S. Panchapakesan

Some locally optimal subset selection rules for comparison with a control

Abstract Let π0,π1,…,πk be k+1 independent populations. For i=0,1,…,k,πi has the densit f(x,θi), where the (unknown) parameter θi belongs to an interval of the real line. Our goal is to select from π1,… πk (experimental treatments) those populations, if any, that are better (suitably defined) than π0 which is the control population. A locally optimal rule is derived in the class of rules for which Pr(πi is selected)γi, i=1,…,k, when θ0=θ1=⋯=θk. The criterion used for local optimality amounts to maximizing the efficiency in a certain sense of the rule in picking out the superior populations for specific configurations of θ=(θ0,…,θk) in a neighborhood of an equiparameter configuration. The general result is then applied to the following special cases: (a) normal means comparison — common known variance, (b) normal means comparison — common unknown variance, (c) gamma scale parameters comparison — known (unequal) shape parameters, and (d) comparison of regression slopes. In all these cases, the rule is obtained based on samples of unequal sizes.

Is the Selected Multinomial Cell the Best

Abstract Suppose observations are taken sequentially from a multinomial distribution with k cells until the count in one of the cells reaches a predetermined number M. With a view to select the cell with the largest probability, we select the cell that has the largest count M. The problem of testing if the selected multinomial cell is the best is being considered in this paper. We propose the test procedure and show that the supremum of the probability of error for our procedure can be written as a single integral involving the gamma distribution. Exact values of the supremum of the probability of error are tabulated and its approximation formula for large values of M is provided.

A Note on a Subset Selection Procedure for the Most Probable Multinomial Event

Abstract Panchapakesan (1971) proposed and investigated a subset selection procedure for selecting the most probable cell in a multinomial distribution on k (≥2) cells. He showed that the least favorable configuration (LFC) for the probability of a correct selection (PCS) is that of equal cell-probabilities. He showed that this result holds exactly for k = 2 and asymptotically for k ≥ 3. Later, Chen (1986) and Liu and Lin (1991) showed that the result holds exactly for k ≥ 3. Their proofs involve differentiation of type-2 Dirichlet integrals with the restriction that the cell-probabilities add up to unity. We now give a fairly simple proof of this result by obtaining the PCS as a single integral involving the gamma distribution. This was the proof behind the claim of the LFC result made without details in the abstract by Panchapakesan (1973). Recommended by N. Mukhopadhyay

Advances in statistical decision theory and applications

Bayesian inference decision theory point and interval estimation - classical approach test of hypotheses ranking and selection distribution and applications industrial applications.

Selecting among the multinomial losers

In a multinomial setting with a fixed number k of cells. the problem of screening out cells to find the best cell, i.e., the one with the smallest cell probability, or looking for a (small) subset of cells containing the best cell is revisited. An inverse sampling procedure is used, unlike past work on this problem ([l], [2], [3], and [4]). Finding the cell with the smallest cell probability is clearly more difficult than finding the one with the argest cell probability. The proposed procedure takes one observation at a time (as usual) and igns a zero to all those (and only those) k - 1 cells into which the observation does not fall Sampling continues sequentially and stops as soon as any one cell has accumulated r zeros. For any given integer c (with 0 ≤ c < r), we put into the selected subset (SS) all those cells with at least r - c zeros and assert that this selected subset contains the best cell. It is important to note that for the slippage configuration (SC) we can attain any specified lower bound...

Discussion on “Two-Stage Procedures for High-Dimensional Data” by Makoto Aoshima and Kazuyoshi Yata

Abstract Professors Mokoto Aoshima and Kaziyoshi Yata have considered several types of statistical inference problems for multivariate high-dimensional data in the so-called large p, small n situation. For this purpose, in their article they have developed the theory for asymptotic normality when p → ∞. Their work opens up new directions for further investigations and generalizations. We briefly comment on these and discuss a multinomial problem.

A review of robustness of selection procedures

The robustness of selection procedures against deviations from model assumptions is an important aspect. Although moderate investigations have been made in the past in this regard, recent years have witnessed a renewed and growing interest. As a framework for further investigations, a review of significant results on robustness of selection procedures is given.