Some theorems, counterexamples, and conjectures in multinomial selection theory
Titel:
Some theorems, counterexamples, and conjectures in multinomial selection theory
Auteur:
Robert, W. Chen Frank, K. Hwang
Verschenen in:
Communications in statistics
Paginering:
Jaargang 13 (1984) nr. 10 pagina's 1289-1298
Jaar:
1984
Inhoud:
Consider the problem of selecting the tcells of largest probability in a multinomial distribution with unknown cell probabilities P1P2…,Pk where 1 ≤ t< kand k≥ 3. The procedure we are concerned with is a single-stage fixed sample size one which selects the tcells of highest count in the sample, with ties broken by randomization. The preference zone is [image omitted] where δ is a constant in [image omitted] and p[1] ≤ p[2] ≤…≤p[k] denote the ranked multinomial cell probabilities. For a given sample size n the probability vector p in D(t, k, δ) which minimizes the probability of a correct selection over D(t, k, delta) is called a least favorable configuration (LFC) over the preference zone D(t, k,δ). In this paper, we first give some theorems which tell us that a least favorable configuration over the preference zone D(t, k, δ) must be in a certain subset D0(t, k, δ) of D(t, k, delta). Then we use these theorems to construct examples which disprove the conjecture that the usual slippage Configuration [image omitted] is a least favorable configuration over the preference zone D(t, k, δ). Finally, we make some conjectures of our own which would be very interesting and useful if true.
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