1. Introduction

Before a specific distribution model is applied to fit the real data, it is crucial to check whether the given distribution satisfies the requirements by its characterization. Recently, a developing concentration has been focused on characterization of probability distributions. Practically, the characterization results are applied widely in insurance, economic, quality control and queuing theory. It has attracted the attention of many researchers in many fields by different methods. For example, Nanda [10] studied the characterization of exponential distribution and Rayleigh distribution through failure rate and mean residual life functions. Afify, et al. [1] used the s ^th conditional expectation of order statistics in terms of their failure rate to characterize the exponential and power function distributions. Kilany [8] introduced a new theorem for characterization of the continuous distribution based on truncated moments of order statistics and employ it to characterize Lindley distribution. Moreover, a characterization of Lindley distribution using the truncated moments of order statistics is presented. Ahsanullah & Hamedani [3] characterized beta of the first kind and power function distribution using order statistics. Many other papers in the literature dealing with characterization, see for instance, Huang & Su [7], Su & Huang [11] and Ahsanullah et al. ([2], [4], [5]).

This paper introduces the characterization of Benktander distribution type II by truncated moments and truncated moments of order statistics. The organization of the paper is; In Section 2, Benktander type II distribution and its main statistical properties are summarized. Section 3 includes the characterization of Benktander type II by left and right truncated moments. Moreover, in Section 4, the characterization through truncated moments of order statistics of Benktander type II is derived. Finally, a simulation study is given to enable the practitioner to identify the present distribution in Section 5.

2. Benktander Type II Distribution

One of the vital size distributions is called Benktander distribution. Starting from the observation that empirical mean excess functions point to distributions that are intermediate between Pareto and exponential distributions. In 1970, Gunnar Benktander discussed two types of distributions called Benktander Type I & Type II distributions (see Kleiber & Kotz [9]). These distributions used to model heavy tailed losses commonly found in non-life and causality actuarial science using various forms of mean excess functions. The first type of Benktander is close to lognormal distribution and Benktander of the second type asymptotically resembles the mean excess function of Weibull distribution. Benktander type II (second type) is sometimes called Benktander Weibull distribution.

The main statistical functions concerning Benktander Type II distribution are defined as follows

Ÿ The probability density function (p.d.f) of Benktander Type II distribution is defined by

(1)

Ÿ The cumulative distribution function (c.d.f)

(2)

Ÿ The failure rate function

(3)

Ÿ The reversed failure rate function

(4)

3. Characterization of Benktander Type II Distribution by Truncated Moments

Henceforward, let X be an absolutely continuous random variable with c.d.f. and p.d.f . Assume and Define the failure rate function

and the reversed failure rate function

3.1. Characterization of Benktander Type II by Left Truncated Moments

The following Lemma is needed to perform the characterization procedure by left truncated moments. See, Ahsanullah et al. [5].

Lemma 1.

Let be a continuous function and If

where is a differentiable function in then

where is a normalization constant.

Theorem 1.

A random variable, X, has Benktander Type II distribution with parameters

if and only if

where, is the hazard function of Benktander Type II distribution.

Proof.

Necessity: Suppose X has Benktander Type II distribution with parameters

Using Equations (1) and (3)

Hence,

Sufficiency: Let and

is a differentiable function in

After some simplifications,

Now for all

By using Lemma 1, for all we get

where is the normalizing constant, Then, the proof is completed.

3.2. Characterization of Benktander Type II by Right Truncated Moments

The following Lemma is needed to obtain the characterization of Benktander Type II by right truncated moments. See, Ahsanullah et al. [5].

Lemma 2.

Let be a continuous function in and

If

where, is a differentiable function in then

where, is a normalization constant.

Theorem 2.

A random variable, X, has Benktander Type II distribution with parameters

if and only if

where is the Benktander Type II reversed hazard function.

Proof.

Necessity. Suppose X has a Benktander Type II distribution with parameters Then by using Equation (1) and (4)

Sufficiency. Let and

is a differentiable function in and

After some simplifications,

For all

Then by using Lemma 2, the following result can be obtained

where is a normalization constant, This complete the proof.

4. Characterization of Benktander Type II Distribution by Truncated Moments of Order Statistics

Suppose be a random sample of size n from a continuous distribution with c.d.f. p.d.f. and survival function Let be the corresponding order statistics. Then the p.d.f. of and the joint p.d.f. of and are given, respectively, as follows (Arnold et al. [6])

(5)

(6)

From Equations (5) and (6) the conditional p.d.f. of given is given by

(7)

In this Section, Benktander type II distribution is to be characterized through truncated moments of order statistics given by

(8)

4.1. Characterization Results

The following theorem is presented for characterization of Benktander type II distribution based on the r^th-truncated moments of the order statistics.

Theorem 3.

Suppose 𝑋 be a continuous random variable with distribution function and survival function Let denote the order statistics of a random sample of size n from Then X has the Benktander type II distribution with parameters if and only if

(9)

Proof.

Necessity. Suppose X has Benktander Type II distribution with parameters

Using Equations (2), (7) and (8), the following results are abtianed

(10)

where,

Substituting in integrations and by and using the definition of incomplete gamma function,

(11)

and

(12)

Substituting from Equations (11) and (12) into Equation (10), the result is

Sufficiency: Let be the right hand side of Equation (9), i.e.

where Then, Equation (9) can be rewritten as follows

(13)

Differentiating both sides of Equation (13) with respect to x, then

(14)

where is the survival function.

Dividing both sides of Equation (14) by then

Or equivalently,

(15)

From the fact that then and will be

and

Hence, Equation (15) becomes

Now, for integration both sides of the above Equation with respect to x gives

Hence,

which is the survival function of the Benktander type II distribution. This completes proof of the theorem.

4.2. Simulation Study

The purpose of the simulation study is to give a validation procedure for simple and easily checks using the simulated data. This enables the experimenter to test whether the available data follows a specified distribution or not, instead of going to sophisticated analysis of hypothesis test.

To validate the correctness of the mathematical results obtained for characterization of Benktander type II, a simulation study is showed by generating samples They are randomly classified into m samples, each of containing n observations. To satisfy more accurate consequence with less variability and more stability in characterization results, the study is conducted with large sample size, for example; Choosing the number of samples are and different values of the order r with various choices of the parameter estimates a and b. Table (1) shows the L.H.S and R.H.S of Equation (9) and the absolute relative difference between them. The results indicate the closeness of both sides of Equation (9) which supporting the precision of the characterization results of Benktander distribution.

Table 1. Verification of the characterization results through truncated order statistics