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ESTIMATION OF PARAMETERS OF PARETO DISTRIBUTION
USING DIFFERENT LOSS FUNCTIONS
DEVYA MAHAJAN & PARMIL KUMAR
Department of Statistics, University of Jammu, Jammu and Kashmir, India
ABSTRACT
The aim of this paper is to derive, the exact analytical expression for estimation of Parameters of Pareto
distribution, using entropy loss functions. Our purpose is to obtain, bias estimator and the associated risk function of
different types of loss function, namely SELF absolute loss function, Linex loss function, Precautionary Loss function and
entropy loss function. The purpose is to find out the most suitable loss function, amongst these five loss functions. In this
paper, parameters of Pareto distribution have been estimated, by using the method of moments. The workability of the
estimator is then compared, on the basis of their risks obtained under different loss functions. The relative efficiency of the
estimator is also obtained. In the end, Monte-Carlo simulation has been performed, to compare performances of the bias
estimates, under different situations.
KEYWORDS: Entropy, Pareto Distribution, Loss Functions, Root Mean Square Error, Efficiency
4.1. INTRODUCTION
Pareto distribution was applied by Pareto, to model the allocation of wealth among individuals and the distribution
of incomes. It has been widely used in economics, insurance, geography, clusters of a Bose Einstein condensate near
absolute zero, physical sciences, chemical sciences. Asrabadi (2015), established the UMVUEs, for the PDF and
cumulative distribution function (CDF) of Pareto distribution. Asrabadi et. al (2015), further studied the MSE of MLEs and
UMVUEs, of PDF and CDF.
The applications of entropy, originated in the nineteenth century, in the field of Statistical Mechanics and
Thermodynamics. In this chapter, we have derived analytical expressions, for estimation of Parameters of Pareto
distribution, using entropy loss function and also have obtained bias of the estimator and the associated risk function, for
other different types of loss functions, namely SELF, Absolute loss function, Linex loss function and Precautionary Loss
function. The objective is to find out the most suitable loss function, amongst these five loss functions. In this chapter,
parameters of Pareto distribution have also been estimated. In this chapter, the entropy expression for Pareto (II)
distribution is derived. The workability of the estimator is then compared, on the basis of their risks, obtained under
different loss functions. These distributions have important roles, as parametric models in reliability, actuarial science,
economics, finance and telecommunications. Analytical expressions, for the entropy of bivariate distributions are
discussed, in references like Hui He (2014), G.H Yari (2010).
The random variable X is said to have two Parameter Pareto distributions, if its density function is given by,
.
=
=
;>β> 0, α > 0
International Journal of Applied Mathematics
& Statistical Sciences (IJAMSS)
ISSN(P): 2319-3972; ISSN(E): 2319-3980
Vol. 6, Issue 5, Aug – Sep 2017; 99-108
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