Transmuted Zeghdoudi Distribution: Theory and Applications

Received: 16/Mar/2020, Accepted: 13/Apr/2020, Online: 30/Apr/2020 AbstractThe Transmuted Zeghdoudi distribution(TZD) is derived using the Quadratic Rank Transmutation Map (QRTM) approach. Some mathematical properties of the proposed distribution are derived such as moments, skewness, kurtosis, mean residual lifetime, hazard rate, order statistics and reliability analysis etc. Estimation Properties like maximum likelihood estimation procedure is derived to obtain the estimates of parameters. A data set are analyzed to illustrate its applicability and to compare that this distribution performs better than many other distributions.


Introduction
Several Statistical distributions plays a very important role in prediction and analysis of real lifetime data. Data obtained from various fields such as financial, biological, physical, actuarial, engineering are fitted with the statistical distribution for prediction and analysis. Some well-known continuous distribution such as Exponential, Gamma, Lindley, Weibull, that have been applied to model the lifetime data sets. For the last 30 years, several new distributions were introduced that has a wide range of application in the field of data modelling. Due to the use of sophisticated technology, more and more data are obtained from various sources. So, research is going to develop new distribution that fits the data better and gives much more reliable estimates. Some of them one parameter mixture distribution such as Beta-Pareto [1], Lindley [2], Beta-Laplace [3], Xgamma [4], Weibull-Pareto [5], Beta Exponentiated-Pareto [6],Transmuted Pranav Distribution [7], Transmuted Lindley Distribution [8], and Gamma-Lindley Distribution [9] and so on were used to fit the lifetime data better and have high flexibility. The Exponential families of distributions have played a vital role in probability distribution for modelling the real world phenomenon.
Recently, on parameter exponential families of distribution which is the mixture of Gamma(2, ) and Gamma (3, ) with mixing probability ) have been introduced and named as Zeghdoudi distribution [10]. The probability density function (pdf) of Zeghdoudi distribution is defined as (1) and its cumulative distribution function (cdf) is given by . ( We use the transmutation map approach which was suggested by Shaw et. al. [11] , where is the cdf of the base distribution. When we have the distribution of the Zeghdoudi random variable. In this paper, we have introduced a transmuted version of the Zeghdoudi Distribution using Quadratic Rank Transmutation Map (QRTM) approach. The uses of this transmutation approach have played a vital role in modeling the lifetime data. We named the distribution as Transmuted Zeghdoudi Distribution (TZD).
The paper is organized as follows, Section II contains the introduction of pdf and cdf of Transmuted Zeghdoudi Distribution. Section III contains the mathematical properties of the distribution such as moments, moment generating function and second, third and fourth order moment. Section IV contains the measures of skewness and kurtosis and plotted for different values. Section V contains the order statistics. Section VI contains Reliability analysis and Hazard Function, Section VII contains simulation and parameter estimation using Maximum Likelihood Estimation., Section VIII contains the application to two real lifetime data and compare with two other distributions using related measures. Section IX concludes research work.

III. Moments and quantiles
Let x ~TG(θ, ), then the r th raw moment about the origin is given by And the first, second, third and fourth order moment about origin are given by: As variance, skewness & kurtosis calculative expressions are so complicate, therefore the graph of skewness & kurtosis is plotted. Figure 3: The graph of skewness and kurtosis for from -1 to +1.

IV. Moment Generating Function
The moment generating function in terms of the moment are given by

V. Order Statistics
Let be the ordered random sample drawn from the Transmuted Zeghdoudi Distribution, then the c.d.f. and p.d.f. of the order statistics is given by So, the p.d.f. and c.d.f. of and is given by

VI. Survival Properties
The hazard rate function or failure rate function for a continuous distribution with pdf f(x), cdf F(x) and survival function (sf) S(x) is defined as, .
For the transmuted zeghdoudi distribution, the hazard rate function is given by

VII. Estimation of the parameters
Let be a random sample of size n drawn from the pdf in . The log likelihood equation is as follows To get the maximum likelihood estimator of and , we have to solve and Hence, we have and .
(9) To get the MLE of and , we use the R-statistical package to solve these equation numerically.

VIII. Simulation Study
To generate the random sample from transmuted zeghdoudi distribution, we proceed as follows: Let , where follows . i.e.
i.e. √ We will choose that solution which lies in . Here is the cdf of zeghdoudi random variable. Then, we follow the approach of generating sample from zeghdoudi distribution. Hence the algorithm is • Generate , .

IX. Data Analysis
In this section,we use a real lifetime data and shown that the flexibility of Transmuted Zeghdoudi Distribution.

X. Conclusion
In this article, we introduce a new two parameter Transmuted Zeghdoudi Distribution(TZD) which is an extension of one parameter Zeghdoudi Distribution. We derived some statistical properties, estimation of parameters and application to some real lifetime data sets. An application of the proposed distribution to the "waiting times before customer service in bank" data are given to show that the new distribution provides better fit than other models available in the paper. We expect that this article will serve as a reference for many researchers for future research in this field.

XI. Aknowledgements
The author is grateful to Prof. Sudhansu Sekhar Maiti, Visva Bharati University who helped him a lot and also thakful to the unknown referee and Editor-In Chief for their helpful and valuable comments and suggestions for publishing the paper.