Journal of Probability and Statistics

Volume 2015, Article ID 393608, 11 pages

http://dx.doi.org/10.1155/2015/393608

## The Type I Generalized Half-Logistic Distribution Based on Upper Record Values

^{1}Department of Statistics, Amity Institute of Applied Sciences, Amity University, Noida 201 303, India^{2}Department of Statistics, PGDAV College, University of Delhi, Delhi 110007, India

Received 1 June 2015; Revised 9 July 2015; Accepted 26 July 2015

Academic Editor: Shein-chung Chow

Copyright © 2015 Devendra Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the type I generalized half-logistic distribution and derive some new explicit expressions and recurrence relations for marginal and joint moment generating functions of upper record values. Here we show the computations for the first four moments and their variances. Next we show that results for record values of this distribution can be derived from our results as special cases. We obtain the characterization result of this distribution on using the recurrence relation for single moment and conditional expectation of upper record values. We obtain the maximum likelihood estimators of upper record values and their confidence intervals. Also, we compute the maximum likelihood estimates of the parameters of upper record values and their confidence intervals. At last, we present one real case data study to emphasize the results of this paper.

#### 1. Introduction

The probability distribution which is a member of the family of logistic distribution is the half-logistic distribution with cumulative distribution function and probability density function that are given, respectively, byBalakrishnan [1] considered half-logistic probability models obtained as the models of the absolute value of the standard logistic models. Some key references about the half-logistic distribution include Balakrishnan and Aggarwala [2], Balakrishnan and Wong [3], and Balakrishnan and Chan [4]. Balakrishnan and Puthenpura [5] obtained the best linear unbiased estimators of location and scale parameters of the half-logistic distribution through linear functions of order statistics. Balakrishnan and Wong [6] obtained approximate maximum likelihood estimates for the location and scale parameters of the half-logistic distribution with type II right-censoring. Torabi and Bagheri [7] gave the estimators of parameters for the extended generalized half-logistic distribution based on complete and censored data.

Record values are found in many situations of daily life as well as in many statistical applications. Often we are interested in observing new records and in recording them, for example, Olympic records or world records in sport. Record values are also used in reliability theory. Moreover, these statistics are closely connected with the occurrence times of some corresponding nonhomogeneous Poisson process used in shock models. The statistical study of record values started with Chandler [8]; he formulated the theory of record values as a model for successive extremes in a sequence of independently and identically random variables. Feller [9] gave some examples of record values with respect to gambling problems. Resnick [10] discussed the asymptotic theory of records. Theory of record values and its distributional properties have been extensively studied in the literature; for example, Ahsanullah [11], Arnold et al. [12, 13], Nevzorov [14], and Kamps [15] can be seen for reviews on various developments in the area of records.

We will now consider the situations in which the record values (e.g., successive largest insurance claims in nonlife insurance, highest water-levels, or highest temperatures) themselves are viewed as “outliers” and hence the second or third largest values are of special interest. Insurance claims in some nonlife insurance can be used as one of the examples. Observing successive th largest values in a sequence, Dziubdziela and Kopocinski [16] proposed the following model of th record values, where is some positive integer.

Let be a sequence of identically independently distributed random variables with and . Let denote the th order statistic of a sample . For a fixed we define the sequence of th upper record times of as follows:The sequence , where , is called the sequences of th upper record values of the sequence . For convenience, we define . Note that for we have , , which are record values of [11].

Let be the sequence of th upper record values. Then the of , , is as follows:Also the joint density function of and , , , as discussed by Grudzień [17] is given bywhere Kumar [18] established recurrence relations for moment generating function of th record values from generalized logistic distribution. Recurrence relations for moment generating function of record values from Pareto, Gumble, power function, and extreme value distributions are derived by Ahsanullah and Raqab [19] and Raqab and Ahsanullah [20, 21], respectively. Recurrence relations for single and product moments of th record values from Weibull, Pareto, generalized Pareto, Burr, exponential, and Gumble distribution are derived by Pawlas and Szynal [22–24]. Sultan [25] established recurrence relations for moments of record values from modified Weibull distribution. Kumar [26] and Kumar and Kulshrestha [27] have established recurrence relations for moments of th record values from exponentiated log-logistic and generalized Pareto distributions, respectively.

In the next section, we present some explicit expressions and recurrence relations for marginal moment generating functions of th upper record values from type I generalized half-logistic distribution and results for record values are deduced as special case. The obtained relations were used to compute mean and variance, upper record values. In Section 3, we discuss joint moment generating function of th upper record values from type I generalized half-logistic distribution and results for upper record values are deduced as special case. In Section 4, we present a characterization of this distribution by using recurrence relation for single moment and conditional expectation of record values. In Section 5, we obtain maximum likelihood estimators of th upper record values from type I generalized half-logistic distribution and the confidence intervals for their estimation. Section 6 consists of simulation study based on the maximum likelihood estimates of the parameters based on upper record values of true values of parameters. In Section 7, the analysis of one real data example is provided to illustrate the performance of maximum likelihood estimates of type I generalized half-logistic distribution. Some final comments in Section 8 conclude the paper.

#### 2. Type I Generalized Half-Logistic Distribution

Olapade [28] proposed of three-parameter type I generalized half-logistic distribution and obtained some basic properties such as moments, median, and mode and also estimated its parameters by maximum likelihood approach. The three-parameter type I generalized half-logistic distribution has the Therefore, type I generalized half-logistic distribution has Here is the shape parameter and and are the location and scale parameter, respectively. Plotted are the probability density function (Figure 1), hazard rate function (Figure 2), and survival function (Figure 3) for some values of parameters.