An Efficient Estimator Improving the Searls Normal Mean Estimator for Known Coefficient of Variation

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1 ISSN: , A Effcet Estmator Improvg the Searls Normal Mea Estmator for Kow Coeffcet of Varato Ashok Saha Departmet of Mathematcs & Statstcs, Faculty of Scece & Techology, St. Auguste Campus The Uversty of the West Ides, Trdad & Tobago; WEST INDIES E-mal: saha.ashok@gmal.com Abstract: Ths paper addresses the ssue of fdg a optmal estmator of the ormal populato mea whe the coeffcet of varato s kow ad s expected to be rather hgh, as per the plot surveys of the populato at had. The paper proposes a Effcet Estmato Improvg the well-kow Searles Normal Mea Estmator. The Relatve Effceces [as compared to the usual ubased sample-mea estmator y ] estmator per the proposed strategy has o smple algebrac form, ad hece s ot ameable to a aalytcal study determg ts relatve gafuless, as compared to the usual ubased sample mea estmator. Nevertheless, we exame these relatve effceces of our estmator wth respect to the usual ubased estmator y, usg a llustratve smulato study wth hgh replcato. MATLAB R203a s used programmg ths llustratve Smulated Emprcal Numercal Study. Keywords: MVUE, MMSE, Complete Suffcet Statstc, Smulated Emprcal Numercal Study I. INTRODUCTION Ths paper addresses the ssue of fdg a optmal estmator of the ormal populato mea whe the coeffcet of varato s kow ad s expected to be rather hgh, as per the plot surveys of the populato at had. Ths mght well happe to be the case partcularly f the populato mea θ s expected to be postve eve though the populato stadard devato, say σ happes to be relatvely large. Such cases are kow to be relevat practcally the statstcal modelg applcatos the areas of astroomy, stock-markets, bodversty, ad evrometal sceces, etc. Searls (964) [7], Kha (968) [3], Gleser & Healy (976) [2], Arholt & Hebert (995) [], Modrag M Lovrc & Saha, Ashok (20), [4], Saha, Ashok (20) [5], Wsto A. Rchards, Rob Atoe, Ashok Saha, ad M. Raghuadh Acharya (200) [9], Skrepek, Grat H; Saha, Ashok; Atoe, Rob (205) [0], ad Wrght, Kmberly & Raghuadh M. Acharya (2009) [] cosdered the problem of estmatg the ormal populato mea ad varace, whe ts coeffcet of varato (c. v.) s kow/ukow. I the cotext of ths paper, the reader would beeft by perusg [6] Samuel-Cah, E. (994), Combg Ubased Estmators, The Amerca Statstca, 48, & Searls, Doald T. & Itarapach, P. (990) [8]. It s very well kow that the Searles (964) estmator for the ormal populato mea gets to be: Say, SE = y / (+a 2 /) Where, a s the Kow Coeffcet of Varato (σ/θ) [ ] & s the sze of the radom sample [ 30] from the Normal populato N (θ, a 2.θ 2 ) , IJAERA - All Rghts Reserved 70

2 Icdetally, Wsto A. Rchards, Rob Atoe, Ashok Saha, ad M. Raghuadh Acharya (200) cosdered the sample couterpart of Searls Estmator of effcet estmato. Also, Modrag M Lovrc & Ashok Saha (20) cosdered usg the sample coeffcet of varato for effcet estmato of Normal populato varace. The major fact s that the statstc ( y, S 2 ); Where, S 2 = ( y y ) 2 / (-) Sample-Varace {Well-kow Estmator of σ 2 } based o the radom sample of sze ; from the Normal populato N (θ, a 2.θ 2 ) s suffcet but NOT complete for {θ, σ 2 }; rederg the uavalablty of the Rao- Blackwellzato. Hece, essetally, we are gropg dark for the UMVU & MMSE estmators for θ, or ay fucto of θ, as such. II. THE PROPOSITION OF PROPOSED UNBIASED (EFFICIENT) ESTIMATOR OF NORMAL MEAN (PPDUEFFESTROMEAN) FOR INCREASING THE RELATIVE EFFICIENCY OF THE ORDINARY SEARLE S ESTIMATOR (OSE) We start by cosderg the Mmum Mea Square (MMSE)/Ordary Searle s Estmator (OSE), usg the kow Coeffcet of Varato ( C.V. ) a σ/μ σ/θ, ad the Usual Ubased Estmator Sample Mea y. The well-kow Searles Estmator of the Normal mea wth kow coeffcet of varato a s as follows: SE = y / (+a 2 /) OSE SEARLESESTROMEAN; Say. (2.) Where, S 2 = ( y ) 2 / (-) Sample-Varace {Well-kow Estmator of σ 2 }. (2.2) y The Relatve Effcecy of ay Estmator (Relatve to the Usual Ubased Estmator Sample Mea y ) are defed as: Reff ( ) = [MSE ( )/V ( x )] x00% [Defed %]. (2.3) Preparatory to the proposto of our Proposed Ubased (Effcet) Estmator Of Normal Mea (PPDUEFFESTROMEAN), we ote a elemetary result: Lemma 2.: For a radom sample of sze from the Normal Populato N (θ, a 2.θ 2 ); say, y, y2, y3,... y we have E [S] = C*σ C*(a. θ); where, C = { 2 /( )}* { / 2}/ {( ) / 2}. (2.4) Hece, a Ubased Effcet Estmator of θ : S/{C.a)} Proof: It follows from the well-kow fact that{( ) * S / } ( ). Q. E. D. I vew of the above smple result our Proposed Ubased (Effcet) Estmator of Normal Mea (PPDUEFFESTROMEAN) gets to be as follows: 205, IJAERA - All Rghts Reserved 7

3 PPDUEFFESTROMEAN = S/{C.a}. Where; a = σ/θ, S 2 = = { 2 /( )}* { / 2}/ {( ) / 2}. (2.5) III. THE SIMULATION EMPIRICAL NUMERICAL STUDY ( y ) 2 / (-) [S Sqrt (S 2 )], ad C I the precedg secto, t s apparet, that the extet of Relatve Ga Effcecy of Estmato wll be algebracally rather too trcate, as the ssue wll deped o the values of the parameters lke, a, θ, ad σ. Cosequetly, the aswer to the questo as to what s the relatve ga/achevemet pursug the Proposed Ubased (Effcet) Estmator Of Normal Mea (PPDUEFFESTROMEAN) of the ormal populato mea les tryg to kow t through a llustratve Smulato Emprcal Numercal Study, as s attempted ths secto. These Relatve Effceces (Relatve to the Usual Ubased Estmator Sample Mea y ); Say Reff ( ) s have bee calculated for SIX llustratve values of the sample sze : 35, 50, 75, 00, 50 & 200 & THREE llustratve values of the Populato Stadard Devato σ : 8, & 5. We could assume that, wthout ay loss of geeralty [Usg the traslato of the Paret Normal Populato Data by 0- y wthout chagg the Populato Stadard Devato σ a.θ ], & for the smplcty of the llustrato, the NORMAL populato mea θ = 6 [.e. postve]. The values of the actual MSE s are calculated by cosderg the radom samples of sze usg 55,555 replcatos (pseudo-radom ormal samples of the sze ) for the TWO estmators as also for the usual ubased estmator, amely the sample mea y. Hece values of Reff ( ) s calculated as per (2.3). These Reff ( ) s reported to the closest three decmal places of ther respectve actual values three tables the APPENDIX. MATLAB R203a s used programmg the calculatos ths llustratve Smulated Emprcal Numercal Study. IV. CONCLUSIONS As expected, the Relatve Effcecy of the proposed Proposed Ubased (Effcet) Estmator Of Normal Mea (PPDUEFFESTROMEAN) estmator of the Normal Populato Mea s way above that of the Mmum Mea Square (MMSE)/Ordary Searle s Estmator (OSE) for all sample szes, as also for all the llustratve value-combatos: {θ, σ}! As oted earler, t s very sgfcat to ote aga that, the fact that the Rao-Blackwellzato s uavalable the absece of a complete-suffcet statstc for θ, ad hece, essetally, we are gropg dark for the UMVU & MMSE estmators for θ, or ay fucto of θ, as such. Ths fact has bee semal to the motvatoal zeal for us to look for a more effcet estmator of the Normal Populato Mea whe Y ~ N (θ, a 2.θ 2 ). Our proposed estmator s ubased, but NOT UMVUE. We had a fulfllg success, but t could NOT be ruled out that the bettermet s of our proposto s ruled ad t s a ope problem to work out, f at all, a UMVU /UMMSE estmator of θ, ad we are workg o t. y 205, IJAERA - All Rghts Reserved 72

4 V. REFERENCES [] Arholt, A. T. & Hebert, J. E. (995), Estmatg the Mea wth kow Coeffcet of Varato, The Amerca Statstca, 49, [2] Gleser, L. J., & Healy, J.D. (976), Estmatg the Mea of a Normal Dstrbuto wth Kow Coeffcet of varato, Joural of the Amerca Statstcal Assocato, 7, [3] Kha, R. A. (968), A Note o Estmatg the Mea of a Normal Dstrbuto wth Kow Coeffcet of Varato, Joural of the Amerca Statstcal Assocato, 63, [4] Modrag M Lovrc & Ashok Saha (20). A Iteratve Algorthm for Effcet Estmato of Normal Varace Usg Sample Coeffcet of Varato, IterStat. # 00. [5] Saha, Ashok (20), Effcet Estmator of Populato Varace of Normal Dstrbuto wth Kow Coeffcet of Varato, IterStat. # 00. [6] Samuel-Cah, E. (994), Combg Ubased Estmators, The Amerca Statstca, 48, [7] Searls, Doald T. (964), The Utlzato of a kow Coeffcet of Varato the Estmato Procedure, Joural of the Amerca Statstcal Assocato, 59, [8] Searls, Doald T. & Itarapach, P. (990), A Note o a Estmator for the varace that Utlzes the Kurtoss, The Amerca Statstca, 44, [9] Wsto A. Rchards, Rob Atoe, Ashok Saha, ad M. Raghuadh Acharya (200), O Effcet Iteratve Estmato Algorthm Usg Sample Couterpart of the Searles Normal Mea Estmator wth Exceptoally Large But Ukow Coeffcet of Varato, IterStat. # 003. [0] Skrepek, Grat H; Saha, Ashok; Atoe, Rob (205), A Improved Mmum Mea Squared Error Estmate of the Square of the Normal Populato Varace Usg Computatoal Itellgece, Joural of Appled Mathematcs ad Boformatcs 5., pp: [] Wrght, Kmberly & Raghuadh M. Acharya (2009), O Effcet Varace Estmato for Normal Populatos, IterStat; November , IJAERA - All Rghts Reserved 73

5 APPENDIX: TABLE A. RELATIVE EFFICIENCIES [W.R.T y ] OF SERALES & OUR PROPOSED MEAN ESTIMATOR { = 35; μ = 6}. SEARLESESTROMEAN PPDUEFFESTROMEAN RELATIVE EFFICIENCIES [W.R.T y ] OF SERALES & OUR PROPOSED MEAN ESTIMATOR { = 50; μ = 6}. SEARLESESTROMEAN PPDUEFFESTROMEAN RELATIVE EFFICIENCIES [W.R.T y ] OF SERALES & OUR PROPOSED MEAN ESTIMATOR { = 75; μ = 6}. PPDUEFFESTROMEAN PPDUEFFESTROMEAN RELATIVE EFFICIENCIES [W.R.T y ] OF SERALES & OUR PROPOSED MEAN ESTIMATOR { =00; μ = 6}. SEARLESESTROMEAN PPDUEFFESTROMEAN RELATIVE EFFICIENCIES [W.R.T y ] OF SERALES & OUR PROPOSED MEAN ESTIMATOR { = 50; μ = 6}. SEARLESESTROMEAN PPDUEFFESTROMEAN RELATIVE EFFICIENCIES [W.R.T y ] OF SERALES & OUR PROPOSED MEAN ESTIMATOR { = 200; μ = 6}. SEARLESESTROMEAN PPDUEFFESTROMEAN σ s: σ =6 σ =7 σ =8 σ =0 σ = , IJAERA - All Rghts Reserved 74