Characterizations of Certain Doubly Truncated Distribution Based on Order Statistics

In this paper, we characterize doubly truncated classes of absolutely continuous distributions by considering the conditional expectation of functions of order statistics. Specific distributions considered as a particular case of the general class of distributions are Weibull, Pareto, Power function, Rayleigh and Inverse Weibull. Citation: Shawky AI, Badr MM (2016) Characterizations of Certain Doubly Truncated Distribution Based on Order Statistics. J Appl Computat Math 5: 329. doi: 10.4172/2168-9679.1000329

Let X 1:n ≤ X 2:n ≤ …≤ X n:n be the first n order statistics based on distribution with probability density function (pdf) f(x) and cumulative distribution function (cdf) F(x). Then the pdf of the r th order statistics, X r:n , 1 ≤ r ≤ n, is given by (see David (1981)) ( ) ( ) ( ) ( ) 1 1 , x r n r = −∞ < < ∞ − − and the joint pdf of two order statistics X r:n and X s:n , 1 ≤ r ≤ s ≤ n is given by The doubly truncated case of a distribution is the most general case since it includes the right truncated, left truncated and non-truncated distributions as special cases, Joshi [15], Balakrishnan and Joshi [16], Khan and Ali [17] and Ahmad [18], among others, investigated doubly truncated distributions.
Suppose that the random variable X has a cdf F(x) and pdf f(x), where α ≤ x ß. Let, for given ε and Then the doubly truncated pdf of X, say g(x), and cdf, say G(x), are given respectively by The conditional density function of X s:n= y, given that X r:n= x is given Also, the conditional density function of X r:n= x, given that X s:n= y is given by

Main Results
In this section, we characterize three general classes of distributions, , and a ≠ 0, c 0, b are finite constants.
Let X be an absolutely continuous random variables with pdf g(x), cdf G(x) and ∅(x) is a monotonic, continuous and differentiable function on (ε,γ).

Special case:
Return to the (2.10), if we put l=-1, v=1we get Integrating by parts, we get After some simplification, we get (2.16).
Taking the derivative of (2.24) with respect to x, we get ( where V(x) is defined in (2.5).

Special case
Return to (2.28), then put l=-1, v=1, we get it is before doubly truncated case.

Proof
As before in Theorem (4) it is before doubly truncated case (Table 2).

Proof
Similarly as given in Theorems (4) and (8), we easily prove it.

Conclusion
It was obtained recurrence relations based on order statistics without truncated and doubly truncated, and have been getting function of various distributions new by using certain parameters.