The Distribution Functions of Tsao's Truncated Smirnov Statistics

• Published in: The Annals of mathematical statistics Vol. 38; no. 4; pp. 1208 - 1215
• Main Author: Conover, W. J.
• Format: Journal Article
• Language: English
• Published: Institute of Mathematical Statistics 01.08.1967; The Institute of Mathematical Statistics
• Subjects: Distribution functions; Indices of summation; Line segments; Probability theory; Random sampling; Random variables; Sampling distributions; Statistical theories; Statistics
• ISSN: 0003-4851; 2168-8990
• DOI: 10.1214/aoms/1177698789

Tsao (1954) deficumulative distribution function Sn(x) = k/n if $X_k \leqq x < X_{k + 1}$, where X0= -∞ and Xn + 1= ∞. 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 drand 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 N1/2dr, N1/2d'r, and N1/2d"rare given, where N = mn/(m + n). Also, for m = n, the exact closed form of the distribution functions of dr, d'r, and d"rare 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(dr≤ x) and P(d"r≤ x) are superfluous while tables for P(d'r≤ x) exist. Epstein (1955) compared the power of Tsao's d'rwith 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'ras a special case in the general scheme of censored rank order statistics.