on the expected values of the sample maximum of order statistics from a discrete uniform...
TRANSCRIPT
Applied Mathematics and Computation 157 (2004) 695–700
www.elsevier.com/locate/amc
On the expected values of the samplemaximum of order statistics
from a discrete uniform distribution
Sinan C�alik *, Mehmet G€ung€or
Department of Mathematics, Firat University, 23119 Elazı�g, Turkey
Abstract
In this paper, the expected values of the sample maximum of order statistics from a
discrete uniform distribution are given by using the sum SðN � 1; nÞ. For n up to 15,
algebraic expressions for the expected values are obtained.
� 2003 Elsevier Inc. All rights reserved.
Keywords: Order statistics; Sum; Discrete uniform distribution; Expected values
1. Introduction
Let X1;X2; . . . ;Xn be a random sample of size n from a discrete distributions
with probability mass function pðxÞ (x ¼ 0; 1; 2; . . .) and cumulative distribu-
tion function P ðxÞ, and let X1:n 6X2:n 6 � � � 6Xn:n be the order statistics ob-
tained from the above sample. Let us denote the expected values EðXr:nÞ by lð1Þr:n
(16 r6 n). For convenience, let us denote lð1Þr:n simply by lr:n.
In this paper, the expected values of the sample maximum of order statistics
from a discrete uniform distribution are given by using the sum SðN � 1; nÞ as
given in (5.1). For n up to 15, algebraic expressions for the expected values areobtained.
* Corresponding author.
E-mail address: [email protected] (S. C�alik).
0096-3003/$ - see front matter � 2003 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2003.08.056
696 S. C�alik, M. G€ung€or / Appl. Math. Comput. 157 (2004) 695–700
2. Marginal distributions of order statistics
Let Fr:nðxÞ (r ¼ 1; 2; . . . ; n) denote the cumulative distribution function (cdf)
of Xr:n. Then it is easy to see that
Fr:nðxÞ ¼ PfXr:n 6 xg¼ Pfat least r of X1;X2; . . . ;Xn are at most xg
¼Xn
i¼r
Pfexactly i of X1;X2; . . . ;Xn are at most xg
¼Xn
i¼r
ni
� �½PðxÞi½1 � P ðxÞn�i
¼Z PðxÞ
0
n!ðr � 1Þ!ðn� rÞ! t
r�1ð1 � tÞn�rdt ð2:1Þ
for �1 < x < 1.
For discrete population, the probability mass function (pmf) of Xr:n
(r ¼ 1; 2; . . . ; n) may be obtained from (2.1) by differencing as
fr:nðxÞ ¼ Fr:nðxÞ � Fr:nðx� 1Þ
¼ n!ðr � 1Þ!ðn� rÞ!
Z PðxÞ
Pðx�1Þtr�1ð1 � tÞn�r
dt; ð2:2Þ
see [2–4].
In particular, we also have
f1:nðxÞ ¼ f1 � P ðx� 1Þgn � f1 � P ðxÞgn
and
fn:nðxÞ ¼ fP ðxÞgn � fP ðx� 1Þgn:
3. Expected value of Xr:n
The mth raw moments of Xr:n can be immediately written down as
lðmÞr:n ¼
X1x¼0
xmfr:nðxÞ;
where fr:nðxÞ is as given in (2.2).
We can use the transformation Xr:n¼d F �1ðUr:nÞ to be express the expected
value of Xr:n as
S. C�alik, M. G€ung€or / Appl. Math. Comput. 157 (2004) 695–700 697
lr:n ¼n!
ðr � 1Þ!ðn� rÞ!
Z 1
0
F �1ðuÞur�1ð1 � uÞn�rdu;
see [2]. However, since F �1ðuÞ does not have a nice from for most of the dis-
crete distributions, this approach is often impractical. When the support B is asubset of nonnegative integers, which is the case with several standard discrete
distributions, one can use the cdf Fr:nðxÞ directly to obtain the expected values
of Xr:n.
Theorem 3.1. Let B, the support of the distribution, be a subset of nonnegativeintegers. Then
lr�n ¼X1x¼0
1½ � Fr:nðxÞ ð3:1Þ
whenever the expected values on the left-hand side is assumed to exist.
Proof. Let us note that if lr:n exists, mPfXr�n > mg ! 0 as m ! 1. Now con-
sider
Xmx¼0
xfr:nðxÞ ¼Xmx¼0
x½PfXr:n > x� 1g � PfXr:n > xg
¼Xm�1
x¼0
½ðxþ 1Þ � xPfXr:n > xg � mPfXr:n > mg:
On letting m ! 1, we obtain
lr:n ¼ limm!1
Xm�1
x¼0
PfXr�n > xg ¼X1x¼0
½1 � Fr:nðxÞ;
which establishes (3.1). Thus, we obtain (3.1). h
These expected values are obtained by Khatri [6] and Arnold et al. [2].
4. Order statistics from a discrete uniform distribution
Let the population random variable X be discrete uniform with support
B ¼ f1; 2; . . . ;Ng. We then write, X is discrete uniform ½1;N . Note that its pmf
is given by pðxÞ ¼ 1N, and its cdf is P ðxÞ ¼ x
N, for x 2 B. Consequently the cdf of
the r-th order statistics is given by
Fr:nðxÞ ¼Xn
i¼r
ni
� �xN
� �i1
�� xN
�n�i; x 2 B:
698 S. C�alik, M. G€ung€or / Appl. Math. Comput. 157 (2004) 695–700
5. Special sums
In the theory of nonparametric statistics, particularly when we deal with
rank sums, we often need for the sums of powers of the first n positive integers;
namely, expression for
SðN � 1; nÞ ¼ 1n þ 2n þ � � � þ ðN � 1Þn ¼XN�1
x¼0
xn ð5:1Þ
for n ¼ 0; 1; 2; . . . The following theorem, we provide a convenient way of
obtaining these sums.
Theorem 5.1
Xk�1
n¼0
kn
� �SðN � 1; nÞ ¼ Nk � 1
for any positive N and k (see, [5]).
A disadvantage of this theorem is that we have to find the sums SðN � 1; nÞone at a time, first for n ¼ 0, then n ¼ 1, then n ¼ 2 and so forth. For instance,
for k ¼ 1 we get
1
0
� �SðN � 1; 0Þ ¼ N � 1
and, hence, SðN � 1; 0Þ ¼ 10 þ 20 þ � � � þ ðN � 1Þ0 ¼ N � 1. Similarly, for
k ¼ 2 we get
2
0
� �SðN � 1; 0Þ þ 2
1
� �SðN � 1; 1Þ ¼ N 2 � 1;
N � 1 þ 2SðN � 1; 1Þ ¼ N 2 � 1
and, hence, SðN � 1; 1Þ ¼ 11 þ 21 þ � � � þ ðN � 1Þ1 ¼ 12ðN � 1ÞN . Using the
same technique, we can find the sums
SðN � 1; 2Þ ¼ 1
6ðN � 1ÞNð2N � 1Þ;
SðN � 1; 3Þ ¼ 1
4ðN � 1Þ2N 2 and so on:
S. C�alik, M. G€ung€or / Appl. Math. Comput. 157 (2004) 695–700 699
6. The expected values of the sample maximum of order statistics
In general, (3.1) expected values are not easy to evaluate analytically.
Sometimes the moments of sample extremes are tractable. Let us see what
happens in the case of discrete uniform distribution.
When X is a discrete uniform ½1;N random variable in the case of the
sample maximum, (3.1) yields
Table 1
The expected values of the sample maximum of order statistics
n ln:n
0 1
1N þ 1
2
24N 2 þ 3N � 1
6N
33N 2 þ 2N � 1
4N
424N 4 þ 15N 3 � 10N 2 þ 1
30N 3
510N 4 þ 6N 3 � 5N 2 þ 1
12N 3
636N 6 þ 21N 5 � 21N 4 þ 7N 2 � 1
42N 5
721N 6 þ 12N 5 � 14N 4 þ 7N 2 � 2
24N 5
880N 8 þ 45N 7 � 60N 6 þ 42N 4 � 20N 2 þ 3
90N 7
918N 8 þ 10N 7 � 15N 6 þ 14N 4 � 10N 2 þ 3
20N 7
1060N 10 þ 33N 9 � 55N 8 þ 66N 6 � 66N 4 þ 33N 2 � 5
66N 9
1122N 10 þ 12N 9 � 22N 8 þ 33N 6 � 44N 4 þ 33N 2 � 10
24N 9
122520N 12 þ 1365N 11 � 2730N 10 þ 5005N 8 � 858N 6 þ 9009N 4 � 4550N 2 þ 691
2730N 11
13390N 12 þ 210N 11 � 455N 10 þ 1001N 8 � 2145N 6 þ 3003N 4 � 2275N 2 þ 691
420N 11
1484N 14 þ 45N 13 � 105N 12 þ 273N 10 � 715N 8 þ 1287N 6 � 1365N 4 þ 691N 2 � 105
90N 13
1545N 14 þ 24N 13 � 60N 12 þ 182N 10 � 572N 8 þ 1287N 6 � 1820N 4 þ 1382N 2 � 420
48N 13
700 S. C�alik, M. G€ung€or / Appl. Math. Comput. 157 (2004) 695–700
ln:n ¼XNx¼0
½1 � Fn:nðxÞ ¼XN�1
x¼0
1h
� xN
� �ni¼ N � 1
Nn
XN�1
x¼1
xn
¼ N � SðN � 1; nÞNn
: ð6:1Þ
The sum on the right-hand side of (6.1) can be evaluated easily. For n up to15, algebraic expressions for the expected values are given in Table 1. Abra-
mowitz and Stegun [1, pp. 813–817] have tabulated it for several n and Nvalues.
Example 6.1. For N ¼ 100 and n ¼ 3, using the value of l3:3 in the Table 1, we
obtain
l3:3 ¼3N 2 þ 2N � 1
4N¼ 3ð100Þ2 þ 2ð100Þ � 1
4ð100Þ ¼ 75:4975:
References
[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Function with Formulas, Graphs,
and Mathematical Tables, Dover, New York, 1965.
[2] B.C. Arnold, N. Balakrishnan, H.N. Nagaraja, A First Course in Order Statistics, John Wiley
and Sons, New York, 1992.
[3] N. Balakrishnan, C.R. Rao, Handbook of Statistics 16––Order Statistics: Theory and Methods,
Elsevier, New York, 1998.
[4] H.A. David, Order Statistics, John Wiley and Sons, New York, 1981.
[5] J.E. Freud, R.E. Walpole, Mathematical Statistics, Prestice-Hall International, Inc., London,
1962.
[6] C.G. Khatri, Distribution of order statistics for discrete case, Ann. Inst. Statist. Math. 14 (1962)
167–171.