Smirnov test

Smirnov -samples test

A non-parametric (or distribution-free) statistical test for testing hypotheses about the homogeneity of two samples.

Let and be mutually-independent random variables, where each sample consists of identically continuously distributed elements, and suppose one wishes to test the hypothesis that both samples are taken from the same population. If

are the order statistics corresponding to the given samples, and and are the empirical distribution functions corresponding to them, then can be written in the form of the identity:

Further, consider the following hypotheses as possible alternatives to :

To test against the one-sided alternatives and , and also against the two-sided , N.V. Smirnov proposed a test based on the statistics

respectively, where it follows from the definitions of and that under the hypothesis , and have the same distribution. Asymptotic tests can be based on the following theorem: If , then the validity of implies that

where is the Kolmogorov distribution function (cf. Statistical estimator). Asymptotic expansions for the distribution functions of the statistics and have been found (see [4]–[6]).

Using the Smirnov test with significance level , may be rejected in favour of one of the above alternatives , when the corresponding statistic exceeds the -critical value of the test; this value can be calculated using the approximations obtained by L.N. Bol'shev [2] by means of Pearson asymptotic transformations.

See also Kolmogorov test; Kolmogorov–Smirnov test.

References

[1]	N.V. Smirnov, "Estimates of the divergence between empirical distribution curves in two independent samples" Byull. Moskov. Gosudarstv. Univ. (A) , 2 : 2 (1939) pp. 3–14
[2]	L.N. Bol'shev, "Asymptotically Pearson transformations" Theor. Probab. Appl. , 8 (1963) pp. 121–146 Teor. Veroyatnost. i Primenen. , 8 : 2 (1963) pp. 129–155
[3]	L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)
[4]	V.S. Korolyuk, "Asymptotic analysis of the distribution of the maximum deviation in the Bernoulli scheme" Theor. Probab. Appl. , 4 (1959) pp. 339–366 Teor. Veroyatnost. i Primenen. , 4 (1959) pp. 369–397
[5]	Li-Chien Chang, "On the exact distribution of A.N. Kolmogorov's statistic and its asymptotic expansion (I and II)" Matematika , 4 : 2 (1960) pp. 135–139 (In Russian)
[6]	A.A. Borovkov, "On the two-sample problem" Izv. Akad. Nauk SSSR Ser. Mat. , 26 : 4 (1962) pp. 605–624 (In Russian)

References

[a1]	D.B. Owen, "A handbook of statistical tables" , Addison-Wesley (1962)
[a2]	E.S. Pearson, H.O. Hartley, "Biometrika tables for statisticians" , 2 , Cambridge Univ. Press (1972)