Some theorems, counterexamples, and conjectures in multinomial selection theory
W. Chen Robert, K. Hwang Frank · Communication in Statistics- Theory and Methods · 1984
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 where δ is a constant in 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 D 0(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 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.