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 xi:n (1≤i≤n) is given by David (1981).
Let us denote the single moments
1≤i≤n and the product moments E(xi:nYi:n) by μi,j:n,
RECURRENCE RELATIONS OF SINGLE MOMENTS
Consider (1.4), the expected value of xi: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
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
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
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,
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,
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 xr, yr. 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
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
Using (3) we have
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
a) Put m = 1, in (6) we have
The variance of xi:n is given by
Dividing (44) by (43) we obtain
Equating the right hand side of (45) to
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,
b) From (5) and for m = 1, α = 1, we have
For m = 2, we have
The variance of Xi;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.
|| The mean of order statistics from pareto distribution
|| The variance of the order statistics from pareto distribution
|| CV of the order statistics from pareto distribution