On mode(s) and shape of the pdfs of the number of runs and of the maximum of runs by type
Titel:
On mode(s) and shape of the pdfs of the number of runs and of the maximum of runs by type
Auteur:
Schuster, Eugene F. He, Jie
Verschenen in:
Communications in statistics
Paginering:
Jaargang 27 (1988) nr. 1 pagina's 115-126
Jaar:
1988
Inhoud:
Let R = R(m, n) = Rs + RF be the total number of runs in a randomly ordered vector [image omitted] of length N = m+n consisting of m letters of type 1 (successes S) and n letters of type 2 (failures F) where Rs and RF are the number of success and failure runs, respectively. When n = m, we show that the distribution of R is mound-shaped, symmetric and unimodal with mode n + 1 when n is odd and trimodal with modes n, n + 1, and n + 2 when n is even. If m ≠ n, the distribution of R is in general neither mound-shaped nor symmetric and (contrary to common belief) can be quite oscillatory. In this case, we show that the exact number and location of the modes (at most 3) can be determined without using the pdf of R by computing four numbers as follows: Assume m ≥ n and take [image omitted] and [image omitted] . Then the modes of R are located at 2r - 1 + i for any i ∈ {1,2,3,4} with νi = max{ν1, ν2, ν3, ν4}. If RM = max{RS, RF}, then 0 ≤ 2RM - R ≤ 1, so the asymptotic distributions of the standard scores of R and RM, are the same. However, in contrast to R, we show that the pdf of RM is mound-shaped and unimodal and hence better behaved in all cases than the pdf of R which can undulating.
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