A Comparative Study of Traditional Estimation Methods and Maximum Product Spacings Method in Generalized Inverted Exponential Distribution
|
|
- Marcus Richardson
- 5 years ago
- Views:
Transcription
1 J. Stat. Appl. Pro. 3, No. 2, (2014) 153 Joural of Statistics Applicatios & Probability A Iteratioal Joural A Comparative Study of Traditioal Estimatio Methods ad Maximum Product Spacigs Method i Geeralized Iverted Expoetial Distributio Umesh Sigh, Sajay Kumar Sigh ad Rajwat Kumar Sigh Departmet of Statistics ad DST-CIMS. Baaras Hidu Uiversity, Varaasi , Idia Received: 22 Feb. 2014, Revised: 28 Apr. 2014, Accepted: 30 Apr Published olie: 1 Jul Abstract: I this paper, we propose the method of Maximum product of spacigs for poit estimatio of parameter of geeralized iverted expoetial distributio (GIED). The aim of this paper is to aalyse the small sample behaviour of proposed estimators. Further, we have also proposed asymptotic cofidece itervals of the parameters ad the estimates of reliability ad hazard fuctio usig Maximum Product Spacigs (MPS) method ad compared with correspodig asymptotic cofidece itervals ad the estimates of reliability ad hazard fuctio of Maximum Likelihood estimatio (MLEs). A comparative study amog the method of MLE, method of least square (LSE) ad the method of maximum product of spacigs (MPS) is performed o the basis of simulated sample of GIED. The MPS method outperforms the method of MLE ad the method of LSE. Furthermore, compariso of differet estimatio method have bee proposed o the basis of K-S distace ad AIC. For umerical illustratio oe real data set has bee cosidered. Keywords: GIED, Reliability characteristic, method of Maximum Product Spacigs, method of Maximum Likelihood Estimatio, method of Least Squares Estimates ad Iterval Estimatio. 1 Itroductio I statistical iferece problem, we are give a set of observatios x 1,x 2,,x. These are the values take by some radom pheomea about whose distributio we have some kowledge. For parameter estimatio, various estimatio methods are widely discussed i literature. Oe ofte uses traditioal estimatio methods such as the method of momets, method of least square, method of weighted least square ad maximum likelihood estimatio (MLE). Each of them havig their ow advatages ad limitatios but amog these methods the most popular method of estimatio is maximum likelihood estimatio method. Which ca be justified o the groud of its various useful properties like cosistecy, sufficiecy, ivariace ad asymptotic efficiecy ad its easy computatios. The MLE method works efficietly if each cotributio to the likelihood fuctio is bouded above. It is the situatio with all discrete distributios. However, havig such ice properties ad better applicability it also has some weakess as metioed by various authors i differet cotext. Its greatest weakess is that it ca ot work for heavy tailed cotiuous distributio with ukow locatio ad scale parameters (Pitma, 1979, p. 70). It also creates problem i situatios where there is oly mixture of cotiuous distributio ad the MLE method ca break dow. It was established by some authors that MLE does ot always provide precise estimates for certai distributios such as gamma, Weibull, ad log ormal distributios. I all these cases the critical difficulty is that there are paths i parameter space with locatio parameter teds to smallest observatio alog which the likelihood becomes ifiite. Ufortuately i such situatios estimates of other parameters becomes icosistet. Harter ad Moore [5] suggests a alterative way to use local maxima as a alterative of global maxima, this ca be effective but ot full proof there beig some weakess as poited out by Cheg for this see [1]. I the cotext of Harter ad Moore, Huzurbazar [19] has show that o statioary poit (ad hece o local maximum) ca provide a cosistet estimator, whe the cocer distributio is J-shaped, for example i the case of Correspodig author rajwat37@gmail.com
2 154 U. Sigh et. al. : A Comparative Study of Traditioal Estimatio Methods ad Maximum... Weibull ad gamma distributios whe the shape parameter is less tha uity. Thus whether we cosider a global or a local maximum, Maximum Likelihood estimatio is boud to fail. The practical problem is that eve if the distributio is ot J-shaped, so that parameters ca i priciple be cosistetly estimated by local Maximum Likelihood estimatio as the sample size teds to ifiity, it ca happe that, with fixed sample size, a particular radom sample gives rise to a likelihood fuctio with o local maximum at all (Griffiths, [20]), this occur maily whe shape parameter equal to uity. Several authors has suggested alterative methods to MLE, either ivolvig modificatio to MLE method or method of momets or percetiles. Despite of the above problems whe MLE is applied it outperforms the alterative methods. I order to overcome these shortcomigs ad havig better applicability i such types of situatios which possess properties similar to MLE, Cheg ad Ami [1] itroduced the Maximum Product of Spacigs (MPS) method as a alterative to MLE for the estimatio of parameters of cotiuous uivariate distributios. Cheg ad Ami proposed to replace the likelihood fuctio by a product of spacigs ad cojectured that it retais most of the properties of the method of maximum likelihood. Raeby [3] idepedetly developed the same method as a approximatio to the Kullback-Leibler measure of iformatio. The approach of Cheg ad Ami is more ituitively attractive ad ca, to some extet, be regarded as a practical solutio to the problems liked with likelihood (Titterigto, [15]), but that of Raeby is more powerful theoretically ad allows the derivatio of the properties of MPS estimators. It may be oted that MPS method is especially suited to the cases where oe of the parameter has a ukow shifted origi, as it is the case i three parameter logormal, gamma ad Weibull distributios or to the distributios havig J-shape.. I order to make a geeral idea of advatages of MPS estimatio over MLE, we first list some good properties of MPS estimatio, which were showed by Cheg ad Ami [1], icludig sufficiecy, cosistecy ad asymptotic efficiecy. I certai cases, it is possible to obtai the distributioal behaviour of a MPS estimator for all sample sizes. Thus, for the uiform distributio with ukow edpoits, the MPS estimators are precisely the MVU estimators ad so their distributio is kow exactly solved by Cheg ad Ami [1]. The cosistecy of MPS estimators have bee discussed i detail by Cheg ad Ami [16]. I brief, asymptotically MPS are at least as efficiet as MLE estimators whe they exit. For distributio where the ed poits are ukow ad the desity is J-shaped the MLE is boud to fail, but MPS gives asymptotically efficiet estimators. MPS estimators will ot ecessarily be fuctio of sufficiet statistics i geeral. However, for the case whe the support of desity fuctios are kow, MPS estimator will show the same asymptotic properties as ML estimators icludig the oe of asymptotic sufficiecy. 1.1 The Model The radom variable X has a geeralized iverted expoetial distributio with two parameter α ad λ if it has a probability desity fuctio of the form: f(x)= ( )( αλ x 2 exp λ )[ ( 1 exp λ (α 1), x 0, λ,α > 0 (1) x x)] Where α is shape parameter ad λ is scale parameter, ad its CDF is give by [ ( F(x)=1 1 exp x)] λ α, α,λ > 0 (2) The model ca be cosidered as aother useful two-parameter geeralizatio of the Iverted expoetial distributio (IED). This lifetime distributio ca model various shapes of failure rates ad hece various shapes of ageig criteria. It is oted that the GIED is reduced to the IED for α = 1. I literature, estimatio of parameters i the two parameter GIED is discussed extesively, but o oe has performed compariso of MLE ad MPS. Readers are referred to the followig refereces: Abouammah ad Alshigiti [12], Gupta ad Kudu [14], Gupta ad Kudu [13]. Various properties of the GIED like reliability ad hazard fuctio, mea ad mode is discussed extesively by Abouammah ad Alshigiti [12]. I this paper, the method of product of spacigs is applied for estimatig the parameters i a two parameter GIED. The purpose here is to examie MPS estimates of the parameters of the GIED ad we also costruct 95% cofidece iterval usig MLE ad MPS. The method of product of spacigs is compared with the method of Least squares estimates (LSE) ad the method of MLE usig simulatio. MSE ad K-S distace are calculated ad o the basis of K-S distace through maximum product of spacigs method is better fitted tha MLE to the cosidered real data. AIC is
3 J. Stat. Appl. Pro. 3, No. 2, (2014) / calculated for MPS ad MLE ad both are compared. The mai objective of this paper is to aalyse the small sample behaviour of MPS. As we all kow that it is impossible to aalyse the whole data set due various reasos like cost factor, time factor etc. The orgaisatio of the paper is as follows: I sectio 2, Differet estimatio procedures are metioed ad estimates of reliability ad hazard fuctios usig MPS method is proposed ad compared with MLE. I sectio 3, asymptotic cofidece itervals of the parameters usig MPS method is proposed ad compared with MLE. I sectio 4, real data illustratio ad its applicatio is discussed, ad compariso of estimatio procedure based o K-S statistics is proposed. I sectio 5, a compariso is coducted usig simulatio study. Fially cocludig remarks are preseted i sectio 6. 2 Parameter estimatio For the cosidered distributio, we use two very kow ad popular method amely least squares method ad the maximum likelihood estimatio method ad oe which is ot very commo i.e MPS method for estimatig the parameters α ad λ. 2.1 Least square estimatio Let x 1 < x 2 < <x be ordered radom sample of ay distributio with CDF F(x), we get The least squares estimates are obtaied by miimizig Puttig the cdf of GIED i equatio (4) we get P(α,λ)= P(α,λ)= E(F(x i ))=i/() (3) (F(x i ) i/()) 2 (4) ( ( ( 1 1 exp λ )) α 2 i/()) (5) x i I order to miimize Equatio (5), we have to differetiate it with respect to λ ad α, which gives the followig equatio: )( )) α 1 ( ( )) α ) (α) exp ( λ 1 exp ( λ 1 1 exp ( λ xi xi xi i/() = 0 (6) x i i 1 i 1 [( ( ( 1 1 exp λ )) α )( ( i/() 1 exp λ )) α ( ))] l 1 exp ( λxi = 0 (7) x i x i The above Likelihood equatio caot be solved aalytically therefore we ca use ay iterative procedure such as Newto- Rapso method, to get the solutio. 2.2 Maximum likelihood estimators The likelihood fuctio for a sample of size from GIED (1) is give by: ( L(θ)=(α λ ( )exp λ (1/x i )) (1/x 2 i [(1 exp ) λ ))] α 1,t 0, α,λ > 0 (8) x i ad the log likelihood fuctio is give as M = ll(θ)=lλ + lα+ l ( 1/x 2 ) i λ l(1/x i )+(α 1) [ ) ] l (1 exp ( λxi ), (9)
4 156 U. Sigh et. al. : A Comparative Study of Traditioal Estimatio Methods ad Maximum... After differetiatig the above equatio with respect to parameter α ad λ ad the equatig them to zero we got the ormal equatio as follows: [ )] (/α) + l 1 exp ( λxi = 0 (10) (/λ) (1/x)+(α 1) (1/x i )exp( λ/x i ) [1 exp( λ/x i )] = 0 (11) The above ormal equatios are ot i ice closed form, therefore we ca use ay iterative procedure such as Newto- Rapso method, to get the solutio. 2.3 Maximum Product of spacigs estimators Here, the method of maximum product of spacigs is described briefly as follows: Cosiderig a uivariate distributio F(x θ) with desity f(x θ) where it is required to estimate θ. The desity is assumed to be strictly positive i a iterval (α, β ) ad zero elsewhere, α ad β may also be elemets of θ, α= - ad β = are icluded. That is F(x θ)=0 ad f(x θ)=0 for x<α:, F(x θ)=1. ad f(x θ)=0 for x>β. Let x 1 < x 2...<x be a complete ordered sample, further defie x 0 = α, x = β. The spacigs are defied as follows: D 1 = F(x 1:,θ), D = 1 F(x :,θ), D i = F(x i:,θ) F(x i 1:,θ),i = 2,3,, as the spacigs of the sample. Clearly the spacigs sum to uity i.e D i = 1. The MPS method is to choose θ which maximizes the geometric mea of the spacigs i.e G = ( D i) 1/, or equivaletly, its logarithm S = log G. The mai aim for maximizig G (or S) is that the maximum, which is bouded above because of the coditio D i = 1, is foud oly whe all D is are equal. Cheg ad Ami [1] showed that maximizig S as a method of parameter estimatio is as efficiet as ML estimatio. Additioally, they showed that ties preset i data would ot be a matter of cocer i parameter estimatio. The CDF of the GIED is give by the equatio (2) ad the spacigs are defied as follows: D 1 = F(x 1 )=1 [ (1 exp( λ/x 1 )) α] (12) Ad the geeral term of spacigs is give by, D i = F(x i ) F(x (i 1) )= D () = 1 F(x )= [ (1 exp( λ/x )) α] (13) [( ) α ] [( ) α ] 1 exp λ/x (i 1) 1 exp λ/x i ) Such that D i = 1, MPS method choose θ which maximizes the product of spacigs or i other words to maximize the geometric mea of the spacigs i.e Takig the logarithm of G we get, Or we may write S as G= ( (14) ) 1/ D i (15) S=1/() ld i (16) { } S= 1 ld 1 + () ld i + ld i=2 { = 1 [ l 1 (1 e λ/x 1 ) α] [ + () l (1 e λ/x i 1 ) α (1 e λ/x i ) α]} i=2 + 1 { l [(1 e x) λ α]} () (17)
5 J. Stat. Appl. Pro. 3, No. 2, (2014) / After differetiatig the above equatio with respect to parameters ad the equatig them to zero we get the ormal equatio as follows: [ S α = 1 1 i=2 S λ = (1 e λ/x 1) α l(1 e λ/x 1) 1 (1 e λ/x 1 ) α l(1 e λ/x )) (1 e λ x i 1 ) α l(1 e λ x i 1 ) (1 e λ x i ) α l((1 e λ (α/x 1)((1 e λ x 1 ) α 1 )(e λ i=2 1 (1 e λ x 1 ) α (1 e λ x i 1 ) α (1 e λ x i ) α x 1 ) ] + x i )) =0 (α/x i 1)((1 e λ x i 1 ) α 1 )(e λ x i 1 ) (α/x i )((1 e λ x i ) α 1 )(e λ [ (α/x )((1 e λ x ) α 1 )(e λ (1 e λ x ) α x (1 e λ i 1 ) α (1 e λ x i ) α ] x ) The above ormal equatios caot be solved aalytically therefore we ca use Newto-Rapso method, i order to get the solutio. = 0 x i 1 ) (18) (19) 2.4 Reliability ad hazard fuctio I this sectio, we propose the estimatio of reliability ad hazard fuctio usig MPS for specified value of time say (t=4). Cheg ad Ami [1] ad Coole ad Newby [17] had metioed i their paper that MPS also shows the ivariace property just like MLE. So o this basis usig the ivariace property we estimate the reliability ad hazard fuctio. The MPS estimates of the reliability ad hazard fuctio is give as: ( ) ˆα ˆR MPS (t)= 1 e ˆλ t, α,λ,t > 0 (20) ( ) ˆα ˆλ e ˆλ t t Ĥ MPS (t)= (1 e 2 ), α,λ,t > 0 (21) ˆλ t respectively, where ˆα = ˆα mp ad ˆλ = ˆλ mp are the MPS estimates of the parameter α ad λ respectively. I equatio (20) ad (21) puttig the estimates of MLE, we ca get the expressio for the reliability ad hazard fuctio usig MLE. 3 Asymptotic cofidece itervals I this sectio, we propose the asymptotic cofidece itervals usig MPS, as it was metioed by Cheg ad Ami [1] ad Staislav Aatolyevi ad Grigory Koseok [18] i their papers that the MPS method also shows asymptotic properties like the Maximum likelihood estimator. Keepig this i mid, we may propose the asymptotic cofidece itervals usig MPS. The exact distributio of the MPS caot be obtaied explicitly. Therefore, the asymptotic properties of MPS ca be used to costruct the cofidece itervals for the parameters. I( ˆα, ˆλ) is the observed Fishers iformatio matrix ad is defie as: [ I( ˆα, ˆλ)= ] S αα S αλ S λ α S λ λ (α= ˆα mp,λ=ˆλ mp ) (22)
6 158 U. Sigh et. al. : A Comparative Study of Traditioal Estimatio Methods ad Maximum... Fig. 1 Variatio of sample size with respect to MSE for differet choices of α ad for fixed value of λ = 1 Fig. 2: Mea Square Error of the estimates for α = 0.5,1 ad λ = 1 with variatio of sample size ()
7 J. Stat. Appl. Pro. 3, No. 2, (2014) / The first derivatives of the product of spacigs i.e the fuctio S with respect to parameter α ad λ are give by Equatios (18) ad (19) ad hece the secod derivatives are calculated as follows: The secod derivative of the fuctio S with respect to α is give as: S αα = [ F(x 1,α,λ)F αα (x 1,α,λ) F α (x 1,α,λ) 2 F(x 1,α,λ) 2 { } {F(x i,α,λ) F(x i 1,α,λ)} F αα(x i,α,λ) F αα(x i 1,α,λ) {F(x i,α,λ) F(x i 1,α,λ)} 2 { } F α(x i,α,λ) F α(x 2 i 1,α,λ) i=2 {F(x i,α,λ) F(x i 1,α,λ)} 2 { 1 {1 F(x,α,λ)}F αα(x,α,λ)+ F α(x,α,λ) {1 F(x,α,λ)} 2 ] } 2 (23) The secod derivative of the fuctio S with respect to λ is give as, S λ λ = 1 F(x 1,α,λ)F λ λ (x 1,α,λ) F λ (x 1,α,λ) 2 F(x 1,α,λ) {F(x i,α,λ) F(x i 1,α,λ)} i=2 {F(x i,α,λ) F(x i 1,α,λ)} 2 { } 2 F λ (x i,α,λ) F λ (x i 1,α,λ) 1 {F(x i,α,λ) F(x i 1,α,λ)} 2 { 1 {1 F(x,α,λ)} F λ λ (x,α,λ)+ F λ (x,α,λ) {1 F(x,α,λ)} 2 { F λ λ (x i,α,λ) F λ λ (x i 1,α,λ) ad the secod derivative of the fuctio S with respect to α,λ is give as: [ S αλ = S λ α = 1 F(x1,α,λ)F αλ (x 1,α,λ) F α (x 1,α,λ)F λ (x ] 1,α,λ) F(x 1,α,λ) 2 { } + 1 {F(x i,α,λ) F(x i 1,α,λ)} F αλ (x i,α,λ) F αλ (x i 1,α,λ) i=2 {F(x i,α,λ) F(x i 1,α,λ)} 2 { }{ } 1 F α (x i,α,λ) F α (x i 1,α,λ) F λ (x i,α,λ) F λ (x i 1,α,λ) {F(x i,α,λ) F(x i 1,α,λ)} 2 { }{ } 1 {1 F(x,α,λ)}F αλ (x,α,λ)+ F α (x,α,λ) F λ (x,α,λ) {1 F(x,α,λ)} 2 } 2 } (24) (25) Where F α(x,α,λ)= (1 e λ/x ) α l(1 e λ/x )
8 160 U. Sigh et. al. : A Comparative Study of Traditioal Estimatio Methods ad Maximum... F αα (x,α,λ)=(1 e λ/x ) α (l(1 e λ/x )) 2 F λ (x,α,λ)= α x (1 e λ/x ) α 1 e λ/x F λ λ (x,α,λ)= α { α 1 (1 e λ/x ) α 2 (e λ/x ) 2 1 } x x x (1 e λ/x ) α 1 e λ/x F e λ/x λ α (x,α,λ)= (1 e λ/x ) α 1{ } α(α 1)l(1 e λ/x )+1 x The asymptotic cofidece itervals of the parameters of GIED usig MLE is already calculated by A. M.Abouammoh ad Arwa M. Alshigiti [12]. the first derivatives of the log likelihood fuctio of GIED usig MLE with respect to parameters are give by equatio (10) ad (11), ad the secod derivatives are as follows: ad the secod derivative with respect to α,λ is give as: (ll) αα = α 2 (26) e λ x i (ll) λ λ =(α 1) x 2 λ i (1 e x i ) λ 2 (27) (ll) αλ = e λ x i x i e λ x i itervals for the parameters α ad λ is give as, ˆα± γ β 2 V( ˆα) ad ˆλ ± γ β 2 (28) So o the basis of these derivatives, we obtai the iformatio matrix I(α,λ ). The approximate(1 β)100% cofidece V(ˆλ) respectively, where γ β is the upper 2 ( β 2 ) percetile of stadard ormal distributio, ˆα = ˆα mp ad ˆλ = ˆλ mp are the MPS estimates of the parameter α ad λ ad V( ˆα) ad V(ˆλ) are elemets of I 1 ( ˆα, ˆλ). 4 Real data illustratio The data set cosidered i this sectio for illustratio, cotais stregths of glass polished aeroplae widow. The use of this data set is described by Fuller et al. (1994) to predict the lifetime for a glass aeroplae widow. Sice the data were used i a study to predict failure times, such type of study is a form of reliability aalysis. The data are as follows: 18.83, 20.8, , 23.03, 23.23, 24.05, , 25.5, 25.52, 25.8, 26.69, 26.77, 26.78, 27.05, 27.67, 29.9, 31.11, 33.2, 33.73, 33.76, 33.89, 34.76, 35.75, 35.91, 36.98, 37.08, 37.09, 39.58, , 45.29, The above data set is already fitted to GIED ad the K-S statistics, betwee the fitted ad the empirical distributio is also calculated ad estimates of the parameter usig MLE method is calculated by Abouammoh ad Alshigiti [12]. MLE of the parameters of the GIED ˆα ad ˆλ are ad respectively ad for the same data set we have calculated the estimates of the parameter through MPS method ad the estimates are ˆα mp = ad ˆλ mp = respectively, ad correspodig K-S distace calculated usig estimates of MLE ad MPS respectively, ad it comes out ad respectively. So o the basis of estimates ad K-S statistics, for the cosidered data set MPS fits better as compared to MLE. The result of these distaces shows that MPS serve better tha MLE i this data set. As K-S statistics, is extesively used for differet model compariso ad is cosidered oe of the best way of model compariso, but o oe had paid attetio o compariso of differet estimatio procedure based o K-S statistics. I literature several authors have discussed K-S methodology for differet model compariso i terms of distaces for a give data set. Cosiderig the similar approach o the basis of the K-S statistics, here, we propose compariso of estimatio procedure or method as least K-S distace provides the better method of estimatio for give data set. For the above data set we otice that K-S distace through MPS is smaller tha K-S distace through MLE. I the support of the above propositio empirical cumulative distributio fuctio (ECDF) plot has bee give below for MPS ad MLE. AIC
9 J. Stat. Appl. Pro. 3, No. 2, (2014) / Fig. 3: Figure 2: Mea Square Error of the estimates for α = 1.5,2 ad λ = 1 with variatio of sample size () is also calculated usig both MLE ad MPS for this real data set ad it comes out smaller whe estimates were provided from MPS as compare to MLE. Sice MPS provides lesser K-S distace ad AIC, we may say that MPS serve better tha MLE for the cosidered data set. Thus we propose to estimatio method compariso o the basis of K-S statistic. AIC MLE = AIC MPS = Simulatio studies I order to compare the MPS, the MLE ad the LSE methods i parameter estimatio, a set of simulatio was doe based o GIED. We have geerated five thousad samples from GIED for differet parameters settigs. It may be metioed here that the exact expressio of MSE ca ot be obtaied because estimates are ot foud i ice close forms. It may be also oted here that MSE will deped o sample size, scale parameter λ ad shape parameter α respectively. I this study differet variatio of sample size() say (=20,40,60,80,100,120), shape parameter α say α( = 1,2,3,4,5,6,7) ad scale parameter λ say λ (=1,2,3,4,5,6,7) have bee cosidered. To study the effect of variatio of the sample size we have cosidered the simulated MSE for α ad λ. Firstly, sample size is varied i.e differet values of are take for fixed value of scale parameter λ take as 1 ad 2, for differet choices of shape parameter α as α(= 0.5,1,1.5,2,3,4). Secodly, we varied shape parameter α i.e differet values of α for two differet choices of sample sizes (=30 ad =50) for fixed values of scale parameter λ as λ ( = 1,2), thirdly, we varied the scale parameter λ i.e differet values of λ is take for two differet choices of sample sizes (=30 ad =50) for fixed values of shape parameter α as (α = 2,3), correspodig graphs are attached. For all the above cosidered choices graph of MSE is plotted ad attached.
10 162 U. Sigh et. al. : A Comparative Study of Traditioal Estimatio Methods ad Maximum... Fig. 4: Mea Square Error of the estimates for α = 3,4 ad λ = 1 with variatio of sample size ()
11 J. Stat. Appl. Pro. 3, No. 2, (2014) / Fig. 5: Variatio of sample size with respect to MSE for differet choices of α ad λ = 2 We have chose sample size = 30 i secod ad third case because at = 30 chages are more visible i terms of MSE. We have also estimates the reliability ad hazard fuctios for α ad λ = 1,2 ad 3, for sample size = 20,30,50,80 cosiderig MLE ad MPS both, for this see Table 1, 2. Further we have costructed the cofidece iterval ad coverage probability for differet choices of α ad λ = 1,2 ad 3 ad for sample size =20,30,50,80 for this see table 2,3,4 ad 5. We have also compared the average legth of cofidece itervals of MPS with correspodig MLEs. R software is used i all computatios. O the basis of the results summarized i graphs ad table, some coclusios ca be draw which are stated as follows: 1: It is observed that as sample size icreases for fixed values of α ad λ the mea square error of the estimates decreases i all the three cosidered methods ad for large size of i.e after =80 all of them are early equivalet but MPS performs better tha other two cosidered method, see Figures 1 ad 2. It is also observed that as icreases MSE of reliability ad hazard fuctio follows the similar tred as stated above ad here also MPS perform better tha MLE (see Table 1). Furthermore, it is oticed that the average legth of cofidece iterval decreases as sample size icreases i both the cosidered case of MLE as well as MPS but the average legth is smaller i case of MPS as compared to MLE. The coverage probability obtaied here fairly attais the prescribed cofidece iterval. This is also true for small sample size, see Tables : From graph ad table, it is observed that as shape parameter α icreases for fixed sample size ad λ the MSE of the estimates of shape parameter α icreases for all the three cosidered cases but MSE of MPS is smaller tha LSE as well as MLE (see figure 3). It is oticed that for smaller values of shape parameter all of them are equally good but for larger value of shape parameter MPS is better tha other two. Geerally, shape parameters are difficult to estimate i most of the cases but here i case of GIED, oe should use MPS for all choices of λ ad. It is also observed that for fixed value of λ as the shape parameter α icreases MSE of reliability decreases but reverse tred is obtaied i the case
12 164 U. Sigh et. al. : A Comparative Study of Traditioal Estimatio Methods ad Maximum... Fig. 6: Variatio of α for differet choices of sample size (=30,50) ad λ = 1,2 of hazard, it is true for MPS ad MLE both but here also MPS serve better tha MLE i terms of MSE. The average legth of cofidece iterval icreases as we icrease the shape parameter α i both the case of MPS ad MLE. Further more, it is also observed that the coverage probability obtaied here fairly attais the prescribed cofidece iterval. 3: It is observed that as scale parameter λ icreases for fixed sample size ad shape parameter α the MSE of the estimates of scale parameter λ icreases ad MPS performs better i terms of MSE i compariso to other two methods. It is also observed that for smaller value of λ all of them are equivalet but for larger value of λ MPS is far better tha LSE as well as MLE see figure 4. It is also observed that for fixed value of α as the scale parameter λ icreases MSE of reliability decreases but reverse tred is obtaied i the case of hazard i case of MLE ad MPS both see Table 1. The average legth of cofidece iterval icreases as we icrease the scale parameter λ for fixed value of shape parameter α similar tred has bee observed i both the case of MPS ad MLE (see Table 2-5). Further more, it is also observed that the coverage probability obtaied here fairly attais the prescribed cofidece iterval ad o ay specific tred has bee observed. 4: From the plot of ECDF ad the value of AIC, it is observed that estimates of MPS fits better tha the estimates of MLE. 6 Coclusio This paper ivolved the compariso of estimates obtaied by MPS, MLE, ad LSE method usig GIED. For smaller sample size it is advised to use MPS as it perform better tha LSE as well as MLE. I this paper we have cosidered the problem of poit estimatio, cofidece iterval ad it also itroduces compariso of differet estimatio procedure o
13 J. Stat. Appl. Pro. 3, No. 2, (2014) / Fig. 7: Variatio of λ for differet choices of sample size (=30,50) ad α = 2,3 the basis of K-S statistic, for this a real data has bee icorporated ad ECDF plot is also give. From the graphs, it is observed that all the estimates appears to be cosistet. The average legth of cofidece iterval usig MPS is smaller tha that of MLE. We have foud that MPS method outperforms the other two method with smaller mea square error. The fidigs of this paper will be very useful to researchers, statisticia ad egieers where such types of thigs were required ad also i cases where we have small sample size to aalyse ad exclusively where GIED is used.
14 Table 1: Average estimates ad correspodig MSEs of the reliability ad hazard fuctio usig MLE ad MPS respectively at specified time say(t=4) for differet parameter settigs ad sample size. Reliability ad hazard usig MLE,t=4 =20 =30 =50 =80 Para rt ht rt ht rt ht rt ht 1, , , , , , , , , MAXIMUM PRODUCT SPACINGS R(t) H(t) (MPS) 1, , , , , , , , , U. Sigh et. al. : A Comparative Study of Traditioal Estimatio Methods ad Maximum...
15 J. Stat. Appl. Pro. 3, No. 2, (2014) / Table 2: Average estimates, coverage probability (i the brackets) ad correspodig cofidece itervals of the parameters α ad λ usig MPS for differet choices of parameter ad for sample sie =20 ad 30. CI AND COVERAGE PROBABILITY MPS =20 =30 Para α(cp) λ(cp) α(cp) λ(cp) 1, (0.9654) (0.964) (0.9636) (0.9548) (0.2395,1.4505) (0.1284,1.2788) (0.8975,1.8539) (0.5718,1.4983) 1, (0.966) (0.9628) (0.966) (0.953) (0.1571,1.4442) (0.1775,2.5652) (0.6109,1.5836) (1.8005,3.6929) 1, (0.9656) (0.9618) (0.965) (0.956) (0.4953,1.7380) (2.0397,5.5381) (0.0923,1.0677) (0.9450, ) 2, (0.9642) (0.9482) (0.9634) (0.9528) (0,2.8277) (0.1480,1.1370) (0.2349,2.4415) (0.5779, ) 2, (0.9714) (0.9566) (0.9648) (0.95) (0,3.0538) (0.6993,2.6476) (0.7629,3.0160) (1.0253,2.6279) 2, (0.966) (0.9596) (0.9666) (0.9484) (1.1893,4.1223) (1.6403,4.6007) (0.5983,2.8156) (0.5318,2.8675) 3, (0.9668) (0.9596) (0.9682) (0.95) (0.3334,5.4452) (0.7535,1.6584) (2.2181,5.9873) (0.6909,1.4533) 3, (0.963) (0.953) (0.967) (0.9474) (0,3.9963) (0.5667,2.3851) (0,3.6252) (0.9020,2.3533) 3, (0.963) (0.953) (0.9698) (0.9456) (0,4.9593) (1.7279,4.5179) (0.7889,4.5274) (1.5604,3.7566) Refereces [1] Cheg, R. C. H. ad Ami, N. A. K. (1983). Estimatig parameters i cotiuous uivariate distributios with a shifted origi. Joural of the Royal Statistical Society B 45, [2] D. L. Fitzgerald (1996). Maximum product of spacigs estimators for the geeralized Pareto ad log-logistic distributios. Stochastic Hydrology ad Hydraulics 10, Table 3: Average estimates, coverage probability (i the brackets) ad correspodig cofidece itervals of the parameters α ad λ usig MPS for differet choices of parameter ad for sample sie =50 ad 80. CI AND COVERAGE PROBABILITY MPS =50 =80 Para α(cp) λ(cp) α(cp) λ(cp) 1, (0.9572) ( ) (0.9518) (0.9450) (0.5423,1.2488) (0.3020,1.0117) (0.5543,1.1285) (0.7195,1.2810) 1, (0.9608) (0.9530) (0.9544) (0.9468) (0.7531,1.4718) (1.2761,2.7254) (0.4942,1.0670) (1.0908,2.2410) 1, (0.9624) (0.9470) (0.9534) (0.9446) (0.6232,1.3199) (2.1661,4.2704) (0.5486,1.1047) (1.8874,3.5803) 2, (0.9632) (0.9480) (0.9550) (0.9438) (0.9364,2.6279) (0.6742,1.2819) (0.7514,2.0445) (0.5774,1.0650) 2, (0.9616) (0.9434) (0.9498) (0.9440) (1.8147,3.4827) (1.9248,3.1392) (1.5378,2.8039) (1.6764,2.6076) 2, (0.9604) (0.9468) (1.9003,0.9538) (2.8732,0.9392) (1.0153,2.7006) (1.2994,3.1588) (1.4479,2.7118) (2.4640,3.8683) 3, (0.9636) (0.9434) (0.9540) (0.9426) (1.3852,4.0453) (0.5759,1.1409) (1.1708,3.2430) (0.5245,0.9668) 3, (0.9606) (0.9448) (0.9600) (0.9460) (0.8577,3.4972) (1.1978,2.3377) (1.8999,4.0433) (1.5154,2.4169) 3, (0.9628) (0.9382) (0.9556) (0.9428) (1.9603,4.6485) (2.0974,3.7866) (1.4215,3.5387) (2.1025,3.4448)
16 168 U. Sigh et. al. : A Comparative Study of Traditioal Estimatio Methods ad Maximum... Table 4: Average estimates, coverage probability (i the brackets) ad correspodig cofidece itervals of the parameters α ad λ usig MLE for differet choices of parameter ad for sample size =20 ad 30. CI AND COVERAGE PROBABILITY MLE =20 =30 Para α(cp) λ(cp) α(cp) λ(cp) 1, (0.9706) (0.9538) (0.9568) (0.9502) (0.4667,1.8570) (0.5008,1.7484) (0.5712,1.6290) (0.5865,1.5788) 1, (0.9724) (0.9476) (0.9622) (0.947) (0.4680,1.8740) (1.0174,3.5427) (0.5745,1.6408) (1.1859,3.1846) 1, (0.9642) (0.9556) (0.9598) (0.9396) (0.4661,1.8591) (1.5103,5.2748) (0.5741,1.6402) (1.7709,4.7559) 2, (0.965) (0.9418) (0.9666) (0.9484) (0.7223,4.1396) (0.5803,1.6307) (1.0060,3.5580) (0.6529,1.4897) 2, (0.9636) (0.9516) (0.9666) (0.943) (0.7196,4.1907) (1.1558,3.2442) (1.0009,3.5325) (1.3039,2.9780) 2, (0.9622) (0.9486) (0.9686) (0.9506) (0.7244,4.1698) (1.7454,4.8951) (1.0042,3.5456) (1.9530,4.4575) 3, (0.9678) (0.9526) (0.9616) (0.94) (0.8246,6.7882) (0.6092,1.5765) (1.3339,5.6402) (0.6762,1.4504) 3, (0.9658) (0.9486) (0.9654) (0.9526) (0.8276,6.7213) (1.2178,3.1552) (1.3316,5.6016) (1.3458,2.8888) 3, (0.964) (0.942) (0.962) (0.9498) (0.8252,6.7323) (1.8293,4.7372) (1.3351,5.6322) (2.0271,4.3473) Table 5: Average estimates, coverage probability (i the brackets) ad correspodig cofidece itervals of the parameters α ad λ usig MLE for differet choices of parameter ad for sample size =50 ad 80. CI AND COVERAGE PROBABILITY MLE =50 =80 Para α(cp) λ(cp) α(cp) λ(cp) 1, (0.9606) (0.9522) (0.9532) (0.9522) (0.6670,1.4436) (0.6708,1.4196) (0.7368,1.3371) (0.7397,1.3274) 1, (0.9602) (0.9482) (0.9524) (0.9458) (0.6675,1.4450) (1.3465,2.8486) (0.7356,1.3355) (1.4749,2.6470) 1, (0.9668) (0.9556) (0.9568) (0.9486) (0.6668,1.4429) (2.0217,4.2784) (0.7355,1.3346) (2.2057,3.9595) 2, (0.9636) (0.9494) (0.9534) (0.9486) (1.2431,3.0811) (0.7256,1.3617) (1.3882,2.7690) (0.7743,1.2698) 2, (0.9532) (0.9478) (0.9556) (0.957) (1.2354,3.0575) (1.439,2.7029) (1.3928,2.7792) (1.5506,2.5420) 2, (0.9582) (0.9446) (0.958) (0.9538) (1.2390,3.0698) (2.1725,4.0796) (1.3953,2.7856) (2.3286,3.8173) 3, (0.9626) (0.9486) (0.964) (0.953) (1.7397,4.7783) (0.7407,1.3280) (2.0087,4.3095) (0.7942,1.2555) 3, (0.9666) (0.9482) (0.9566) (0.9492) (1.7410,4.7812) (1.4922,2.6750) (2.0102,4.3160) (1.5849,2.5055) 3, (0.9602) (0.9462) (0.956) (0.9514) (1.7368,4.7663) (2.2213,3.9826) (2.0133,4.3237) (2.3809,3.7631) [3] Raeby, B., (1984). The Maximum Spacigs Method. A Estimatio Method Related to the Maximum Likelihood Method. Scad. J. Stat., 11, [4] Shah, A. ad Gokhale, D. V., (1993). O Maximum Product of Spacigs Estimatio for Burr XII Distributios. Commu. Stat. Simulat. Computat., 22(3), [5] Harter, H. L. ad Moore, A. H. (1965). Maximum likelihood estimatio of the parameters of Gamma ad Weibull populatios from complete ad from cesored samples. [6] Ghosh, S.R. Jammalamadaka (2001) A geeral estimatio method usig spacigs. Joural of Statistical Plaig ad Iferece 93.
17 J. Stat. Appl. Pro. 3, No. 2, (2014) / [7] Shao, Y. (2001). Cosistecy of the maximum product of spacigs method ad estimatio of a uimodal distributio. Statistica Siica 11, [8] Wog, T. S. T. ad Li, W. K. (2006). A ote o the estimatio of extreme value distributios usig maximum product of spacigs. IMS Lecture Notes Moograph Series 52, [9] Mezbahur Rahma ad Larry M. Pearso (2002): Estimatio i two-parameter Expoetial distributios. Joural of Stat. Computat. Simulat.,70(4), [10] Mezbahur Rahma ad Larry M. Pearso (2003): A ote o estimatig parameters i two-parameter Pareto distributios, Iteratioal Joural of Mathematical Educatio i Sciece ad Techology, 34:2, [11] Mezbahur Rahma, Larry M. Pearso ad Uros R Martiovic (2007): Method of product of spacigs i the two parameter gamma distributio, Joural of Statistical Research Bagladesh Vol. 41, [12] A.M. Abouammoh ad A.M. Alshigiti, Reliability estimatio of geeralized iverted expoetial distributio, J. Statist. Comput. Simul. 79(11) (2009), pp. 1301?1315. [13] R.D. Gupta ad D.Kudu, Geeralized expoetial distributio: Differet methods of estimatios, J. Statist. Comput. Simul. 69(4) (2001), pp. 315?338. [14] R.D. Gupta ad D. Kudu, Geeralized expoetial distributio, Aust. N. Z. J. Statist. 41(2) (1999). [15] Titterigto, D.M. (1985) Commet o Estimatig parameters i cotiuous uivariate distributios. Joural of the Royal Statistical Society Series B 47. [16] Cheg R.C.H.; Ami, N.A.K. 1979: Maximum product-of-spacigs estimatio with applicatios to the logormal distributio, Uiversity of Wales IST, Math Report79-1. [17] F.P.A Coole ad M.J Newby, A ote o the use of the product of spacigs i bayesia iferece. [18] Aatolyev Staislav ad Koseok Grigory, A alterative to maximum likelihood based o spacigs. [19] Huzurbazar, V. S. (1948) The likelihood equatio, cosistecy ad the maxima of the likelihood fuctio. A. Eugeics, 14, [20] Griffiths, D. A. (1980) Iterval estimatio for the three-parameter logormal distributio via the likelihood fuctio. Appl. Statist., 29 (1),
Double Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationConfidence interval for the two-parameter exponentiated Gumbel distribution based on record values
Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp. 61-68 Witer 2014 Joural homepage: www.ijorlu.ir Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More information1.010 Uncertainty in Engineering Fall 2008
MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval
More informationANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY
ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY Sulema Nasiru, MSc. Departmet of Statistics, Faculty of Mathematical Scieces, Uiversity for Developmet Studies, Navrogo, Upper East Regio, Ghaa,
More informationDepartment of Mathematics
Departmet of Mathematics Ma 3/103 KC Border Itroductio to Probability ad Statistics Witer 2017 Lecture 19: Estimatio II Relevat textbook passages: Larse Marx [1]: Sectios 5.2 5.7 19.1 The method of momets
More informationMaximum likelihood estimation from record-breaking data for the generalized Pareto distribution
METRON - Iteratioal Joural of Statistics 004, vol. LXII,. 3, pp. 377-389 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN Maximum likelihood estimatio from record-breakig data for the geeralized Pareto distributio
More informationEECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1
EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationThe Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution
Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 1263-1277 HIKARI Ltd, www.m-hikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationBayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function
Iteratioal Joural of Statistics ad Systems ISSN 973-2675 Volume 12, Number 4 (217), pp. 791-796 Research Idia Publicatios http://www.ripublicatio.com Bayesia ad E- Bayesia Method of Estimatio of Parameter
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationMathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution
America Joural of Theoretical ad Applied Statistics 05; 4(: 6-69 Published olie May 8, 05 (http://www.sciecepublishiggroup.com/j/ajtas doi: 0.648/j.ajtas.05040. ISSN: 6-8999 (Prit; ISSN: 6-9006 (Olie Mathematical
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
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
More informationDirection: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
More informationPOWER AKASH DISTRIBUTION AND ITS APPLICATION
POWER AKASH DISTRIBUTION AND ITS APPLICATION Rama SHANKER PhD, Uiversity Professor, Departmet of Statistics, College of Sciece, Eritrea Istitute of Techology, Asmara, Eritrea E-mail: shakerrama009@gmail.com
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationLecture Note 8 Point Estimators and Point Estimation Methods. MIT Spring 2006 Herman Bennett
Lecture Note 8 Poit Estimators ad Poit Estimatio Methods MIT 14.30 Sprig 2006 Herma Beett Give a parameter with ukow value, the goal of poit estimatio is to use a sample to compute a umber that represets
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationBayesian inference for Parameter and Reliability function of Inverse Rayleigh Distribution Under Modified Squared Error Loss Function
Australia Joural of Basic ad Applied Scieces, (6) November 26, Pages: 24-248 AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN:99-878 EISSN: 239-844 Joural home page: www.ajbasweb.com Bayesia iferece
More informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationModeling and Estimation of a Bivariate Pareto Distribution using the Principle of Maximum Entropy
Sri Laka Joural of Applied Statistics, Vol (5-3) Modelig ad Estimatio of a Bivariate Pareto Distributio usig the Priciple of Maximum Etropy Jagathath Krisha K.M. * Ecoomics Research Divisio, CSIR-Cetral
More informationInternational Journal of Mathematical Archive-5(7), 2014, Available online through ISSN
Iteratioal Joural of Mathematical Archive-5(7), 214, 11-117 Available olie through www.ijma.ifo ISSN 2229 546 USING SQUARED-LOG ERROR LOSS FUNCTION TO ESTIMATE THE SHAPE PARAMETER AND THE RELIABILITY FUNCTION
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationMOMENT-METHOD ESTIMATION BASED ON CENSORED SAMPLE
Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:
More informationApproximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece
More informationStatistical Theory MT 2008 Problems 1: Solution sketches
Statistical Theory MT 008 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. a) Let 0 < θ < ad put fx, θ) = θ)θ x ; x = 0,,,... b) c) where α
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationx = Pr ( X (n) βx ) =
Exercise 93 / page 45 The desity of a variable X i i 1 is fx α α a For α kow let say equal to α α > fx α α x α Pr X i x < x < Usig a Pivotal Quatity: x α 1 < x < α > x α 1 ad We solve i a similar way as
More informationStatistical Theory MT 2009 Problems 1: Solution sketches
Statistical Theory MT 009 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. (a) Let 0 < θ < ad put f(x, θ) = ( θ)θ x ; x = 0,,,... (b) (c) where
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationEstimation of Gumbel Parameters under Ranked Set Sampling
Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article 11-2014 Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, abuyaza_o@yahoo.com
More informationA goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality
A goodess-of-fit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationEfficient GMM LECTURE 12 GMM II
DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationExponential Families and Bayesian Inference
Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a d-parameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where
More informationA Method of Proposing New Distribution and its Application to Bladder Cancer Patients Data
J. Stat. Appl. Pro. Lett. 2, No. 3, 235-245 (2015) 235 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.12785/jsapl/020306 A Method of Proposig New Distributio
More informationECE 8527: Introduction to Machine Learning and Pattern Recognition Midterm # 1. Vaishali Amin Fall, 2015
ECE 8527: Itroductio to Machie Learig ad Patter Recogitio Midterm # 1 Vaishali Ami Fall, 2015 tue39624@temple.edu Problem No. 1: Cosider a two-class discrete distributio problem: ω 1 :{[0,0], [2,0], [2,2],
More informationImproved Class of Ratio -Cum- Product Estimators of Finite Population Mean in two Phase Sampling
Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 4 Issue 2 Versio.0 Year 204 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA
More informationA New Distribution Using Sine Function- Its Application To Bladder Cancer Patients Data
J. Stat. Appl. Pro. 4, No. 3, 417-47 015 417 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://dx.doi.org/10.1785/jsap/040309 A New Distributio Usig Sie Fuctio- Its Applicatio To
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationEstimation of Backward Perturbation Bounds For Linear Least Squares Problem
dvaced Sciece ad Techology Letters Vol.53 (ITS 4), pp.47-476 http://dx.doi.org/.457/astl.4.53.96 Estimatio of Bacward Perturbatio Bouds For Liear Least Squares Problem Xixiu Li School of Natural Scieces,
More informationA Note on Box-Cox Quantile Regression Estimation of the Parameters of the Generalized Pareto Distribution
A Note o Box-Cox Quatile Regressio Estimatio of the Parameters of the Geeralized Pareto Distributio JM va Zyl Abstract: Makig use of the quatile equatio, Box-Cox regressio ad Laplace distributed disturbaces,
More informationMinimax Estimation of the Parameter of Maxwell Distribution Under Different Loss Functions
America Joural of heoretical ad Applied Statistics 6; 5(4): -7 http://www.sciecepublishiggroup.com/j/ajtas doi:.648/j.ajtas.654.6 ISSN: 6-8999 (Prit); ISSN: 6-96 (Olie) Miimax Estimatio of the Parameter
More informationBootstrap Intervals of the Parameters of Lognormal Distribution Using Power Rule Model and Accelerated Life Tests
Joural of Moder Applied Statistical Methods Volume 5 Issue Article --5 Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests Mohammed Al-Ha Ebrahem
More informationStochastic Simulation
Stochastic Simulatio 1 Itroductio Readig Assigmet: Read Chapter 1 of text. We shall itroduce may of the key issues to be discussed i this course via a couple of model problems. Model Problem 1 (Jackso
More informationLecture 33: Bootstrap
Lecture 33: ootstrap Motivatio To evaluate ad compare differet estimators, we eed cosistet estimators of variaces or asymptotic variaces of estimators. This is also importat for hypothesis testig ad cofidece
More informationSummary. Recap ... Last Lecture. Summary. Theorem
Last Lecture Biostatistics 602 - Statistical Iferece Lecture 23 Hyu Mi Kag April 11th, 2013 What is p-value? What is the advatage of p-value compared to hypothesis testig procedure with size α? How ca
More informationIntroductory statistics
CM9S: Machie Learig for Bioiformatics Lecture - 03/3/06 Itroductory statistics Lecturer: Sriram Sakararama Scribe: Sriram Sakararama We will provide a overview of statistical iferece focussig o the key
More information( θ. sup θ Θ f X (x θ) = L. sup Pr (Λ (X) < c) = α. x : Λ (x) = sup θ H 0. sup θ Θ f X (x θ) = ) < c. NH : θ 1 = θ 2 against AH : θ 1 θ 2
82 CHAPTER 4. MAXIMUM IKEIHOOD ESTIMATION Defiitio: et X be a radom sample with joit p.m/d.f. f X x θ. The geeralised likelihood ratio test g.l.r.t. of the NH : θ H 0 agaist the alterative AH : θ H 1,
More informationThe new class of Kummer beta generalized distributions
The ew class of Kummer beta geeralized distributios Rodrigo Rossetto Pescim 12 Clarice Garcia Borges Demétrio 1 Gauss Moutiho Cordeiro 3 Saralees Nadarajah 4 Edwi Moisés Marcos Ortega 1 1 Itroductio Geeralized
More informationA NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES. Dennis D. Boos
.- A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES by Deis D. Boos Departmet of Statistics North Carolia State Uiversity Istitute of Statistics Mimeo Series #1198 September,
More informationBayesian Control Charts for the Two-parameter Exponential Distribution
Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com 2 Uiversity of the Free State Abstract By usig data that are the mileages
More informationSome Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation
; [Formerly kow as the Bulleti of Statistics & Ecoomics (ISSN 097-70)]; ISSN 0975-556X; Year: 0, Volume:, Issue Number: ; It. j. stat. eco.; opyright 0 by ESER Publicatios Some Expoetial Ratio-Product
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationLECTURE NOTES 9. 1 Point Estimation. 1.1 The Method of Moments
LECTURE NOTES 9 Poit Estimatio Uder the hypothesis that the sample was geerated from some parametric statistical model, a atural way to uderstad the uderlyig populatio is by estimatig the parameters of
More informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More informationSome Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation
Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA 70504-1010,
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More informationThis is an introductory course in Analysis of Variance and Design of Experiments.
1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class
More informationIntroduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT
Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory
More informationEconomics 241B Relation to Method of Moments and Maximum Likelihood OLSE as a Maximum Likelihood Estimator
Ecoomics 24B Relatio to Method of Momets ad Maximum Likelihood OLSE as a Maximum Likelihood Estimator Uder Assumptio 5 we have speci ed the distributio of the error, so we ca estimate the model parameters
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More informationSince X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain
Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the
More informationNUMERICAL METHODS COURSEWORK INFORMAL NOTES ON NUMERICAL INTEGRATION COURSEWORK
NUMERICAL METHODS COURSEWORK INFORMAL NOTES ON NUMERICAL INTEGRATION COURSEWORK For this piece of coursework studets must use the methods for umerical itegratio they meet i the Numerical Methods module
More informationThe (P-A-L) Generalized Exponential Distribution: Properties and Estimation
Iteratioal Mathematical Forum, Vol. 12, 2017, o. 1, 27-37 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.610140 The (P-A-L) Geeralized Expoetial Distributio: Properties ad Estimatio M.R.
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More informationApproximations to the Distribution of the Sample Correlation Coefficient
World Academy of Sciece Egieerig ad Techology Iteratioal Joural of Mathematical ad Computatioal Scieces Vol:5 No:4 0 Approximatios to the Distributio of the Sample Correlatio Coefficiet Joh N Haddad ad
More informationOn Marshall-Olkin Extended Weibull Distribution
Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 O Marshall-Olki Exteded Weibull Distributio Haa Haj Ahmad Departmet of Mathematics, Uiversity of Hail Hail, KSA haaahm@yahoo.com Omar
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationMath 312 Lecture Notes One Dimensional Maps
Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5
CS434a/54a: Patter Recogitio Prof. Olga Veksler Lecture 5 Today Itroductio to parameter estimatio Two methods for parameter estimatio Maimum Likelihood Estimatio Bayesia Estimatio Itroducto Bayesia Decisio
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationAccess to the published version may require journal subscription. Published with permission from: Elsevier.
This is a author produced versio of a paper published i Statistics ad Probability Letters. This paper has bee peer-reviewed, it does ot iclude the joural pagiatio. Citatio for the published paper: Forkma,
More informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013. Large Deviations for i.i.d. Random Variables
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 2 9/9/2013 Large Deviatios for i.i.d. Radom Variables Cotet. Cheroff boud usig expoetial momet geeratig fuctios. Properties of a momet
More informationEstimation of a population proportion March 23,
1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More information