Extended Poisson Inverse Weibull Distribution : Theoretical and Computational Aspects

Extended Poisson Inverse Weibull Distribution: Theoretical and Computational Aspects Saeed Al-mualim1,2 1 Management Information System Department, Taibah University, Saudi Arabia 2 Department of Statistics, Sana’a University, Yemen Correspondence: Saeed Al-mualim, Management Information System Department, Taibah University, Saudi Arabia; Department of Statistics, Sana’a University, Yemen. E-mail: s almoalim@hotmail.com


Introduction
A certain continuous random variable (rv) Z is said to have the Inverse Weibull distribution (IWD) with scale parameter δb β −1 > 0 and and shape parameter β > 0 if its probability density function (PDF) is given by ) .
In this work we shall propose a new version (generalization) of the the Topp Leone Inverse Weibull distribution (TL-IWD) via using the discrete zero truncated Poisson distribution (ZTPD).The PDF and CDF of TL-IWD are given by f [θ,b,β,δ] and respectively, where β, b, and θ > 0 are shape parameters.The probability mass function (PMF) of ZTPD is given by If we have a system has N subsystems functioning independently at a given time t where N has ZTPD with expected value E(N) and variance Var(N|α) are, respectively, given by Assume that the failure time of each subsystem has the TLEIW model defined by PDF and CDF in (1) and (2).Let Y i denote the failure time of the i th subsystem and let then the conditional CDF (CCDF) of X given N is then, the marginal CDF (MCDF) of X is will be equation ( 5) is called the CDF of the PTLIWD, where Φ=α, θ, b, β, δ.The corresponding PDF of (5) reduces to Some useful extensions of the IWD can be found in the statistical literature and cited such as the beta IW distribution (B-IWD) (see Barreto-Souza et al. (2011)) , the Marshall-Olkin IW distribution (MO-IWD) (see Krishna et.al. (2013)), the transmuted IW distribution (T-IWD) (see Mahmoud and Mandouh (2013)), the transmuted exponentiated IW distribution (TE-IWD) (see Elbatal et al. (2014)), the transmuted Marshall-Olkin IW distribution (TMO-IWD) (see Afify et al. (2015)), the transmuted exponentiated generated IW distribution (TEG-IWD) (see Yousof et al. (2015)), the beta expo- Expanding A i via the power series, we have inserting ( 7) into (6), we get Consider the power series Using (9) we get where and is the IWD with scale parameter δ {[(1 + τ) θ + w] b} β −1 and shape parameter β and is the IWD with scale parameter δ {[(1 + τ) θ + w + 1] b} β −1 and shape parameter β.Via integrating (10), we get ] , Vol. 8, No. 2; 2019 and shape parameter β.
The PTLIWD is a suitable for fitting the unimodal and right skewed data sets (see Figure 1(left panel)).The PTLIWD provide adequate fits as compared to other IWDs in both applications with smallest values of AI C and BI C .

Moments
The r th non central moment of X is given by then we have where Setting r = 1, 2, 3 and 4 in (11), we have which is the mean of X, and

Incomplete Moments (IM)
The Then where

Probability Weighted Moments (PWMs)
The (s, r)th PWM of a rv X following the PTLIWD is derived by Using ( 5) and ( 6), we have ] , where ) .
The (s, r)th PWM will be

Residual Life and Reversed Residual Life Functions (MRL) & (MRRL)
The n th MRL given as Then th MRL of X can derived as where The n th MRRL given as X≤t,n=1,2,...

]
we obtain Then, the n th MRRL can written as where q τ,w = a τ,w (−1) r t n−r and q ⋆ τ,w = a ⋆ τ,w (−1) r t n−r .

Order Statistics
The PDF of i th order statistic, say X i:n , can be where B(i, n − i + 1) is the beta function.using ( 5), ( 6) and ( 13) we get where and The PDF of X i:n will be The moments of X i:n will be

Maximum Likelihood Estimation
The log-likelihood function The above ℓ(Φ) can be maximized numerically by using R (optim).The components of the score vector where

Conclusions
A new extension of the Poisson Inverse Weibull distribution is derived and studied in details.Number of structural mathematical properties are derived.We used the well-known maximum likelihood method for estimating the model parameters.The new model is applied for modeling some real data sets to prove its importance and flexibility empirically.We also conclude that: 1-The PTLIWD provide sufficient fits as compared to other IW extensions with the smallest values of AI C and BI C in both applications.From application 1, the PTLIWD is much better than the MO-IWD, B-IWD, Kum-IWD, MO-IRD, MO-IED, E-IWD and IWD.

Figure 2 .
Figure 2. TTT plots for 1 st data set

Figure 3 .
Figure 3. Fitted PDF, PP Plot, and Kaplan-Meier survival plot and estimated HRF for the 1 st data

Figure 5 .
Figure 5. Fitted PDF, PP Plot, and Kaplan-Meier survival plot and estimated HRF for the 2 nd data

Table 1 .
The AI C and BI C statistics for the survival times for Guinea pigs

Table 3 .
The AI C and BI C statistics for the repair times data

Table 4 .
MLEs and their standard errors (in parentheses) for the repair times