“Mircea cel Batran” Naval Academy Scientific Bulletin, Volume XVIII – 2015 – Issue 2 Published by “Mircea cel Batran” Naval Academy Press, Constanta, Romania // The journal is indexed in: PROQUEST SciTech Journals, PROQUEST Engineering Journals, PROQUEST Illustrata: Technology, PROQUEST Technology Journals, PROQUEST Military Collection PROQUEST Advanced Technologies & Aerospace THE TRANSMUTED GENERALIZED PARETO DISTRIBUTION. STATISTICAL INFERENCE AND SIMULATION RESULTS Romică TRANDAFIR 1 Vasile PREDA 2 Sorin DEMETRIU 3 Ion MIERLUS-MAZILU 4 1 Department of Mathematics and Computer Science Technical University of Civil Engineering Bucharest 2 Faculty of Mathematics and Computer Science University of Bucharest 3 Department of Structural Mechanics Technical University of Civil Engineering Bucharest 4 Department of Mathematics and Computer Science Technical University of Civil Engineering Bucharest Abstract: In this paper for the Transmuted Generalized Pareto Distribution introduced in [14] we estimate the distribution parameters using the method of moments, maximum likelihood method and Method of probability-weighted moments. For different values of parameters we generate samples of volume 1000 and determine from these samples the mean and standard deviation comparing them to the theoretical. We study the performance of the estimating methods used considering the bias and the root mean squar error and we conclude that are adequate. For the consolidated presentation of the subject approached the paper contains important part of the paper [14]. Some mathematical properties of the new distribution are presented in this paper. Mathematics Subject Classification (2010): 62P30, 62N02, 62N05 Key words: transmuted probability distributions, quadratic rank transmutation map, distribution of order statistics, estimating of parameters, numerical simulation. Introduction In recent years there have been considerable efforts in finding statistical models, not necessarily symmetrical to represent real world phenomena. Given that many of these phenomena are not symmetrical, the efforts were directed towards skewed distributions from other popular distributions symmetrical or not. Asymmetrical patterns that express different degrees of asymmetry are a useful tool in modeling real world phenomena. Starting from a symmetrical distribution with cumulative distribution function ) x ( G and probability density function ) x ( g , Azzalini [3] proposes asymmetric distribution whose probability density function is ) x ( G ) x ( g ) x ( f β 2 = , where β is the parameter the asymmetry. Shaw and Buckley [12] investigate a novel technique for introducing skewness or kourtosis into symetric or other distribution. Aryal and Tsokos [1] use quadratic rank transmutation map to generate a flexible family of probability distribution starting from extreme value distribution and generalize the two parameter Weibull distribution [2 ]. Merovci [9], Merovci and Elbatal [8], Elbatan and Elgarhy [4], M Merovci and Puka [10] , generalize different kind of Lindley distribution and Pareto distribution using quadratic rank transmutation map obtaining new distributions with applications in reliability. Elbatal et al. [5] consider like base distribution linear exponential distribution and by quadratic rank transmutation map obtain transmuted generalized linear exponential distribution which can use in modeling of life time data. Khan and King [1 1] introduce transmuted modified Weibull distribution as an important competitive model for life time distributions. The transmuted additive Weibull distribution introduced by Elbatal and Aryal [6] can be used to model lifetime data. The purpose of this paper is to investigate a probability distribution that can be obtained from an asymmetric distribution, namely generalized Pareto distribution and that can be used for modeling and analyzing real-world data. Quadratic rank transmutation map Definition 1. [1] A functional composition of the cumulative distribution function of one probability distribution ) x ( F with the inverse cumulative distribution function of another ) x ( G , ( ) ) u ( G F ) u ( GF 1 − = R (1) is called the transmutation map, where G is considered as the base distribution and F as the modulated distribution. Obviously, one can also define mutual representation ( ) ) u ( F G ) u ( FG 1 − = R (2) thus obtaining a pair of rank transmutation maps. Note that the inverse cumulative distribution function also known as quantile function is defined as [ ] { } 1 0 1 , y y ) x ( F inf ) y ( F R x ∈ ≥ = ∈ − (3) The functions ) u ( FG R and ) u ( GF R both map the unit interval [ ] 1 0, I = into itself, and under 411