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

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.

Devendra Kumar 1 and Neetu Jain 2 and Shivani Gupta 1

Academic Editor:Shein-chung Chow

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; 0

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.

1. Introduction

The probability distribution which is a member of the family of logistic distribution is the half-logistic distribution with cumulative distribution function [figure omitted; refer to PDF] and probability density function [figure omitted; refer to PDF] that are given, respectively, by [figure omitted; refer to PDF] Balakrishnan [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 [figure omitted; refer to PDF] th largest values in a sequence, Dziubdziela and Kopocinski [16] proposed the following model of [figure omitted; refer to PDF] th record values, where [figure omitted; refer to PDF] is some positive integer.

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

Let [figure omitted; refer to PDF] be the sequence of [figure omitted; refer to PDF] th upper record values. Then the [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , is as follows: [figure omitted; refer to PDF] Also the joint density function of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , as discussed by Grudzien [17] is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Kumar [18] established recurrence relations for moment generating function of [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] record values from modified Weibull distribution. Kumar [26] and Kumar and Kulshrestha [27] have established recurrence relations for moments of [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] [figure omitted; refer to PDF] Therefore, type I generalized half-logistic distribution has [figure omitted; refer to PDF] [figure omitted; refer to PDF] Here [figure omitted; refer to PDF] is the shape parameter and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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.

Figure 1: Probability density function of the type I generalized half-logistic distribution for the indicated values of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 2: Hazard rate function of the type I generalized half-logistic distribution for the indicated values of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 3: Survival function of the type I generalized half-logistic distribution for the indicated values of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

2.1. Hazard Rate Function

This function is used in analysis of time relating to the event and describes the current chance of failure for the population that has not been failed yet. Hazard rate function plays an important role in reliability analysis, survival analysis, and demography and in defining and formulating a model when dealing with lifetime data.

For the type I generalized half-logistic distribution, hazard rate function takes the form [figure omitted; refer to PDF]

2.2. Survival Function

In engineering science, it is called reliability analysis. In fact the survival function is the probability of failure by time [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] represents survival time. We use survival function to predict quantiles of the survival time. Survival function is given by [figure omitted; refer to PDF] We assume through this study, without loss of generality, that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , in which case the [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are, respectively, reduced to [figure omitted; refer to PDF] and the corresponding [figure omitted; refer to PDF] is [figure omitted; refer to PDF] If [figure omitted; refer to PDF] , then the type I generalized half-logistic distribution reduces to the half-logistic distribution.

A recurrence relation for single and product moments of upper record values from the type I generalized half-logistic distribution is obtained by making use of the following differential equation (obtained from (10) and (11)): [figure omitted; refer to PDF] Let us denote the marginal moment generating functions of [figure omitted; refer to PDF] by [figure omitted; refer to PDF] and its [figure omitted; refer to PDF] th derivative by [figure omitted; refer to PDF] . Similarly, let [figure omitted; refer to PDF] denote the joint moment generating function of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and its [figure omitted; refer to PDF] th partial derivatives by [figure omitted; refer to PDF] with respect to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively.

We will first establish the explicit expressions for marginal moment generating function of [figure omitted; refer to PDF] th upper record values [figure omitted; refer to PDF] by the following theorem.

Theorem 1.

For distribution as given in (11) and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Proof.

From (3), we have [figure omitted; refer to PDF] By making use of the transformation [figure omitted; refer to PDF] in (14), we get the desired result as (13).

Remark 2.

Setting [figure omitted; refer to PDF] in (13) we deduce the explicit expression of marginal moment generating function of upper record values from the type I generalized half-logistic distribution.

Recurrence relations for marginal moment generating function of [figure omitted; refer to PDF] th upper record values from [figure omitted; refer to PDF] (11) are derived in the following theorem.

Theorem 3.

For a positive integer [figure omitted; refer to PDF] and for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Proof.

From (3), we have [figure omitted; refer to PDF] Integrating by parts, taking [figure omitted; refer to PDF] as the part to be integrated and the rest of the integrand for differentiation, we get [figure omitted; refer to PDF] The constant of integration vanishes since the integral considered in (16) is a definite integral. On using (12), we obtain [figure omitted; refer to PDF] Differentiating both sides of (18) [figure omitted; refer to PDF] times with respect to [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] The recurrence relation in (15) is derived simply by rewriting the above equation.

By differentiating both sides of (15) with respect to [figure omitted; refer to PDF] and then setting [figure omitted; refer to PDF] , we obtain the recurrence relations for single moment of [figure omitted; refer to PDF] th upper record values from type I generalized half-logistic distribution in the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Remark 4.

Setting [figure omitted; refer to PDF] in (15), we deduce the recurrence relation for marginal moment generating function of upper record values from the type I generalized half-logistic distribution.

The mean and variances of upper record values of a type I generalized half-logistic distribution for different values of [figure omitted; refer to PDF] are calculated in Tables 1 and 2, respectively.

Table 1: Mean of upper records.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1	1.3862	0.7725	0.5451	0.4237
2	2.5507	1.4305	1.0189	0.7982
3	3.6252	2.0259	1.4483	1.1396
4	4.6601	2.5852	1.8487	1.4579
5	5.6769	3.1228	2.2298	1.7597

Table 2: Variance of upper records.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
1	0.43784	0.22676	0.14147
2	0.71407	0.37174	0.23488
3	0.94243	0.48383	0.30581
4	1.15894	0.58331	0.36593
5	1.37612	0.67789	0.42086

It appears from the results that the mean of upper record values increases with size being increased. In addition, depending on the values of [figure omitted; refer to PDF] the mean of upper record values of the distribution can be greater than its variance.

3. Relations for Joint Moment Generating Functions

The explicit expression for the joint moment generating functions of [figure omitted; refer to PDF] th upper record values is derived by the following theorem.

Theorem 5.

For the distribution as given in (10), [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Proof.

From (4), we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] By setting [figure omitted; refer to PDF] and simplifying the resulting expression we get [figure omitted; refer to PDF] Substituting the value of [figure omitted; refer to PDF] in equation (23) and simplifying we get result given in (22).

Remark 6.

Setting [figure omitted; refer to PDF] in (22) we deduce the explicit expression for joint moment generating functions of upper record values for the type I generalized half-logistic distribution.

Recurrence relations for joint moment generating functions of [figure omitted; refer to PDF] th upper record values [figure omitted; refer to PDF] (6) can be derived in the following theorem.

Theorem 7.

For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Proof.

From (4) for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Integrating [figure omitted; refer to PDF] by parts, taking [figure omitted; refer to PDF] for integration and the rest of the integrand for differentiation, and substituting the resulting expression in (27), we get [figure omitted; refer to PDF] The constant of integration vanishes since the integral in [figure omitted; refer to PDF] is a definite integral. On using the relation (10), we obtain [figure omitted; refer to PDF] Differentiating both sides of (30) [figure omitted; refer to PDF] times with respect to [figure omitted; refer to PDF] and then [figure omitted; refer to PDF] times with respect to [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] and hence the result is given in (26).

By differentiating both sides of (30) with respect to [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and then setting [figure omitted; refer to PDF] , we obtain the recurrence relations for product moments of [figure omitted; refer to PDF] th upper record values from type I generalized half-logistic distribution in the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Remark 8.

Setting [figure omitted; refer to PDF] in (26), we deduce the recurrence relation for joint moment generating functions of upper record values for the type I generalized half-logistic distribution.

4. Characterizations

This section contains characterizations of type I generalized half-logistic distribution based on recurrence relation of marginal moment generating functions of [figure omitted; refer to PDF] th upper record values and conditional expectation of upper record values.

Let [figure omitted; refer to PDF] stand for the space of all integrable functions on [figure omitted; refer to PDF] . A sequence [figure omitted; refer to PDF] is called complete on [figure omitted; refer to PDF] if, for all functions [figure omitted; refer to PDF] , the condition [figure omitted; refer to PDF] implies [figure omitted; refer to PDF] a.e. on [figure omitted; refer to PDF] . We start with the following result of Lin [29].

Proposition 9.

Let [figure omitted; refer to PDF] be any fixed nonnegative integer, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] an absolutely continuous function with [figure omitted; refer to PDF] a.e. on [figure omitted; refer to PDF] . Then the sequence of functions [figure omitted; refer to PDF] is complete in [figure omitted; refer to PDF] iff [figure omitted; refer to PDF] is strictly monotone on [figure omitted; refer to PDF] .

Using the above proposition we get a stronger version of Theorem 1.

Theorem 10.

Let [figure omitted; refer to PDF] be a nonnegative random variable having an absolutely continuous distribution function [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] : [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF]

Proof.

The necessary part follows immediately from (15). On the other hand if the recurrence relation in (35) is satisfied, then on using (3), we have [figure omitted; refer to PDF] Integrating the first integral on the right hand side of (37), by parts, we get [figure omitted; refer to PDF] It now follows from Proposition 9 that [figure omitted; refer to PDF] which proves that [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] be a sequence of [figure omitted; refer to PDF] continuous random variables with [figure omitted; refer to PDF] [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be the [figure omitted; refer to PDF] th upper record value; then the conditional [figure omitted; refer to PDF] of [figure omitted; refer to PDF] given [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , in view of (3) and (4), for [figure omitted; refer to PDF] is [figure omitted; refer to PDF]

Theorem 11.

Let [figure omitted; refer to PDF] be an absolutely continuous [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] on the support [figure omitted; refer to PDF] ; then, for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF]

Proof.

From (41), we have [figure omitted; refer to PDF] By setting [figure omitted; refer to PDF] from (4) in (44), we obtain [figure omitted; refer to PDF] Simplifying the above expression, we derive the relation given in (42).

To prove sufficient part, we have from (41) and (42) that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Differentiating both sides of (46) with respect to [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] or [figure omitted; refer to PDF] which proves that [figure omitted; refer to PDF]

5. Estimation

In this section, we obtain the maximum likelihood estimators of the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] of type I generalized half-logistic distribution by using the method of least squares based on the upper record values.

Let [figure omitted; refer to PDF] be a sequence of [figure omitted; refer to PDF] random variables with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] on positive support. Let [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . The observation [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , is an upper record value of this sequence, if it is greater than all preceding observations; that is, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Suppose we observe [figure omitted; refer to PDF] upper record values [figure omitted; refer to PDF] from a sequence of [figure omitted; refer to PDF] random variables from type I generalized half-logistic distribution with [figure omitted; refer to PDF] (3). The likelihood function based on the random sample of size [figure omitted; refer to PDF] is obtained from [figure omitted; refer to PDF] By using (7), (51) can be rewritten as [figure omitted; refer to PDF]

5.1. Maximum Likelihood Estimation

The log-likelihood function for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF] By using the method of least squares, the normal equations are [figure omitted; refer to PDF] The maximum likelihood estimators [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] of the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively, are obtained from the above nonlinear equations and can be computed by using numerical methods.

5.2. Asymptotic Confidence Interval Estimation

In this section, we obtain the asymptotic confidence intervals for the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] of type I generalized half-logistic distribution.

For obtaining confidence intervals, we define Fisher information matrix as [figure omitted; refer to PDF] where the elements are given in the Appendix. The expectations in the Fisher information matrix can be obtained numerically. Let [figure omitted; refer to PDF] be the maximum likelihood estimate of [figure omitted; refer to PDF] . Now assuming that the usual regularity conditions hold true and that the parameters are in the interior of the parameter space, but not on the boundary, we get [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the expected Fisher information matrix. The asymptotic behaviour is still valid if [figure omitted; refer to PDF] is replaced by the observed information matrix evaluated at [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] . Now the multivariate normal distribution [figure omitted; refer to PDF] , where the mean vector [figure omitted; refer to PDF] , can be used to construct confidence intervals and confidence regions for the individual parameters.

The asymptotic [figure omitted; refer to PDF] two-sided confidence intervals for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are given by [figure omitted; refer to PDF] respectively, where [figure omitted; refer to PDF] is the upper [figure omitted; refer to PDF] th percentile of a standard normal distribution.

6. Simulation Study

In this section, we are carrying out simulation procedure for computing maximum likelihood estimates of the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] of the type I generalized half-logistic distribution based on upper record values for different sample sizes ( [figure omitted; refer to PDF] ). We proceed by using 1000 iterations for the true values of the parameters: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Table 3 provides the MLEs of the three parameters and their respective mean squared errors (MSEs) in the parentheses associated with them, respectively. From the results obtained, we infer that the MLE of parameter [figure omitted; refer to PDF] is 0.5 irrespective of sample sizes and hence the MSE is 0.25. Also we see that as the sample size increases, the estimates of the other two parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] tend to be closer to their true parameter values and their MSEs decrease as the sample size increases which quantifies the consistency of the estimation procedure.

Table 3: Simulation results: MLEs of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with their respective MSEs.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
5	0.5(0.25)	1.787478(2.094613)	0.566887(0.418088)
7	0.5(0.25)	1.328763(1.562644)	0.689088(0.273315)
10	0.5(0.25)	1.021207(0.510523)	0.815709(0.255344)
15	0.5(0.25)	0.870377(0.496804)	0.960903(0.212186)

7. Real Data Analysis

To illustrate the results of this paper, we analyze one real data set. Consider the following data which represent failure times, in minutes, for a specific type of electrical insulation in an experiment in which the insulation was subjected to a continuously increasing voltage stress (Balakrishnan and Puthenpura [5], Lawless [30]): [figure omitted; refer to PDF] From this data set, we extract [figure omitted; refer to PDF] upper record values 21.8, 70.7, 138.6, and 151.9. By using the method of maximum likelihood described in Section 5, we compute the maximum likelihood estimates as well as 95% confidence intervals for the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The maximum likelihood estimates of the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are 0.1295051, 11.67251, and 4.439656, respectively. The 95% confidence intervals for the parameters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively.

8. Concluding Remarks

(i) In this study, some new explicit expressions and recurrence relations for marginal and joint moment generating functions of [figure omitted; refer to PDF] th upper record values from the type I generalized half-logistic distribution have been established. Further, characterization of this distribution has also been obtained on using the conditional expectation of record values. Finally, we obtain the maximum likelihood estimators of upper record values and their confidence intervals.

(ii) The recurrence relations for moments of ordered random variables are important because they reduce the amount of direct computations for moments, evaluate the higher order moments, and can be used to characterize distributions.

(iii): The recurrence relations of higher order joint moments enable us to derive single, product, triple, and quadruple moments which can be used in Edgeworth approximate inference.

(iv) In this paper, we computed moments, maximum likelihood estimates of the parameters of upper record values, and their confidence intervals to infer the main characteristics of the type I generalized half-logistic distribution.

Acknowledgment

The authors are grateful to the anonymous reviewers for their valuable comments and suggestions in improving the paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Appendix

Appendix

Consider the following: [figure omitted; refer to PDF]