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in this paper, we discuss generating a random sample from gamma distribution using generalized exponential distribution.

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Abstract: Depending on the type of distribution, estimation of parameters are not sometimes simple in practice. In particular, this is the case for Birnbaum-Saunders distribution (BS). In this article, we present four different methods for estimating the parameters of a BS distribution. First, a simple graphical technique, analogous to probability plotting, is used to estimate the parameters and check for goodness-of-fit of failure times following a Birnbaum-Saunders distribution. Then, the maximum likelihood estimators and a modification of the moment estimators of a two-parameter Birnbaum–Saunders distribution are discussed. Finally, The jackknife technique is considered as another method which is appropriate for the small sample size case. Monte Carlo simulation is also used to compare the performance of all these estimators.

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Mathematical methods and statistical distributions present exact results in the climate calculations and hydrological processes. Awareness of the rainfall probability distribution provides the appropriate conditions for water resource planning. Many studies have been done to estimate probability of rainfall by various methods due to the importance of rainfall distribution in the economic, social and particularly agriculture studies. In these studies, the various probabilistic models have been used and the results of the most investigations show that the bivariate gamma distribution branches of gamma model are compatible for rainfall data. The bivariate gamma distribution is used in the hydrological processes modeling. In the present paper, supposing that the X and Y follow the crovelli’s bivariate gamma model, at first a brief description was given in the case of the exact distributions of the functions U=X+Y, P=XY and Q=X⁄((X+Y)) as well as their respective moments, then the validity of this model was evaluated for Rasht airport weather station data. The results showed that rainfall data of this region also confirms The suitability of the crovelli’s bivariate gamma model.

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‎Generalized Lambda Distribution is an extension of Tukey's lambda distribution‎, ‎that is very flexible in modeling information and statistical data‎. ‎In this paper‎, ‎We introduced two parameterization of this distribution‎. ‎Then We estimate parameters by moment matching‎, ‎percentile‎, ‎starship and maximum likelihood methods and compare two parameterization and parameter estimation methods with Kolmogorov-Smirnov test‎.

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‎In this paper‎, ‎first‎, ‎we investigate probability density function and the failure rate function of some families of exponential distributions‎. ‎Then we present their features such as expectation‎, ‎variance‎, ‎moments and maximum likelihood estimation and we identify the most flexible distributions according to the figure of probability density function and the failure rate function and finally we offer practical examples of them‎.  

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‎In this paper‎, ‎a new distribution is introduced‎, ‎which is a generalization of a well-known distribution‎. ‎This distribution is flexible and applies to income data modeling‎. ‎We first provide some of the mathematical and distributional properties of this new model and then‎, ‎to demonstrate the flexibility the new distribution‎, ‎we will present the applications of this distribution with real data‎. ‎Data fitting results confirm the appropriateness of this new model for the real data set‎.

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In the analysis of Bernoulli's variables, an investigation of the their dependence is of the prime importance. In this paper, the distribution of the Markov logarithmic series is introduced by the execution of the first-order dependence among Bernoulli variables. In order to estimate the parameters of this distribution, maximum likelihood, moment, Bayesian and also a new method which called the expected Bayesian method (E-Bayesian) are employed. In continuation, using a simulation study, it is shown that the expected Bayesian estimator out performed over the other estimators.

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In this paper, a generalization of the Gumbel distribution as the cubic transmuted Gumbel distribution based on ‎the ‎cubic ranking transmutation map is introduced. It is shown that for some of the parameters, the proposed density function is mesokurtic and for others parameters the density function is platykurtic function. The statistical properties of new distribution, consist of survival function, hazard function, moments and moment generating function have been studied. The parameters of cubic transmuted Gumbel distribution are estimated using the maximum likelihood method. Also, the application of the cubic transmuted Gumbel distribution is shown with two numerical examples and compared with Gumbel distribution and transmuted Gumbel distribution. Finally, it is shown that for a data set, the proposed cubic transmuted Gumbel distribution is better than Gumbel distribution and transmuted Gumbel distribution.

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The empirical distribution function is used as an estimate of the cumulative probability distribution function
of a random variable. The empirical distribution function has a fundamental role in many statistical inferences, which are
little known in some cases. In this article, the empirical probability function is introduced as a derivative of the empirical
distribution function, and it is shown that moment estimators such as sample mean, sample median, sample variance, and
sample correlation coefficient result from replacing the random variable density function with the empirical probability
function in the theoretical definitions. In addition, the kernel probability density function estimator is used to estimate the
population parameters and a new method for bandwidth estimation in the kernel density estimation is introduced.
Keywords: Empirical distribution function, moment estimate, kernel estimator, bandwidth.

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In this paper, the concept of unit generalized Gompertz (UGG) distribution will be introduced as a new transformed model of the unit Gompertz distribution, which contains the unit Gompertz distribution as a special case. We calculate explicit expressions for the moments, moment generating, quantile, and hazard functions, and Tsallis and R'{e}nyi entropy. Some different methods for estimation and inference about model parameters are presented too. To estimate the unknown parameters of the model, the maximum likelihood, maximum product spacings, and bootstrap sampling have been discussed, and also approximate confidence interval is presented. Finally, a simulation study and an application to a real data set are given.

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Actuarial studies treat insurance losses as random variables, and appropriate probabilistic models are sought to model them. Since losses are evaluated in terms of a unitary amount, distributions with positive support are typically used to model them. However, in practice, losses are often bounded due to policyholder conditions, which must be considered when modeling. While this is not a problem for univariate cases, it becomes complicated for multivariate cases. Copulas can be helpful in such situations, but studying the correlation is crucial in the first step. Therefore, this paper addresses the problem of investigating the effect of restricted losses on correlation in multivariate cases.
The Pearson correlation coefficient is a widely used measure of linear correlation between variables. In this study, we examine the correlation between two random variables and investigate the estimator of the correlation coefficient. Furthermore, we analyze a real-world dataset from an Iranian insurance company, including losses due to physical damage and bodily injury as covered by third-party liability insurance.
Upper and lower limits for both the Pearson correlation coefficient and its estimator were derived. The Copula method was employed to obtain the bounds for the correlation parameters, while order statistics were used to obtain the bounds for the sample correlation coefficient. Furthermore, two methods were used to determine the correlation between physical damage and bodily injury, and the results were compared.
Our findings suggest that the commonly used upper and lower bounds of -1 and +1 for the Pearson correlation coefficient may not always apply to insurance losses. Instead, our analysis reveals that narrower bounds can be established for this measure in such cases. The results of this study provide important insights into modeling insurance losses in multivariate cases and have practical implications for risk management and pricing decisions in the insurance industry.