^{1}Department of Statistics, Faculty of Science, King Abdulaziz University, Saudi Arabia
^{2}Department of Statistics, Faculty of Science for Girls, King Abdulaziz University, Saudi Arabia
Received date: October 24 , 2016; Accepted date: November 23, 2016; Published date: November 29, 2016
Citation: Shawky AI, Badr MM (2016) Characterizations of Certain Doubly Truncated Distribution Based on Order Statistics. J Appl Computat Math 5:329. doi: 10.4172/21689679.1000329
Copyright: © 2016 Shawky AI, et al. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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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.
Double truncated; Order statistics; Conditional expectation; Weibull, Pareto; Rayleigh; Inverse Weibull distributions
AMS 2000 Subject Classification: 62E10; 62G30
The order statistics arise naturally in many real life applications and it is considered as an increasingly important subject. Articles relating to this area have appeared in numerous different publications. Many authors have studied order statistics; for example, David [1], Balakrishnan and Cohen [2], Arnold et al. [3], David [4], David and Nagaraja [5] and Mahmoud et al. [6,7]. Several authors discussed conditional expectations, for example, Balakrishnan and Sultan [8], Mohie ElDin et al. [9], AbuYoussef [10], Abd ElMougod [11], Shawky and AbuZinadah [12], Shawky and Bakoban [13] and Pushkarna et al. [14].
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))
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 nontruncated 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
where
l1=Q – P, G(ε)=0 and G(γ)=1.
The conditional density function of X_{s:n}=y, given that X_{r:n}=x is given [3] by
Also, the conditional density function of X_{r:n}=x, given that X_{s:n}=y is given by
Let
where ? (.) is a monotonic, continuous and differentiable function on the interval (α,ß). For abbreviation, we will denote
In this section, we characterize three general classes of distributions,
where
v=1–P, l=P – Q, P=F(ε), Q=F(γ), G(ε)=0, G(γ)=1,
F(α)=0, F(ß)=1, and a ≠ 0, c 0, b are finite constants.
F(x)=1 – [a – b ∅(x)]c, α < x < ß, i.e.,
where
F(α)=0, F(ß)=1, and b ≠ 0, c ≠ 0, a are finite constants.
where
v=1 – P, l=P – Q, P=F(ε), Q=F(γ), G(ε)=0, G(γ)=1,
F(α)=0, F(ß)=1, and C ≠ 0, a > 0, b > 0 are finite constants.
Note: If we put l=–1, v=1, thus G(x) reduces to complete cdf of x, i.e. F(x), α < x < ß.
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).
Theorems 14 given below characterize the general class given by (2.1), Theorems 58 characterize the general class given by (2.2), while Theorems 912 characterize the general class given by (2.3).
Referring to (1.6), (1.7) and (2.1), then
where
Proof
It is clear from (1.6) and (1.7) that
Integrating (2.6) by parts, we get
Differentiating (2.1) with respect to x, we have
From (2.7) and (2.8), we obtain
(2.9)
Simplifying (2.9), we get (2.4). Thus, the theorem is proved.
Referring to (1.6), (1.7), then (2.1) if and only if
Proof
It is clear that
Integrating by parts, we get
Compensation for (2.8) in (2.11), we have
Expand and compensation for (2.12), after some simplification, we get (2.10). Thus (2.1) implies (2.10). Now from (1.6) and (2.10), we obtain
Taking the derivative, we get
which gives
Integrate (2.14), hence G(y) has the form (2.1), and so (2.10) implies (2.1).
Special case:
Return to the (2.10), if we put l=–1, v=1we get
the relation (2.15) is before doubly truncated case.
Referring to (1.5), (1.7) and (2.1), then
where
It is clear from (1.5) and (1.7) that
Integrating by parts, we get
Substituting (2.7) in (2.18), we get
After some simplification, we get (2.16).
Referring to (1.5), (1.7), then (2.1) if and only if
where N(y) is defined in (2.17).
Proof
It is clear that
Integrating by parts, we get
Compensation for (2.8) in (2.22), we have
Simplifying (2.23), we obtain (2.20). Thus (2.1) implies (2.20), i.e. the necessary condition is proved. To prove the sufficient condition, from (2.20) and (1.7), we have
Taking the derivative of (2.24) with respect to x, we get (2.8), and integrate it we have (2.1), thus (2.20) implies (2.1). Then, the Theorem is proved.
Special case
Return to (2.17), if we put l=–1, v=1 we get
it is before doubly truncated case (Table 1).
Name  [lG(x)+ν]  φ(x)  (l,ν)  (a,b,c) 

Weibull  x^{p} qx^{p} 
(–1,0,θ) (–1,0,1) 

Pareto  θ^{p} x^{p}; α ≤ ε < x < γ ≤ β, ε = 0, γ → ∞  ln(x)? ln[x–p] 
(θ^{p}(γ–p–ε–p), θ^{p} ε–p)  (–θ,0,p) (–θ^{p},0,–1) 
Power function  1–θ–p x^{p},α ≤ ε < x < γ ≤ β, ε = 0, γ → ∞  (θ^{–p} (ε^{p}–γ^{p}), 1–θ^{–p} ε^{p})  (1,1,1) (θ^{–p},1,1) 

Rayleigh  x^{2}  (–1,0,θ)  
Inverse Weibull  θx–p  (–1,0,1) 
Table 1: Example of distributions.
Referring to (1.6), (1.7) and (2.2), then
where V(x) is defined in (2.5).
Proof
As before in Theorem (1), differentiate (2.2) with respect to x, we have
Compensation for (2.26) in (2.7), we get
Simplifying (2.27), we obtain (2.25). Thus, the Theorem is proved.
Referring to (1.6), (1.7), then (2.2) if and only if
Where V(x) is defined in (2.5).
Proof
As before in Theorem (2), from (2.26) and (2.11), we have
Therefore, we get (2.28), then (2.2) implies (2.28). To prove the sufficient condition, from (2.28) and (1.7), we obtain
Taking the derivative, we get (2.26) and we obtain, after integration, (2.2). Thus (2.28) implies (2.2).
Special case
Return to (2.28), then put l=–1, v=1, we get
it is before doubly truncated case.
Referring to (1.5), (1.7) and (2.2), then
where N(y) is defined in (2.17).
Proof
As before in Theorem (3), compensation for (2.26) in (2.18), we have
After simplification, we get (2.31). Then (2.2) implies (2.31).
Referring to (1.5), (1.7), then (2.2) if and only if
where N(y) is defined in (2.17).
Proof
As before in Theorem (4), compensation for (2.26) in (2.23), we have
Then, we obtain (2.35). Thus (2.2) implies (2.33). Now from (1.7) and (2.33) we get
Taking the derivative with respect to x, we obtain (2.26), and integrate it we have (2.2), thus (2.33) implies (2.2).
Hence, the Theorem is proved. Special caseReturn to (2.33), then put l=–1, v=1, we get
it is before doubly truncated case (Table 2).
Name  [lG(x) + v]  j(x)  (l,v)  (a,b,c) 

Weibull  (1,0,θ)  
Power function  1– θ^{–p} x^{p}, α ≤ ε < x < γ ≤ β, ε = 0, γ → ∞ 
x^{p} 
(θ^{–p} (ε^{p} + γ^{p}), 1–θ^{–p}ε^{p})  (1,1,1) (q^{–p},1,1) 
Rayleigh  (–1,0,θ)  
Inverse Weibull  (0,–1,1) 
Table 2: Example of distributions.
Referring to (1.6), (1.7) and (2.3), then
Proof
As given in Theorems (1) and (5), , differentiate (2.3) with respect to x, we have
Compensation for (2.37) in (2.7), we get
which gives (2.36). Thus, (2.3) implies (2.36).
Referring to (1.6) and (1.7), then (2.3) if and only if
Proof
As given previously of Theorems (2) and (6), substituting from (2.26) in (2.11), we have
After some simplification, we get (2.36). Then (2.3) implies (2.36). Now from (1.7) and (2.36), we obtain
Taking the derivative with respect to y, we get
Integrate (2.41), we obtain (2.3).
Special case
Return to (2.36), then put l=–1, v=1, we get
it is before doubly truncated case.
Referring to (1.5), (1.7) and (2.3), then
Proof
As previously in Theorems (3) and (7), from (2.37) in (2.18), we have
which gives (2.42).
Referring to (1.5) and (1.7), then (2.3) if and only if
Proof
Similarly as given in Theorems (4) and (8), we easily prove it.
Special case
Return to (2.43), then put l=–1, v=1, we get
The relation before doubly truncated case (Table 3).
Name  [lG(x) + v]  j(x)  (l,v)  (a,b,c) 

Power distribution  1–θ^{–p} x^{p},α ≤ ε < x < γ ≤ β, ε = 0, γ = θ  Ln[1θ^{–p}x^{p}]  (θ^{–p}(ε^{p}+ γ^{p}), 1θ^{–p}ε^{p})  (1,0,–1) 
Weibull  e–θx^{p}, α ≤ ε < x < γ ≤ β, ε = 0, γ → ∞  θx^{p}  (–1,0,1)  
Burr  (1+θx^{p})^{–γ}, α ≤ ε < x < γ ≤ β, ε = 0, γ→∞  ln[1 + qx^{p}]  (θ^{γ}^{–p}) + θ^{εp} +1), 1+θ^{εp})  (–1,0,γ) 
Inverse Weibull  (–1,0,1) 
Table 3: Example of distributions.
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.
This paper was founded by the Deanship of Scientific Research (DSR), King Abdulaziz University,Jeddah, under grant No. (65363D1431).The authors, therefore, acknowledge with thanks DSR technical and financial support.
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