Abstract
In this paper, we present recurrence relations between the single and the product moments for order statistics from doubly truncated Makeham distribution. Characterizations for the Makeham distribution are studied.
moments of order statistics; Makeham distribution; doubly truncated distribution; recurrence relations
Recurrence relations between moments of order statistics from doubly truncated Makeham distribution
A.W. Aboutahoun; N.M. Al-Otaibi
Department of Mathematics, King Saud University, Riyadh, Kingdom of Saudi Arabia E-mail: tahoun@ksu.edu.sa
ABSTRACT
In this paper, we present recurrence relations between the single and the product moments for order statistics from doubly truncated Makeham distribution. Characterizations for the Makeham distribution are studied.
Mathematical subject classification: 62G30, 65C60.
Keywords: moments of order statistics, Makeham distribution, doubly truncated distribution, recurrence relations.
1 Introduction
Many researchers have studies the moments of order statistics of several distributions. A number of recurrence relalations satisfied by these moments of order statistics are available in literature. Balakrishnan and Malik [2] derived some identities involving the density functions of order statistics. These identities are useful in checking the computation of the moments of order statistics. Bala-krishnan and Malik [3] established some recurrence relations of order statistics from the liear-expoential distribution. Balakrishnan et al. [4] reviewed several recurrence relations and identities for the single and product moments of order statistics from some specific distributions. Mohie El-Din et al. [9, 10] presented recurrence relations for the single and product moments of order statistics from the doubly truncated parabolic and skewed distribution and linear-exponential distribution. Hendi et al. [1] developed recurrence relations for the single and product moments of order statistics from doubly truncated Gompertz distribution. Khanetal. [7] established general result about recurrence relations between product moments of order statistics. They used that result to get the recurrence relations between product moments of some doubly truncated distributions (Weibull, expoential, Pareto, power function, and Cauchy). Several recurrence relations satisfied by these moments of order statistics are also available in Khan and Khan [5], [6].
The probability density function (pdf) of the Makeham distribution is given by
The doubly truncated pdf of continuous rv is given by
where
The cumulative distribution function c.d.f. is given by
where
Let X be a continuous random variable having a c.d.f. (1.2) and p.d.f.. Let X1, X2, ..., Xn be a random sample of size n from the Makeham distribution and X1:n<X2:n<· · · <Xn:n be the corresponding order statistics obtained from the doubly truncated Makeham distribution (1. 1), then
where
The expected value of any measurable function h (x) can be obtained as follows:
and the expected value of any measurable joint function h (x, y) can be calculated by
where the joint density function of Xr:s and Xs:n, (1 <r <s <n) is given by
where
The rest of this paper is organized as follows: In Section 2 the recurrence relations for the single moments of order statistics from doubly truncated Makeham distribution is obtained. In Section 3 the recurrence relations for the product moments of order statistics from doubly truncated Makeham distribution is developed. Two results that characterize Maheham distribution are presented in Section 4. Some numerical results illustrating the developed recurrence relations are given in Section 5.
2 Recurrence relations for single moments of order statistics
Recurrence relations for the single moments of order statistics from the doubly truncated Maheham distribution are given by the following theorem.
Theorem 1.Let Xi:n<Xi + 1:n, (1 < i < n) be an order statics, Q1 < Xr;n< P1, 1 < r < n, n > 1 and for any measurable function h (x), then
where
Proof. From (1.4), we find
By using integration by parts, we get
Using (1 .2) in the previous equation, we obtain
Similarily, we can show that,
From (2.2) and (2.3), we obtain
Since
Then
By substituting for αr-1:n- 2 from the previous equation into Equation (2.4) we get the relation (2.1).
Remark 1. Let h (x) = xk in Equation (2.1), we obtain the single moments of the Makeham distribution
where
Remark 2. For the special case r = 1, n = 1, we can find
where , and
3 Recurrence relations for product moments of order statistics
Recurrence relations for the single moments of order statistics from the doubly truncated Maheham distribution are given by the following theorem.
Theorem 2.Let Xr:n<Xr+1:n, r = 1 , 2, . . . , n - 1 be an order statistics from a random sample of size n with pdf (1.1),
where
Proof. From (1.5)
Suppose that
then
By using integration by parts in the following integration
Hence,
By using (1.2)
Similarily, we can find that
Using the previous result in Equation (3.2)
then
which completes the proof.
Remark 3. If h (x, y) = xjyk, then (3.1) takes the form
which represents the identities for the product moments for doubly truncated Makeham distribution.
Khan et al. [7, 8] established the following results
Remark 4. For 1 <r <s <n and j > 0
4 Characterization of Makeham distribution
We discuss in this section two theorems that characterize the truncated Makeham distribution using the properties of the order statistics.
The pdf of (s -ρ) th order statistics of a sample of size (n -ρ) is given by (x <y)
where, f (Xs:n |Xr:n= x) is the conditional density of Xs:n given Xr:n x and the sample drawn from population with
which is obtained from the truncated paraent distribution F () at x. In the case of the left truncation at x, we have
and by putting s = r + 1, then (4.1) takes the form
Similarily, if the parent distribution truncated from the right at y (x < y and r < s), then
In the case of the right truncation at y, we have
and by putting r = 1, s = 2 then (4.3) takes the form
Theorem 3.If F (x)< 1,(0 < x∞) is the cummulative distribution function of a random variable X and F (0) = 0, then
Proof. The proof of the necessity condition starts by subsituting h (x) = x + θ(x + e-x-1), r = 1 in (2.1)
which means that
In the case of the left truncation at x, we get
To prove the sufficient condition, we use (4.2) and (1.2)
By differentiating both side w.r.t. x, we get
Theorem 4.If F (x) < 1, (0 < x < ∞) is the cummulative distribution function of a random variable X and F (0) = 0, then
Proof. To prove the necessity condition, let n = 1 , r = 1 in (2.1 )
which means that
To prove the sufficient condition, by using (4.2) and (1.2)
Differentiating both sides w.r.t. x
Then
5 Some numerical results
According to Khan et al. [7, 8], we have the following special cases of the moments of order statistics for any distribution
These special cases are used as initial conditions for generating nuerical values for the moments.
We implemented the two recurrence relations (2.1) and (3.1) using Matlab. The first table gives the numerical results for the single moments of order statistics for a random sample of size n = 10 from the doubly truncated Makeham distribution.
Acknowledgement. The authors ould like to thank the anonyous referee and the associate editor for their comments and suggestions, which were helpful in improving the manuscript.
Received: 18/IX/08.
Accepted: 15/VI/09.
#CAM-20/08.
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Publication Dates
-
Publication in this collection
05 Nov 2009 -
Date of issue
2009
History
-
Accepted
15 June 2009 -
Received
18 Sept 2008