1. Introduction

The probability density function of a continuous random variable X that follows an half logistic distribution is

(1.1)

while its cumulative distribution function is

(1.2)

The half logistic distribution had been studied by many authors. Balakrishnan (1985) studied ordered statistics of the half logistic distribution. Balakrishnan and Puthenpura (1986) obtained best linear unbiased estimator of location and scale parameters of the distribution through linear functions of order statistics while Balakrishnan and Wong (1991) obtained approximate maximum likelihood estimates for the location and scale parameters of the distribution with Type II Right-Censoring. Olapade (2003) stated and proved some theorems that characterized the half logistic distribution. Olapade (2009) obtained a generalized form of the distribution with probability density function (pdf)

(1.3)

and its corresponding cumulative distribution function (cdf)

(1.4)

The pdf in equation (1.3) is called type II generalized half logistic distribution while the cdf is as in equation (1.4). The moment, median, mode and the 100p-percentile point including some theorems that characterize the type II generalized half logistic distribution were presented in Olapade (2009). In this paper, we want to obtain the distributions of the range and quasi-range of a sample from the type II generalized half logistic distribution as was done for extended type I generalized logistic distribution in Olapade (2010).

2. Distribution of the Range

Given a set of random variables of size n coming from the type II generalized half logistic distribution and let be the corresponding order statistics. Let and be the cdf and pdf of the r^th order statistics respectively. David (1970) obtained the pdf of as

(2.0)

Let us define the sample range by The cdf of can be written as shown in Gupta and Shah (1965)

(2.1)

By expanding , we have

(2.2)

Substituting (1.3) and (1.4) in (2.2) we have

Since

and

Therefore,

(2.3)

Now, we consider

with the following transformations. Let , where , then

Let then with the intervals:

Let and , then we have

Since

We have

We then integrate and substitute the intervals to get,

(2.4)

Substitute into equation (2.4) to get

(2.5)

By differentiating the distribution function of the sample range in (2.5) with respect to w, we derive the pdf of as

(2.6)

3. Distribution of the Quasi-Range

The sample r^th quasi-range denoted by W, is defined as

(3.1)

where n is odd. Thus the joint pdf of is

(3.2)

Since we have

(3.3)

Integrating the expression in braces r times by parts, we have

(3.4)

Let

Substitute equations (1.3) and (1.4) in (*), we have

Since

and

so, we have

(3.5)

Now consider

Let where then

Let and then with the intervals when and when

we have

Let and then we have

Since

and

We then integrate and substitute the intervals to get,

(3.6)

Substitute (3.6) into (3.5) to get

(3.7)

Substitute (3.7) into to get

(3.8)

Substitute into (3.8) to get

(3.9)

By differentiating the distribution function of the sample quasi-range in (3.9) with respect to w, we derive the pdf of W as

(3.10)

4. Conclusions

The distribution functions of the range and the quasi-range of the type II generalized half logistic distribution have been theoretically established in this paper.