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Research Article
 

Order Statistics from Pareto Distribution



El desoky E. Afify
 
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ABSTRACT

In this study, we derive some recurrence relations of single and product moments of order statistics from Pareto distribution. We estimate the parameters of the distribution using the moment of order statistics. We compute the mean, variance and coefficient of variation of order statistics from Pareto distribution.

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  How to cite this article:

El desoky E. Afify , 2006. Order Statistics from Pareto Distribution. Journal of Applied Sciences, 6: 2151-2157.

DOI: 10.3923/jas.2006.2151.2157

URL: https://scialert.net/abstract/?doi=jas.2006.2151.2157

INTRODUCTION

Pareto distribution has a wide use in economic studies. It has played a major part in investigation of several economic phenomina. Arnald (1983) gives an extensive historical survey of its use in context of income distribution. Khan and Abu ammoh (1999) characterized Pareto distribution by conditional expectation of order statistics. Galambos and Kotz (1978) and Ahmed (1991) have used the concept of left truncated moments to characterized some probability distributions like exponential, Pareto, gamma, negative binomial, beta, binomial and Piossion. Adeyemi (2002) derived some recurrence relations for moments of order statistics from a symmetric general log logistic distribution. Balakrishnan et al. (1988) reviewed recurrence relations for product moments of order statistics in case of specific distribution concerning exponential, power function, Pareto, Burr, Rayleigh and logistic distributions. Mohie et al. (1996) obtained a general identity for product moments of order statistics in a class of distribution functions, including Pareto, Weibull, exponential, Rayleigh and Burr distributions. In this paper, we derive some recurrence relations of single and product moments of order statistics from Pareto distribution. We estimate the parameters of the distribution by using the moment of the first order statistics and the mean, variance and the coefficient of variation are also computed.

Image for - Order Statistics from Pareto Distribution
(1)

Image for - Order Statistics from Pareto Distribution
(2)

From (1) and (2) we have

Image for - Order Statistics from Pareto Distribution
(3)

Probability distribution function of xi:n (1≤i≤n) is given by David (1981).

Image for - Order Statistics from Pareto Distribution
(4)

Let us denote the single moments Image for - Order Statistics from Pareto Distribution 1≤i≤n and the product moments E(xi:nYi:n) by μi,j:n, 1≤i<j≤n.

RECURRENCE RELATIONS OF SINGLE MOMENTS

Consider (1.4), the expected value of xi:n is given by

Image for - Order Statistics from Pareto Distribution

From (1) and (2) we have

Image for - Order Statistics from Pareto Distribution

Image for - Order Statistics from Pareto Distribution
(5)

Theorem 1: Replace I-1 for I in (5) for (1≤i≤n-1) we have,

Image for - Order Statistics from Pareto Distribution

Then we have

Image for - Order Statistics from Pareto Distribution

Theorem 2: For (1≤i≤n-1), let i = 1 in (5) we have

Image for - Order Statistics from Pareto Distribution
(6)

Put i = n we have

Image for - Order Statistics from Pareto Distribution
(7)

From (6) and (7) we have

Image for - Order Statistics from Pareto Distribution
(8)

Theorem 3: Replace i+1 for i in (5) for (1≤i≤n-1) we have,

Image for - Order Statistics from Pareto Distribution

We have that

Image for - Order Statistics from Pareto Distribution
(9)

And

Image for - Order Statistics from Pareto Distribution
(10)

Theorem 4: From (5) and for m = 1, we have

Image for - Order Statistics from Pareto Distribution
(11)

From (5) and (11) we obtain

Image for - Order Statistics from Pareto Distribution
(12)

Also we obtain

Image for - Order Statistics from Pareto Distribution
(13)

Theorem 5: In (5) replace i-1 for i and n-1 for n we obtain

Image for - Order Statistics from Pareto Distribution
(14)

Dividing (5) by (14) we have

Image for - Order Statistics from Pareto Distribution
(15)

Theorem 6: For 1< i <n-1

Image for - Order Statistics from Pareto Distribution

Where

Image for - Order Statistics from Pareto Distribution

Proof: Using (5) we have

Image for - Order Statistics from Pareto Distribution
(16)

Adding the last equation to (5) and simplifying, we readily obtain the result (16).

Theorem 7: For 1< i <n-1,

Image for - Order Statistics from Pareto Distribution
(17)

Proof: From (5), we have

(i)
Image for - Order Statistics from Pareto Distribution

(ii)
Image for - Order Statistics from Pareto Distribution

Subtracting (i) from (ii) and simplifying we will obtain the result (17).

Theorem 8: For 1< i <n-1

Image for - Order Statistics from Pareto Distribution
(18)

Proof:

Image for - Order Statistics from Pareto Distribution

Using (3), we may written for 1 = i = n-1

(iii)
Image for - Order Statistics from Pareto Distribution

Integrating the RHS of (iii) by parts, treating x for integrating and the rest of integrand for differentiating, we obtain for 1≤i≤n-1, the equation

Image for - Order Statistics from Pareto Distribution

If we split the first integral of the RHS of the last equation into two and combine with the second integral, the last equation may be written as:

Image for - Order Statistics from Pareto Distribution

This equation, when simplified, yields the relation (18).

Theorem 9: For 1<i<n-1,

Image for - Order Statistics from Pareto Distribution
(19)

Proof:

Image for - Order Statistics from Pareto Distribution

Using (3) we obtain

Image for - Order Statistics from Pareto Distribution

Integrating by parts we have

Image for - Order Statistics from Pareto Distribution

This equation, when simplified, we obtain the relation (19).

Theorem 10: For 1< i <n-1,

Image for - Order Statistics from Pareto Distribution
(20)

Proof:

Image for - Order Statistics from Pareto Distribution

using Eq. (3), the last equation becomes

Image for - Order Statistics from Pareto Distribution

Integrating by parts treating x, we obtain the result (20).

RECURRENCE RELATIONS OF PRODUCT MOMENTS

The probability density function of Image for - Order Statistics from Pareto Distribution is given by

Image for - Order Statistics from Pareto Distribution

is defined by xr, yr. The expected value of

Image for - Order Statistics from Pareto Distribution
(21)

We let r = s = 1, in Eq. (21), we obtain the well known results (Johnson and Kotz, 1970).

Image for - Order Statistics from Pareto Distribution
(22)

Theorem 11: For 1≤i<j≤n

Image for - Order Statistics from Pareto Distribution
(23)

Proof:

Image for - Order Statistics from Pareto Distribution
(24)

Using (3) we have

Image for - Order Statistics from Pareto Distribution
(24a)

Integrating (24a) by parts then subsititiute in (24), we have

Image for - Order Statistics from Pareto Distribution

On simplifying the last equation, the result (23) will obtained.

Theorem 12: For 1≤i<j≤n

Image for - Order Statistics from Pareto Distribution
(25)

Proof

Image for - Order Statistics from Pareto Distribution

Using (3) we have

Image for - Order Statistics from Pareto Distribution
(26)

Let

Image for - Order Statistics from Pareto Distribution
(26a)

Integrating (26a) by parts, then subsititute in equation (26) and simplifying the result (25) will obtained.

Theorem 13: For 1≤i<j≤n

Image for - Order Statistics from Pareto Distribution
(27)

Proof: From (21) we have

Image for - Order Statistics from Pareto Distribution
(28)

From (21) we have

Image for - Order Statistics from Pareto Distribution
(29)

Adding (28) to (29) and simplifying we obtain the result (27). Also we obtain that

Image for - Order Statistics from Pareto Distribution
(30)

Theorem 14: For 1≤i<j≤n

Image for - Order Statistics from Pareto Distribution
(31)

Proof: From (22) we have

Image for - Order Statistics from Pareto Distribution
(32)

Subtracting (22) into (32) and simplifying we obtain the result (31).

Theorem 15: For 1≤i<j≤n-1 for

Image for - Order Statistics from Pareto Distribution
(33)

Proof: Replace j+1 with j in (22) we can easily obtain

Image for - Order Statistics from Pareto Distribution
(34)

From (22) and (34) we obtain the relation (33).

Using (21), some recurrence relations for product moments of order statistics from Pareto distribution can be obtained by simple rearrangement and manipulation, they are

Image for - Order Statistics from Pareto Distribution
(35)

Image for - Order Statistics from Pareto Distribution
(36)

Image for - Order Statistics from Pareto Distribution
(37)

Image for - Order Statistics from Pareto Distribution
(38)

Image for - Order Statistics from Pareto Distribution
(39)

Image for - Order Statistics from Pareto Distribution
(40)

Image for - Order Statistics from Pareto Distribution
(41)

Image for - Order Statistics from Pareto Distribution
(42)

APPLICATION

a) Put m = 1, in (6) we have

Image for - Order Statistics from Pareto Distribution
(43)

The variance of xi:n is given by

Image for - Order Statistics from Pareto Distribution
(44)

Dividing (44) by (43) we obtain

Image for - Order Statistics from Pareto Distribution
(45)

Equating the right hand side of (45) to Image for - Order Statistics from Pareto Distribution we have

Image for - Order Statistics from Pareto Distribution
(46)

Where, Xl:n is the smallest value of the sample and S2 is the variance of the sample. We solved Eq. (46) for α then subsititute in (43) to obtain γ.

We generate a data set from Pareto distribution with parameters α = 1 and γ = 2

And a sample of size 20, using equations (46) and (43) we obtain the estimates for the parameters α and γ as 1 and 1.9997, respectively.

b) From (5) and for m = 1, α = 1, we have

Image for - Order Statistics from Pareto Distribution

For m = 2, we have

Image for - Order Statistics from Pareto Distribution

The variance of Xi;n is given by

Image for - Order Statistics from Pareto Distribution

The coefficient of variation (CV) is given by

Image for - Order Statistics from Pareto Distribution

We compute the mean, variance and coefficient of variation of order statistics from Pareto distribution up to 10. The results are shown in Table 1-3.

Table 1: The mean of order statistics from pareto distribution
Image for - Order Statistics from Pareto Distribution

Table 2: The variance of the order statistics from pareto distribution
Image for - Order Statistics from Pareto Distribution

Table 3: CV of the order statistics from pareto distribution
Image for - Order Statistics from Pareto Distribution

REFERENCES

1:  Adeyemi, S., 2002. Some recurrence relations for moments of order statistics from a symmetric generalized log logistic distribution. http://interstat.statjournals.net/YEAR/2002/articles/0212001.pdf.

2:  Ahmed, A., 1991. Characterization of beta, binomial and Poisson distributions. IEEE Trans. Reliability, 40: 290-295.
CrossRef  |  

3:  Arnold, B.C., 1983. The Pareto Distribution. International Cooperative Pupllishing House, Fairland, Maryland

4:  Balakrishnan, N., H.J. Malik and S.E. Ahmed, 1988. Recurrence relations and identities for moments of order statistics II: Specific continuous distribution. Commmun. Statist. Theor-Meth., 17: 2657-2694.
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5:  David, H.A., 1981. Order Statistics. 2nd Edn., John Wiely, New York

6:  Galambos, J. and S. Kotz, 1978. Characterization of Probability Distributions. Springer Verlag, Berlin

7:  Khan, A.H. and A.M. Abuammoh, 1999. Characterization of distributions by conditional expectation of order statistics. J. Applied Sci., 9: 159-168.

8:  El-Din, M., M.M. Mahmoud and S.E. Abu-Youssef, 1996. An identity for the product moments of order statistics. Metrika, 44: 95-100.
CrossRef  |  

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