Difference between revisions of "Rank vector"
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− | < | + | A vector statistic (cf. [[Statistics|Statistics]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r0775401.png" /> constructed from a random observation vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r0775402.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r0775403.png" />-th component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r0775404.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r0775405.png" />, defined by |
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− | + | <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/r/r077/r077540/r0775406.png" /></td> </tr></table> | |
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− | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r0775407.png" /> is the characteristic function (indicator function) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r0775408.png" />, that is, | |
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− | + | <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/r/r077/r077540/r0775409.png" /></td> </tr></table> | |
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− | + | The statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754010.png" /> is called the rank of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754011.png" />-th component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754013.png" />, of the random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754014.png" />. This definition of a rank vector is precise under the condition | |
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− | + | <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/r/r077/r077540/r07754015.png" /></td> </tr></table> | |
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− | + | which automatically holds if the probability distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754016.png" /> is defined by a density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754017.png" />. It follows from the definition of a rank vector that, under these conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754018.png" /> takes values in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754019.png" /> of all permutations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754021.png" /> and the realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754022.png" /> of the rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754023.png" /> is equal to the number of components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754024.png" /> whose observed values do not exceed the realization of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754025.png" />-th component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754027.png" />. | |
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− | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754028.png" /> be the vector of order statistics (cf. [[Order statistic|Order statistic]]) constructed from the observation vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754029.png" />. Then the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754030.png" /> is a [[Sufficient statistic|sufficient statistic]] for the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754031.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754032.png" /> itself can be uniquely recovered from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754033.png" />. Moreover, under the additional assumption that the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754035.png" /> is symmetric with respect to permutations of the arguments, the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754037.png" /> of the sufficient statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754038.png" /> are independent and | |
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− | + | <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/r/r077/r077540/r07754039.png" /></td> </tr></table> | |
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In particular, if | In particular, 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/r/r077/r077540/r07754040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table> | |
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− | + | that is, the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754041.png" /> are independent identically-distributed random variables (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754042.png" /> stands for the density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754043.png" />), then | |
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− | + | <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/r/r077/r077540/r07754044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table> | |
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− | + | for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754045.png" />. | |
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− | + | If (1) holds, there is a joint density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754047.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754049.png" />, defined by the formula | |
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− | + | <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/r/r077/r077540/r07754050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table> | |
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− | + | <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/r/r077/r077540/r07754051.png" /></td> </tr></table> | |
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− | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754052.png" /> is the distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754053.png" />. It follows from (2) and (3) that the conditional density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754055.png" /> given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754056.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754057.png" />) is expressed by the formula | |
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− | + | <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/r/r077/r077540/r07754058.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table> | |
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− | + | <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/r/r077/r077540/r07754059.png" /></td> </tr></table> | |
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− | + | The latter formula allows one to trace the internal connection between the observation vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754060.png" />, the rank vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754061.png" /> and the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754062.png" /> of order statistics, since (4) is just the probability density of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754063.png" />-th order statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754065.png" />. Moreover, it follows from (3) that the conditional distribution of the rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754066.png" /> is given by the formula | |
− | = | ||
− | + | <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/r/r077/r077540/r07754067.png" /></td> </tr></table> | |
− | + | <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/r/r077/r077540/r07754068.png" /></td> </tr></table> | |
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− | Finally, under the assumption that the moments | + | Finally, under the assumption that the moments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754070.png" /> exist and that (1) holds, (2) and (3) imply that the correlation coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754071.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754073.png" /> is equal to |
− | and | ||
− | exist and that (1) holds, (2) and (3) imply that the correlation coefficient | ||
− | between | ||
− | and | ||
− | is equal to | ||
− | + | <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/r/r077/r077540/r07754074.png" /></td> </tr></table> | |
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− | In particular, if | + | In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754075.png" /> is uniformly distributed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754076.png" />, then |
− | is uniformly distributed on | ||
− | then | ||
− | + | <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/r/r077/r077540/r07754077.png" /></td> </tr></table> | |
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− | If | + | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754078.png" /> has the [[Normal distribution|normal distribution]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754079.png" />, then |
− | has the [[Normal distribution|normal distribution]] | ||
− | then | ||
− | + | <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/r/r077/r077540/r07754080.png" /></td> </tr></table> | |
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− | and | + | and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077540/r07754081.png" /> does not depend on the parameters of the normal distribution. |
− | does not depend on the parameters of the normal distribution. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Hoeffding, " "Optimum" nonparametric tests" , ''Proc. 2nd Berkeley Symp. Math. Stat. Probab. (1950)'' , Univ. California Press (1951) pp. 83–92</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> F.P. Tarasenko, "Non-parametric statistics" , Tomsk (1976) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Hoeffding, " "Optimum" nonparametric tests" , ''Proc. 2nd Berkeley Symp. Math. Stat. Probab. (1950)'' , Univ. California Press (1951) pp. 83–92</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> F.P. Tarasenko, "Non-parametric statistics" , Tomsk (1976) (In Russian)</TD></TR></table> |
Revision as of 14:53, 7 June 2020
A vector statistic (cf. Statistics) constructed from a random observation vector
with
-th component
,
, defined by
![]() |
where is the characteristic function (indicator function) of
, that is,
![]() |
The statistic is called the rank of the
-th component
,
, of the random vector
. This definition of a rank vector is precise under the condition
![]() |
which automatically holds if the probability distribution of is defined by a density
. It follows from the definition of a rank vector that, under these conditions,
takes values in the space
of all permutations
of
and the realization
of the rank
is equal to the number of components of
whose observed values do not exceed the realization of the
-th component
,
.
Let be the vector of order statistics (cf. Order statistic) constructed from the observation vector
. Then the pair
is a sufficient statistic for the distribution of
, and
itself can be uniquely recovered from
. Moreover, under the additional assumption that the density
of
is symmetric with respect to permutations of the arguments, the components
and
of the sufficient statistic
are independent and
![]() |
In particular, if
![]() | (1) |
that is, the components are independent identically-distributed random variables (
stands for the density of
), then
![]() | (2) |
for any .
If (1) holds, there is a joint density ,
, of
and
, defined by the formula
![]() | (3) |
![]() |
where is the distribution function of
. It follows from (2) and (3) that the conditional density
of
given
(
) is expressed by the formula
![]() | (4) |
![]() |
The latter formula allows one to trace the internal connection between the observation vector , the rank vector
and the vector
of order statistics, since (4) is just the probability density of the
-th order statistic
,
. Moreover, it follows from (3) that the conditional distribution of the rank
is given by the formula
![]() |
![]() |
Finally, under the assumption that the moments and
exist and that (1) holds, (2) and (3) imply that the correlation coefficient
between
and
is equal to
![]() |
In particular, if is uniformly distributed on
, then
![]() |
If has the normal distribution
, then
![]() |
and does not depend on the parameters of the normal distribution.
References
[1] | W. Hoeffding, " "Optimum" nonparametric tests" , Proc. 2nd Berkeley Symp. Math. Stat. Probab. (1950) , Univ. California Press (1951) pp. 83–92 |
[2] | J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967) |
[3] | F.P. Tarasenko, "Non-parametric statistics" , Tomsk (1976) (In Russian) |
Rank vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_vector&oldid=48436