INTRODUCING DIFFERENT ESTIMATORS OF TWO PARAMETERS KAPPA DISTRIBUTION

Transcription

1 International INTERNATIONAL Journal of Advanced JOURNAL Research OF ADVANCED in Engineering RESEARCH and Technology IN (IJARET), ENGINEERING ISSN 0976 AND TECHNOLOGY (IJARET) ISSN (Print) ISSN (Online) Volume 5, Issue 12, December (2014), pp IAEME: IJARET.asp Journal Impact Factor (2014): (Calculated by GISI) IJARET I A E M E INTRODUCING DIFFERENT ESTIMATORS OF TWO PARAMETERS KAPPA DISTRIBUTION Dhwyia S. Hassan 1, InamAbdulrahmanNoaman 2, Layla M. Nassir 3 1 Department of Statistics, College of Administration & Economic, Baghdad University, Iraq 2 Department Statistics, College of Administration & Economic, University of Diyala, Iraq 3 Departement of Electricity, College of Engineering, Al Mustansiriyah University, Iraq ABSTRACT This paper deals with introducing three methods, for estimating the two parameters (,), Kappa distribution, these methods are maximum likelihood, maximum entropy, and L moments. The non central moments is derived, all the required formula for integrals needed explained, especially for maximum entropy method, since it requires four steps to be applied, using Lagrange multipliers and certain constraints. Keywords: Two Parameters Kappa Distribution, Maximum Likelihood Estimators, Maximum Entropy Estimators, L Moment Estimators. 1. INTRODUCTION Many statistical data follow skewed positive or negative distribution, especially data with a hydrological application and data of rainfall, and time to failure of hydrologic engineering system, for treating the skewed of these data, many researchers works on introducing kappa distribution with three and four parameters, they use the principle of maximum entropy under constraints as a tool, for reducing the uncertainty as explained by [NadarajahandKotz (2006) and Z. Hradil, J. Rehacek (2006)] [15]. The maximum entropy (ME) ia a tool for inference under uncertainty, many authors [Singh and Deng [13] ], [BungonKumphon (2012) [4] ] applied (ME) to estimate the four and three parameters kappa distribution. Here we introduce two parameter kappa, and investigate how to estimate its two parameters (,)using (ME) and maximum likelihood method, also deriving a formula for L moments, also as a tool of estimation.the principleof maximum entropy (ME) is a tool for under uncertainty, where this approach produces the most suitable probability distribution given the available information and work on maximizing the information constraints (Lagrange Multiplier), for a family positively skewed distribution (Kappa Distribution) which was introduced 107

2 by [Mielke, and, MielkeandJhonson] [7&8]. The papers introduced by researcher include various methods of estimation, such as L moment, moments and maximum likelihood (ML) techniques. Here we introduce the two parameters Kappa distribution [Kappa (,)] and work on estimating the two parameters (,) using maximum likelihood and principle of maximum entropy, and L - moment. Let () be a r.v distributed as two parameters Kappa distribution defined by; = +,,>0 (1) 0 is shape parameter, is scale parameter, the corresponding cumulative distribution function (C.D.F) is; = + (2) 2. ESTIMATION OF PARAMETERS 2.1 Maximum Likelihood To estimate the two parameters (,), the firstmethod used is maximum likelihood, from equation (1) the likelihood function is; Log likelihood function of (3) can be written as; = + log=log log +1 log+ 4 3 log = [ log = [ log +log+ 1 ] 1+ log + 1 log+ +log+ ] = log 5 Equation (5) can be solved numerically to obtain ( ), by using fixed point method where; 108

3 = [ log log+ (6) And; log = +1 + = ++1 = =0 = Which is an implicate function of (,), can be solved numerically. 2.2 Maximum Entropy The second method of estimation is the maximum entropy (ME), which depends on four steps; a. Specification of appropriate constraints. b. Construction of Lagrange Multipliers. c. Derivation of the entropy function of the distribution. d. Derivation of the relation between Lagrange Multipliers and constrains. e. Step one: Since the.. of the two parameters Kappa is; = + ln=ln ln+ (8) = ln[] = ln +1 ln+ 109

4 The (S) function is subject to constraints; +1 ln+ = ln+ (9) Also; =exp =1 10,=1,2,, exp ln+ =1 Such that; = + = exp ln+ (11) =+ = = = = = 1 = 1 1 According to the formula;,= 1 110

5 = = 1+ 1 = 1+,= = Therefore;,= 1+ = Γ Γ (12) Γ Taking logarithms; = 1 =ln+ 1 1 ln+lnγ1 +lnγ 1 lnγ1 + 1 (13) Since; =exp ln+ = Γ Γ Γ (14) Also taking natural logarithms of (14) we get; ln=lnγ1 + 1 lnγ1 lnγ 1 ln 1 1 ln ln+ Using definition of entropy; = lnthen; = lnγ lnγ1 +lnγ 1 +ln+1 1 ln+ ln+ For simplification, let; = 1 = 1 111

6 =1 =1 + 1 = 1 lnγ= = = 1 lnγ1 = lnγ1 + 1 = 1 lnγ1 + 1 = = ln+ln+ = ln 1 Putting =0; = Where; And from =0 we obtain; Then; 1+ 1ln + + =1 + + = + (15) ln+1 1+ = 1+ ln ln+1+1+ (16) Which can be solved numerically to find ( ) =0 112

7 = (17) 2.3 L Moment Estimator The set of L moments equation used in estimation is defined by; = = = Then we will set this equal to an unbiased estimate of ( ) which is; = 1 Where are the ordered observation of (,,, ). = + = + + = = = = [+ ] = = = 1 = =[] =[] [+] = = = =[] [1+] =[] [1+] 113

8 Since; = From (18); = =, = [1+],= 1+ (18) = [1+] = = +2,+1 Γ Γ (19) Γ Γ Γ Γ (20) = therefore, from solving [ = ], Γ Γ = 1 Γ (21) Taking (=1,=2), we can obtain the estimator of ( ). Also from ( = ); Γ Γ = Γ = 1 2 (22) CONCLUSION In this paper we have considered one of time to failure, which is a family that is positively skewed, we investigate the theoretical back ground for parameter estimation by maximum likelihood method and the L moments were derived and used also in estimation, also the researcher apply principle of maximum entropy (ME), which is a measure of uncertainty (i.e) the (ME) is a tool for inference under uncertainty. The estimation by (ME) need certain integrals values, and minimize the measure of uncertainty due to using maximum entropy (ME) as a tool of inference as we refer, also the estimators by L moments were explained. 114

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