Weibull Inverse Lomax Distribution

A four-parameter lifetime model, named as Weibull inverse Lomax (WIL) is presented and studied. Some structural properties are derived. The estimation of the model parameters is performed based on Type II censored samples. Maximum likelihood estimators along with asymptotic confidence intervals of population parameters and reliability function are constructed. The property of consistency of maximum likelihood estimators is verified on the basis of simulated samples. Further, the results are applied on two real data.


Introduction
In the past, a number of probability distributions has been found to be useful in the fields of insurance, engineering, medicine, economics and finance (among others). However, generalizing these probability distributions provided several new models that are more flexible compared to the baseline distributions. Several attempts have been made to extend new families of distributions by utilizing a number of techniques. One may refer to Eugene et al.( 2002), Jones (2004), Alzaatreh et al. (2013) and Bourguignon et al. (2014), for some more interesting results on this topic. Out of the several families of distributions, of interest to us in this study is the Weibull-generated (W-G) family of distributions presented by Bourguignon et al. (2014). The cumulative distribution function (cdf) of the W-G is defined as follows, The probability density function (pdf) of two-parameter IL distribution is specified by , , , 0, where,  and  are the scale and shape parameters respectively. The cdf corresponding to (2) is It is common practice in a life testing experiment to terminate the experiment before all the units have failed as there is limited availability of test time. The observations obtained in such situation are called censored samples. Censoring is done to save time and cost associated with testing. Type-I and Type-II censoring (TIIC) schemes are the two most popular censoring schemes used in the reliability and life testing experiments. In TIIC scheme, the experiment of n items is placed on test and the number of uncensored data r is predetermined. Instead of continuing experiments until all n items have failed, the experiment is terminated when the r th item fails. The remaining n−r items are regarded as censored data.
The main aim here is to introduce a four-parameter Weibull inverse Lomax (WIL) distribution as well as study the estimate of its population parameters depending on TIIC samples. The rest of the paper contains the following sections. In Section 2, the pdf, cdf, reliability function, hazard rate function and reversed hazard rate function of the WIL distribution are defined. Statistical properties of the WIL distribution include; moments, Rényi entropy, quantile function and distribution of order statistics are derived in Section 3. The maximum likelihood (ML) estimators of the model parameters are derived in Section 4. Simulation study is carried out in Section 5. Analysis to real data sets is presented in Section 6. The article ends with concluding remarks.

Weibull Inverse Lomax Distribution
In this section, the WIL distribution is defined depending on the W-G family. The expansion for the WIL density function is provided. The reliability characterizations of WIL distribution are provided. Taking the base-line distribution in (1) to be the cdf of IL defined in (3), hence the cdf of WIL distribution is specified by For 2, b = the pdf (5) reduces to Rayleigh inverse Lomax distribution as a new model. For 1, b = the pdf (5) reduces to exponential inverse Lomax distribution as a new model. A random variable X with density function (5) will be denoted by X~WIL ( ; ).
x 

Useful Expansion
Here, an explicit expression for the WIL pdf is provided. By using the exponential function in (5), we obtain Since, the binomial expansion, for real non-integer value of k, is given Then, by applying the binomial expansion (7) in (6), then we have Or, it can be written as follows where ( ), () b bi j gx  ++ denotes the pdf of IL distribution with parameters () b bi j  ++ and . 

Reliability Characterization
The reliability function and hazard rate function (hrf) of X are given, respectively, as follows: Plots of the pdf and hrf of the WIL distribution are displayed in Figures 1 and 2, respectively, for different values of parameters. As seen the shapes of pdf take different forms as symmetric, right skewed, and unimodel. Also, it is clear from Figure 2 that the shapes of the hrf are J-shaped, reversed J-shaped, decreasing and increasing at some selected values of parameters.

Moments and Inverse Moments
The s th moment of the WIL distribution about zero is derived by using pdf (9) as follows , ( ), ,0 0 ( ) .  Therefore, the s th moment of the WIL distribution can be obtained as follows The k th inverse moment, for the WIL distribution is derived by using pdf (9) as follows ,0 The harmonic mean of the WIL distribution can be obtained by using the first inverse moment.

Rényi Entropy
The entropy of a random variable provides an excellent tool to quantify the amount of information (or uncertainty) contained in a random observation regarding its parent distribution (population). A large value of entropy implies the greater uncertainty in the data (Rényi, 1961). The concept of entropy is important in various situations in science, engineering and economics. The Rényi entropy of a random variable X is defined by for 0,   and 1.
  The Rényi entropy of the WIL distribution is obtained by inserting the pdf (5) in (11) as follows After simplification, the Rényi entropy of WIL can be expressed as follows

Distribution of Order Statistics
A closed form expression for the pdf of the u th order statistics of the WIL distribution is derived. Let X(1)< X(2),…, <X(n) be the order statistics for a random sample X1, X2,…, Xn of size n from WIL distribution. It is known that, the pdf of the u th order statistic is defined by Inserting pdf (9) and cdf (4) in (12) and after simplification, we obtain the pdf of u th order statistic of WIL distribution as follows  (13) The pdf of the smallest and largest order statistic of WIL distribution is obtained by setting u =1 and u = n in (13), respectively, as follows (1)

Quantile Function
The quantile function of X, where X~WIL ( ; ) x  is obtained by inverting XQ = F -1 (Q) as Setting Q =0.5 in (14) gives the median of X. Simulating the WIL random variable is straightforward. If Q is a uniform variate on the unit interval (0,1), then the random variable X follows (5).

Estimation of the Model Parameters
In this section, ML estimators of the population parameters for WIL distribution are derived based on Type II censored samples. ML, reliability function estimators and approximate confidence interval (CI) for the population parameters are obtained.
Let n items, whose life time's follow the WIL distribution (5) are put on test. The test is terminated when the r th item fails for some fixed value of r. The lifetimes of these first r failed items, say X(1)< X(2) <…< X(r) are observed. The log likelihood function of the WIL distribution is obtained as follows

Simulation Study
In this section, numerical study is presented to illustrate the performance of the estimates for different parameter values and at two censoring levels. The performance of the estimates of unknown parameters and the reliability function are measured in terms of their mean square errors (MSEs), bias, standard errors (SEs), lower confidence bound (LCB), upper confidence bound (UCB), and the length of 95%, CI. The numerical procedures are defined through the following algorithm.
Step (1): 1000 random samples of size n = 50, 100 and 150 are selected; these random samples are generated from the WIL distribution.
Step (2): The number of failure items; r, based on two levels of censoring schemes, is selected at 90% and 70% censoring levels.
Step ( Step (4): For all sample sizes and for parameters values; the MSE, bias, and SE are computed.
Step (5) Tables 2 to 7. From these tables, the following observations can be detected about the performance of the estimated parameters: ▪ For all set of parameters, SE of all estimated parameters decreases as n increases.
Also, the SEs for all set of parameters at censoring level 90% are smaller than the corresponding at censoring level 70% (see Tables 2 and 3). ▪ The MSE and bias of ,, ab  and  decrease as n increases for all values of parameters. Also, the MSE and bias for the numbers of the failure items at r =0.7n for all parameters values are the larger than the corresponding at r = 0.9n (see Tables 2 and 3).  Tables 2 and 3). ▪ For the same sample size, for fixed value of 0.5, 0.7  == and as the value of a increases, the MSE and bias of a increase (see Table 2). ▪ The length of CIs for the population parameters decreases as n increases (see Tables 4 and 5). ▪ The MSE for the Set 4 has the smallest values corresponding to other sets. ▪ The ML estimates of reliability function decrease as the mission time's increase for all values of parameters. ▪ The length of CI gets shorter when n increases (see Tables 4, 5, 6 and 7). ▪ The MSE of all ML estimates decreases as the censoring level increases for all sets of parameters (see for examples Figures 3 and 4). ▪ The ML estimates decrease as the censoring level increases. Also, the Set 1 has the smallest MSE compared to the other sets. ▪ The MSE of ML estimates of Set 3 at censoring level 90% decreases as n increases (see Figure 5). Also, the MSE of ML estimates of Set 2 at censoring level 70% decreases as n increases (see Figure 6).

Applications
In this section, two real data sets are provided to illustrate the importance of the WIL distribution. The first data set refers to Hinkley (1977) Figure 7: Estimated pdf and cdf of the WIL model for the first data Furthermore, the ML estimates of the population parameters, reliability function and their SEs for both real data sets based on TIIC are listed in Table 8.

Concluding Remarks
In this paper, a new distribution with four parameters, called the Weibull inverse Lomax distribution is introduced. Some of the statistical properties of the new distribution such as quantile function, moments, order statistics and Rényi entropy are obtained. Maximum likelihood estimators of population parameters are obtained based on Type II censored samples. The approximate confidence intervals for the model parameters and reliability function are derived. Simulation study is presented to illustrate the performance of estimates for two censoring schemes. Two real data sets are presented to illustrate theortical results.