Journal of Applied Sciences

Year: 2006 | Volume: 6 | Issue: 10 | Page No.: 2151-2157
DOI: 10.3923/jas.2006.2151.2157

Order Statistics from Pareto Distribution

El desoky E. Afify

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.

Keywords: Pareto distribution, recurrence relations, single moment, order statistics, product moment, estimation and coefficient of variation

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.

From (1) and (2) we have

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

Let us denote the single moments 1≤i≤n and the product moments E(x_i:nY_i:n) by μ_i,j:n, 1≤i<j≤n.

RECURRENCE RELATIONS OF SINGLE MOMENTS

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

From (1) and (2) we have

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

Then we have

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

Put i = n we have

From (6) and (7) we have

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

We have that

And

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

From (5) and (11) we obtain

Also we obtain

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

Dividing (5) by (14) we have

Theorem 6: For 1< i <n-1

Where

Proof: Using (5) we have

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

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

Proof: From (5), we have

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

Theorem 8: For 1< i <n-1

Proof:

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

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

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:

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

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

Proof:

Using (3) we obtain

Integrating by parts we have

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

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

Proof:

using Eq. (3), the last equation becomes

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

RECURRENCE RELATIONS OF PRODUCT MOMENTS

The probability density function of is given by

is defined by x^r, y^r. The expected value of

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

Theorem 11: For 1≤i<j≤n

Proof:

Using (3) we have

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

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

Theorem 12: For 1≤i<j≤n

Proof

Using (3) we have

Let

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

Theorem 13: For 1≤i<j≤n

Proof: From (21) we have

From (21) we have

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

Theorem 14: For 1≤i<j≤n

Proof: From (22) we have

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

Theorem 15: For 1≤i<j≤n-1 for

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

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

APPLICATION

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

The variance of x_i:n is given by

Dividing (44) by (43) we obtain

Equating the right hand side of (45) to we have

Where, X_l: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

For m = 2, we have

The variance of X_i;n is given by

The coefficient of variation (CV) is given by

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.

REFERENCES