The Distribution Functions of Tsao's Truncated Smirnov Statistics

Abstract

Tsao (1954) deficumulative distribution function $S_n(x) = k/n$ if $X_k \leqq x < X_{k + 1}$, where $X_0 = -\infty$ and $X_{n + 1} = \infty$. Let $Y_1 < Y_2 < \cdots < Y_m$ represent an ordered random sample from the continuous distribution function $G(x)$, with the empirical cumulative distribution function $S'_m (x)$. As test statistics for testing $H_0:F(x) \equiv G(x)$ against $H_1:F(x) \not\equiv G(x)$, Tsao (1954) proposed $d_r = \max_{x \leqq X_\tau} |S_n(x) - S'_m(x)|,\quad r \leqq n,$ and $d'_r = \max_{x \leqq \max(X_r, Y_r)} |S_n(x) - S'_m(x)|,\quad r \leqq \min (m, n).$ It seems natural to consider also the test statistic $d"_r = \max_{x \leqq \min (X_r, Y_r)} |S_n(x) - S'_m(x)|,\quad r \leqq \min (m, n)$ Tsao described a counting procedure to obtain the probabilities associated with the distribution functions of $d_r$ and $d'_r$, and illustrated this procedure in the relatively simple case where $m = n$. Tables were compiled using the procedure for various values of $r$ and $m(= n)$. In this paper the asymptotic distributions of $N^{\frac{1}{2}} d_r, N^{\frac{1}{2}} d'_r$, and $N^{\frac{1}{2}} d''_r$ are given, where $N = mn/(m + n)$. Also, for $m = n$, the exact closed form of the distribution functions of $d_r, d'_r$, and $d''_r$ are derived under the null hypothesis. Also shown are the relationships \begin{align*}P(d_r \leqq x) = \frac{1}{2}P(d'_r \leqq x) + \frac{1}{2}P(d"_r \leqq x); \notag \\ P(d"_r \leqq x) = P(d'_{r - c} \leqq x),\quad\text{for} c < r, \text{where} c = \lbrack nx\rbrack, \notag \\ = 1, \text{for} c \geqq r,\end{align*} and therefore \begin{align*}P(d_r \leqq x) = \frac{1}{2}P(d'_r \leqq x) + \frac{1}{2}P(d'_{r - c} \leqq x), \quad\text{for} c < r, \notag \\ = \frac{1}{2}P(d'_r \leqq x) + \frac{1}{2}, \text{for} c \geqq r,\end{align*} illustrating that tables for $P(d_r \leqq x)$ and $P(d''_r \leqq x)$ are superfluous while tables for $P(d'_r \leqq x)$ exist. Epstein (1955) compared the power of Tsao's $d'_r$ with three other nonparametric statistics on the basis of 200 pairs of random samples of size 10 drawn from tables of normal random numbers. Rao, Savage, and Sobel (1960) considered $d'_r$ as a special case in the general scheme of censored rank order statistics.