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Characterizations of Certain Doubly Truncated Distribution Based on Order Statistics | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Characterizations of Certain Doubly Truncated Distribution Based on Order Statistics

Shawky AI1* and Badr MM2

1Department of Statistics, Faculty of Science, King Abdulaziz University, Saudi Arabia

2Department of Statistics, Faculty of Science for Girls, King Abdulaziz University, Saudi Arabia

*Corresponding Author:
Shawky AI
Department of Statistics, Faculty of Science
King Abdulaziz University, P.O. Box 80203
Jeddah 21589, Saudi Arabia
Tel: +966 800 116 9528
E-mail: [email protected]

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: 10.4172/2168-9679.1000329

Copyright: © 2016 Shawky AI, et al. This is an open-access 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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Abstract

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.

Keywords

Double truncated; Order statistics; Conditional expectation; Weibull, Pareto; Rayleigh; Inverse Weibull distributions

AMS 2000 Subject Classification: 62E10; 62G30

Introduction

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 El-Din et al. [9], Abu-Youssef [10], Abd- El-Mougod [11], Shawky and Abu-Zinadah [12], Shawky and Bakoban [13] and Pushkarna et al. [14].

Let X1:n ≤ X2:n ≤ …≤ Xn: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 rth order statistics, Xr:n, 1 ≤ r ≤ n, is given by (see David (1981))

equation

equation

and the joint pdf of two order statistics Xr:n and Xs:n, 1 ≤ r ≤ s ≤ n is given by

equation

equation

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

equation

Then the doubly truncated pdf of X, say g(x), and cdf, say G(x), are given respectively by

equation

equation

where

l1=Q – P, G(ε)=0 and G(γ)=1.

The conditional density function of Xs:n=y, given that Xr:n=x is given [3] by

Also, the conditional density function of Xr:n=x, given that Xs:n=y is given by

equation

Let

equation

where ? (.) is a monotonic, continuous and differentiable function on the interval (α,ß). For abbreviation, we will denote

equation

Main Results

In this section, we characterize three general classes of distributions,

equation

equation

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.,

equation

where

equation

F(α)=0, F(ß)=1, and b ≠ 0, c ≠ 0, a are finite constants.

equation

equation

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 1-4 given below characterize the general class given by (2.1), Theorems 5-8 characterize the general class given by (2.2), while Theorems 9-12 characterize the general class given by (2.3).

Theorem 1

Referring to (1.6), (1.7) and (2.1), then

equation

where

equation

Proof

It is clear from (1.6) and (1.7) that

equation

Integrating (2.6) by parts, we get

equation

Differentiating (2.1) with respect to x, we have

equation

From (2.7) and (2.8), we obtain

equation (2.9)

Simplifying (2.9), we get (2.4). Thus, the theorem is proved.

Theorem 2

Referring to (1.6), (1.7), then (2.1) if and only if

equation

Proof

It is clear that

equation

Integrating by parts, we get

equation

Compensation for (2.8) in (2.11), we have

equation

Expand equation 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

equation

Taking the derivative, we get

which gives

equation

equation

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

equation

the relation (2.15) is before doubly truncated case.

Theorem 3

Referring to (1.5), (1.7) and (2.1), then

equation

where

equation

It is clear from (1.5) and (1.7) that

equation

Integrating by parts, we get

equation

equation

Substituting (2.7) in (2.18), we get

equation

After some simplification, we get (2.16).

Theorem 4

Referring to (1.5), (1.7), then (2.1) if and only if

equation

where N(y) is defined in (2.17).

Proof

It is clear that

equation

Integrating by parts, we get

equation

Compensation for (2.8) in (2.22), we have

equation

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

equation

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

equation

it is before doubly truncated case (Table 1).

 Name [lG(x)+ν] φ(x) (l,ν)  (a,b,c)
Weibull xp
qxp
(–1,0,θ)
(–1,0,1)
Pareto θp x-p; αε < x < γβ, ε = 0, γ ln(x)?
ln[xp]
(θp(γp–ε–p), θp ε–p) (–θ,0,p)
(–θp,0,–1)
Power function 1–θ–p xp,α ε < x < γ β, ε = 0, γ    ∞   (θpp–γp), 1–θp εp) (1,1,1)
(θp,1,1)
Rayleigh x2 (–1,0,θ)
Inverse Weibull θxp (–1,0,1)

Table 1: Example of distributions.

Theorem 5

Referring to (1.6), (1.7) and (2.2), then

equation

where V(x) is defined in (2.5).

Proof

As before in Theorem (1), differentiate (2.2) with respect to x, we have

equation

Compensation for (2.26) in (2.7), we get

equation

Simplifying (2.27), we obtain (2.25). Thus, the Theorem is proved.

Theorem 6

Referring to (1.6), (1.7), then (2.2) if and only if

equation

Where V(x) is defined in (2.5).

Proof

As before in Theorem (2), from (2.26) and (2.11), we have

equation

Therefore, we get (2.28), then (2.2) implies (2.28). To prove the sufficient condition, from (2.28) and (1.7), we obtain

equation

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

equation

it is before doubly truncated case.

Theorem 7

Referring to (1.5), (1.7) and (2.2), then

equation

where N(y) is defined in (2.17).

Proof

As before in Theorem (3), compensation for (2.26) in (2.18), we have

equation

After simplification, we get (2.31). Then (2.2) implies (2.31).

Theorem 8

Referring to (1.5), (1.7), then (2.2) if and only if

equation

where N(y) is defined in (2.17).

Proof

As before in Theorem (4), compensation for (2.26) in (2.23), we have

equation

Then, we obtain (2.35). Thus (2.2) implies (2.33). Now from (1.7) and (2.33) we get

equation

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

equation

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– θp xp,
α ε < x < γβ, ε = 0, γ → ∞

xp
(θp (εp + γp), 1–θpεp) (1,1,1)
(q–p,1,1)
Rayleigh (–1,0,θ)
Inverse Weibull (0,–1,1)

Table 2: Example of  distributions.

Theorem 9

Referring to (1.6), (1.7) and (2.3), then

equation

Proof

As given in Theorems (1) and (5), , differentiate (2.3) with respect to x, we have

equation

Compensation for (2.37) in (2.7), we get

equation

which gives (2.36). Thus, (2.3) implies (2.36).

Theorem 10

Referring to (1.6) and (1.7), then (2.3) if and only if

equation

Proof

As given previously of Theorems (2) and (6), substituting from (2.26) in (2.11), we have

equation

After some simplification, we get (2.36). Then (2.3) implies (2.36). Now from (1.7) and (2.36), we obtain

equation

equation

Taking the derivative with respect to y, we get

equation

Integrate (2.41), we obtain (2.3).

Special case

Return to (2.36), then put l=–1, v=1, we get

equation

it is before doubly truncated case.

Theorem 11

Referring to (1.5), (1.7) and (2.3), then

equation

Proof

As previously in Theorems (3) and (7), from (2.37) in (2.18), we have

equation which gives (2.42).

Theorem 12

Referring to (1.5) and (1.7), then (2.3) if and only if

equation

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

equation

The relation before doubly truncated case (Table 3).

Name [lG(x) + v] j(x) (l,v) (a,b,c)
Power distribution 1–θp xp,α ε < x < γ β, ε = 0, γ = θ Ln[1θpxp] (θp(εp+ γp), 1θ–pεp) (1,0,–1)
Weibull eθxp, αε < x < γβ, ε = 0, γ → ∞ θxp (–1,0,1)
Burr (1+θxp)γ, αε < x < γβ, ε = 0, γ→∞ ln[1 + qxp] (θγp) + θεp +1), 1+θεp) (–1,0,γ)
Inverse Weibull (–1,0,1)

Table 3: Example of distributions.

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.

Acknowledgement

This paper was founded by the Deanship of Scientific Research (DSR), King Abdulaziz University,Jeddah, under grant No. (65-363-D1431).The authors, therefore, acknowledge with thanks DSR technical and financial support.

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