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A test for the verification of a hypothesis  $  H _ {0} $
according to which a random vector of frequencies  $  \nu = ( \nu _ {1} \dots \nu _ {k} ) $
has a given polynomial distribution, characterized by a vector of positive probabilities  $  p = ( p _ {1} \dots p _ {k} ) $, 
$  p _ {1} + \dots + p _ {k} = 1 $. 
The  "chi-squared"  test is based on the Pearson statistic

$$ 
X  ^ {2}  = \ 
\sum _ {i = 1 } ^ { k } 

\frac{( \nu _ {i} - np _ {i} )  ^ {2} }{np _ {i} }
  = \ 
{
\frac{1}{n}
 } \sum

\frac{\nu _ {i}  ^ {2} }{p _ {i} }
 - n,\ \ 
n = \nu _ {1} + \dots + \nu _ {k} ,
$$

which has in the limit, as  $  n \rightarrow \infty $, 
a [[Chi-squared distribution| "chi-squared"  distribution]] with  $  k - 1 $
degrees of freedom, that is,

$$ 
\lim\limits _ {n \rightarrow \infty } \ 
{\mathsf P} \{ X  ^ {2} \leq 
x \mid  H _ {0} \}  = \ 
{\mathsf P} \{ \chi _ {k - 1 }   ^ {2} \leq  x \} .
$$

According to the  "chi-squared"  test with significance level  $  \approx \alpha $, 
the hypothesis  $  H _ {0} $
must be rejected if  $  X  ^ {2} \geq  \chi _ {k - 1 }   ^ {2} ( \alpha ) $, 
where  $  \chi _ {k - 1 }   ^ {2} ( \alpha ) $
is the upper  $  \alpha $-
quantile of the  "chi-squared"  distribution with  $  k - 1 $
degrees of freedom, that is,

$$ 
{\mathsf P} \{
\chi _ {k - 1 }   ^ {2} \geq  \chi _ {k - 1 }   ^ {2} ( \alpha )
\}  =  \alpha .
$$

The statistic  $  X  ^ {2} $
is also used to verify the hypothesis  $  H _ {0} $
that the distribution functions of independent identically-distributed random variables  $  X _ {1} \dots X _ {k} $
belong to a family of continuous functions  $  F ( x, \theta ) $, 
$  x \in \mathbf R  ^ {1} $, 
$  \theta = ( \theta _ {1} \dots \theta _ {m} ) \in \Theta \subset  \mathbf R  ^ {m} $, 
$  \Theta $
an open set. After dividing the real line by points  $  x _ {0} < \dots < x _ {k} $, 
$  x _ {0} = - \infty $, 
$  x _ {k} = + \infty $, 
into  $  k $
intervals  $  ( x _ {0} , x _ {1} ] \dots ( x _ {k - 1 }  , x _ {k} ) $, 
$  k > m $, 
such that for all  $  \theta \in \Theta $,

$$ 
p _ {i} ( \theta )  = \ 
{\mathsf P} \{ X _ {i} \in
( x _ {i - 1 }  , x _ {i} ]
\}  >  0,
$$

$  i = 1 \dots k $; 
$  p _ {1} ( \theta ) + \dots + p _ {k} ( \theta ) = 1 $, 
one forms the frequency vector  $  \nu = ( \nu _ {1} \dots \nu _ {k} ) $, 
which is obtained as a result of grouping the values of the random variables  $  X _ {1} \dots X _ {n} $
into these intervals. Let

$$ 
X  ^ {2} ( \theta )  = \ 
\sum _ {i = 1 } ^ { k } 

\frac{[ \nu _ {i} - np _ {i} ( \theta )]  ^ {2} }{np _ {i} ( \theta ) }

$$

be a random variable depending on the unknown parameter  $  \theta $. 
To verify the hypothesis  $  H _ {0} $
one uses the statistic  $  X  ^ {2} ( \widetilde \theta   _ {n} ) $, 
where  $  \widetilde \theta   _ {n} $
is an estimator of the parameter  $  \theta $, 
computed by the method of the minimum of  "chi-squared" , that is,

$$ 
X  ^ {2} ( \widetilde \theta   _ {n} )  = \ 
\min _ {\theta \in \Theta } \ 
X  ^ {2} ( \theta ).
$$

If the intervals of the grouping are chosen so that all  $  p _ {i} ( \theta ) > 0 $, 
if the functions  $  \partial   ^ {2} p _ {i} ( \theta )/ \partial  \theta _ {j} \partial  \theta _ {r} $
are continuous for all  $  \theta \in \Theta $, 
$  i = 1 \dots k $; 
$  j, r = 1 \dots m $, 
and if the matrix  $  \| \partial  p _ {i} ( \theta )/ \partial  \theta _ {j} \| $
has rank  $  m $, 
then if the hypothesis  $  H _ {0} $
is valid and as  $  n \rightarrow \infty $, 
the statistic  $  X  ^ {2} ( \widetilde \theta   _ {n} ) $
has in the limit a  "chi-squared"  distribution with  $  k - m - 1 $
degrees of freedom, which can be used to verify  $  H _ {0} $
by the  "chi-squared"  test. If one substitutes a maximum-likelihood estimator  $  \widehat \theta   _ {n} $
in  $  X  ^ {2} ( \theta ) $, 
computed from the non-grouped data  $  X _ {1} \dots X _ {n} $, 
then under the validity of  $  H _ {0} $
and as  $  n \rightarrow \infty $, 
the statistic  $  X  ^ {2} ( \widehat \theta   _ {n} ) $
is distributed in the limit like

$$ 
\xi _ {1}  ^ {2} + \dots +
\xi _ {k - m - 1 }   ^ {2} +
\mu _ {1} \xi _ {k - m }   ^ {2} + \dots +
\mu _ {m} \xi _ {k - 1 }   ^ {2} ,
$$

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

In [[#References|[3]]]–[[#References|[8]]] there are some recommendations concerning the  $  \chi  ^ {2} $-
test in this case; in particular, in the normal case [[#References|[3]]], the general continuous case [[#References|[4]]], [[#References|[8]]], the discrete case [[#References|[6]]], [[#References|[8]]], and in the problem of several samples [[#References|[7]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.G. Kendall,   A. Stuart,   "The advanced theory of statistics" , '''2. Inference and relationship''' , Griffin  (1983)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L.N. Bol'shev,   M.S. Nikulin,   "A certain solution of the homogeneity problem"  ''Serdica'' , '''1'''  (1975)  pp. 104–109  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  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)</TD></TR></table>

====Comments====
The  "chi-squared"  test is also called the  "chi-square"  test or  $  \chi  ^ {2} $-
test.