A New Formula for $P(R_i \leqq b_i, 1 \leqq i \leqq m \mid m, n, F = G^k)$

Abstract

Let $X_1 \leqq X_2 \leqq\cdots \leqq X_m$ and $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_n$ be independent samples of i.i.d. random variables from continuous distributions $F$ and $G$, respectively, and suppose $F(x) = \lbrack G(x)\rbrack^k$ or $F(x) = 1 - \lbrack 1 - G(x)\rbrack^k, k > 0.$ Let $R_i$ and $S_j$ denote the ranks of $X_i$ and $Y_j$, respectively, in the ordered combined sample. We express $P(R_i \leqq b_i$, all $i$) as the determinant of a simple $m \times m$ matrix. We also show that for increasing sequences $\{a_i\}$ and $\{b_i\}, P(a_i \leqq R_i \leqq b_i$, all $i\mid F, G) = P(\alpha_j \leqq S_j \leqq \beta_j$, all $j\mid F, G)$, where $\{\alpha_j\} = \{b_i\}^c$ and $\{\beta_j\} = \{a_i\}^c$ and complementation is with respect to the set $\{i\mid 1 \leqq i \leqq m + n\}$, for any pair of continuous distributions $F$ and $G$.