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Research Article Article OpenOpen Access Access A Class of Scale Mixtured Normal Distribution Rezaul Karim Md* Department of Statistics, Jahangirnagar University, Bangladesh
Abstract A class of scale mixtured normal distribution with lifetime probability distributions has been proposed in this article. Different moments, characteristic function and shape characteristics of these proposed probability distributions have also been provided. The scale mixture of normal distribution is extensively used in Biostatistical field. In this article, some new lifetime probability distributions have been proposed which is the scale mixture of normal distribution. Different properties such as moments, characteristic function, shape characteristics of these probability distributions have also been mentioned.
Keyword: Scale mixture distribution; Inverted probability Main results distribution; Moments; Skewness; Kurtosis Here, the definition of scale mixture of normal distribution with Introduction well known lifetime distributions such as inverted chi-square, gamma, exponential, Rayleigh distributions are mentioned. The important A mixture distribution is a convex function of a set of probability characteristics of these distributions such as characteristic functions, distributions [1]. At first, Pearson considered a mixture distribution moments, and shape characteristics are also mentioned. The main of two normal distributions in 1894 and he studied the estimation results of the paper are presented in the form of definitions and of the parameters of his proposed distribution [2]. After around half theorems. century, some characteristics of mixture distributions were mentioned Definition 3.1 by Robins (1948). Some of researchers like Rider, Blichke, Karim [1,3-5] studied the various mixture of distributions. However, they A continuous random variable X is said to have a scale mixture of studied only for location mixture of the distributions though the scale normal distribution if its probability density function is given by 2 ∞ 1 x−µ mixture of the distributions have widely used in biostatistical field. The − 2 1 2σ location testing for Normal scale mixture distributions obtains in some fx(;µσ , )=∫ e h()τ d τ; −∞< x <∞ (3.1) 0 τσ2 π fastidious types of situations where the student’s t-test is inapplicable. Where, the parameters µ and σ2 satisfy − ∞ < µ < ∞ and σ2 > 0. Specifically, these cases permit tests of location to be prepared where Here h(τ) is a probability function of a random variable τ. The variable the assumption that sample observations obtain form populations X whose density function (3.1) is called a scale mixture normal variate having a Gaussian distribution can be replaced by assumption which can be written as X τ ~ N στµ 22 ),( that they arise from a Normal scale mixture distribution. A few researchers [3,6-8] mentioned the definition of scale mixture of normal Definition 3.2 distributions with its applications. There are many applications of scale If a random variable X has a probability density function mixture of normal distribution with lifetime probability distribution 11 = −∞< <∞ (3.2) in biostatistical field whereas; only the scale mixture of normal fx() n+1 ; x n 1 2 distribution is familiar for us. The scale mixture of normal distribution nσβ (,)x − µ 2 with lifetime distributions are not only used in biological field but also 22n σ used in fisheries, agricultural, psychological, paleontological field. It is also widely used in electrophoresis, and communication theories. In then it said to be scale mixture of normal distribution with inverted 2 this paper, the definition of the scale mixture of normal distribution chi-square distribution with location parameter µ, scale parameter σ with a set of lifetime probability distributions is defined initially and and shape parameter β. Various moments and shape characteristic of then scale mixture of normal distribution with inverted Chi-square, this probability distribution are presented by following theorem. Rayleigh, Exponential, Gamma and Erlang distributions are studied. Theorem 3.1 The different properties of these distributions are also presented. The moments and shape characteristics of scale mixture of normal Preliminaries A mixture distribution is the probability distribution of a random *Corresponding author: Rezaul Karim Md, Department of Statistics, Jahangirnagar variable whose values can be taken in a simple way from an underlying University, Bangladesh, Tel: 880-2-7791045-51; E-mail: [email protected] set of other random variables, with a certain probability of selection being associated with each. Suppose the random variable X has a Received February 27, 2016; Accepted March 01, 2016; Published March 08, 2016 probability density function (Pdf) f (x|θ). The sampling distribution of a statistic of the parameter θ is τ(θ)where Θ Citation: Rezaul Karim Md (2016) A Class of Scale Mixtured Normal Distribution. is the parameter space J Biom Biostat 7: 285. doi:10.4172/2155-6180.1000285 of θ. And then, a continuous mixture of densities f (x|θ) with weight functions τ(θ) is expressed by following: Copyright: © 2016 Rezaul Karim Md. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits mx()()()= ∫ f x|θτθ d θ. unrestricted use, distribution, and reproduction in any medium, provided the Θ original author and source are credited.
J Biom Biostat ISSN: 2155-6180 JBMBS, an open access journal Volume 7 • Issue 2 • 1000285 Citation: Rezaul Karim Md (2016) A Class of Scale Mixtured Normal Distribution. J Biom Biostat 7: 285. doi:10.4172/2155-6180.1000285
Page 2 of 2 distribution with inverted chi-square distribution are given by Where µ, σ2 and β represent location, scale and shape parameters respectively. n 1 nrrσ 2 −+ rr 22 Theorem 3.4 µµ2rr= ;21+ = 0; n 1 The main characteristics of scale mixture normal distribution with 22 inverted exponential distribution are
n 2 (2!r) Mean = 0, Variance = σ µ= σθ2rr1 −=r ; µ 0; n − 2 2rrr 21+ 32(n − ) 2!r Skewness: β = 0 , Kurtosis: β = . 2 1 2 n − 4 Mean=0 and variance = σ θ . Definition 3.3 Definition 3.6 A random variable X said to be scale mixture of normal distribution Let X be a continuous random variable. The probability density with inverted Gamma distribution if its probability density function is function of X is said to be a scale mixture normal distribution with following inverted Erlang distribution if its follows: nβ 1 = −∞< <∞ λθ fx() 1 ; x (3.3) 1 + = −∞ < < ∞ (3.6) 1 2 n fx() 1 ; x 2 θ + σ 2,Bnx − µ 1 2 2 21nβ + σ 2,Bnx − µ 2 σ 2 21λθ + σ Where, µ, σ2 and β represent location, scale and shape parameters respectively. Theorem 3.5 The different characteristics such as moments, skewnbess and Theorem 3.2 Kurtosis of the scale mixture normal distribution with inverted Erlang The main features such asr th central moment, Skewness, kurtosis, distribution are following: etc. of scale mixture of normal distribution with inverted Gamma θ − r distribution are following: (2!r) 2r r µ= σ( λθ) ; µ + = 0; Mean= 0, variance = 2,rm 2!r r λ 2r 1 (2!r) r nr− µ= σβ2r µ= 2rrr (n ) ;21+ 0; λθ 2!r n σ 2 θ −1 nβ 2 Mean= 0, variance = σ 3(θ −1) n −1 Skewness = 0, Kurtosis = . (θ − ) 3(n −1) 2 Skewness = 0, Kurtosis = . (n − 2) Concluding Remarks Definition 3.4 This article has been proposed a class of scale mixtured of normal distribution with different lifetime probability distribution such as The probability density function of a random variableX is said inverted Rayleigh distribution, Erlang distribution and explained to have a scale mixture of normal distribution with inverted Rayleigh some basic properties of these proposed probability distributions with distribution if its probability density function is defined by 1 applications in various fields of applied science and business. I believe, = −∞< <∞ fx() 3 ; x (3.4) this paper will help and encourage young researchers to stimulate 2 2 further research. 2 x − µ 2 σθ + 2 σ θ References Where θ is the parameter which satisfiesθ >0. 1. Karim MR, Hossain MP, Begum S, Hossain MF (2011) Rayleigh Mixture Distribution. Journal of Applied Mathematics.
Theorem 3.3 2. Pearson K (1894) Contribution to the mathematical theory of evolution. The main properties of scale mixture normal distribution with Philosophical Transaction Royal Society. 185: 71-110. inverted Rayleigh distribution are given by following: 3. Andrews DF, Mallows CL (1974) Scale Mixtures of Normal Distribution. Journal 1− r of the Royal Statistical Society 36: 99-102. (2!r) 2r µ2rr= σµ;21+ = 0; 2!rrr θ 2 4. Rider PR (1961) The Method of Moments Applied To A Mixture of Two Exponential Distributions. The Annals of Mathematical Statistics 32: 143-147. σ 2 Mean=0 and variance = . θ 2 5. Blichke WR (1965) Mixtures of Discrete Distribution, Classical and Contagious Discrete Distributions. Calcutta Statistical Publishing Society Definition 3.5 Ed GP Patil 351- 372.
The probability density function of scale mixture normal 6. Choy STB, Jennifer SKC (2008) Scale Mixtures Distributions in Statistical distribution with inverted exponential distribution is as follows Modelling. Australian & NewZealand Journal of Statistics 50: 135-146. θ = −∞< <∞ 7. West M (1987) On Scale Mixtures of Normal Distributions. Biometricka 74: fx() 3 , x (3.5) 2 646-648. x − µ 2 σθ+ 2 8. Monahan JF, Stefanski LA (1989) Normal scale mixture approximations to the σ logistic distribution with applications. Institute of Statistics Mineograph Series.
J Biom Biostat ISSN: 2155-6180 JBMBS, an open access journal Volume 7 • Issue 2 • 1000285