The Fréchet Topp Leone-G Family of Distributions: Properties, Characterizations and Applications

A new family of continuous distributions which ensure model flexiblity, is introduced based on the Fréchet distribution and Topp Leone-G family. Two special sub-models of the new family are discussed. We provide some distributional properties of this family in the general setting such as the series expansions of density, moments, generating function, stress strength model, Rényi and Shannon entropies, probability weighted moments and order statistics. Certain characterizations of the proposed family are presented. The maximum likelihood estimates and the observed information matrix are obtained for the model parameters. We assess the performance of the maximum likelihood estimators by means of a graphical simulation study. The potentiality of the new class is shown via two applications to real data sets.

Authors and Affiliations

1. Department of Information Systems and Production Management, Qassim University, Buraydah, Kingdom of Saudi Arabia

   Hesham Reyad

2. Department of Measurement and Evaluation, Artvin Çoruh University, Artvin, Turkey

   Mustafa Ç. Korkmaz

3. Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt

   Ahmed Z. Afify

4. Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, USA

   G. G. Hamedani

5. Department of Applied Statistics, Cairo University, Giza, Egypt

   Soha Othman

Corresponding author

Correspondence to Hesham Reyad.

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Appendix A

Theorem 1

Let \( (\varOmega \,,\,\,F,\,\,{\rm P}) \) be a given probability space and let \( H = [a,b] \) be an interval for some \( d < b \) \( (a = - \infty ,\,\,b = \infty \,\,{\text{might}}\,\,{\text{as}}\,\,{\text{well}}\,\,{\text{be}}\,\,{\text{allowed)}} . \) Let \( X:\varOmega \to H \) be a continuous random variable with the distribution function \( F \) and let \( q_{1} \) and \( q_{2} \) be two real functions defined on \( H \) such that

is defined with some real function \( \xi . \) Assume that \( q_{1} ,\,q_{2} \in C^{1} (H),\,\,\xi \in C^{2} (H) \), \( F \) is twice continuously differentiable and strictly monotone function on the set \( H. \) Finally, assume that the equation \( \xi \,q_{1} = q_{2} \) has no real solution in the interior of \( H. \) Then \( F \) is uniquely determined by the functions \( q_{1} ,\,\,q_{2} \) and \( \xi , \) particularly

where the function \( s \) is a solution of the differential equation \( s^{\prime} = \frac{{\xi^{\prime}q_1}}{q_{1} - q_{2}}\) and \( C \) is the normalization constant, such that \( \int_{H} {dF = 1.} \)

We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence (see [13]), in particular, let us assume that there is a sequence \( \left\{ {X_{n} } \right\} \) of random variables with distribution function \( \left\{ {F_{n} } \right\} \) such that the functions \( q_{1n} ,\,\,q_{2n} \) and \( \xi_{n} \,\,(n \in N) \) satisfy the conditions of Theorem 1 and let \( q_{1n} \to q_{1} ,\,\,q_{2n} \to q_{2} \) for some continuously differentiable real functions \( q_{1} \) and \( q_{2} . \) Let, finally, \( X \) be a random variable with distribution \( F. \) Under the condition that \( q_{1n} (X) \) and \( q_{2n} (X) \) are uniformly integrable and the family \( \left\{ {F_{n} } \right\} \) is relatively compact, the sequence \( X_{n} \) converges to \( X \) in distribution if and only if \( \xi_{n} \) converges to \( \xi , \) where

This stability theorem makes sure that the convergence of distribution function is reflected by corresponding convergence of the function \( q_{1} ,\,\,q_{2} \) and \( \xi , \) respectively. It guarantees, for instance, the convergence of characterization on the Wald distribution to that of the Levy-Smirrnov distribution if \( \alpha \to \,\,\infty . \)

A further consequence of the stability property of Theorem 1 is the application of this theorem to special tasks in statistical practice such as the estimation of the parameters of discrete distributions. For such purpose, the functions \( q_{1} ,\,\,q_{2} \) and, specially, \( \xi \) should be as simple as possible. Since the function triplet is not uniquely determined it is often possible to choose \( \xi \) as a linear function. Therefore, it is worth analyzing some special cases which helps to find new characterizations reflecting the relationship between individual continuous univariate distributions and appropriate in other areas of statistics.

In some case, one can take \( q_{1} \equiv 1 \) which reduces the condition of Theorem 1 to \( E\left[ {\left. {q_{2} (X)} \right|X \ge x} \right] = \xi (x),\,\,\,x \in H. \) We, however, believe that employing three functions \( q_{1} ,\,\,q_{2} \) and \( \xi \) will enhance the domain of applicability of Theorem 1.

Appendix B

The elements of the observed information matrix are given below

where, \( g^{{\prime \prime }} (x_{i} ,\phi ) = {{\partial^{2} g(x_{i} ,\phi )} \mathord{\left/ {\vphantom {{\partial^{2} g(x_{i} ,\phi )} {\partial \phi^{2} }}} \right. \kern-0pt} {\partial \phi^{2} }}\;{\text{and}}\;G^{{\prime \prime }} (x_{i} ,\phi ) = {{\partial^{2} G(x_{i} ,\phi )} \mathord{\left/ {\vphantom {{\partial^{2} G(x_{i} ,\phi )} {\partial \phi^{2} }}} \right. \kern-0pt} {\partial \phi^{2} }}. \)

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