The Annals of Mathematical Statistics

The Distribution Functions of Tsao's Truncated Smirnov Statistics

W. J. Conover

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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.

Article information

Source
Ann. Math. Statist., Volume 38, Number 4 (1967), 1208-1215.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177698789

Digital Object Identifier
doi:10.1214/aoms/1177698789

Mathematical Reviews number (MathSciNet)
MR212906

Zentralblatt MATH identifier
0155.26502

JSTOR
links.jstor.org

Citation

Conover, W. J. The Distribution Functions of Tsao's Truncated Smirnov Statistics. Ann. Math. Statist. 38 (1967), no. 4, 1208--1215. doi:10.1214/aoms/1177698789. https://projecteuclid.org/euclid.aoms/1177698789


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