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ON AN OPTIMAL BOUND FOR THE
VARIANCE OF SAMPLE MAXIMUM
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ˇ
Andrius Ciginas
Let X = {x1 , . . . , xN } denote measurements of the study variable x of
the population {1, . . . , N }. Let X = {X1 , . . . , Xn } denote measurements of
units of the simple random sample of size n < N drawn without replacement
from the population. Let X1:n ≤ · · · ≤ Xn:n be the order statistics of X. We
are interesting in an optimal bound of the form Var Xn:n ≤ an;N Var X1 ,
where an;N may depend on n and N only. Here optimality means that there
exists a nontrivial population where equality is attained.
Let us mention few known results, which we extend to samples drawn
without replacement. In the case of independent identically distributed
(i.i.d.) observations Papadatos (1995) showed that an;N = n. The same
constant appears in the case of arbitrarily dependent identically distributed
observations, see Rychlik (2008). For i.i.d. samples, additionally assuming
that X1 has a symmetric distribution, Moriguti (1951) obtained an;N = n/2.
We note that for samples drawn without replacement the optimality
constant an;N is the same for the sample minimum. Similar bounds for the
variance of other order statistics will be also discussed at the conference.
References
Moriguti, S.: Extremal properties of extreme value distributions. Ann.
Math. Statist. 22, 523–536 (1951)
Papadatos, N.: Maximum variance of order statistics. Ann. Inst. Statist.
Math. 47, 185–193 (1995)
Rychlik, T.: Extreme variances of order statistics in dependent samples.
Stat. Probab. Lett. 78, 1577–1582 (2008)
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Vilnius University, Lithuania