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''Smirnov <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s0858502.png" />-samples test''
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''Smirnov  $  2 $-
 +
samples test''
  
 
A non-parametric (or distribution-free) statistical test for testing hypotheses about the homogeneity of two samples.
 
A non-parametric (or distribution-free) statistical test for testing hypotheses about the homogeneity of two samples.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s0858503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s0858504.png" /> be mutually-independent random variables, where each sample consists of identically continuously distributed elements, and suppose one wishes to test the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s0858505.png" /> that both samples are taken from the same population. If
+
Let $  X _ {1} \dots X _ {n} $
 +
and $  Y _ {1} \dots Y _ {m} $
 +
be mutually-independent random variables, where each sample consists of identically continuously distributed elements, and suppose one wishes to test the hypothesis $  H _ {0} $
 +
that both samples are taken from the same population. If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s0858506.png" /></td> </tr></table>
+
$$
 +
X _ {(} 1)  \leq  \dots \leq  X _ {(} n) \ \
 +
\textrm{ and } \  Y _ {(} 1)  \leq  \dots \leq  Y _ {(} m)
 +
$$
  
are the order statistics corresponding to the given samples, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s0858507.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s0858508.png" /> are the empirical distribution functions corresponding to them, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s0858509.png" /> can be written in the form of the identity:
+
are the order statistics corresponding to the given samples, and $  F _ {n} ( x) $
 +
and $  G _ {m} ( x) $
 +
are the empirical distribution functions corresponding to them, then $  H _ {0} $
 +
can be written in the form of the identity:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585010.png" /></td> </tr></table>
+
$$
 +
H _ {0} :\  {\mathsf E} F _ {n} ( x)  \equiv  {\mathsf E} G _ {m} ( x) .
 +
$$
  
Further, consider the following hypotheses as possible alternatives to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585011.png" />:
+
Further, consider the following hypotheses as possible alternatives to $  H _ {0} $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585012.png" /></td> </tr></table>
+
$$
 +
H _ {1}  ^ {+} :\  \sup _
 +
{| x | < \infty }  {\mathsf E} [ G _ {m} ( x) - F _ {n} ( x) ]  > 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585013.png" /></td> </tr></table>
+
$$
 +
H _ {1}  ^ {-} : \  \inf _ {| x | < \infty }
 +
{\mathsf E} [ G _ {m} ( x) - F _ {n} ( x) ]  < 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585014.png" /></td> </tr></table>
+
$$
 +
H _ {1} : \  \sup _ {| x | < \infty }  |
 +
{\mathsf E} [ G _ {m} ( x) - F _ {n} ( x) ] |  > 0 .
 +
$$
  
To test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585015.png" /> against the one-sided alternatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585017.png" />, and also against the two-sided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585018.png" />, N.V. Smirnov proposed a test based on the statistics
+
To test $  H _ {0} $
 +
against the one-sided alternatives $  H _ {1}  ^ {+} $
 +
and $  H _ {1}  ^ {-} $,  
 +
and also against the two-sided $  H _ {1} $,  
 +
N.V. Smirnov proposed a test based on the statistics
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585019.png" /></td> </tr></table>
+
$$
 +
D _ {m,n}  ^ {+}  = \sup _
 +
{| x | < \infty }  [ G _ {m} ( x) - F _ {n} ( x) ] =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585020.png" /></td> </tr></table>
+
$$
 +
= \
 +
\max _ {1 \leq  k \leq  m }  \left (
 +
\frac{k}{m}
 +
- F _ {n} ( Y _ {(} k) ) \right )  = \max _ {1 \leq  s \leq  n }
 +
\left ( G _ {m} ( X _ {(} s) ) - s-
 +
\frac{1}{n}
 +
\right ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585021.png" /></td> </tr></table>
+
$$
 +
D _ {m,n}  ^ {-}  = - \inf _ {| x| < \infty }  [ G _ {m} ( x) - F _ {n} ( x) ] =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585022.png" /></td> </tr></table>
+
$$
 +
= \
 +
\max _ {i \leq  k \leq  m }  \left ( F _ {n} ( Y _ {(} k) ) - k-
 +
\frac{1}{m}
 +
\right )  = \max _ {1 \leq  s \leq  n }
 +
\left (
 +
\frac{s}{n}
 +
- G _ {m} ( X _ {(} s) ) \right ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585023.png" /></td> </tr></table>
+
$$
 +
D _ {m,n}  = \sup _ {| x | < \infty }  |
 +
G _ {m} ( x) - F _ {n} ( x) |  = \max  ( D _ {m,n}  ^ {+} , D _ {m,n}  ^ {-} ),
 +
$$
  
respectively, where it follows from the definitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585025.png" /> that under the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585028.png" /> have the same distribution. Asymptotic tests can be based on the following theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585029.png" />, then the validity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585030.png" /> implies that
+
respectively, where it follows from the definitions of $  D _ {m,n}  ^ {+} $
 +
and $  D _ {m,n}  ^ {-} $
 +
that under the hypothesis $  H _ {0} $,
 +
$  D _ {m,n}  ^ {+} $
 +
and $  D _ {m,n}  ^ {-} $
 +
have the same distribution. Asymptotic tests can be based on the following theorem: If $  \min ( m , n ) \rightarrow \infty $,  
 +
then the validity of $  H _ {0} $
 +
implies that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585031.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {m \rightarrow \infty }  {\mathsf P}
 +
\left \{ \sqrt {
 +
\frac{mn}{m+}
 +
n } D _ {m,n}  ^ {+} < y \right \}
 +
= 1 - e ^ {- 2 y  ^ {2} } ,\  y > 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585032.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {m \rightarrow \infty }  {\mathsf P} \left \{ \sqrt
 +
{
 +
\frac{mn}{m+}
 +
n } D _ {m,n} < y \right \}  = K ( y) ,\  y > 0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585033.png" /> is the Kolmogorov distribution function (cf. [[Statistical estimator|Statistical estimator]]). Asymptotic expansions for the distribution functions of the statistics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585035.png" /> have been found (see [[#References|[4]]]–[[#References|[6]]]).
+
where $  K ( y) $
 +
is the Kolmogorov distribution function (cf. [[Statistical estimator|Statistical estimator]]). Asymptotic expansions for the distribution functions of the statistics $  D _ {m,n}  ^ {+} $
 +
and $  D _ {m,n}  ^ {-} $
 +
have been found (see [[#References|[4]]]–[[#References|[6]]]).
  
Using the Smirnov test with significance level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585037.png" /> may be rejected in favour of one of the above alternatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585039.png" /> when the corresponding statistic exceeds the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085850/s08585040.png" />-critical value of the test; this value can be calculated using the approximations obtained by L.N. Bol'shev [[#References|[2]]] by means of Pearson asymptotic transformations.
+
Using the Smirnov test with significance level $  \alpha $,  
 +
$  H _ {0} $
 +
may be rejected in favour of one of the above alternatives $  H _ {1}  ^ {+} $,  
 +
$  H _ {1}  ^ {-} $
 +
when the corresponding statistic exceeds the $  \alpha $-
 +
critical value of the test; this value can be calculated using the approximations obtained by L.N. Bol'shev [[#References|[2]]] by means of Pearson asymptotic transformations.
  
 
See also [[Kolmogorov test|Kolmogorov test]]; [[Kolmogorov–Smirnov test|Kolmogorov–Smirnov test]].
 
See also [[Kolmogorov test|Kolmogorov test]]; [[Kolmogorov–Smirnov test|Kolmogorov–Smirnov test]].
Line 45: Line 132:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.A. Borovkov,  "On the two-sample problem"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''26''' :  4  (1962)  pp. 605–624  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.A. Borovkov,  "On the two-sample problem"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''26''' :  4  (1962)  pp. 605–624  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.B. Owen,  "A handbook of statistical tables" , Addison-Wesley  (1962)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.S. Pearson,  H.O. Hartley,  "Biometrika tables for statisticians" , '''2''' , Cambridge Univ. Press  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.B. Owen,  "A handbook of statistical tables" , Addison-Wesley  (1962)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.S. Pearson,  H.O. Hartley,  "Biometrika tables for statisticians" , '''2''' , Cambridge Univ. Press  (1972)</TD></TR></table>

Latest revision as of 08:14, 6 June 2020


Smirnov $ 2 $- samples test

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

Let $ X _ {1} \dots X _ {n} $ and $ Y _ {1} \dots Y _ {m} $ be mutually-independent random variables, where each sample consists of identically continuously distributed elements, and suppose one wishes to test the hypothesis $ H _ {0} $ that both samples are taken from the same population. If

$$ X _ {(} 1) \leq \dots \leq X _ {(} n) \ \ \textrm{ and } \ Y _ {(} 1) \leq \dots \leq Y _ {(} m) $$

are the order statistics corresponding to the given samples, and $ F _ {n} ( x) $ and $ G _ {m} ( x) $ are the empirical distribution functions corresponding to them, then $ H _ {0} $ can be written in the form of the identity:

$$ H _ {0} :\ {\mathsf E} F _ {n} ( x) \equiv {\mathsf E} G _ {m} ( x) . $$

Further, consider the following hypotheses as possible alternatives to $ H _ {0} $:

$$ H _ {1} ^ {+} :\ \sup _ {| x | < \infty } {\mathsf E} [ G _ {m} ( x) - F _ {n} ( x) ] > 0 , $$

$$ H _ {1} ^ {-} : \ \inf _ {| x | < \infty } {\mathsf E} [ G _ {m} ( x) - F _ {n} ( x) ] < 0 , $$

$$ H _ {1} : \ \sup _ {| x | < \infty } | {\mathsf E} [ G _ {m} ( x) - F _ {n} ( x) ] | > 0 . $$

To test $ H _ {0} $ against the one-sided alternatives $ H _ {1} ^ {+} $ and $ H _ {1} ^ {-} $, and also against the two-sided $ H _ {1} $, N.V. Smirnov proposed a test based on the statistics

$$ D _ {m,n} ^ {+} = \sup _ {| x | < \infty } [ G _ {m} ( x) - F _ {n} ( x) ] = $$

$$ = \ \max _ {1 \leq k \leq m } \left ( \frac{k}{m} - F _ {n} ( Y _ {(} k) ) \right ) = \max _ {1 \leq s \leq n } \left ( G _ {m} ( X _ {(} s) ) - s- \frac{1}{n} \right ) , $$

$$ D _ {m,n} ^ {-} = - \inf _ {| x| < \infty } [ G _ {m} ( x) - F _ {n} ( x) ] = $$

$$ = \ \max _ {i \leq k \leq m } \left ( F _ {n} ( Y _ {(} k) ) - k- \frac{1}{m} \right ) = \max _ {1 \leq s \leq n } \left ( \frac{s}{n} - G _ {m} ( X _ {(} s) ) \right ) , $$

$$ D _ {m,n} = \sup _ {| x | < \infty } | G _ {m} ( x) - F _ {n} ( x) | = \max ( D _ {m,n} ^ {+} , D _ {m,n} ^ {-} ), $$

respectively, where it follows from the definitions of $ D _ {m,n} ^ {+} $ and $ D _ {m,n} ^ {-} $ that under the hypothesis $ H _ {0} $, $ D _ {m,n} ^ {+} $ and $ D _ {m,n} ^ {-} $ have the same distribution. Asymptotic tests can be based on the following theorem: If $ \min ( m , n ) \rightarrow \infty $, then the validity of $ H _ {0} $ implies that

$$ \lim\limits _ {m \rightarrow \infty } {\mathsf P} \left \{ \sqrt { \frac{mn}{m+} n } D _ {m,n} ^ {+} < y \right \} = 1 - e ^ {- 2 y ^ {2} } ,\ y > 0 , $$

$$ \lim\limits _ {m \rightarrow \infty } {\mathsf P} \left \{ \sqrt { \frac{mn}{m+} n } D _ {m,n} < y \right \} = K ( y) ,\ y > 0 , $$

where $ K ( y) $ is the Kolmogorov distribution function (cf. Statistical estimator). Asymptotic expansions for the distribution functions of the statistics $ D _ {m,n} ^ {+} $ and $ D _ {m,n} ^ {-} $ have been found (see [4][6]).

Using the Smirnov test with significance level $ \alpha $, $ H _ {0} $ may be rejected in favour of one of the above alternatives $ H _ {1} ^ {+} $, $ H _ {1} ^ {-} $ when the corresponding statistic exceeds the $ \alpha $- 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)

Comments

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)
How to Cite This Entry:
Smirnov test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smirnov_test&oldid=48739
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article