Chi-squared test

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A test for the verification of a hypothesis according to which a random vector of frequencies has a given polynomial distribution, characterized by a vector of positive probabilities , . The "chi-squared" test is based on the Pearson statistic

which has in the limit, as , a "chi-squared" distribution with degrees of freedom, that is,

According to the "chi-squared" test with significance level , the hypothesis must be rejected if , where is the upper -quantile of the "chi-squared" distribution with degrees of freedom, that is,

The statistic is also used to verify the hypothesis that the distribution functions of independent identically-distributed random variables belong to a family of continuous functions , , , an open set. After dividing the real line by points , , , into intervals , , such that for all ,

; , one forms the frequency vector , which is obtained as a result of grouping the values of the random variables into these intervals. Let

be a random variable depending on the unknown parameter . To verify the hypothesis one uses the statistic , where is an estimator of the parameter , computed by the method of the minimum of "chi-squared" , that is,

If the intervals of the grouping are chosen so that all , if the functions are continuous for all , ; , and if the matrix has rank , then if the hypothesis is valid and as , the statistic has in the limit a "chi-squared" distribution with degrees of freedom, which can be used to verify by the "chi-squared" test. If one substitutes a maximum-likelihood estimator in , computed from the non-grouped data , then under the validity of and as , the statistic is distributed in the limit like

where are independent standard normally-distributed random variables, and the numbers lie between 0 and 1 and, generally speaking, depend upon the unknown parameter . From this it follows that the use of maximum-likelihood estimators in applications of the "chi-squared" test for the verification of the hypothesis leads to difficulties connected with the computation of a non-standard limit distribution.

In [3][8] there are some recommendations concerning the -test in this case; in particular, in the normal case [3], the general continuous case [4], [8], the discrete case [6], [8], and in the problem of several samples [7].


[1] M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 2. Inference and relationship , Griffin (1983)
[2] D.M. Chibisov, "Certain chi-square type tests for continuous distributions" Theory Probab. Appl. , 16 : 1 (1971) pp. 1–22 Teor. Veroyatnost. i Primenen. , 16 : 1 (1971) pp. 3–20
[3] M.S. Nikulin, "Chi-square test for continuous distributions with shift and scale parameters" Theory Probab. Appl. , 18 : 3 (1973) pp. 559–568 Teor. Veroyatnost. i Primenen. , 18 : 3 (1973) pp. 583–592
[4] K.O. Dzhaparidze, M.S. Nikulin, "On a modification of the standard statistics of Pearson" Theor. Probab. Appl. , 19 : 4 (1974) pp. 851–853 Teor. Veroyatnost. i Primenen. , 19 : 4 (1974) pp. 886–888
[5] M.S. Nikulin, "On a quantile test" Theory Probab. Appl. , 19 : 2 (1974) pp. 410–413 Teor. Veroyatnost. i Primenen. : 2 (1974) pp. 410–414
[6] L.N. Bol'shev, M. Mirvaliev, "Chi-square goodness-of-fit test for the Poisson, binomial and negative binomial distributions" Theory Probab. Appl. , 23 : 3 (1974) pp. 461–474 Teor. Veroyatnost. i Primenen. , 23 : 3 (1978) pp. 481–494
[7] L.N. Bol'shev, M.S. Nikulin, "A certain solution of the homogeneity problem" Serdica , 1 (1975) pp. 104–109 (In Russian)
[8] P.E. Greenwood, M.S. Nikulin, "Investigations in the theory of probabilities distributions. X" Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. Steklov. , 156 (1987) pp. 42–65 (In Russian)


The "chi-squared" test is also called the "chi-square" test or -test.

How to Cite This Entry:
Chi-squared test. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article