Double Truncated Transmuted Fréchet Distribution: Properties and Applications

Muhammad Zafar Iqbal , Muhammad Zeshan Arshad , Munir Ahmad, Iftikhar Ahmad, Taswar Iqbal Muhammad Arslan Bhatti 1, 5, 6 Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan 2, 3 National College of Business Administration and Economics, Lahore, Pakistan University of Gothenburg, Department Mathematical Sciences, Sweden E-mails: *mzts2004@hotmail.com, profarshad@yahoo.com, munirahmaddr@yahoo.co.uk, ifti_ch2002@yahoo.com, therisers1@gmail.com and bhatti.stat.agri@gmail.com Abstract


Introduction
In numerous continuous probability distributions, particularly extreme value theory is an important part of statistical literature and one of the special cases (inverse Weibull, inverse Rayleigh, inverse Exponential or Gumbel type-II) is Fréchet distribution. Maurice Frechet (1878-1973) a French mathematician developed a significant relation with Pareto distribution in 1927 when he discovered a limiting distribution for higher order statistic. The vital role of Fréchet distribution is observed in applied fields, for instance, through accelerated lifetesting to engineering, geology, hydrology, horse racing, insurance, meteorology, sea currents, wind speed and many other diverse problems of life. Several generalizations and modifications, as well as progressive expansions over the last two decades, have been studied and a lot more is about to happen. Nadarajah and Kotz (2003) developed an Exponentiated Fréchet distribution. Krishna    In distribution theory, truncation is referred to as conditional distribution that provides more constructive and reliable results. The present distribution is initiated on this motivation that it has not been studied earlier and will present more flexible estimates on the skewed datasets. Furthermore, it is applicable to many diverse problems other than income and wealth studies.
Rest of the article is arranged into several sections as follows: CDF, PDF, graphical representation alongside special cases are developed in Section 2. In Section 3, we illustrate the moments and various reliability measures in Section 4. Quantiles function along with several descriptive statistics, Rényi entropy, The Mellin 13 transformation and order statistics is discussed in Section 5. Estimation of the parameters by the maximum likelihood, the study of simulation and application is developed in Section 6 and lastly, the conclusion is reported in Section 7.

Double Truncated Transmuted Fréchet Distribution
Here we establish a model by applying the technique of double truncation to Transmuted Fréchet distribution, originally developed by Mahmoud and Mandouh (2013). It is referred to a Double Truncated Transmuted Fréchet (DTTF) distribution. The CDF of DTTF distribution is followed by and PDF where > 0, , > 0 , | | < 1 , A = − g − , B = − − , m and g are lower and upper truncation points.    (2), r-th moment of DTTF distribution is written as Following the above procedure, we obtain the simplified form of 2 hence the r-th moment of DTTF distribution is obtained by placing equation (5) and equation (6) in equation (4) where where > 0, , 3.4. Fractional negative moments of DTTF distribution, just replace (m/n) with (-m/n) in equation (7), we get 3.5. Lower incomplete moments of DTTF distribution, we replace the upper limit g to w in equation (7) ( ) = ≤ ( ) = ∫ ( ) hence reduced and simplified form of lower incomplete moments is given by 3.6. Upper incomplete moments of DTTF distribution are obtained by incorporating equation (7) ( ) = > ( ) = ∫ ( ) g hence upper incomplete moments of DTTF distribution is followed by 3.7. Factorial moments of DTTF distribution, we achieve by equation (7) [ ] = ∑ / =0 where , equation (14) can be defined as 3.9. Central moments can be obtained by using a relation between ordinary and central moments. It is defined based on equation (7), central moments of DTTF distribution

Cumulants generating function based on a relation between ordinary moments and cumulants is defined
cumulants generating function of DTTF distribution can be written as

Kurtosis of DTTF distribution is identified as
where

Reliability Measures of DTTF Distribution
In reliability engineering, reliability analysis through probability distribution is the most extensively exercised method which pays significant contribution in studying and predicting the survival or hazard life of the component during a particular interval of time.

Survival function
Survival or reliability function is used to measure the risk of occurrence of some event at a specific time. It is denoted by S(x). For DTTF distribution it can written as

Hazard function
Hazard function H(x) is used to measure the failure rate of some components in a particular period of time x. For DTTF distribution it is illustrated by

Reverse hazard function
Hr(x) of DTTF distribution is obtained by incorporating equation (1) and equation (20) (1) and equation (20)  It is referred to a quantile function.

Mills ratio M(x) of DTTF distribution is obtained by equation
q-th quantile function of DTTF distribution is given by equation (1) =  To generate random numbers, we suppose that CDF of DTTF distribution follow to uniform distribution u= U (0, 1).

Random numbers of DTTF distribution is calculated by
and

Entropy of DTTF distribution
The degree of disorder or unpredictability / randomness in a system is defined as entropy.
By definition, Rényi (1961) entropy is described as

Rényi entropy of DTTF distribution is obtained by incorporating equation (2)
let's simplify first ( )

The Mellin Transformation of DTTF distribution
In the theory of statistics, the Mellin transformation is well-known since it is a distribution of the product and proportion for independent r.v.'s.
The Mellin transformation is defined as The Mellin transformation of DTTF distribution, we replace r by n-1 in equation (7), we get

i-th Order Statistic PDF is defined as
by equation (1) and equation (2), i-th order statistics PDF of DTTF distribution may obtain by .

minimum order statistic of DTTF distribution is given by
.
(44) 26 maximum order statistic of DTTF distribution is given by

Maximum Order Statistic PDF is defined as
Median Order Statistic PDF is defined as . (46)

Joint Distribution of DTTF distribution
The joint distribution of i-th and j-th order statistics of DTTF distribution is

Estimation of DTTF distribution
Parameters of the DTTF distribution are derived by the method of maximum likelihood. Here equation (2) is presented in a simplified way likelihood function of DTTF distribution can be written as log of equation (48) provides the log-likelihood function of DTTF distribution partial derivatives of equation (49) w.r.t , , and C yield where . (53) Since m and g are the lower and upper truncation points of density function of DTTF distribution, as a result minimum and maximum value of the sample will be considered the estimate of m and g.

Simulation Study of DTTF distribution
The study of simulation is conducted to assess the behavior of a finite sample.

Application of DTTF distribution
In this section, flexibility and potentiality of DTTF distribution is demonstrated by experiencing and integrating two suitable lifetime datasets.   hesitate anymore to declare that DTTF distribution is a better fitted model on both the datasets as compared to its competing models. Furthermore, one can see the empirically fitted PDF ( Fig. 9 and Fig. 11) and CDF ( Fig. 10 and Fig. 12) plots of DTTF distribution display the close fit to the empirical histogram. Fig.13 and Fig. 14 represents the plots of total test time (TTT), proposed by Aarset (1987) may be used as a tool for obtaining empirical behavior of failure rate of the DTTF distribution.

Conclusion
The present study is conducted to provide supportive and more fixable results than its competing models on skewed datasets. Diverse probabilistic and reliability measures along with Rényi entropy and order statistics are developed and discussed. The method of MLE is suggested to derive the estimates and execution of the estimates is assessed by simulation study. The application of DTTF distribution is illustrated by two real-time datasets.