A New Three-parameter Xgamma Fréchet Distribution with Different Methods of Estimation and Applications

In this article an attempt is made to introduce a new extension of the Fréchet model called the Xgamma Fréchet model. Some of its properties are derived. The estimation of the parameters via different estimation methods are discussed. The performances of the proposed estimation methods are investigated through simulations as well as real life data sets. The potentiality of the proposed model is established through modelling of two real life data sets. The results have shown clear preference for the proposed model compared to several know competing ones.


Introduction and genesis
The probability density function (PDF) and cumulative distribution function (CDF) of Fréchet (Fr) distribution are given, respectively, by The PDF corresponding to (3) For = 1, the XG-Fr reduces to the two-parameter XG-Fr distribution ). The XG-Fr density in (4) can be expressed as , (5) and the power series raised to a positive integer (Gradshteyn and Ryzhik, 2002, Section 0.314) where the coefficients ( , ) (for = 1,2, … ) can be easily determined from the recurrence equation The coefficient ( , ) can be calculated from ( ,0) , … , ( , −1) and hence from the quantities 0 , 1 , … , . For ) .
Next, the quantity ( ; , ) follows using (6)  and shape parameter . So, the density of is a linear combination of Fr densities. The CDF of follows by integrating (8) as where ( ) is the Fr density with scale parameter 1 and shape parameter . Equations (8) and (9) are the main results of this section. We provide some plots of the PDF and HRF of the XG-Fr model to show its flexibility. The HRF plots of the XG-Fr distribution can be "upside down" or "monotonically increasing". For more details and useful extension of the Fr model see Harlow (2002) Then, where ( ,1+ ) = (1 + ) and  (10) gives the mean of .

Incomplete moments
The ℎ incomplete moment of is defined by We can write from (8) Then,

Moment generating function
The moment generating function (MGF) of , say ( ) = ( ) , is obtained from (8) as Then, where ( ) is the generating function of . The last two integrals can be computed numerically for most parent distributions. Consider the Wright generalized hypergeometric function (see Wright (1935)) defined by Combining expressions (8) and (12), we obtain the MGF of , say ( ), as .
Numerical analysis for the E(X), variance (Var (X)). skewness (Ske (X)) and kurtosis (Ku (X)) measures. In this section we explore the some commonly used descriptive measure numerically. Based on table 1, we note that:

I-
The value of ( ) of the new model decreases as and increases.

II-
The value of ( ) of the new model increases as increases.

III-
The skewness of the new model can be only positive in the interval (0.00004, 31.43).

IV-
The parameter has a very little effect on Ske( ) ) and Ku ( ).

V-
The Ku ( ) of new model can be only more than 3.

Estimation 3.1 Maximum likelihood estimation (MLE)
Here, we consider the estimation of the unknown parameters of the new family from complete samples by maximum likelihood. Let 1 , 2 , ⋯ , be a random sample from the XG-Fr model. Then, the log-likelihood function for is given by where The log-likelihood function in (13) can be maximized numerically by using R (optim), SAS (PROC NLMIXED) or Ox program (sub-routine MaxBFGS), among others. For interval estimation of the parameters, the elements of the 3 × 3 observed information matrix ( ) can be evaluated numerically.

Ordinary and weighted least-squares estimators
The theory of least square estimation and weighted least square estimation was proposed by Swain et al. (1988) to estimate the parameters of the Beta distribution. It is based on the minimization of the sum of the square of differences of theoretical cumulative distribution function and empirical distribution function. Suppose , , ( ) denotes the distribution function of XG-Fr distribution and if 1 < 2 < ⋯ < be the ordered random sample. The ordinary least square estimates (OLSEs) are obtained by minimizing The, least square estimators (LSE) of the parameters are obtained by simultaneously solving the following nonlinear equations: A New Three-parameter Xgamma Fréchet Distribution with Different Methods of Estimation and Applications 296 The WLSE of the parameters are obtained by solving the following non-linear equations: where, | , , ( ), | , , ( ) and | , , ( ) are the values of first derivatives of the CDF of XG-Fr distribution.

Method of Cramer-Von-Mises estimation
The Cramer-Von-Mises estimation (CVME) method of the parameters is based on the theory of minimum distance estimation. It was proposed and justified that the bias of the estimator is smaller than the other minimum distance estimators. Thus, The Crammer-Von Mises estimates of the parameter , , are obtained by minimizing the following expression w.r.t. to the parameters , , respectively. .
The CVME of the parameters are obtained by solving the following non-linear equations where | , , ( ), | , , ( ) and | , , ( ) are the values of the first derivatives of the CDF of XG-Fr distribution with respect to , , respectively.

Maximum product spacing distance estimators
The maximum product of spacings distance (MPSD) method has been originally proposed by Cheng and Amin (1983) as an alternative to MLE for the estimation of parameters of continuous univariate models. Let | , , ( : ) be the uniform spacings of a random sample from the XG-Fr model where where ⋅| , , ( ⋅ ) is defined before.

Bootstrapping method
Bootstrapping method is a powerful statistical technique. It is especially useful when the sample size that we are working with is small. Under the usual circumstances, sample sizes of less than 40 cannot be dealt with by assuming a normal or a distributions. Bootstrap techniques work quite well with samples that have less than 40 elements. The reason for this is that bootstrapping involves resampling. These kinds of techniques assume nothing about the distribution of our data. Bootstrapping has become more popular as computing resources have become more readily available. This is because for bootstrapping to be practical a computer must be used. We will see how this works in the following Sections.  Kundu and Raqab, 2009) (2017) Figure 1 gives the total time test (TTT) plot (see Aarset (1987)) for the two data sets. The TTT is an important graphical approach to verify whether the data can be applied to a specific distribution or not. 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. This plot indicates that the empirical HRFs of the two data sets are "monotonically increasing".  Tables 2 and 4. Tables 3 and 5 list the MLEs and their corresponding standard errors (SEs) (in parentheses) as well as the confidence intervals (CIs) in [square brackets] of the model parameters. We proved empirically that the XG-Fr distribution provides better fits to two real data sets than other seven extended Fr distributions (see Table 3 and Table 5), that results is supported by Figure 2 and Figure 3.     Figure 2: Estimated PDF and estimated CDF for data set I. Figure 3: Estimated PDF and estimated CDF for data set II.

Comparing methods 5.1 Using real data
In this subsection, we consider the two real data sets to compare the classical methods for the purpose of comparison. We consider W* and A* statistics.  Table 6 we conclude that the ML method is the best method for modelling the data I. However, all other methods performed well.  Table 7 we conclude that the CVM method is the best method for modelling the data II. However, all other methods performed well.  Tables 8-13, we observe that all the estimates show the property of consistency, i.e., the MSEs decrease as sample size increase.

Concluding remarks
A new extension of the Fréchet model called the Xgamma Fréchet model is proposed. Some of its properties such as ordinary moments, incomplete moments and moment generating function are derived. The estimation of the parameters is carried out via different methods. The performances of the proposed estimation methods are studied through Monte Carlo simulations and real-life data sets. The potentiality of the proposed model is analyzed through modelling of two real life data sets. As a future work, we can use and apply many new goodness-of-fit statistic tests for right censored distributional validation such as the "Nikulin-Rao-Robson goodness-of-fit statistical test", the "modified Nikulin-Rao-Robson goodness-of-fit statistical test", the "Bagdonavicius-Nikulin goodness-of-fit statistical test" and the "modified Bagdonavicius-Nikulin goodness-of-fit statistical test" as performed by Ibrahim