Abstract
Suppose $X_1, X_2, \cdots, X_n$ are known to be independently and identically distributed, each with the density function $f(x),$ with $\int^1_0f(x) dx = 1.$ Let $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_n$ be the ordered values of $X_1, X_2, \cdots, X_n,$ and define $W_1 = Y_1, W_2 = Y_2 - Y_1, \cdots, W_n = Y_n - Y_{n -1},$ and $W_{n + 1} = 1 - Y_n,$ so that $W_1 + \cdots + W_{n + 1} = 1.$ Finally, define $Z_1, \cdots, Z_{n + 1}$ as the ordered values of $W_1, \cdots, W_{n + 1},$ so that $0 \leqq Z_1 \leqq Z_2 \leqq \cdots \leqq Z_{n + 1},$ with $Z_1 + \cdots + Z_{n+1} = 1.$ We are going to test the hypothesis that $f(x) = 1$ for $0 < x < 1,$ and we are going to consider only tests based on $Z_1, Z_2, \cdots, Z_n.$ The intutitive justification for this is that, roughly speaking, deviations from the hypothesis on any part of the unit interval are treated alike. Several authors have discussed tests based on $Z_1, \cdots, Z_n.$ (See references [1], [2], [3].) If $u$ is a number greater than unity, it is shown that the test of the form "reject the hypothesis if $Z^u_1 + \cdots + Z^u_{n+1} > K"$ is consistent against a very wide class of alternatives. When $u = 2,$ the resulting test has some desirable properties with respect to alternatives with linear density functions.
Citation
Lionel Weiss. "A Certain Class of Tests of Fit." Ann. Math. Statist. 27 (4) 1165 - 1170, December, 1956. https://doi.org/10.1214/aoms/1177728085
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