MOMENTS OF NONIDENTICAL ORDER STATISTICS FROM BURR XII DISTRIBUTION WITH GAMMA AND NORMAL OUTLIERS

There are some distributions with no simple closed form for distribution functions such as the Normal and Gamma distributions. This will be the problem if weant to find moments of nonidentical order statist ics in the presence of Gamma and Normal outliers observati ons. We used the idea of approximating Normal and Gamma distributions with Burr type XII distribution . We get single moments for order statistics from sample of independent nonidentically distributed Bu rr XII random variables that contains p-outlier fro m Normal or Gamma distributions. Approximating these distributions with Burr XII distribution and then w e compared the results by previous method.


INTRODUCTION
There are no simple closed form exists for the normal distribution function and the gamma distribution function so that approximations to G (x) must be used to find moments of the rth identical order statistic. An approach of obtaining close approximation to the normal distribution was presented by Burr (1942). Burr (1967;1973), Burr recalculated the values of shape parameters for Burr XII distribution. These values give a closer approximation to the normal distribution. Another approach based on approximating gamma distribution with the Burr family distribution has been presented by Tadikamalla and Ramberg (1975) and Wheeler (1975), who approximate Gamma with two parameters with Burr with two parameters. Tadikamalla (1977) used the generalized four parameter B-distribution to approximate gamma distribution and put a table for selected integral values of shape parameter and other parameters which Burr XII distribution function approximate to the exact Gamma distribution function.
In this chapter we used the idea of approximating Normal and Gamma distributions with Burr XII to get single moments for order statistics from sample of independent nonidentically distributed Burr XII random variables that contains p-outlier from normal or Gamma distributions. approximating these distributions with Burr XII distribution and then we compared the results by those using the Barakat and Abdelkader (2004). Barnett and Lewis (1994) have defined an outlier in a set of data to be "an observation" or subset of observations" which appears to be inconsistent with the remainder of the set of data". They also describe several models for outlier; see (Moshref and Sultan, 2007). Density functions and joint density functions of order statistics arising from a sample of a single outlier have been given by Shu (1978) and Hartley and David (1978). One may also refer to Vaughan and Venables (1972) for more general expressions of distributions of order statistics using permanent expressions. Arnold and Balakrishnan (1989) have obtained the density function of Xr:n when the sample of size n contains unidentified Science Publications JMSS single outlier. They also obtained the joint density function of X r:n and X s:n ,1≤r<s≤n. Balakrishnan and Balasubramanian (1995) has derived some recurrence relations satisfied by the single and product moments of order statistics from the right truncated exponential distribution. Also he has deduced the recurrence relations for the multiple outlier models (with slippage of observations), see also Balakrishnan (1994). Childs et al. (2001) have derived some recurrence relations for the single and product moments of order statistics from n independent and non-identically distributed Lomax and the right-truncated Lomax random variables.
We assume X 1 , X 2 ,…,X p are independent with probability density function f (x) while X n -p+1,…,X n are independent were arise from some modified version of f (x) which call g(x) in which the shape parameters have been shifted in value. Finally, some special cases are deduced.
The probability density function of the rth order statistics X r:n , under the multiple outlier model can be written as, see Childs (1996) Setting p = 1 in (1) we obtain the corresponding pdf's in case of the single outlier given by Shu (1978).
In this study, we consider the case when the variable X 1 , X 2 ,…, X n-p are independent observations from Burr XII with four parameters distribution with density Equation 3: and X n-p+1 ,…, X n arise from the same distribution with density Equation 4: The corresponding cumulative distribution functions F (x) and G(x) are given as Equation 5 and 6: The relation between f (x) and F (x) is given by Similarly, the relation between g(x) and G(x) is In the following section, we use (3) and (4) to derive the single and product moments of order statistics from Burr XII distribution under the multiple outlier models. This situation is known as a multiple outlier model with Science Publications JMSS slippage of p observations; Barnett and Lewise (1994). This specific multiple outlier model was introduced by Launer and Bills (1979).

Single Moments
We derive the k th moment of the r th order statistic under multiple outlier models (with a slippage of p observations). Let µ (k) [p]; (1≤r≤n) denote the k th single moments of order statistics in the presence of p-outlier observations from BurrXII distribution The following theorem gives an explicit form of µ(k) [p].

Proof
Starting from (1), we have Equation 11: ( k ) k r :n r :n 0 min ( n p 1, r 1) k s r s 1 1 0 s max ( 0 , r p 1) n p s 1 p r s 1 min ( n p , r 1) k s r s 1 2 0 s max ( 0 , r p ) Using the relations Equation 12: We get Equation 13 min ( n p 1, r 1) ( k ) r :n s max ( 0 , r p 1) c k c 1 1 1 a s r s 1 n p s p r s 1 min ( n p , r 1) Now by writing: (14) We know that Equation 16-18: c c min ( n p 1, r 1) ( k ) r :n s max ( 0 , r p 1) Le: ( 1) 1 y b ( ) a y [ ( n p s l)| y dy Let: k r :n s max ( 0 , r p 1)

Remark
If we put a = 0, b = 1 in (9) and consider 0 0 = 1 we get moments of order statistics from Burr XII distribution with two parameters in the presence of outlier observations as Equation 20: min ( n p 1, r 1) ( k ) r :n s max ( 0 , r p 1)

Special Cases
We deduce some special cases from the single moments given in (9) • If we put p = n, we have the same relations above but with parameter τ • If we put p = 1, we have the relations for single outlier

Moments of Order Statistics from Burr XII with Four Parameters Using Theorem of Barakat and Abdelkader (2004)
Let X 1 , X 2 ,…, X n be independent nonidentically distributed r.v , .s. The k th moment of all order statistics, ( k ) r:n µ for 1≤r≤n and k = 1,2,… is given by (Barakat and Abdelkader, 2004)  G i (x) = 1-F i (x), with (i 1 , I 2 ,…, i n ) is a permutation of (1, 2, …, n ) for which i 1 ≤i 2 <…< i n .
We consider the case when the variable X 1 , X 2 ,…, X n-p are independent observations from Burr XII with four parameters distribution with density Equation 25: c c 1 c x a 1 f ( x ) ( The corresponding cumulative distribution function F (x) is given as: