Recurrence relations between moments of order statistics from doubly truncated Makeham distribution

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: [e-mail]

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, ..., X_n be a random sample of size n from the Makeham distribution and X_1:n<X_2:n<· · · <X_n: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 X_r:s and X_s: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 X_i:n<X_i + _1:n, (1 < i < n) be an order statics, Q1 < X_r;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 α_r-_{1:n- 2} from the previous equation into Equation (2.4) we get the relation (2.1).

Remark 1. Let h (x) = x^k 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 X_r:n<X_r+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) = x^jy^k, 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 (X_s:n |X_r:n= x) is the conditional density of X_s:n given X_r: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.

• [1] M.I. Hendi, S.E. Abu-Youssef and A.A. Alraddadi, Order Statistics from Doubly Truncated Gompertz Distribution and its Characterizations. The Egyptian Statistical Journal, 50(1)(2006),21-31.
• [2] N. Blakrishnan and H.J. Malik, Some general identities involving order statistics. Commun. Statist. Theor. Meth., 14(2) (1985), 333-339.
• [3] N. Balakrishnan and H.J. Malik, Order statistics from the linear-exponential distribution, part I: Increasing hazard rate case. Commun. Statist. Theor. Meth., 15(1)(1986), 179-203.
• [4] N. Blakrishnan, H.J. Malik and S.E. Ahmed, Recurrence relations and identities for moments of order statistics, II: Specific continuous distributions. Commun. Statist. Theor. Meth., 17(8) (1988),2657-2694.
• [5]A.H Khan and I.AKhan, Moments of order statistics from Burr distribution and its characterizations. Metron, 45 (1987), 21-29.
• [6] A.H. Khan and I.A. Khan, Recurrence relations between moments of order statistics from generalized gamma distribution. Journal of Statistical Research, 17(1-2)(1983), 75-82.
• [7] A.H. Khan, S. Parvez and M. Yaqub, Recurrence relations between product moments of order statistics. J. Statist. Plan. Inf., 8 (1983a), 175-183.
• [8] A.H. Khan, M. Yaqub and S. Parvez Recurrence relations between moments oforder statistics. Naval Res. Logist. Quart., 30 (1983b), 419-441.
• [9] M.M. Mohie El-Din, M.A.W. Mahmoud, S.E. Abu-Youssef and K.S. Sultan, Order Statistics from the doubly truncated linear exponential distribution and its characterizations. Commun. Statist-Simula., 26 (1997), 281-290.
• [10] M.M. Mohie El-Din, M.A.W. Mahmoud and S.E. Abu-Youssef, Moments of order statistics from parabolic and skewed distributions and characterization ofWeilbull distribution. Commun. Statist. Simul. Comput., 20(2,3) (1991), 639-645.

Publication Dates

• Publication in this collection
  05 Nov 2009
• Date of issue
  2009

History

• Accepted
  15 June 2009
• Received
  18 Sept 2008