Extended Poisson-Log-Logistic Distribution

Abstract In this work, we introduce a new Poisson-log-logistic distribution with a physical interpretation and some applications. Some essential properties are derived. Modeling of four real data sets are provided to illustrate the wide applicability of the new model in differnt fields like finance, reliability, economy and medicine. The new compound model is better than other well-known competitive models which have at least the same number of parameters.


Introduction and Physical Motivation
The cumulative distribution function (CDF) of Burr type XII (BXII) is given as both α and β are shape parameters (for more details about the model of BXII and other related models see Tadikamalla (1980), Rodriguez (1977), Burr and Cislak (1968), Burr (1942 and1973)).
Setting β = 1 we obtain the well known one parameter log-logistic (LL) model Scale parameter can easily be added for getting other versions of the LL models as follows and G (α,ϕ) (x) = 1 − 1 , the corresponding probability density function (PDF) of G (α) (x) is given by (2) Upon following Yousof et al. (2016), we propose a new model called the Burr type X LL (BXLL) model defined by the CDF given by where ϑ > 0 is a shape parameter.Suppose that we have a system has N subsystems functioning independently at a given time where N has zero truncated Poisson (ZTP) distribution with parameter λ.The probability mass function (PMF) of N is given by Note that for ZTP random variable (r.v.), the expected value E(N|λ) and the variance Vαr(N|λ) are, respectively, given by and Assume that the failure time of each subsystem has the BXLL(ϑ, α).Let Y i denote the failure time of the i th subsystem, let Then, the conditional CDF of X | N is with the corresponding PDF as The PBXLL reduces to Poisson Rayleigh log-logistic when ϑ = 1.After some algebra the PDF of the PBXLL can be expressed as where and π [α,(1+r)] (x) is the LL density with parameters α and (1 + r).Similarly, the CDF of the PBXLL can also be expressed as where Π [α,(1+r)] (x) is the LL CDF with parameters α and (1 + r).The hazard rate function (HRF) of the new model can be calculated from ] .The PBXLL density can be left-skewed, right-skewed, unimodal and symmetric (see Figure 1(a)) while the PBXLL HRF can be bathtub or unimodal or unimodal then bathtub or decreasing or unimodal then increasing (see Figure 1(b)).
The nth ordinary moment of X, say µ ′ r , follows from (7) as Setting n = 1 in (9) gives the mean of X.The nth incomplete moment of X is defined by where are beta and incomplete beta functions from the second type, respectively.

Parameter Estimation
The log-likelihood function (ℓ n (ϕ)) for ϕ is given by where The ℓ n (Φ) in ( 10) can be numerically maximized via SASor R or Ox programs.The components of the score vector,

∂α
) are easily to be derived.

Applications
For the four data sets, we will compare the PBXLL distribution with other well-known generalizations of the LL model such as the BXII, Zografos-Balakrishnan BXII (ZBBXII), Marshall-Olkin BXII (MOBXII), the Five Parameters beta BXII (FBBXII), BBXII, Beta exponentiated BXII (BEBXII), Five Parameters Kumaraswamy BXII (FKumBXII), Topp Leone BXII (TLBXII) and KumBXII distributions (for more details about the competitive models see Altun et al. 2018 a, b andYousof et al. 2018 a, b).Data Set I called breaking stress data.This data set consists of 100 observations of breaking stress of carbon fibres (in Gba) given by Nichols and Padgett (2006).Data Set II called survival times.In this application, we work with the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, originally observed and reported by Bjerkedal (1960).Data Set III called taxes revenue data.The actual taxes revenue data (in 1000 million Egyptian pounds).Data set IV called leukaemia data.This real data set gives the survival times, in weeks, of 33 patients suffering from acute Myelogeneous Leukaemia (see the appendix).
The total time test (T.T.T.) plots for the four real data sets is presented in Figure 2.This plot indicates that the empirical HRFs of the four data sets are increasing, increasing, increasing and U-shaped (for more details about the T.T.T. see Aarset (1987)).

Figure 1 .
Figure 1.PDFs and HRFs plots for the PBXLL model 2-Bayesian Information Criterion (BI c ); 3-Hannan-Quinn Information Criterion (HQI c ); Data set I. Data set II.Data set III. Data set IV.

Table 1 .
MLEs and standard errors, confidence interval (in parentheses) with AI c , BI c , CAI c and HQI c values for the data set I AI c , BI c , CAI c , HQI c BXII

Table 2 .
MLEs and standard errors, confidence interval (in parentheses) with AI c , BI c , CAI c and HQI c values for the data set II AI c , BI c , CAI c , HQI c BXII

Table 3 .
MLEs and standard errors, confidence interval (in parentheses) with AI c , BI c , CAI c and HQI c values for the data set III