The Distribution Functions of Tsao's Truncated Smirnov Statistics

Tsao (1954) deficumulative distribution function Sₙ(x) = k/n if Xₖ x < Xk + 1, where X₀ = -∞ and Xn + 1 = ∞. Let Y₁ < Y₂ < ⋯ < Yₘ 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 H₀:F(x) ≡ G(x) against H₁:F(x) ≡ G(x), Tsao (1954) proposed dᵣ = maxx X_τ |Sₙ(x) - S'ₘ(x)|, r n, and d'ᵣ = maxx max(Xᵣ, Yᵣ) |Sₙ(x) - S'ₘ(x)|, r min (m, n). It seems natural to consider also the test statistic d"ᵣ = maxx min (Xᵣ, Yᵣ) |Sₙ(x) - S'ₘ(x)|, r min (m, n) Tsao described a counting procedure to obtain the probabilities associated with the distribution functions of dᵣ 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 N1/2 dᵣ, N1/2 d'ᵣ, and N1/2 d''ᵣ are given, where $N = mn/(m + n)$. Also, for $m = n$, the exact closed form of the distribution functions of dᵣ, d'ᵣ, and d''ᵣ are derived under the null hypothesis. Also shown are the relationships {align*}P(d_r x) = 1/2P(d'_r x) + 1/2P(d"_r x); \\ P(d"_r x) = P(d'r - c x), c < r, where c = nx, \\ = 1, for c r,{align*} and therefore {align*}P(d_r x) = 1/2P(d'_r x) + 1/2P(d'r - c x), c < r, \\ = 1/2P(d'_r x) + 1/2, for c r,{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'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.