Abstract

The Inverse Weibull distribution has been applied to a wide range of situations including applications in medicine, reliability, and ecology. It can also be used to describe the degradation phenomenon of mechanical components. We introduce Inverse Generalized Weibull and Generalized Inverse Generalized Weibull (GIGW) distributions. GIGW distribution is a generalization of several distributions in literature. The mathematical properties of this distribution have been studied and the mixture model of two Generalized Inverse Generalized Weibull distributions is investigated. Estimates of parameters using method of maximum likelihood have been computed through simulations for complete and censored data.

1. Introduction

The Generalized Weibull (GW) distribution possessing bathtub failure rate was introduced by Mudholkar and Srivastava [1]. Mudholkar et al. [2] and Mudholkar and Hutson [3] applied GW distribution for analysis of data relating to bus motor failure, head and neck cancer, and flood.

Inverse distributions, namely, Inverse Gamma, Inverse Generalized Gamma, Inverse Weibull, and Inverse Rayleigh, have been studied in literature [4–8]. Keller et al. [9] studied shapes of density and failure rate function for the basic inverse model and Drapella [6] worked on Inverse Weibull (IW) distribution. Drapella [6] and Mudholkar and Kolia [10] suggested the names complementary Weibull and reciprocal Weibull. These distributions have applications in reliability engineering and medical sciences and are used for modelling infant mortality, wear-out periods, degradation of mechanical components [11], times of breakdown of an insulating fluid subject to the action of constant tension [5], and load-strength relationship for a component [12]. Aleem and Pasha [13] studied some distributional properties of IW.

In this paper, we first introduce a three-parameter continuous probability distribution on the positive real line, known as Inverse Generalized Weibull (IGW) distribution. It is the distribution of reciprocal of a variable distributed according to the Generalized Weibull distribution. It can also be called Complementary or Reciprocal Generalized Weibull distribution. Using IGW distribution, a four-parameter distribution named as Generalized Inverse Generalized Weibull (GIGW) distribution is introduced and its properties are studied. The mixture of two GIGW distributions has been investigated. Empirical estimates of parameters have been found using maximum likelihood method for complete and censored data. An application to real data set has been provided to illustrate the potentiality of the proposed models. The paper is organized as follows.

In Section 2, we introduce the Inverse Generalized Weibull (IGW) and Generalized Inverse Generalized Weibull (GIGW) distributions. Moments of GIGW distribution have been derived in Section 2.1. A mixture of two GIGW distributions and its properties have been discussed in Section 3. In Section 4, the nonlinear equations for finding the maximum likelihood estimates of the parameters have been derived for both uncensored and censored case. To check the theoretical results, simulations have been carried out in Section 5 and an application to a real data set has been provided. In case of censoring, the elements of the information matrix are included in the appendix for estimation and testing purposes.

2. The Generalized Inverse Generalized Weibull (GIGW) Distribution

The cumulative distribution function (cdf) and the probability density function (pdf) of GW distribution [1] are respectively.

For introducing GIGW distribution, we first propose Inverse Generalized Weibull (IGW) distribution with cdf and pdf written as where and .

The corresponding survival and hazard rate functions are given by respectively.

A generalization of IGW distribution is proposed as Generalized Inverse Generalized Weibull (GIGW) distribution with cdf where is the scale parameter and , , and are shape parameters.

This gives the density function as

By substituting it can be easily shown that (7) is a density function.

The survival and hazard rate functions of GIGW distribution have the forms respectively.

The plots of density function are displayed in Figure 1 for different values of the parameters. In this figure, the plots for correspond to IGW distribution whereas the other two plots for and >1 depict the pdfs of GIGW distribution.

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Figure 1 

Plots for GIGW density function.

It is depicted by Figure 1 that IGW () and GIGW are a positively skewed distribution.

The hazard rate functions of IGW () and GIGW distributions are plotted in Figure 2 for different combinations of parametric values.

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Figure 2 

Plots for GIGW hazard function.

Figure 2 indicates that(i)for ,(a)hazard rate function for IGW distribution lies in between the hazard rate functions of GIGW distribution when is less than or greater than 1;(b)for , hazard rate of GIGW distribution decreases at a faster rate than that of IGW distribution;(ii)for ,(a)hazard rate function for IGW distribution lies below the hazard rate functions of GIGW distribution when is less than or greater than 1;(b)hazard rate of GIGW distribution decreases at a faster rate than that of IGW distribution.

If is a random variable (r.v.) with pdf given by (7), we use the notation .

Special Cases. (1) When , Generalized Inverse Generalized Weibull (GIGW) reduces to Generalized Inverse Weibull (GIW) [14] with shape parameters and and scale parameter .

(2) Putting , we get IGW as a particular case of GIGW.

(3) and give Inverse Weibull (IW) distribution with parameters and .

Remark 1. Observations from distribution can be simulated using the transformation where follows Uniform distribution on (0,1).

We simulate a data set from GIGW distribution with parameters , , , and and plot the exact and empirical cdfs of GIGW in Figure 3 to check the correctness of procedure for simulations.

Figure 3 

Plots for comparison of exact and empirical cdfs of GIGW.

The closeness of two cdfs implies the validity of the procedure for generating observations from GIGW.

2.1. Moments of GIGW Distribution

Moments of a distribution are important characteristics of any distribution. These are helpful in finding average, dispersion, coefficients of skewness, and kurtosis.

For , the th order moment around zero is written as Taking , using binomial expansion with .

Hence gives the general expression for th moment of GIGW distribution.

Remark 2. For in (14), we get the th moment of GIW distribution given by de Gusmão et al. [14].
The moment generating function (mgf) of can be written as
The cumulant generating function of is given by
Using (14), the mean and variance of GIGW can be written as

3. Mixture of Two GIGW (MGIGW) Distributions and Properties

Everitt and Hand [15], Maclachlan and Krishnan [16], Maclachlan and Peel [17], Al-Hussaini and Sultan [18], Jiang et al. [19], and Sultan et al. [20] discussed various mixture distributions and found them useful in complex problems. The density of mixture of two GIGW distributions called Mixed Generalized Inverse Generalized Weibull (MGIGW) distribution is given by where for and 2 (i);(ii) corresponds to th component of the mixture and is given by (7);(iii) and .

The density function of MGIGW distribution is plotted in Figure 4 for some combinations of values of and with , , and and , , and .

Figure 4 

MGIGW density plots.

The corresponding cdf is given by where is the cdf corresponding to for and 2.

3.1. Properties of MGIGW Distribution

The corresponding survival and failure rate functions are respectively.

The th moment of MGIGW distribution can be written as

Remark 3. For , the results are the same as those derived by de Gusmão et al. [14] for GIW.

3.2. Identifiability

Let be a family of cdfs with transforms and for (the domain of definition) such that the mapping is linear and one-to-one. If there exists a total ordering of such that implies(i),(ii)the existence of some ( being independent of ) such that ,then the class of all finite mixtures of is identifiable [21].

Using Chandra’s approach [22] and the results by Sultan et al. [20], we prove the following proposition.

Proposition 4. The class of all finite mixing distributions relative to the GIGW distribution is identifiable.

Proof. If follows GIGW distribution, the th moment is
Using the cdf of , we have
Let , , and .
From (24), we have
Hence
This proves the identifiability of MGIGW distribution.

In the next section, the maximum likelihood estimation (MLE) of the parameters of GIGW distribution is explored.

4. Estimation for GIGW Distribution

Let and be the parameter vector.

The equations for finding maximum likelihood estimates (MLEs) of parameters of GIGW distribution have been derived in the following subsections for uncensored and censored case.

4.1. Uncensored Case

A complete data set without any missing observation is termed as uncensored.

The log likelihood for a single observation of can be written as

The unit score vector is given by .

Writing we have

The MLE of can be obtained by solving the nonlinear equations using numerical methods.

4.2. Censored Case

In survival analysis and reliability studies, we are generally encountered with censored data. Censoring occurs when the information about the survival time of the observations under study is incomplete. It represents a particular type of missing data [23]. Let and be independent random variables where is the lifetime of th individual and is the censoring time and for . The distribution of does not depend on any of the unknown parameters of where each follows GIGW distribution with .

Let be the number of failures and let and denote the sets of uncensored and censored observations, respectively. The likelihood function for the censored case can be written as where and are density and survival functions of GIGW distribution, respectively, and are given in (7) and (9).

Hence the censored log likelihood for GIGW distribution can be written as

For uncensored data, while writing the log likelihood function, the values of are from the complete sample but in case of censored data set, the log likelihood is split into two parts, one corresponding to censored observations and the other corresponding to those observations that are not censored.

Writing the elements of the score vector are given by

The observed information matrix for interval estimation and hypothesis testing for parameters in is given by

Note that is observed and not the expected information matrix because for writing the elements of the expected information matrix, the expressions turn out to be very complicated. The expressions for the elements of are given in appendix for censored case. The elements of for uncensored case can be analogously written using (27).

Under certain regularity conditions (fulfilled for parameters in the interior of the parameter space but not on the boundary), is the expected information matrix used for construction of tests of hypotheses and appropriate confidence regions for the parameters and can be replaced by the observed information matrix . The asymptotic normality is useful for testing goodness of fit of GIGW distribution versus some of its submodels.

5. Simulations and Application

5.1. Estimation Based on Simulations

For checking the theoretical results given in Section 4, we simulate data by generating observations from GIGW distribution with , , , and . The assumed values for the parameters are arbitrary. The values of these parameters are estimated using quasi-Newton method in . The sample sizes considered are , 100, 200, and 300 with number of repetitions as 10,000, a standard followed in literature for carrying out simulations. The estimates of parameters with corresponding root mean square errors (RMSEs) are given in Table 1.

It is observed from Table 1 that estimates get closer to assumed values of parameters as sample size increases. It has also been checked that if we change the assumed values of parameters, the results will not vary. We consider another set of parametric values as , , , and and the results are reported in Table 2.

It is observed from the values in Table 2 that, even on assuming different parametric values, the conclusions are similar to those drawn from Table 1.

5.2. Real Data Illustration

To check the superiority of GIGW distribution over GIW and IGW distributions, we use the censored data discussed in Sickle-Santanello et al. [24] and given in Klein and Moeschberger [23]. The data consist of death times (in weeks) of patients with cancer of tongue with aneuploid DNA profile. The observations are 1, 3, 3, 4, 10, 13, 13, 16, 16, 24, 26, 27, 28, 30, 30, 32, 41, 51, , 65, 67, 70, 72, 73, , 77, , , , , , , , 91, 93, , 96, , 100, , 104, , , , , , , 157, 167, , , and , where asterisks denote censored observations.