Bayes Estimator for Coefficient of Variation and Inverse Coefficient of Variation for the Normal Distribution

Size: px

Start display at page:

Download "Bayes Estimator for Coefficient of Variation and Inverse Coefficient of Variation for the Normal Distribution"

Julius Johnston
5 years ago
Views:

1 Iteratioal Joural of Statistics ad Systems ISSN Volume, Number 4 (07, pp Research Idia Publicatios Bayes Estimator for Coefficiet of Variatio ad Iverse Coefficiet of Variatio for the Normal Distributio Arua Kalkur T. St.Aloysius College (Autoomous Magaluru, Karataka Idia. Arua Rao Professor (Retd. Magalore Uiversity, Karataka, Idia Abstract I this paper Bayes estimators of Coefficiet of Variatio (CV ad Iverse Coefficiet of Variatio (ICV of a ormal distributio are derived usig five objective priors amely Left ivariat, Right ivariat, Jeffry s Rule prior, Uiform prior ad Probability matchig prior. Mea Square Error (MSE of these estimators are derived to the order of O( ad the expressios are ew. Numerical results idicate that Maximum Likelihood Estimators (MLE of CV ad ICV have smaller MSE compared to Bayes estimators. It is further show that i a certai class of estimators MLE of CV ad ICV are ot admissible. Key words: Bayes estimators, Objective priors, Mea Square Error, Admissibility.. Itroductio The estimatio ad cofidece iterval for Coefficiet of Variatio (CV of a ormal distributio is well addressed i the literature. The origial work dates back to 93 whe Mckay derived a cofidece iterval for the CV of a ormal distributio usig trasformed CV. (Also see the refereces cited i this paper. It was Sigh (983 who emphasized that Iverse Sample Coefficiet of Variatio

2 7 Arua Kalkur T. ad Arua Rao (ISCV ca be used to derive cofidece iterval for the CV of a distributio. He derived the first four momets of the ISCV. Sharma ad Krisha (994 developed the asymptotic distributio of the ISCV without makig a assumptio about the populatio distributio. Although may papers have appeared for the estimatio ad the cofidece iterval of a ormal distributio, they are derived usig the classical theory of iferece. I recet years Bayesia iferece is widely used i scietific ivestigatio. Whe objective priors are used the Bayes estimator performs well compared to the classical estimator i terms of mea square error. Not may papers have appeared i the past regardig Bayes estimator of CV of the ormal distributio. I this paper, we discuss Bayes estimator for of the CV ad ICV of the ormal distributio uder squared error loss fuctio. We have used objective priors so that the bias ad Mea Square Error (MSE of these estimators ca be compared with maximum likelihood estimators. The objective priors used are Left ivariat, Right ivariat, Jeffry s Rule prior, Uiform prior ad Probability matchig prior. I Chapter we have listed may papers dealig with the estimatio for the CV of the ormal distributio. I this chapter, we discuss Bayes estimator for the CV ad ICV of the ormal distributio uder squared error loss fuctio. We have used objective priors so that the bias ad MSE of these estimators ca be compared with MLEs. The objective priors used are Right ivariat, Left ivariat, Jeffrey s Rule prior, Uiform prior ad Probability matchig prior. Harvey ad Merwe (0 make a distictio betwee Jeffrey s prior ad Jeffry s Rule prior. Both are proportioal to square root of the determiat of the Fisher Iformatio matrix; but the distictio is that i Jeffrey s prior σ is treated as parameter ad i Jeffrey s Rule prior σ is treated as parameter. The compariso betwee the Bayes estimator, maximum likelihood ad other estimators of the classical iferece are the focal poit of ivestigatio i may research papers i the past. Some of the research refereces are the followig: Pereira ad Ster (000 derived Full Bayesia Test for the CV of the ormal distributio. Kag ad Kim (003 proposed Bayesia test procedure for the equality of CVs of two ormal distributios usig fractioal Bayes factor. Pag et al. (005 derived Bayesia credible itervals for the CVs of three parameters Logormal ad Weibull distributios. D Cuha ad Rao (04 compared Bayesia credible iterval ad cofidece iterval based o MLE i logormal distributio. This chapter is divided ito 6 sectios. Sectio presets the expressios for the Bayes estimators of CV usig five objective priors metioed previously. The bias ad MSE of the Bayes estimator of CV to the order of O(adO( respectively are also derived. Numerical values of bias ad MSE of the Bayes estimator for CVs, alog with the MLE s are preseted i sectio 3. I sectio 4 we preset the expressios for the

3 Bayes Estimator for Coefficiet of Variatio ad Iverse Coefficiet 73 Bayes estimator of ICV alog with the bias ad MSE to the order of O(adO( respectively. Numerical values of bias ad MSE of the Bayes estimator for ICV, alog with the MLE s are preseted i sectio 5. The chapter cocludes i sectio 6.. Bayes Estimators of Coefficiet of Variatio Let x,, x be i.i.d. N (μ, σ. The maximum likelihood estimator of CV of the ormal distributio is give byθ = σ μ, where μ = x = x i= i ad σ =s = ( i= x i x deotes the sample mea ad sample variace of ormal distributio respectively. Five priors are used for the estimatio of CV ad ICV. Let π(μ, σ deote the prior desity for μ ad σ. The expressios for the prior desity are give below. a Right ivariat Jeffrey s prior π(μ, σ = σ ( b Left ivariat prior π(μ, σ = σ ( c Jeffrey s Rule prior π(μ, σ = σ 3 (3 d Uiform prior π(μ, σ = (4 e Probability matchig prior π(μ, σ = σ (5 Sice the distributio of x ad s are idepedet, after some simplificatio we obtai the posterior desity of ( σ as Gamma ((, ( s uder right ivariat prior, Gamma (( +3, ( s uder left ivariat Jeffreys prior,

4 74 Arua Kalkur T. ad Arua Rao Gamma(( +4, ( s uder Jeffry s Rule prior, Gamma (( +, ( s uder Uiform prior ad Gamma ((, ( s uder Probability matchig prior. Bayes estimator of CV is give by E( σ, where the expectatio is take with respect to μ the posterior desity of π(μ, σ data. The posterior desity π(μ, σ x,, x for the right ivariat, left ivariat, Jeffrey s Rule, Uiform ad Probability matchig priors respectively are give by the followig expressios. π(μ, σ x,, x = π σ e (x μ ( σ ( s ( Γ ( + ( σ e ( σ s, (Usig Right ivariat prior < μ <, σ > 0 (6 π(μ, σ x,, x = π σ e (x μ +3 ( ( s Γ( +3 ( σ ( +3 e ( (Usig Left ivariat prior < μ <, σ > 0 σ s, (7 π(μ, σ x,, x = π σ e (x μ +4 ( ( s Γ( +4 ( σ ( +4 e ( σ s, (8 (Usig Jeffrey s Rule prior < μ <, σ > 0 π(μ, σ x,, x = π σ e (x μ + ( ( s Γ( + ( σ ( + e ( σ s, (9 (Usig Uiform prior < μ <, σ > 0 π(μ, σ x,, x = π σ e (x μ σ ( ( s ( Γ( σ ( e ( σ s, (0 (Usig Probability matchig prior < μ <, σ > 0

5 Bayes Estimator for Coefficiet of Variatio ad Iverse Coefficiet 75 The Bayes estimator of CV is E(θ x,, x, where the expectatio is take with respect to the posterior desity of μ ad σ. For the right ivariat prior it is give by θ R = μ σ π σ e (x μ ( ( s Γ( ( σ ( e ( σ s dμdσ ( = Г( ( / ( s x = a θ, wherea = Г( ( / ad θ =( s ( x The followig theorem gives the Bayes estimators. Theorem : The Bayes estimators of CV correspodig to Left Ivariat prior, Right Ivariat prior, Jeffrey s Rule prior, Uiform prior ad Probability Matchig prior are the followig. θ R = Г( ( / ( s x (3 θ L= Г(+ ( / ( s x Г( (4 θ JR = Г(+3 ( / ( s x Г( +4 (5 θ U = (Г( ( / ( s x Г( + (6 ad θ PM = (Г( ( / ( s x Г( respectively. (7 Proof: Straight forward ad is omitted.

6 76 Arua Kalkur T. ad Arua Rao 3. Bias ad Mea Square Error of the Bayes Estimators of Coefficiet of Variatio Theorem : The followig table gives the bias ad mea square error to the order O( ad O( for differet estimators of CV. Prior Bias MSE. Right Ivariat. Left Ivariat 3. Jeffrey s Rule Г( ( / ( s x Г( + ( / ( s x Г( ( σ μ ( ( σ μ ( ( / ( s x Г( Uiform Г( ( / ( s x 5. Probability Matchig Г( + ( σ μ ( ( σ μ ( Г( ( / ( s x Г( ( σ μ ( x 4 s + ( x s Г( +( ( / ( s x ( σ μ x 4 s + ( x s + (Г( +( ( s ( x ( σ μ x 4 s + ( x s +3 (Г( +( ( s ( x x 4 s + ( x s (Г( +( ( s ( x Г( +4 ( σ μ Г( + ( σ μ x 4 s + ( x s (Г( +( ( s ( x Г( ( σ μ Proof: Give i the Appedix. Cosider the class of estimators {a θ }. This class icludes the MLE θ as well as various Bayes estimators for differet choice of a. The followig theorem gives the optimal estimator i this class. Theorem 3: Amog the class of estimators {a θ } of Coefficiet of variatio θ, where θ deotes the maximum likelihood estimator of θ, the estimator with miimum mea square error to the order of O( is θ ( (θ ++ Proof: The expressio for MSE of (a θ is give by MSE of (aθ =a V(θ + Bias of (aθ (7 =a V(θ +(θ (a- Differetiatig with respect to a ad equatig it with zero we get a= ( (θ ++ (8

7 Bayes Estimator for Coefficiet of Variatio ad Iverse Coefficiet 77 Substitutig θ for θ i a we get estimator as i the class of aθ. As, θ ( (θ ++ θ θ ( (θ ++, this is the optimal estimator Table 3. presets the umerical values of Bias ad MSE of the estimators of CV. From the umerical results it follows that maximum likelihood estimators has the smallest MSE compared to other Bayes estimators. Table 3. Bias ad Mea Square Error of the Bays Estimators of CV =0 =30 θ Bias MSE MSE(θ A Bias MSE MSE(θ.Left Ivariat Prior x x x x Right Ivariat Prior x x x0-4.7x Jeffrey s Rule Prior x x x x Uiform Prior x x x x x Probability Matchig Prior x x x x x

8 78 Arua Kalkur T. ad Arua Rao 4. Bayes Estimators of Iverse Coefficiet of Variatio Let x,, x be i.i.d. N(μ, σ. The maximum likelihood estimator of ICV of the ormal distributio is give by θ = μ σ The Bayes estimator of ICV is E(θ x,, x, where the expectatio is take with respect to the posterior desity of μ ad σ. For the right ivariat prior it is give by θ R = μ σ = π σ ( Г( ( / e (x μ =b, θ where b = ( ( s Γ( ( σ ( ( Г( ( / ad θ e ( = ( x s σ s dμdσ (9 Theorem 4: The Bayes estimators of ICV for differet priors are give below.. Right ivariat Jeffrey s prior θ R =. Left Ivariat Prior = θ L ( Г( ( / (0 Г( 3. Jeffrey s Rule Prior θ JR = 4. Uiform Prior θ U = ( ( Г( + ( / Г( +4 ( ( / ( Г( + ( Г( ( / (3

9 Bayes Estimator for Coefficiet of Variatio ad Iverse Coefficiet Probability Matchig Prior θ PM = Г( ( Г( ( / (4 Proof: Straight forward ad is omitted. 5. Bias ad Mea Square Error of the Bayes Estimator of Iverse Coefficiet of Variatio Theorem 5: The followig table gives the bias ad mea square error to the order O(adO( respectively of differet estimators of ICV. Prior Bias MSE. Right Ivariat ( Г( (. Left Ivariat Г( 3. Jeffrey s Rule Г( Uiform Г( + 5. Probability Matchig Г( / - ( μ σ + ( ( Г( + ( / -( μ σ ( ( / -( μ σ ( Г( ( / -( μ σ ( Г( ( / -( μ σ + ( x s + ( x s + ( x s + ( x s x s +3 +( Г( ( Г( ( +( Г( ( Г( + ( +4 +( Г( ( ( + +( Г( / ( μ σ / ( μ σ / ( μ σ ( Г( ( / ( μ σ +( Г( ( Г( ( / ( μ σ Proof: Give i the Appedix. Theorem 6: Amog the class of estimators {b θ } of Iverse Coefficiet of variatio θ, where θ deotes the maximum likelihood estimator of θ, the estimator with miimum mea square error to the order of O( is θ Proof: The expressio for MSE of (bθ is give by ( ( θ + + MSE of (bθ =b V(θ +θ (b- (5

10 730 Arua Kalkur T. ad Arua Rao Differetiatig with respect to b ad equatig it to zero we get b= θ V(θ +θ = ( ( θ + + (6 Substitutig θ for θ i b we get estimator as estimator. As, bθ θ θ ( ( θ + +, which is the optimal Table 5. presets the umerical values of Bias ad MSE of the estimators of ICV. From the umerical results it follows that maximum likelihood estimators have the smallest MSE compared to other Bayes estimators. Table 5.. Bias ad Mea Square Error of the Bays Estimators of ICV =0 =30 θ B Bias MSE MSE(θ B Bias MSE MSE(θ.Left Ivariat Prior Right Ivariat Prior Jeffrey s Rule Prior Uiform Prior Probability Matchig Prior x x

11 Bayes Estimator for Coefficiet of Variatio ad Iverse Coefficiet Coclusio I this chapter we derive five Bayes estimators of coefficiets of variatio (CV as well as Iverse Coefficiet of Variatio (ICV respectively. The bias ad mea square error (MSE of these estimators are derived to the order of O( ad O( respectively. The umerical results idicate that the maximum likelihood estimator (MLE of CV ad ICV has smaller MSE compared to the Bayes estimators of CV ad ICV. Amog the class of estimators {aθ } of CV, the MLE θ of CV is domiated by the estimator θ to the order of O( (. I a parallel result the MLE of θ of ICV is (θ ++ domiated by the estimator θ. ( ( θ + + to the order of O( APPENDIX Proof of Theorem 3: Usig right ivariat prior, the bias of Bayes estimator of CV is give by Bias(θ, θ= E(θ θ σ μ π σ e (x μ ( ( s Γ( ( σ ( The MSE (θ = E((θ θ =V(θ + Bias(θ, θ The MSE of θ usig right ivariat prior is give by e ( σ s dμdσ - σ = Г( ( / μ σ μ MSE (θ = θ 4 + θ +( μ σ π σ e (x μ ( ( s Γ( ( σ ( = θ 4 + θ +( Г( ( / ( σ x θ, where θ = σ μ. e ( σ s dμdσ σ μ

12 73 Arua Kalkur T. ad Arua Rao Proof of Theorem 5 Usig right ivariat prior the bias of Bayes estimator of ICV is give by μ σ π σ e (x μ ( ( s Γ( ( σ ( e ( The MSE of θ usig right ivariat prior is give by σ x dμdσ - μ σ = (x s ( Г( ( / μ σ + π σ e (x μ θ + ( μ σ θ +( ( Г( ( ( ( s Γ( ( σ ( / - θ, where θ = μ σ e ( σ x dμdσ θ = + Similarly the Bias ad MSEs of other Bayes estimators of CV ad ICV ca be obtaied. Refereces [] D Cuha, J.G., &Rao, K.A. (05. Applicatio of Bayesia iferece for the aalysis of stock prices. Advaces ad Applicatios i Statistics, 6 (, [] Harvey, J., & Va der Merwe, A. J. (0. Bayesia cofidece Itervals for Meas ad Variaces of Logormal ad Bivariate logormal distributios. Joural of Statistical Plaig ad Iferece, 4(6, [3] McKay, A.T. (93. Distributio of the Co-efficiet of variatio ad the exteded t distributio. Joural of Royal Statistics Society.95, [4] Pag W.K., Leug P.K., Huag W. K. ad W. Liu. (003. O Bayesia Estimatio of the coefficiet of variatio for three parameter Weibull, Logormal ad Gamma distributios. [5] Pereira, C.A.B., Ster J.M. (999b. Evidece ad Credibility: Full Bayesia Sigificace Test for Precise Hypothesis, Etropy, [6] Sigh M. (993. Behavior of Sample Coefficiet of Variatio draw from several distributios, Sakya, 55, series B., Pt, [7] Sharma ad Krisha (994. Asymptotic samplig distributio of iverse coefficiet of variatio ad its applicatios. IEEE Trasactios o Reliability. 43(4,

5 Statistical Inference

5 Statistical Inference 5 Statistical Iferece 5.1 Trasitio from Probability Theory to Statistical Iferece 1. We have ow more or less fiished the probability sectio of the course - we ow tur attetio to statistical iferece. I statistical