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 – there are some recommendations concerning the -test in this case; in particular, in the normal case , the general continuous case , , the discrete case , , and in the problem of several samples .
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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=15852