G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan


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1 Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity Multa, Pakista Muhammad Aslam Departmet of Statistics Bahauddi Zakariya Uiversity Multa, Pakista Muhammad Javed Departmet of Statistics Bahauddi Zakariya Uiversity Multa, Pakista Abstract I this paper, we preset that how much the variaces of the classical estimators, amely, maximum likelihood estimator ad momet estimator deviate from the miimum variace boud while estimatig for the Maxwell distributio. We also sketch this differece for the egative iteger momet estimator. We ote the poor performace of the egative iteger momet estimator i the said cosideratio while maximum likelihood estimator attais miimum variace boud ad becomes a attractive choice. Keywords: Classical estimator; CramerRao iequality; Maximum likelihood estimator; Maxwell distributio; Miimum variace boud; Momet estimator; Negative iteger momet estimator. 1. Itroductio I the cotext of the kietic molecular theory of gases, a gas cotais a large umber of particles i rapid motios. Each particle has a differet speed, ad each collisio betwee particles chages the speeds of the particles. A uderstadig of the properties of the gas requires a uderstadig of the distributio of particle speeds. The Maxwell distributio describes the distributio of particle speeds i a ideal gas. This distributio has a variety of applicatios i the study of the distributio of speeds of molecules i thermal equilibrium as give by statistical mechaics (see Papoulis, 1984 for more details). The probability desity fuctio of Maxwell distributio is give as x x f ( x; ) = exp ( ), for > 0, x > 0 (1.1) 3 π Pak. j. stat. oper. res. Vol.II No. 006 pp
2 G. R. Pasha, Muhammad Aslam, Muhammad Javed With the applicatios of the Maxwell distributio kietic molecular theory of gases ad a umber of other similar cases, the estimatio for its ukow parameter also become crucial. I the preset paper, we compare two classical estimators, amely, maximum likelihood estimator (MLE) ad estimator by method of momets (MME) with the egative iteger momet estimator (NIME). Mohsi ad Shahbaz (005) also coducted such kid of compariso while estimatig the parameter of iverse Rayleigh distributio but they just compare MLE with NIME. But i our study, i additio of MME, we maily, compute miimum variace boud (MVB) for ay ubiased estimator of ukow parameter of the Maxwell distributio ad compute the deviatio of this MVB form the variaces of the said estimators. I Sectio, we compute the classical estimates of the ukow value of the parameter for desity (1.1) alog with variaces. For this we use MLE ad MME. Sectio 3, dedicates for the estimatio by method of egative iteger momets. I Sectio 4, we give CramerRao iequality ad compute the MVB for estimatio of. I Sectio 5, we discuss the deviatio of the variaces from the MVB ad relative efficiecies while Sectio 6 cocludes.. Classical Estimatio The MLE is oe of the classical estimators i statistical iferece. Efro (198) explaied the method of maximum likelihood estimatio alog with the properties of the estimator. Accordig to Aldrich (1997), the makig of maximum likelihood was oe of the most importat developmets i 0th cetury statistics. I his paper, he cosidered Fisher s chagig justificatios for the method. It ca be foud that the log likelihood of (1.1) is; log (L) = log ( ) 13 log + log ( ) x i  x i, (.1) π 1 1 ad, resultatly, the MLE of is give as with ML = var( ML 1 x 3 i, (.) Γ ) = 3 3 Γ (.3) The method of momet (MM) is also a commoly used method of estimatio. I this method, the sample momets are assumed to be estimates of populatio momets ad thus momet estimates for the ukow values of populatio parameters are foud (see Lehma ad Casella, 1998, for details). For ukow parameter i (1.1), the momet estimator ca be foud as 146 Pak. j. stat. oper. res. Vol.II No. 006 pp
3 Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator 1 π MM = X, (.4) where X is sample mea. also, ( 3π 8 var MM ) = (.5) 8 3. Negative Iteger Momet Estimator Negative iteger momets are useful i applicatios i several cotexts, otably i life testig problems. Bock et al. (1984) illustrated the examples of their use i the evaluatio of a variety of estimators. With the particular referece to Chisquare distributio, i the iverse regressio problem, Oma (1985) gave a exact formula for the mea squared error of Kruutchkoff s iverse estimator by use of egative iteger momets of the ocetral Chisquared variable. The rth egative iteger momet is defied as: r E{( X + c) } where X is a radom variable, c is a costat, ad r is a positive iteger. Before fidig the estimator by the method of egative iteger momets, we have the rth order egative iteger momets of Maxwell distributio is as r 1 3 r μ ( r) = Γ r π For r = 1, the first order egative iteger momet is; (3.1) μ 1 ( 1) = (3.) π Now accordig to method of egative iteger momet (NIM) estimatio, the estimator for ukow parameter for (1.1) is 1 NIM =, (3.3) π m ( 1) 1/ xi 1 where m( 1) =. For variace of estimator i (3.3), we have var( NIM ) 4. Miimum Variace Boud = π (3.4) Let X = (X 1, X,, X ) be a radom sample ad let f(x; ) deotes the probability desity fuctio for some model of the data with ukow parameter. If T(X) be Pak. j. stat. oper. res. Vol.II No. 006 pp
4 G. R. Pasha, Muhammad Aslam, Muhammad Javed ay statistic ad ψ() be its expectatio such that ψ() = E [T(X)]. Uder some regularity coditios (see Lehma ad Casella, 1998), it follows that for all, ψ ( ) Var( T ( X )), (4.1) I ( ) where I () is Fisher s iformatio matrix. This is called CramerRao iequality ad the right had side of (4.1) is called CramerRao lower boud or MVB. I particular, if T(X) is a ubiased estimator of, the the umerator becomes 1, ad MVB is simply 1/I (). Obviously, if the variace of estimator coicides with the MVB, it meas that we are usig a estimator that bears miimum variace i its class of estimators. For the ukow parameter i (1.1), it ca easily be show that for ay ubiased estimator of ; MVB = 6 (4.) 5. Deviatio from MVB ad Relative Efficiecy We compare all the three variaces (.3), (.5), ad (3.4) with the MVB give i (4.). Sice all these variaces cotais the term as a multiplier so we drop it while comparig them all. Fig. 5.1 shows the deviatio of the variaces (actual variaces i the figure should be multiplied by ) of the estimators uder cosideratio from the MVB. These values are draw agaist differet sample sizes. We ote that the variace of MLE has least deviatio from the MVB. As the sample size icreases, the variace of MLE begis to coicide with the MVB. For > 0, variace of MLE equalizes to the MVB. For small samples, the variace of momet estimator otably deviates from the MVB but this differece covers with the icrease i sample size. Whe sample size becomes larger tha 30, this variace begis to coicide with the MVB. I the case of NIM estimatio, the performace of NIME remais quite miserable, especially, for small sample. The variace of NIME remais largely deviat from the MVB as it falls miles away form the MVB. Although, the severity of the deviatio decreases with the icrease i sample size but eve the it does ot appear to make NIME attractive while comparig with its two competitors. As for as relative efficiecy is cocered, we ote the efficiecy of MLE relative to NIME as 3.45 for = 10 ad MLE remais about.5 times more efficiet as compared to NIME for larger sample size. Our fidigs thus also matches to those of Mohsi ad Shahbaz (005) who also declared MLE more efficiet as compared to NIME while estimatig the parameter of iverse Rayleigh distributio. 148 Pak. j. stat. oper. res. Vol.II No. 006 pp
5 Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator We also ote that MME remais about. times more efficiet as compared to NIME. We further ote that this relative efficiecy dose ot chage with the icrease i sample size. Whe we compare MLE with MME for variaces, we fid MLE about 7% more efficiet as compared to the momet estimator Variace MLE MME NIME MVB Sample Size Figure 5.1: Deviatio from MVB 6. Coclusio Although the egative iteger momets are gaiig popularity for their applicatios but NIM estimator performs poorly i the term of efficiecy while estimatig the ukow parameter of the Maxwell distributio. Its variace remai too far form the CramerRao lower boud for variace i.e., MVB. Whe we are dealig with the kietic molecular theory of gases or the matters of thermodyamic with the applicatio of the Maxwell distributio, MLE is the best choice for estimatig the ukow parameter. Eve for large samples ( > 30), momet estimator may equally be utilized. Refereces 1. Aldrich, J. (1997). R. A. Fisher ad the Makig of Maximum Likelihood Statistical Sciece, 3, Bock, M. E., Judge, G. G. ad Yacy, T. A. (1984). A Simple Form for the Iverse Momets of Nocetral ChiSquare ad F Radom Variables ad Certai Cofluet Hypergeometric Fuctios. J. Ecoometrics, 5, Efro, B. (198). Maximum Likelihood ad Decisio Theory. A. Statist. 10, Lehma, E. L., ad Casella, G. (1998). Theory of Poit Estimatio. Spriger, New York. Pak. j. stat. oper. res. Vol.II No. 006 pp
6 G. R. Pasha, Muhammad Aslam, Muhammad Javed 5. Mohsi, M. ad Shahbaz, M.Q., (005). Compariso of Negative Momet Estimator with Maximum Likelihood Estimator of Iverse Rayliegh Distributio, PJSOR, 1, Oma, S. D., (1985). A Explicit Formula for the Mea Squared Error of the Iverse Estimator i the Liear Calibratio Problem, Joural of Statistics Plaig ad Iferece, 11, Papoulis, A. (1984). Probability, Radom Variables, ad Stochastic Processes. McGrawHill, New York. 150 Pak. j. stat. oper. res. Vol.II No. 006 pp
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