. In particular, we prove that if δ_X and δ_Y are two experiments and δ_X≥δ_Y then l_X≥l_y for all of the above measures.     

On the ABLUE of the normal mean from a censored sample

The asymptotically best linear unbiased estimate (ABLUE) of the normal mean is discussed. The estimate is based on k selected order statistics chosen from a singly or doubly censored large sample of size n(>k). The coefficients, the asymptotic relative efficiency of the estimate, and the optimum spacing of k real numbers between 0 and 1 which determines the optimum ranks of order statistics, are provided. A comparison between the ABLUE and the iterated maximum likelihood estimate is made.     

Some sufficient conditions for establishing (M,S)-optimality

Two sufficient conditions are given for an incomplete block design to be (M,S- optimal. For binary designs the conditions are (i) that the elements in each row, excluding the diagonal element, of the association matrix differ by at most one, and (ii) that the off-diagonal elements of the block characteristic matrix differ by at most one. It is also shown how the conditions can be utilized for nonbinary designs and that for blocks of size two the sufficient condition in terms of the association matrix can be attained.     

A survey of nested designs

A nest with parameters (r,k,λ)→(r′,k′,λ′) is a BIBD on (b,v,r,k,λ) where each block has a distinguished sublock of cardinality k, the sublocks forming a (b,v,r,k,λ)-design.These designs are ‘nested’ in the sense of W.T. Federer (1972), who recommended the use of these designs for the sequential addition of periods in marketing experiments in order to retain Youden design properties as rows are added. Note that for a Youden design, the b columns and v treatments are in an SBIBD arrangement with parameters v=b, k=r, and λ.     

A simple test of symmetry about an unknown median

A simple test statistic for testing symmetry of a distribution function about an unknown value is presented. The asymptotic distributions under symmetry and asymmetry are derived. Using the normal as a “calibration” distribution, the critical values of the test are calculated by Monte Carlo methods. Comparisons with other tests indicate that this procedure performs well.