Dual Generalized Order Statistics from Gompertz-Verhulst Distribution and Characterization

The dual generalized order statistics is a unified scheme which contains the well known decreasingly ordered random variables such as (reversed) order statistics, lower record values and lower Pfeifer record values. In this article, characterization results on Gompertz-Verhulst distribution through the conditional expectation of dual generalized order statistics based on non-adjacent dual generalized order statistics are given. These relations are deduced for moments of reversed order statistics, order statistics and lower record values. Further a characterization result through the truncated moment is also derived.


Introduction
Models of ordered random variables are important and received a great attention from many researchers during the past century. These models are order statistics, sequential order statistics, record values, Pfeifer record values, progressively censored order statistics and generalized order statistics, see M. Q. Shahbaz et al. [1]. Kamps [2] has proposed generalized order statistics (gos) as a unified models of ordered random variables which produce several models as a special case. The comprehensive model for ordered random variables in ascending order is generalized order statistics. Often it happen that the sample is arranged in descending order for example the life length of an electric bulb arranged from highest to lowest. In such situations the distributional properties of variables cannot be studied by using the models of ordered random variables. The study of distributional properties of such random variables is studied by using the inverse image of gos and is popularly known as dual generalized order statistics. The dual generalized order statistics (dgos) was introduced by Burkschat et al. [3] as a unified model for descendingly ordered random variables like reverse order statistics, lower record values and lower Pfeifer record values.
The characterization of probability distributions is mainly useful in the area of goodness of fit tests, which is helpful in the construction of statistical tests. The book by Kagan et al. [4] provides the reader a good idea of the different types of properties which have been used for the characterization. Further it is a valuable source of information about the whole field of characterization of probability distributions.
Characterization of probability distributions play an important role in probability and statistics. A probability distribution can be characterized by several methods. In recent years there has been a great interest in the The dual generalized order statistics (dgos) or sometimes called lower generalized order statistics (lgos) is a combined mechanism of studying random variables arranged in descending order. The technique was introduced by Burkschat et al. [3] and is defined in the following. Let F (x) be an absolutely continuous distribution function (df ) with the probability density function (pdf ) Then X ′ (r, n,m, k), r = 1, 2, . . . ., n are called (dgos) if their joint pdf is given by, Here two cases may be considered: The probability density function of the r th − dgos is given by, The joint probability density function of the r th and s th −dgos is given by, The conditional pdf of X ′ (s, n, m, k) given X ′ (r, n, m, k) = x, 1 ≤ r ≤ s ≤ n, is where, The probability density function of the r th − dgos is given by,

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The joint probability density function of the r th and s th −dgos is given by, and a (r) The If m = 0, k =1, then X ′ (r, n, m, k) reduces to (n − r + 1) th order statistic X n−r+1:n from the sample X 1 , X 2 , · · · , X n and when m → −1 then X ′ (r, n, m, k) reduces to r th lower k record values.

Gompertz-Verhulst distribution
A continuous random variable X is said to have Gompertz-Verhulst distribution if its distribution function (df ) F (x) and probability density function (pdf ) f (x) are given, respectively, where β = ln(ρ) λ and ρ, λ and α ≥ 0. The above model was used by Gompertz and Verhulst known to compare human morality tables and to represent populations growth, see Gompertz [22] and Verhulst [23,24,25]. The pdf of Gompertz -Verhulst distribution can take different shapes. It is unimodal for α > 1 and reversed 'J' shaped for α ≤ 1. Further, the density function of Gompertz-Verhulst distribution is log-convex if α ≤ 1 and log-concave if α ≥ 1. It has an increasing hazard function if α > 1 and decreasing hazard function if α < 1 and for α = 1 the hazard function is constant. Putting ρ = 1 in (10) Gompertz -Verhulst distribution reduces to exponentiated exponential distribution. For detailed study and properties of Gompertz-Verhulst distribution one may refer to Ahsanullah et al. [26]. It appears from literature that no attention has been paid on the characterization of Gompertz -Verhulst distribution through conditional expectations of dual generalized order statistics and truncation moment. Throughout the paper, we consider the Case II and then deduce it for Case I.

Characterization Results based on Conditional Expectations
In this section, we derive the conditional expectations of dual generalized order statistics from the Gompertz -Verhulst distribution.

if and only if
where, and ξ (y) = e −λy Proof: We have from (9), After simplification we have, This proves the necessary part.
To prove the sufficiency part. Using the result Khan et al. [16], Integrating (16) on both the sides with respect to x between (β, y), the sufficiency part is proved.
And for order statistics, 805 as X ′ r:n = X n−r+1:n where X r:n is the r th order statistic. iii) Letting m → 1 in the (12), we obtain the characterization results for Gompertz-Verhulst distribution based on lower record values as, where, iv) Putting m = 0, k = 1, s = n and l = n − 1 in equation (12), we get characterization results for Gompertz-Verhulst distribution based on left truncated moment of order statistics as follows, Theorem 2.2: Let X ′ (r, n,m, k) , r = 1, 2, . . . . . . .n be the r th dual generalized order statistics from a continuous population with df F (x) and the pdf f (x). Then for 1 ≤ r ≤ s ≤ t ≤ n, if and only if where Proof: For the necessary part, in view of Theorem 2.1, we have Hence the necessary part is proved. For sufficiency part, we have Differentiating (19) with respect to x and rearranging we get, After noting that That is, or, g t|r (x) − a t|s g s|r (x) = g t|r+1 (x) − a t|s g s|r+1 (x) = . . . g t|s (x) − a t|s g s|s (x) = b t|s .
Noting that, g s|s (x) = e −λx , we have g t|s (x) = a t|s e −λx + b t|s Using the result Khan et al. [16], where, and hence the Theorem.
Remarks v) Putting s = r in (17) vii) It may be seen that when and consequently (5) will reduces to (2) and (6) to (3).

Characterization Results based on Truncated Moments
Following theorem contains characterization result for Gompertz-Verhulst distribution based on truncated moments. Theorem 3.1: Suppose an absolutely continuous (with respect to Lebesgue measure) random variable X has the df F (x) and pdf f (x) for x ≥ 0 such that f ′ (x) and E(X|X ≤ x) exist then where, Proof: We have, Integrating (21) by parts treating e −λu (1 − ρe −λu ) α−1 for integration and rest for the integrand for differentiation, we get, After multiplying and dividing by f (x) in (22), we have the result given (20)