Extended Poisson-Fr´echet Distribution: Mathematical Properties and Applications to Survival and Repair Times

In this paper, a new four parameter zero truncated Poisson Fr´echet distribution is deﬁned and studied. Various structural mathematical properties of the proposed model including ordinary moments, incomplete moments, generating functions, order statistics, residual and reversed residual life functions are investigated. The maximum likelihood method is used to estimate the model parameters. We assess the performance of the maximum likelihood method by means of a numerical simulation study. The new distribution is applied for modeling two real data sets to illustrate empirically its ﬂexibility.


Genesis, physical motivation and justification
Assume that X 1 , X 2 , ... , X n is a finite sequence of independent and identically distributed random variables (iid rvs) with common cumulative distribution function (CDF). One of the most interesting statistics is the sample maximum One is interested in the behavior of M n as the sample size n increases to infinity, then Pr {M n ≤ x} = Pr {X 1 ≤ x, ..., X n ≤ x, } = Pr {X 1 ≤ x} ...p r {X n ≤ x} = G (x) n .

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Suppose there are sequences of constants {a n > 0} and {b n } such that Then, if G (x) is a non-degenerate CDF, then it will belong to one of the three following fundamental types of classic extreme value family: 1-Gumbel (Gum) model (Type I extreme value distribution); 2-Fréchet (Fr) model (Type II extreme value distribution); 3-Weibull (W) model (Type III extreme value distribution). The extreme value theory focuses on the behavior of the block maxima or minima. The extreme value theory was firstly introduced by Fréchet (1927) then followed by Von Mises (1936), Gnedenko (1943), Von Mises (1964), Kotz and Johnson (1992), among others. The Fr distribution is one of the important distributions in extreme value theory and has many applications such as accelerated life testing, earthquakes, floods, horse racing, rainfall, queues in supermarkets, wind speeds and sea waves. For more details about the Fr distribution and its applications, see Kotz and Nadarajah (2000). Moreover, applications of this distribution in various fields are given in Harlow (2002). Recently, some extensions of the Fr distribution were considered. The exponentiated Fr by Nadarajah and Kotz (2003), beta Fr by Nadarajah and Gupta (2004), Nadarajah and Kotz (2008)  The goal of this paper is to propose a new generalization of the Topp Leone Fr (TL-Fr) distribution (Yousof et al. (2018b)) using the zero-truncated Poisson (ZTP) model. the probability density function (PDF) and CDF of TL-Fr distribution are given by and respectively, where δ > 0 is a scale parameter and β, θ > 0 is a shape parameter. The ZTP distribution is a discrete probability model whose support is the set of only the positive integers (I (+) ) with probability mass function (PMF) of N given by Suppose that a system has N subsystems functioning independently at a given time where N follows the ZTP distribution with parameter α. The expected value (E (N |α)) and variance (V(N |α)) are, respectively, given by where ∆ (α) = 1 − exp (−α)and The ZTP is known also as the positive Poisson distribution or the conditional Poisson distribution. It is the conditional probability distribution of a Poisson distributed rv, given that the value of the rv is not zero. Thus it is impossible for a ZTP rv to be zero.
Suppose that the failure time of each subsystem has the TL-Fr model defned 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 of X given N can be written as therefore, the marginal CDF of X can be expressed as equation (5) is called the CDF of the zero truncated Poisson Topp Leone Fr (ZTPTL-Fr) model. The corresponding PDF of (5) reduces to Then we provide a linear mixture for the ZTPTL-Fr density function in (6). Expanding the quantity A (x) in power series, we can write consider the power series which holds for a 1 a 2 < 1 and q > 0 real non-integer. Using the power series in (8) and after some algebra the PDF of the ZTPTL-Fr in (7) can be expressed as where the function π (1+τ )θ+κ (x; β, δ) is the Fr density with scale parameter δ [(1 + τ ) θ + κ] 1 β and shape parameter β and π 1+κ+(1+τ )θ (x; β, δ) is the Fr density with scale parameter δ [1 + κ + (1 + τ ) θ] 1 β and shape parameter β. Equation (9) reveals that the density of X can be expressed as a double linear mixture of Fr densities. So, several of its structural properties can be obtained from Equation (9) and those properties of the Fr distribution. By integrating (9), we obtain the same mixture representation where Π (1+τ )θ+κ (x; β, δ) is the CDF of the Fr model with scale parameter δ [(1 + τ ) θ + κ] 1 β and shape parameter β and Π 1+κ+(1+τ )θ (x; β, δ) is the CDF of the Fr model with scale parameter δ [1 + κ + (1 + τ ) θ] 1 β and shape parameter β. The hazard rate function (HRF) can be derived Figure 1 gives some plots of the ZTPTL-Fr PDF and HRF for some parameter values.
The justification for the practicality of the ZTPTL-Fr lifetime model is based on the wider use of the Fr model. As well as we are motivated to introduce the ZTPTL-Fr lifetime model because it exhibits a unimodal hazard rate as illustrated in Figure 1  exponentiated Fr and Fr models, so the ZTPTL-Fr model is a suitable alternative to these models for modeling survival times data as illustrated in application 1. As well as the proposed ZTPTL-Fr lifetime model is much better than the Topp Leone Generated Fr, Fr, Kumaraswamy Fr, exponentiated Fr, beta Fr, Transmuted Fr, Marshall-Olkin Fr and Mcdonald Fr models, so the ZTPTL-Fr model is a suitable alternative to these models for modeling repair times data as illustrated in application 2.
The rest of the paper is outlined as follows. In Section 2, we derive some statistical properties for the new model. Maximum likelihood estimation of the model parameters is addressed in Section 3. Simulation results are presented in Section 4. Two applications to real data sets to illustrate the importance of the new family are provided in Section 5. Finally, we offer some concluding remarks in Section 6.

Mathematical properties 2.1 Moments and incomplete moments
The r (th) ordinary moment of X is given by then we obtain where The constants c τ,κ and c τ,κ have been defined before in Section 1, and Setting r = 1 in (11), we have the mean of X (µ ). The last integration can be computed numerically for most parent distributions. The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships. The r (th) incomplete moment, say Υ r (t), of X can be expressed from (9) as where is a confluent hypergeometric function, which can be evaluated by statistical software like R software.
2.2 Numerical analysis for the µ , variance (V(X)), skewness (Sk(X)) and kurtosis (Ku(X)) measures Numerical analysis for the µ , V(X), Sk(X) and Ku(X) are listed in Tables 1 and 2 for the ZTPTL-Fr model and for the Fr model respectively for some selected values of parameter α, θ, β and δ using the R software. Based on Table 1 we note that: 1-The Sk(X) of the ZTPTL-Fr model is always positive. 2-The Ku(X) of the ZTPTL-Fr model can be more than 3 or less than 3. Based on Tables 1 and 2 we note that: The skewness of the ZTPTL-Fr distribution can range in the interval (1.704, 99.032), whereas the skewness of the Fr distribution varies only in the interval (1.001, 3.53). Further, the spread for the ZTPTL-Fr kurtosis is ranging from 0.625235 to 148485.5, whereas the spread for the Fr kurtosis only varies from 1.002 to 98.8 with the above parameter values. Table 1:

Moment generating function
The moment generating function (MGF) M X (t) = E (exp (t X)) of X can be derived from equation (9) as Using the Wright generalized hypergeometric (WGH) function which defined as Then, we can write M (t; δ, β) as Combining (9) and the last equation, we obtain the MGF of X in terms of WGH function, say M (t), as

Probability weighted moments
The (s, r) th PWM of X following the ZTPTL-Fr is formally defined by Using equations (5) and (6), we can write then, the (s, r) th PWM of X can be expressed as

Residual life and reversed residual life functions
The n (th) moment of the residual life, say ], uniquely determine F (x). The n (th) moment of the residual life of X is given by The n (th) moment of the reversed residual life, say Then, the n (th) moment of the reversed residual life of X becomes

Order statistics and quantile spread ordering
Let X 1 , . . . , X n be a random sample (RS) from the ZTPTL-Fr distribution and let X 1:n , . . . , X n:n be the corresponding order statistics. The PDF of i (th) order statistic, say X i:n , can be written as where B(·, ·) is the beta function. Substituting (5) and (6) in equation (13) and using a power series expansion,we get The PDF of X i:n can be expressed as The q (th) ordinary moment of X i:n can be expressed as where t The quantile spread (QS) of the rv T ∼ZTPTL-Fr(α, θ, β, δ) is given by is the survival function. The QS of a any probability distribution describes how the probability mass is placed symmetrically about its median and hence it can be used to formalize concepts such as peakedness and tail weight traditionally associated with the kurtosis. So, it allows us to separate concepts of the kurtosis and peakedness for asymmetric models. Let T 1 and T 2 be two rvs following the ZTPTL-Fr model with {QS} T 1 and {QS} T 2 , respectively. Then T 1 is called smaller than T 2 in quantile spread order, denoted as then we have the following results: and 3−Let F T 1 and F T 2 be symmetric, then T 1 ≤ {QS} T 2 if, and only if

4−The order ≤ {QS} implies ordering of the mean absolute deviation around the median, say
where

Estimation
Let x 1 , . . . , x n be a RS from the ZTPTL-Fr distribution with parameters α, θ, β and δ. Let Θ =(α, θ, β, δ) be the 4 × 1 parameter vector. For determining the maximum likelihood estimators (MLEs) of Θ, we have the log-likelihood function = (Θ) = n log 2 + n log α + n log θ + n log β + nβ log δ The above log-likelihood function can be maximized numerically by using R (optim), SAS (PROC NLMIXED) or Ox program (sub-routine MaxBFGS), among others. The components of the score vector, are availabe if needed. Setting the nonlinear system of equations U α = U θ = U β = U δ = 0 and solving them simultaneously yields the MLE Θ = ( α, θ, β, δ) . To solve these equations, it is usually more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize . For interval estimation of the parameters, we obtain the 4 × 4 observed information matrix J(Θ) = { ∂ 2 ∂r ∂s } (for r, s = α, θ, β, δ), whose elements can be computed numerically. Under standard regularity conditions when n → ∞, the distribution of Θ can be approximated by a multivariate normal N 4 (0, J( Θ) −1 ) distribution to construct approximate confidence intervals for the parameters. Here, J( Θ) is the total observed information matrix evaluated at Θ. The method of the resampling bootstrap can be used for correcting the biases of the MLEs of the model parameters. Good interval estimates may also be obtained using the bootstrap percentile method.

Simulation results
We present some simulation experiments for some different sample sizes in order to assess the accuracy of the MLEs. Simulating rvs from well defined probability distributions has been discussed in the literature of computational statistics, e.g. the inverse transformation method, the rejection and acceptance sampling technique, etc. The ideal technique for simulating from the ZTPTL-Fr distribution is the inversion method, we can simulate rv X by where U is a uniform random number in (0, 1). For selected combinations of α, θ, β and δ we generate samples of sizes n = 50, 100, 200, 300, 500 and 1000 from the ZTPTL-Fr distribution. We repeat the simulations N = 1000 times,we use two combinations for the parameter values (α=2.5, θ =1.5, β = 1 and δ=2 ) in order to obtain average estimates (AEs) and mean square errors (MSEs) of the parameters. The empirical results obtained via using the well-known R package are given in Table 3. We observe that our estimates are pretty stable especially when n ≥ 300 and as n increases the MSEs and biases decreases. So, the maximum likelihood method works very well to estimate the model parameters.

Data analysis
In this section we provide applications of the ZTPTL-Fr distribution using two real data sets. In order to compare the distributions, we consider some criteria like Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) value is chosen as the best model to fit the data.
The first data set consists of 72 observations of survival times for Guinea pigs injected with different doses of tubercle bacilli: 12, 15, 43, 44, 263, 297, 341, 34148, 76, 76, 81, 83, 84, 85, 87, 58, 52, 53, 73, 75, 59, 60, 54, 4, 24, 175, 22, 234, 38, 38, 70, 70, 72, 175, 211, 32, 62, 63, 65, 65, 67, 68, 60, 32, 33, 54, 55, 56, 146, 233, 258, 57, 58, 60, 60, 61, 91, 95, 96, 98, 99, 109, 110, 121, 127, 129, 131, 143, 146, 258 and 376.These data were previously studied by Krishna  The total time test (TTT) plot is an important graphical approach to verify whether the data set can be applied to a specific model or not. Due to Aarset (1987), the empirical version of the TTT plot is given by plotting T (rn −1 ) = n j=1 y j:n −1 r j=1 y j:n + (n − r)y j:n , against rn −1 , where r = 1, . . . , n and y j:n | (j=1,...,n) are the order statistics of the sample. Aarset (1987) showed that the HRF is constant if the TTT plot is graphically presented as a straight diagonal. The HRF is increasing (or decreasing) if the TTT plot is concave (or convex). The HRF is U-shaped (bathtub) if the TTT plot is firstly convex and then concave, if not, the HRF is unimodal. The TTT plots the three real data sets is presented in Figure 2. Plots in Figure  2 indicates that the empirical HRFs of the two data sets are "upside down then bathtub" and upside down respectively. We compare the proposed ZTPTL-Fr distribution with other related  All values are obtained using the R program. Figure 3 give the fitted PDF, CDF, HRF, P-P plot and Kaplan-Meier survival plot for data I. Figure 4 give the fitted PDF, CDF, HRF, P-P plot and Kaplan-Meier survival plot for data II.  Tables 4 and 6, we conclude that the ZTPTL-Fr model provide adequate fits as compared to other Fr models in both applications with small values for AIC and BIC. In Application 1, the proposed ZTPTL-Fr model is much better than the B-Fr, E-Fr, MOKw-Fr, MOIE, KwMO-Fr, MO-Fr, Kw-Fr, MOIR and Fr models, so the ZTPTL-Fr model is a good alternative to these models. In Application 2, the proposed ZTPTL-Fr lifetime model is much better than the Fr, T-Fr, Kw-Fr, MO-Fr, TLG-Fr , E-Fr, B-Fr and Mc-Fr models, so the ZTPTL-Fr model a good alternative to these models.

Conclusions
In this paper, a new four parameter zero truncated Poisson Fr distribution called the zerotruncated Poisson Topp Leone Fr (ZTPTL-Fr) model is defined and studied. Various structural mathematical properties of the proposed extreme value model including ordinary and incomplete moments, residual and reversed residual life functions generating functions and order statistics are investigated. The maximum likelihood method is used to estimate the model parameters. The new distribution is applied for modeling two real data sets to illustrate empirically its flexibility. The ZTPTL-Fr model provide adequate fits as compared to other Fr models in both applications with small values for AIC and BIC. The proposed ZTPTL-Fr model is much better than Marshall-Olkin Kumaraswamy Fr, beta Fr, Marshall-Olkin Fr, Kumaraswamy Fr, the Kumaraswamy-Marshall-Olkin -Fr, Marshall-Olkin inverse exponential, Marshall-Olkin inverse Rayleigh, exponentiated Fr and Fr models, so the ZTPTL-Fr model is a good alternative to these models for modeling survival times data. As well as the proposed ZTPTL-Fr lifetime model is much better than the Fr, Transmuted Fr, Kumaraswamy Fr, Topp Leone Generated Fr, exponentiated Fr, beta Fr, Marshall-Olkin Fr and Mcdonald Fr models, so the ZTPTL-Fr model a good alternative to these models for modeling repair times data. We assess the performance of the maximum likelihood method by means of a numerical simulation study, We observe that our estimates are pretty stable especially when n ≥ 300 and as n increases the MSEs decreases. So, the maximum likelihood method works very well to estimate the model parameters.  Figure 3: The fitted PDF, CDF, HRF, P-P plot and Kaplan-Meier survival plot for the first data set. Figure 4: The fitted PDF, CDF, HRF, P-P plot and Kaplan-Meier survival plot for the second data set.