A member of the series of order statistics (also called variational series) based on the results of observations. Let a random vector 
be observed which assumes values 
in an 
-dimensional Euclidean space 
, 
, and let, further, a function 
be given on 
by the rule
where 
is a vector in 
obtained from 
by rearranging its coordinates 
in ascending order of magnitude, i.e. the components 
of the vector 
satisfy the relation
 | (1) |
In this case the statistic 
is the series (or vector) of order statistics, and its 
-th component 
(
) is called the 
-th order statistic.
In the theory of order statistics the best studied case is the one where the components 
of the random vector 
are independent random variables having the same distribution, as is assumed hereafter. If 
is the distribution function of the random variable 
, 
, then the distribution function 
of the 
-th order statistic 
is given by the formula
 | (2) |
where
is the incomplete beta-function. From (2) it follows that if the distribution function 
has probability density 
, then the probability density 
of the 
-th order statistic 
, 
, also exists and is given by the formula
 | (3) |
Assuming the existence of the probability density 
one obtains the joint probability density 
of the order statistics 
, 
, 
, which is given by the formula
 | (4) |
The formulas (2)–(4) allow one, for instance, to find the distribution of the so-called extremal order statistics (or sample minimum and sample maximum)
and also the distribution of 
, called the range statistic (or sample range). For instance, if the distribution function 
is continuous, then the distribution of 
is given by
 | (5) |
Formulas (2)–(5) show that, as in the general theory of sampling methods, exact distributions of order statistics cannot be used to obtain statistical inferences if the distribution function 
is unknown. It is precisely for this reason that asymptotic methods for the distribution functions of order statistics, as the dimension 
of the vector of observations tends to infinity, have been widely developed in the theory of order statistics. In the asymptotic theory of order statistics one studies the limit distributions of appropriately standardized sequences of order statistics 
as 
; moreover, generally speaking, the order number 
can change as a function of 
. If the order number 
changes as 
tends to infinity in such a way that the limit 
exists and is not equal to 
or to 
, then the corresponding order statistics 
of the considered sequence 
are called central or mean order statistics. If, however, 
is equal to 
or to 
, then they are called extreme order statistics.
In mathematical statistics central order statistics are used to construct consistent sequences of estimators (cf. Consistent estimator) for quantiles (cf. Quantile) of the unknown distribution 
based on the realization of a random vector 
or, in other words, to estimate the function 
. For instance, let 
be a quantile of level 
(
) of the distribution function 
about which one knowns that its probability density 
is continuous and strictly positive in some neighbourhood of the point 
. In this case the sequence of central order statistics 
with order numbers 
, where 
is the integer part of the real number 
, is a sequence of consistent estimators for the quantiles 
, 
. Moreover, this sequence of order statistics 
has an asymptotically normal distribution with parameters
i.e. for any real 
 | (6) |
where 
is the standard normal distribution function.
Example 1. Let 
be a vector of order statistics based on a random vector 
. The components of this vector are assumed to be independent random variables having the same probability distribution with a probability density that is continuous and positive in some neighbourhood of the median 
. In this case the sequence of sample medians 
, defined for any 
by
has an asymptotically normal distribution, as 
, with parameters
In particular, if
that is, 
has the normal distribution 
, then the sequence 
is asymptotically normally distributed with parameters 
and 
. If the sequence of statistics 
is compared with the sequence of best unbiased estimators (cf. Unbiased estimator)
for the mean 
of the normal distribution, then one should prefer the sequence 
, since
for any 
.
Example 2. Let 
be the vector of order statistics based on the random vector 
whose components are independent and uniformly distributed on an interval 
; moreover, suppose that the parameters 
and 
are unknown. In this case the sequences 
and 
of statistics, where
are consistent sequences of superefficient unbiased estimators (cf. Superefficient estimator) for 
and 
, respectively. Moreover,
One can show that the sequences 
and 
define the best estimators for 
and 
in the sense of the minimum of the square risk in the class of linear unbiased estimators expressed in terms of order statistics.
References
| [1] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) |
| [2] | S.S. Wilks, "Mathematical statistics" , Princeton Univ. Press (1950) |
| [3] | H.A. David, "Order statistics" , Wiley (1970) |
| [4] | E.J. Gumble, "Statistics of extremes" , Columbia Univ. Press (1958) |
| [5] | J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967) |
| [6] | B.V. Gnedenko, "Limit theorems for the maximal term of a variational series" Dokl. Akad. Nauk SSSR , 32 : 1 (1941) pp. 7–9 (In Russian) |
| [7] | B.V. Gnedenko, "Sur la distribution limite du terme maximum d'une série aléatoire" Ann. of Math. , 44 : 3 (1943) pp. 423–453 |
| [8] | N.V. Smirnov, "Limit distributions for the terms of a variational series" Trudy Mat. Inst. Steklov. , 25 (1949) pp. 5–59 (In Russian) |
| [9] | N.V. Smirnov, "Some remarks on limit laws for order statistics" Theor. Probab. Appl. , 12 : 2 (1967) pp. 337–339 Teor. Veroyatnost. i Primenen. , 12 : 2 (1967) pp. 391–392 |
| [10] | D.M. Chibisov, "On limit distributions for order statistics" Theor. Probab. Appl. , 9 : 1 (1964) pp. 142–148 Teor. Veroyatnost. Primenen. , 9 : 1 (1964) pp. 159–165 |
| [11] | A.T. Craig, "On the distributions of certain statistics" Amer. J. Math. , 54 (1932) pp. 353–366 |
| [12] | L.H.C. Tippett, "On the extreme individuals and the range of samples taken from a normal population" Biometrika , 17 (1925) pp. 364–387 |
| [13] | E.S. Pearson, "The percentage limits for the distribution of ranges in samples from a normal population ( )" Biometrika , 24 (1932) pp. 404–417 |
References
| [a1] | R.J. Serfling, "Approximation theorems of mathematical statistics" , Wiley (1980) |
