Joural of Moder Applied Statistical Methods Volume 6 Issue Article 4 December 07 Parameter Estimatio I Weighted Rayleigh Distributio M. Ajami Vali-e-Asr Uiversity of Rafsaja Rafsaja Ira m.ajami@vru.ac.ir S. M. A. Jahashahi Uiversity of Sista ad Baluchesta Zaheda Ira mehdi.jahashahi@gmail.com Follow this ad additioal works at: http://digitalcommos.waye.edu/jmasm Part of the Applied Statistics Commos Social ad Behavioral Scieces Commos ad the Statistical Theory Commos Recommeded Citatio Ajami M. & Jahashahi S. M. A. (07). Parameter Estimatio I Weighted Rayleigh Distributio. Joural of Moder Applied Statistical Methods 6() 56-76. doi: 0.37/jmasm/50949540 This Regular Article is brought to you for free ad ope access by the Ope Access Jourals at DigitalCommos@WayeState. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of DigitalCommos@WayeState.
Joural of Moder Applied Statistical Methods November 07 Vol. 6 No. 56-76. doi: 0.37/jmasm/50949540 Copyright 07 JMASM Ic. ISSN 538 947 Parameter Estimatio I Weighted Rayleigh Distributio M. Ajami Vali-e-Asr Uiversity of Rafsaja Rafsaja Ira S. M. A. Jahashahi Uiversity of Sista ad Baluchesta Zaheda Ira A weighted model based o the Rayleigh distributio is proposed ad the statistical ad reliability properties of this model are preseted. Some o-bayesia ad Bayesia methods are used to estimate the β parameter of proposed model. The Bayes estimators are obtaied uder the symmetric (squared error) ad the asymmetric (liear expoetial) loss fuctios usig o-iformative ad reciprocal gamma priors. The performace of the estimators is assessed o the basis of their biases ad relative risks uder the two above-metioed loss fuctios. A simulatio study is costructed to evaluate the ability of cosidered estimatio methods. The suitability of the proposed model for a real data is show by usig the Kolmogorov-Smirov goodess-of-fit test. Keywords: Bayesia estimators estimatio methods goodess-of-fit loss fuctio reliability weighted model Itroductio The Rayleigh distributio has bee used i may areas of research such as reliability life-testig ad survival aalysis. Modelig the lifetime of radom pheomea has bee aother area of study for which the Rayleigh distributio has bee sigificatly used. Beig first itroduced by Rayleigh (880) this statistical model was origially derived i coectio with a problem i acoustics. More details o the Rayleigh distributio ca be foud i Johso et al. (994) ad refereces therei. The Rayleigh distributio has the followig probability desity fuctio (pdf) ad the cumulative distributio fuctio (cdf) respectively M. Ajami is a Assistat Professor i the Departmet of Statistics. Email him at m.ajami@vru.ac.ir. S. M. A. Jahashahi is a Professor i the Departmet of Statistics. Email him at mehdi.jahashahi@gmail.com. 56
AJAMI & JAHANSHAHI x x f x exp x 0 0 F x x exp x 0 0. Weighted distributios are employed maily i research associated with reliability bio-medicie meta-aalysis ecoometrics survival aalysis reewal processes physics ecology ad brachig processes which are foud i Patil ad Rao (978) Gupta ad Kirmai (990) Gupta ad Keatig (985) Oluyede (999) Patil ad Ord (976) ad Zele ad Feileib (969). A weighted form of Rayleigh distributio has bee published by Reshi et al. (04). They itroduced a ew class of Size-biased Geeralized Rayleigh distributio ad also ivestigated the various structural ad characterizig properties of that model. I additio they studied the Bayes estimator of the parameter of the Rayleigh distributio uder the Jeffrey s ad the exteded Jeffrey s priors assumig two differet loss fuctios. They compared four estimatio methods by usig mea square error through simulatio study with varyig sample sizes. I fact weighted distributios arise i practice whe observatios from a sample are recorded with uequal probabilities Suppose X is a o-egative radom variable with its ubiased pdf f(xβ) β is a parameter the g distributio is weighted versio of f ad is defied as E w X w x f x g x where the weight fuctio w(xα) is a o-egative fuctio ad 0 < E(w(Xα)) is a ormalizig costat which is E(w(Xα)) = w(xα)f(xβ)dx. Furthermore α is a parameter which may or may ot deped o β ad E(w(Xα)) = /E g (/w(xα)) is the harmoic mea of w(xα) with the pdf g(.). Whe w(xα) = x α α = 0 the distributio is referred to as weighted distributios of order α. x f x gx. () E X 57
PARAMETER ESTIMATION IN WEIGHTED RAYEIGH DISTRIBUTION For α = or the pdf () are referred to as legth-biased (size-biased) ad area-biased distributios respectively. A weighted Rayleigh (WR) distributio is proposed based o () ad all calculatios are doe based upo this model but i the sectios of umerical simulatios ad applicatio to real data a legth-biased Rayleigh (BR) distributio is used without loss of geerality. Because determiig the value of α depeds o the samplig method so it is ot ecessary to estimate α i practice therefore the focus o estimatig the β parameter. Weighted Rayleigh distributio I the followig the WR(αβ) distributio is itroduced ad the some properties icludig the r th momet the correspodig CDF ad hazard rate fuctio are calculated. Defiitio. A oegative radom variable X is said to have the WR(αβ) distributio provided that the variable s desity fuctio is give by x / g x x e x 0 0. / / / () Remark. Suppose that X follows WR(αβ) ad let U = X /β the U follows Γ(α/+) distributio. Remark. The WR(αβ) distributio belogs to the expoetial family. Therefore T = Σ i=x i is a sufficiet complete statistic. The r th momets are useful for iferece ad model fittig. A result that allows us to compute the momets of the WR(αβ) distributio is give i the followig lemma. emma. If X be a radom variable with desity fuctio () the the r th momet is give by 58
AJAMI & JAHANSHAHI r r r/ r/ E X where r is a positive iteger. Proof. Accordig to () let x /β = u the we have r r x x / E X e dx 0 / / / r r r r u r/ r/ E X u e du. / / 0 / emma cocludes V X CV X / / E X. 59
PARAMETER ESTIMATION IN WEIGHTED RAYEIGH DISTRIBUTION The correspodig CDF of the WR(αβ) distributio is as follows: x / / G x t e dt / x / a where / 0 / z a z t e dt deotes the lower icomplete gamma fuctio. 0 I additio the survival ad the hazard rate fuctios of the WR(αβ) distributio are ad G x / t e t dt / x / / x / / h x a respectively where x / x e / x / / / a z t e dt deotes upper icomplete gamma z fuctio. I special cases if α = correspodig legth-biased distributio is x x / g x e x 0 0 3/ ad if α = correspodig area-biased distributio is 3 x x / g x e x 0 0. Plots of legth-biased ad area-biased (ABR) distributios for some parameter values are displayed i Figure. Some possible shapes of the BR ad ABR hazard rate fuctios are displayed i Figure 60
AJAMI & JAHANSHAHI Figure. The BR(β) (left pael) ad ABR(β) (right pael) desity fuctios for some parameter values. Figure. The BR(β) (left pael) ad ABR(β) (right pael) hazard rate fuctios for some parameter values. Parameter estimatio I this sectio the method of momets the maximum likelihood method uiformly miimum variace ubiased method maximum goodess-of-fit method ad some Bayesia methods are used to estimate the β parameter of the model. 6
PARAMETER ESTIMATION IN WEIGHTED RAYEIGH DISTRIBUTION Method of momets estimator Hereafter let X X be a radom sample from the WR(αβ) distributio. The method of momets estimator (MME) is ˆ X MME. Maximum likelihood estimator The likelihood fuctio ca be writte as / i / / Xi i i ; x x X e x 0 0. Oe ca easily calculate maximum likelihood estimator (ME) of β by takig atural logarithm ad derivative relative to β as ˆ T ME where T = Σ i=x i. To study asymptotic ormality of ˆME calculate the Fisher iformatio I(β) as I (3) So accordig to theorem 8 of Ferguso (996) I ˆ N0 ME D as. 6
AJAMI & JAHANSHAHI 63 Therefore a 00( α)% approximate cofidece iterval of β ca be obtaied as / ˆ ME Z I where Z α/ is the α/ th percetile poit of the stadard ormal distributio. Uiformly miimum variace ubiased estimator Based upo emma. E X Therefore E T ad T E which is a fuctio of the sufficiet ad complete statistic T that is ubiased for β. Thus based o ehma-scheffe theorem we have
PARAMETER ESTIMATION IN WEIGHTED RAYEIGH DISTRIBUTION ˆ T UMVUE. Maximum goodess-of-fit estimators Maximum goodess-of-fit estimators (otherwise kow as miimum distace estimators) of the parameters of the CDF ca be calculated by miimizig ay distace of the empirical distributio fuctio (EDF) statistics regardig to the ukow parameters. As other research has show there is o uique EDF statistic which ca be cosidered the most efficiet for all situatios (Alizadeh ad Arghami 0). Kolmogorov-Smirov Cramer-vo Mises ad Aderso-Darlig statistics seem to be mometous i situatios are D sup G x G x i i i i i G i xi i i i i W G x G x p x G x G x A p x i G x where p(x (i) ) = G(x (i) ) G(x (i) ) is the probability uder H 0 ad cosiderig that G (.) is EDF for G(.). Bayes estimators of β Cosiderig β as a radom variable two differet priors amely Jeffreys ad reciprocal gamma are cosidered for β. Takig ito accout the priors two differet loss fuctios are used for the WR(αβ) model the first oe is the squared error loss (SE) fuctio ad the secod oe is liear expoetial (INEX) loss fuctio. Bayes estimator based o Jeffreys prior Based o (3) the Jeffreys prior is i 64
AJAMI & JAHANSHAHI I / 0 ad the the posterior desity will be i / X i x e / (4) which follows reciprocal gamma distributio as X i i x : rgamma. The Bayesia estimator of β uder the SE fuctio is where the SE fuctio is ˆ T SE ˆ ˆ ad T = Σ i=x i. I the followig Bayesia estimator is calculated uder the INEX loss fuctio. This loss fuctio was proposed by Varia (975) ad Zeller (986). The INEX loss fuctio for scale parameter β is give by a 0 e a a (5) ˆ where ad ˆ is a estimator of β. The sig ad magitude of a represet the directio ad degree of asymmetry respectively (see Solima 000 ad Saku 0). Uder INEX loss fuctio (5) ad usig the posterior (4) the posterior mea of loss fuctio (Δ) is 65
PARAMETER ESTIMATION IN WEIGHTED RAYEIGH DISTRIBUTION ˆ a ˆ a E e E e ae oe ca easily obtai ˆ which miimizes the posterior expectatio of the loss fuctio (5) deoted by ˆJ as ˆ T exp a / ). J a Bayes estimator based o reciprocal gamma prior Suppose β follows reciprocal gamma distributio as prior distributio which is b e The the posterior desity satisfies b/ 0. bi Xi / / x e / so the Bayesia estimator of β uder the SE fuctio is ˆ T b SERG c where c = (α + ) + σ. I special case if we suppose σ = b = 0 the Bayesia estimator of β is ˆ T SERG which is equal to ME. I additio Bayesia estimator of β uder the INEX loss fuctio is 66
AJAMI & JAHANSHAHI ˆ RG d T b where a exp / d. a The risk efficiecy of ˆ SEJ regardig to ˆ J uder INEX ad squared errors loss fuctio based o Jeffreys prior If radom variable X follows the distributio fuctio () so X obeys Γ((α/+)β) the T : Γ((α/+)β) as / / / h t t e T 0. Because the risk fuctios of estimators ˆSEJ ad ˆJ are importat ˆ R ˆ calculate these risk fuctios which are deoted by R J SEJ R ˆ S J ad ˆ S SEJ R where the subject deotes risk relative INEX loss fuctio ad the subject S deotes risk relative to SE. emma. et X : WR(αβ) the risk fuctio of ˆSEJ uder INEX loss fuctio with respect to the Jeffreys prior is a a SEJ / ˆ a / R e a. / / Proof. By defiitio 67
PARAMETER ESTIMATION IN WEIGHTED RAYEIGH DISTRIBUTION ˆ SEJ a ˆ ˆ SEJ R SEJ E e a h tdt 0 ˆ SEJ a ˆ SEJ e htdt a htdt a. 0 0 (6) It is easy to verify ˆ SEJ / a a a (I). e h t dt e 0 / ˆ / SEJ (II). htdt. 0 / Substitutig (I)-(II) ito (6) the result desired follows. Corollary. Based o emma oe ca coclude that emma 3. a a R J e e a ˆ / / /. SE fuctio with respect to the Jeffreys prior is R S ˆ J et X : WR(αβ) the the risk fuctio of ˆJ uder a a e e a. a a a e 68
AJAMI & JAHANSHAHI Proof. By defiitio ˆ ˆ ˆ ˆ RS J. 0 J h t dt E J E J (7) However (I). E ˆ J / / / e a a a / e (II). E ˆ J. a Substitutig (I)-(II) ito (7) the proof is completed. Corollary. I the same procedure of emma 3 the R ˆ S SEJ uder the SE is R S / ˆ SEJ. / / / / Defiitio. The risk efficiecy of ˆ regardig to ˆ uder loss fuctio is defied as RE R ˆ ˆ ˆ. R ˆ 69
PARAMETER ESTIMATION IN WEIGHTED RAYEIGH DISTRIBUTION The risk efficiecy of ˆ SERG regardig to ˆ RG uder INEX ad SE fuctios based o reciprocal gamma s prior I the followig the risk fuctios of estimators ˆSERG ad ˆRG are calculated. ˆ ˆ ˆ R ˆ. Therefore they are deoted by R RG R SERG RS RG ad S SERG Corollary 3. et X : WR(αβ) the the risk fuctio of ˆSERG uder the INEX ad the SE fuctios ad reciprocal gamma prior are R b a c ˆ e a R b a ˆ SERG SERG / c c 4b 4b b c c. Corollary 4. Similar to Corollary 3 uder the INEX ad the SE fuctios ad reciprocal gamma prior we have ad R / k b k a ˆ e RG ˆ RG 4 4 RS d b b d 4 bd where exp a k /. 70
AJAMI & JAHANSHAHI Numerical simulatios I the followig some experimetal results are preseted to ivestigate the effectiveess of the differet estimatio methods which have bee so far performed. Bias ad MSE for o-bayesia estimators are mostly compared for differet estimatio methods. I this study differet sample sizes of = 0 0 (small) 30 40 (moderate) 50 (large) ad 00 (very large) are cosidered. I Table the average estimates of β based o 0000 replicatios are preseted for differet estimatio methods i which the MSEs are oted i the paretheses. As ca be see i Table amog simple estimators the ME ad UMVUE have the smallest values of bias ad MSE for various values of sample size so ME ad UMVUE are the best estimatio methods i terms of bias ad MSE. I additio the other two good methods of estimatio i priority of order are MME ad CVM. Table. Bias ad MSE values of simple estimators for β parameter ME MME UMVUE KS CVM AD 0-0.00550 0.0400-0.00550 0.04430 0.03930 0.030840 (0.000007) (0.000995) (0.000007) (0.00060) (0.00057) (0.00095) 0-0.00770 0.006340-0.00770 0.0050 0.070 0.04600 (0.000006) (0.000040) (0.000006) (0.00045) (0.0005) (0.0003) 30-0.00060 0.005690-0.00060 0.009430 0.00970 0.0480 (0.000003) (0.00003) (0.000003) (0.000088) (0.000084) (0.0003) 40-0.00070 0.003430-0.00070 0.006660 0.00560 0.007670 (0.00000) (0.0000) (0.00000) (0.000044) (0.00003) (0.000058) 50-0.003730-0.000380-0.003730 0.0090 0.00050 0.0080 (0.00000) (0.000000) (0.00000) (0.000000) (0.000000) (0.000000) 00 0.000550 0.00840 0.000550 0.00360 0.0030 0.00490 (0.000000) (0.000000) (0.000000) (0.00000) (0.00000) (0.00000) Bias values ad risk fuctios are computed to compare cosidered Bayesia estimators. A compariso of this type is eeded to check whether a estimator is iadmissible uder some loss fuctio. Therefore if it is so the estimator would ot be used for the losses specified by that loss fuctio. For this purpose the risks of the estimators ad the efficiecy of them are computed. I each case a = a = b = ad σ = are take without loss of geerality. Because comparig differet loss fuctios is ot reasoable compare the results i similar loss fuctio but i differet priors. Accordig to results compiled i Tables 3 5 ad 6 all the four cosidered Bayesia estimators 7
PARAMETER ESTIMATION IN WEIGHTED RAYEIGH DISTRIBUTION based o reciprocal gamma prior have small values of bias. Further the ˆSERG estimator has smaller bias tha ˆRG estimator for a = while ˆRG estimator has smaller bias tha ˆSERG estimator for a =. Accordig to Tables 3 5 ad 6 amog the four cosidered Bayesia risks R ˆ has the smallest values of risk for various values of based SE the S RG sample size. Also amog the four cosidered Bayesia risks based INEX the R ˆ J has the smallest values of risk for various values of sample size. Table. Bias ad risk values of Bayesia estimators for β parameter ad a = bias ˆ bias ˆ R ˆ R ˆ R ˆ R ˆ SEJ J s SEJ 0 0.07-0.090 0.00 0.056 0.047 0.03 0 0.035-0.047 0.04 0.03 0.00 0.06 30 0.04-0.03 0.05 0.0 0.0 0.0 40 0.06-0.05 0.08 0.06 0.009 0.008 50 0.0-0.0 0.04 0.03 0.007 0.007 00 0.005-0.0 0.007 0.007 0.003 0.003 s J SEJ J Table 3. Bias ad risk values of Bayesia estimators for β parameter ad a = bias ˆ bias ˆ R ˆ R ˆ R ˆ R ˆ SEJ J s SEJ 0 0.065-0.039 0.099 0.07 0.037 0.03 0 0.034-0.07 0.04 0.035 0.08 0.06 30 0.0-0.0 0.05 0.03 0.0 0.0 40 0.08-0.007 0.08 0.07 0.009 0.008 50 0.03-0.007 0.04 0.04 0.007 0.007 00 0.007-0.003 0.007 0.007 0.003 0.003 s J SEJ J Table 4. Relative risk values of Bayesia estimators for β parameter 0 0 30 40 50 00 RE ˆ ˆ.54.33.49.0.087.043 J SEJ ˆ ˆ J SEJ ˆ ˆ J SEJ ˆ ˆ J SEJ s a= RE.788.338.4.57.4.060 a= RE.64.078.05.038.03.05 s a=- RE.38.74..083.066.03 a=- 7
AJAMI & JAHANSHAHI Table 5. Bias ad risk values of Bayesia estimators for β parameter ad a = bias SERG ˆ bias R ˆ R ˆ R ˆ R ˆ ˆ RG s SERG s RG SERG 0 0.06-0.083 0.073 0.045 0.504.37 0 0.03-0.046 0.035 0.07 0.50.487 30 0.0-0.03 0.03 0.09 0.500.556 40 0.05-0.05 0.07 0.05 0.500.593 50 0.0-0.0 0.04 0.0 0.500.67 00 0.007-0.009 0.007 0.006 0.500.666 RG Table 6. Bias ad risk values of Bayesia estimators for β parameter ad a = bias SERG ˆ bias R ˆ R ˆ R ˆ R ˆ ˆ RG s SERG s RG SERG 0 0.060-0.03 0.073 0.050 4.96 0.474 0 0.034-0.04 0.035 0.09 5.365 0.45 30 0.0-0.0 0.03 0.00 5.53 0.407 40 0.05-0.009 0.07 0.05 5.66 0.397 50 0.04-0.006 0.04 0.03 5.670 0.39 00 0.007-0.003 0.007 0.006 5.778 0.380 RG Table 7. Relative risk values of Bayesia estimators for β parameter 0 0 30 40 50 00 RE ˆ ˆ.743.33..56.3.060 RG SERG ˆ ˆ RG SERG ˆ ˆ RG SERG ˆ ˆ RG SERG s a= RE 0.8 0.0 0.96 0.93 0.9 0.88 a= RE.5.4.57.7.093.046 s a=- RE 0.37.65 3.593 4.3 4.474 5.05 a=- Applicatio to real data Here i order to display the usage of proposed model i real data it is eeded to aalyze two sets of the seve from the afore preseted data i paper by Beett ad Fillibe (000). Reportedly they have otified miority electro mobility for p-type Ga -x Al x As with seve differet values of mole fractio. To do so two data sets are employed relatig to the mole fractios of 0.5 ad 0.30. The data values are as followed: 73
PARAMETER ESTIMATION IN WEIGHTED RAYEIGH DISTRIBUTION Data Set (belogs to mole fractio 0.5): 3.05.779.604.37.4.045.75.55.96.54.06 0.7948 0.7007 0.69 0.675 0.6449 0.888.5.397.506.58. Data Set (belogs to mole fractio 0.30):.658.434.88.09.959.84.530.366.65.04 0.998 0.74 0.6403 0.576 0.5647 0.5873 0.803.00.50.347.368. To evaluate the fittig quality of the Rayleigh ad BR distributios the Kolmogorov-Smirov (K-S) tests ad AIC ad BIC s criterios are used. The iformatio about comparig both models are give i Table 8. Sice probability values of the BR model are greater tha correspodig values of the Rayleigh model ad the AIC ad BIC criterios of the BR model are less tha correspodig values of the Rayleigh model. Although the values of cosidered statistics are ot sigificatly differet but we it ca be ifered that the BR distributio fits better tha the Rayleigh distributio i both cosidered data. The MEs of β are 0.93 ad 0.7309 ad the 95 percet cofidece itervals of β based o MEs as suggested above uder headig Parameter Estimatio ca be obtaied as (0.6067.577) ad (0.47570.986) respectively. Table 8. Comparig related statistics for Rayleigh ad BR Data Model D p.value AIC BIC Rayleigh 0.4 0.7458 46.0090 47.0540 BR 0.75 0.847 45.960 46.960 Rayleigh 0.354 0.7883 40.3870 4.430 BR 0.3 0.880 39.780 40.860 Coclusio Differet estimatio procedures were studied for estimatig the ukow scale parameter of the WR(αβ) distributio beig the maximum likelihood estimator the method of momet estimator uiformly miimum variace ubiased estimator maximum goodess-of-fit estimators ad the Bayes estimators. Sice it is ot possible to compare differet methods theoretically some simulatios were used for compariso of differet estimators with respect to biases mea squared errors ad risks. All the four cosidered Bayesia estimators based o reciprocal gamma prior have small values of bias. I additio the ˆSERG estimator has smaller bias 74
AJAMI & JAHANSHAHI tha ˆRG estimator for a = but ˆRG estimator has smaller bias tha ˆSERG estimator for a =. Amog the four cosidered Bayesia risks based SE the ˆ S RG smallest values of risk ad based INEX the ˆ J R has the R has the smallest values of risk for various values of sample size. Thus from a Bayesia perspective we suggest usig ˆRG estimator based o SE ad usig ˆJ based o INEX loss fuctio. The performace of the ME ad UMVUE is also quite satisfactory ad i overall o-bayesia estimators are better tha Bayesia estimators thereby employig of the ME ad UMVUE estimators ca be recommed for all practical purposes. Ackowledgemets Portios of this paper are developed from the authors earlier coferece presetatio (Ajami & Jahashahi 06). Refereces Ajami M. & Jahashahi S. M. A. (06 August 4-6). Compariso of various estimatio methods for size-biased Rayleigh distributio. Paper preseted at the 3 th Iraia Statistical Coferece Shahid Bahoar Uiversity of Kerma Ira. Alizadeh N. H. & Arghami N. R. (0). Mote Carlo compariso of five expoetiality tests usig differet etropy estimates. Joural of Statistical Computatio ad Simulatio 8() 579 59. doi: 0.080/00949655.00.496368 Beett H. S. & Fillibe J. J. A. (000). A systematic approach for multidimesioal closed form aalytic modelig: miority electro mobilities i Ga -x Al x As heterostructures Joural of Research of the Natioal Istitute of Stadards ad Techology 05(3) 44 45. doi: 0.608/jres.05.037 Ferguso T. S. (996). A Course I arge Sample Theory. New York: Chapma ad Hall. Gupta R. C. & Keatig J. P. (985). Relatios for reliability measures uder legth biased samplig Scadiavia Joural of Statistics 3 49 56. 75
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