A New Compound Version of the Generalized Lomax Distribution for Modeling Failure and Service Times

The main goal of this article is to introduce a new extension of the continuous Lomax distribution with a strong physical motivation. Some of its statistical properties such as moments, incomplete moments, moment generating function, quantile function, random number generation, quantile spread ordering and moment of the reversed residual life are derived. Two applications are provided to illustrate the importance and flexibility of the new model.


1.Introduction and physical motivation
In this work, we develop and study a new univariate extension of the generalized Lomax (GLx) model by compounding the Rayleigh generalized Lomax (RGLx) model with the zero truncated Poisson (ZTP) distribution. Due to Yousof et al. (2017), a random variable (r.v.) is said to have the RGLx distribution if its cumulative distribution function (CDF) is given For = 1, we have the standard two-parameters Lx distribution (see Lomax (1954)). When = = 1, we have the standard one-parameters Lx distribution. The probability mass function (PMF) of (where ~ZTP( )) is given by ( The unconditional CDF of the PRGLx PDF can be written as with corresponding PDF as The plots of the PRGLx PDF and HRF are displayed in Fig.s 1

Mathematical properties 2.1 Useful expansions
Using the power series the PDF in (6) can be written as If | 1 | < 1 and 2 > 0 is a real non-integer, then Applying (8) to (7) we have Applying the power series to Equation (9) can be written as Applying (11) to Equation (10) becomes This can be written as where * = [2(1 + 1 ) + 2 ] and , and * ( ; ) = * which is the PDF of the GLx model with power parameter * . Similarly where * ( ; , is the CDF of the GLx model with power parameter * .

Moments and incomplete moments
Let be a continuous r.v. having the GLx model with power parameter , then the ℎ ordinary and incomplete moments ( ′ ( ) and ( ) ) of the GLx r.v. (defined in this subsection) given by is the incomplete beta function. The ℎ ordinary moment of , say ′ , follows from (12) as where 1 , 2 , Setting = 1 in (14) gives the mean of as ( * , ℎ−1 + 1) | ( >1) .

3.Estimation
The log-likelihood function for is given by The above log-likelihood function can be maximized numerically by via R (optim) or SAS (PROC NLMIXED) or Ox program (sub-routine MaxBFGS).

4.Modeling data
In this section, we provide two applications to show empirically the potentiality of the PRGLx model. In order to compare the fits of the PRGLx distribution with other competing distributions, we consider the Cramér-Von-Mises ( * ) and the Anderson-Darling ( * ) statistics. We compare the fit of the new model with competitive models namely: The ZTP Burr-X Lomax (ZTPBrXLx); the BrXLx; GLx model; the gamma Lomax (GLx) model; the beta Lomax (BLX) model and Lx model. The MLEs and the corresponding standard errors (in parentheses) of the model parameters are given in Tables 1 and 3. The statistics * and * are listed in Tables 2 and 4. The Estimated PDF, P-P plot, TTT plot, estimated HRF and and and Kaplan-Meier survival plot of the two data sets and the estimated PDF of the proposed model are displayed in Fig.s 3 and 4.

4.1Modeling failure times
This data set represents the data on failure times of 84 aircraft windshield given in application 1. From table 2 and Fig. 3, the PRGLx lifetime model is much better than the ZTP Burr-X Lx, the Burr-X Lx, gamma Lx, beta Lx, GLx and Lx models so the PRGLx model is a good alternative to these models in modeling aircraft windshield data. Table 1: MLEs (standard errors in parentheses) for data set I.

4.2Modeling service times
This real data set represents the data on service times of 63 aircraft windshield given in Murthy et al. (2004). From  table 4 and Fig. 4, the PRGLx lifetime model is much better than the ZTP Burr-X Lx, the Burr-X Lx, the gamma Lx, beta Lx, GLx and Lx models so the new PRGLx model is a good alternative to these models in modeling the service times data.  PRGLx(λ,c,a,b Fig. 4: Estimated-PDF, P-P plot, TTT plot, estimated-HRF and Kaplan-Meier plot for data II.

Concluding remarks
In this paper, we introduced a new extension of the continuous Lomax distribution with a strong physical motivation. Some of its statistical properties such as moments, incomplete moments, moment generating function, quantile function, random number generation, quantile spread ordering and moment of the reversed residual life are derived. Two real data applications are provided to illustrate the importance and flexibility of the new model. The new lifetime model is much better than other competitive models such as the ZTP Burr-X Lomax, the Burr-X Lomax, the gamma Lomax, the beta Lomax, the generalized Lomax and the original Lomax model so the new model is a useful alternative to these models in modeling the failure and service times data.