Chi-squared test
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
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which has in the limit, as , a "chi-squared" distribution with
degrees of freedom, that is,
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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,
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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].
References
[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) |
Comments
The "chi-squared" test is also called the "chi-square" test or -test.
Chi-squared test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chi-squared_test&oldid=28552