Confidence Interval Estimation of the Shape. Parameter of Pareto Distribution. Using Extreme Order Statistics

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1 Applied Mathematical Scieces, Vol 6, 0, o 93, Cofidece Iterval Estimatio of the Shape Parameter of Pareto Distributio Usig Extreme Order Statistics Aissa Omar, Kamarulzama Ibrahim ad Ahmad Mahir Razali School of Mathematical Scieces Uiversiti Kebagsaa Malaysia, Bagi, Selagor, Malaysia Abstract The asymptotic distributios of several estimators of the shape parameter of the Pareto distributio i the case where the scale parameter is ow are derived The study is carried out usig simple radom samplig (SRS, extreme raed set samplig (ERSS ad o radom samples that are based o miimum order statistics oly or maximum order statistics oly These distributios are used to costruct asymptotic cofidece itervals (ACI for A simulatio study the compares these cofidece itervals via their expected legths ad coverage probabilities Keywords: Pareto distributiohape parameterimple radom samplig, extreme raed set samplig, miimum order statistics, maximum order statistics, asymptotic cofidece iterval, expected legth, coverage probability Itroductio For situatios where the variable of iterest is either costly or time cosumig to be measured, but ca easily be raed, raed set samplig (RSS devised by McItyre (95 is geerally useful i a samplig scheme utilizig the iformatio of the ras i additio to measuremets Sice the performace of RSS is susceptible to error i raig, various modificatios of RSS have bee

2 468 A Omar, K Ibrahim ad A M Razali proposed which are more practicable Such modificatios have show to improve the precisio of estimatio i some cases as opposed to simple radom samplig (SRS ad RSS, cf Helu et al (00, Al-Saleh ad Al-Kadiri (000, Al-Saleh ad Al-Omari (00 ad Muttla (997 The Pareto distributios are quite commoly applied i may areas of researchuch as actuarial sciece (cf Hogg ad Klugma (984, ecoomic studies (cf Arold (983, hydrology (cf Nadarajah ad Ali (008 ad reliability studies (cf Harris (968 I this paper, we costruct the asymptotic cofidece itervals for the shape parameter of Pareto distributio based o asymptotic distributios of several estimators of with the scale parameter ow ad the data gathered uder SRS, ERSS ad o radom samples that are based o miimum order statistics oly or maximum order statistics oly A simulatio study is used to compare these cofidece itervals via their expected legths ad coverage probabilities Assume that the radom variable X p( γ,, where > 0 ad γ > 0 are the shape ad scale parameters respectively The the probability desity fuctio (pdf of X is give by γ, x γ f( x = x, 0, otherwise ad the cumulative distributio fuctio (cdf is give by γ F( x = x, x γ We are iterested i maig a iferece for whe γ is ow This is applicable, for example i the area of actuarial scieces, whe the reisurer is iterested i the iformatio about all losses exceedig a certai limit which could for istace be the priority of the excess of loss treaty (see, Rytgaard (990 Thus it ca be assumed without loss of geerality that γ is equal to oe The our pdf ad cdf reduce respectively to ( x (, x f x, > = 0, 0, otherwise (0 ad F( x = x, x (0 The mea ad variace of X are give by EX ( = (, >, (03 var( X, (04 respectively ( (

3 Cofidece iterval estimatio 469 Cofidece Itervals based o SRS Let X,, X p, Next, we cosider asymptotic cofidece itervals of based o momet method (MOME, uiformly miimum variace ubiased estimator (UMVUE ad maximum lielihood estimator (MLE as follows: ACI based o the MOME Whe X ~ p(,, the mea of X is give by (3 Put EX ( = Xsrs ad solve for to get the MOME of as = X X ( where K be a simple radom sample of size from ( X ( mom, s srs srs srs = X i Rytgaard (990 has show that mom i = is asymptotically ormal distributed by N, ( whe > This implies that mom Z ( 0 /, mom Z ( 0 / ( ( ( is a approximate 00( 0% cofidece iterval for, where 0 is the sigificace level, ad Z( 0 / is the upper 00( 0 th percetile of N ( 0, ACI based o the MLE ad UMVUE It is straightforward to show that the MLE of is give by mle, s = l ( xi (3 i= Note that Ε ( mle, s = whe > ad V ( mle = ( ( whe > Accordig to Lehma (983 the asymptotic distributio of mle is N(, [ I srs ( ] where I srs ( is the Fisher iformatio of ad is give by l f( x, K, x, Isrs ( = E =, which implies that mle is asymptotically N, Also, it is a simple exercise to show that the UMVUE of exists ad is give by

4 4630 A Omar, K Ibrahim ad A M Razali umvue = ( l ( xi i= (4 ad umvue is asymptotically N, Thus, the ACIs for MLE ad UMVUE are: mle Z( 0/, mle Z( 0/, (5 umvue Z( 0/, umvue Z( 0/ (6 respectively X 3 Cofidece Itervals based o miimum order statistics Let (: m,, (: m K X be a versio of raed set sample that cosists of miimum order statistics of size where m is the set size Next, we cosider the followig asymptotic cofidece itervals for based o this radom sample 3 ACI based o MOME Note that X (: m, K, X (: m are idepedet idetically distributed (iid with m m mea, m > ad variace σ(:, m = m m ( m ( m > the ( m E X give by = X Solvig with respect to we obtai the momet estimator (: (: m ( ( mome = X m X ( ( ( where X( = X(: mi By cetral limit theorem (CLT we get i= m d X( N( 0, σ (: m, m x The applyig the delta method with g( x = we have mx ( d ( m ( mome( N 0, m( m The ACI based o momet estimator is therefore give by

5 Cofidece iterval estimatio 463 ( m ( m mome( Z( 0/, mome( Z ( 0/ m( m m( m ( 3 ACI based o MLE The pdf of the miimum order statistic X (: m is give by ( g( : ( x; = mx x, m ad log lielihood of X(: m, K, X(: m is ( m j l L( ; x = l m m l x, ( : m ( ( ( : l Lx ( ( : ; m = m l ( x( : m j ( j= Equatig (3 to zero the solvig with respect to gives mle( = m l ( x(: m i i= Let V V(: m j j = = where V (: m j= l( X (: m j, j =, K, iid with a expoetial distributio with mea are m The d ( mle( N 0, thus mle( Z( 0/, mle( Z( 0/ (3 as a approximate 00( 0% cofidece iterval for j= X 4 Cofidece Itervals based o maximum order statistics, K, X be a versio of raed set sample that cosists of maximum Let ( m ( m order statistics of size where m is the set size Next, we cosider the followig asymptotic cofidece itervals for based o this samplig techique 4 ACI based o the MLE The pdf of the maximum order statistic X ( mm : j is give by ( m ( g( mm : x; = m x x The the log-lielihood fuctio is ( = ( ( ( mm : j ( ( ( mm : j l L l m ( m l x m l x j= j=

6 463 A Omar, K Ibrahim ad A M Razali This implies that l L( ; x ( mm : j ( m l ( x( : mm j = ml ( x( mm : j (3 j ( x = ( mm : j Settig (4 equal to zero we get the solutio deoted by mle, m which is the MLE of (Abu-Dayyeh et al (0 The, accordig to Lehma (983, is asymptotically,[ ( ] mle, m ( N I mo where m 3 ( l L 3 ( ; x m ( : mm j Imo ( = E = m ( m 3 Therefore = 0 ( we have ± Z I ( (3 mle, m ( 0 / ( ( mo is a approximate 00( 0% cofidece iterval for 4 ACI based o the ad hoc estimator d m It is easy to show that Y( mm : j = l( X( mm : j, j =,, are iid with mea ad m ( m c variace m where d m = m ad ( m m ( ( m m cm = m m 3 = 0 ( = 0 ( Adhoc, m is asymptotically = 0 σ N,, m where d m Adhoc, m = ; Yrss, m = l X( mm, j ad Yrss, m j= m m ( ( m m σ, m = m m 3 = 0 ( = 0 ( The dm σ, m σ, m Adhocm, Z( 0/, Adhocm, Z( 0/ (33

7 Cofidece iterval estimatio 4633 as a approximate 00( 0% cofidece iterval for 5 Cofidece Itervals based o ERSS Samawi et al (996 itroduced a modificatio of RSS called extreme raed set samplig (ERSS Uder ERSS complete raig is ot required, but oly extremes are determied The idea of ERSS is as follows If the sample size m is eveelect from the first m/ samples the smallest uits ad select from last m/ samples the largest uits If the sample size m is oddelect from the first (m-/ samples the smallest uits ad from the ext (m-/ samples select the largest uits From the last sample, we select the media Next, we will cosider the followig asymptotic cofidece itervals for : 5 ACI based o the MLE X, K, X, X, K, X be a extreme raed set sample of size Let ( m m ( m m ( ( m =m where m is the set size ad m is eve ad Xij(, Xij( m, X( j( m /, i =,,,( m /, j =,, r be a extreme raed set samplig of size =rm where m is the set size ad m is odd The MLE were derived by Omar et al (0 (a The case whe m is eve The joit pdf of Xij( ad X ij( m is give by m ( ( ( m ij( ij( m = ij( ij( ij( m ij( m g x, x ; mx x m x x The the log-lielihood fuctio is give by r l ( ij( ij( m = ( ( ( ij( l L ; x, x mrl m m l x This implies that j= i= r m ( m ( xij( m ( ( xij ( m l l l L mr = m j= i= l r l= m j= i= l ( xij( ( m xij( m l( xij( m r m l ( xij( m = 0 j= i= l x ij( m The (5 has exactly oe root, ie MLE, e is the MLE of sice The estimator mle, e ( r m ( m l( xij( m x ij( m 0 j i= l ( x ij( m l L mr = < caot be obtaied i a closed form Accordig to (4

8 4634 A Omar, K Ibrahim ad A M Razali ( is asymptotically [ ] Lehma (983 mle, e N, I ERSS ( where I ERSS ( is the Fisher iformatio of uder ERSS which ca be foud as follows: l L( x (, ( ; r m l ij xij m mr ( x ( ij m x ij( m IERSS ( = E = ( m E, j i ( x ij( m m 3 ( s m3 mr s = mm ( 3 s= 0 ( s This implies that, ( ( mle e Z ( 0/ IERSS (, mle, e Z( 0/ IERSS ( (4 (b The case whe m is odd The joit pdf of Xij(, Xij( m ad X ( m is give by j ( g x x x mx x m x x m ( m ( ij(, ij( m, ; = ij( ij( m ij( m ij( m j m m m! x x m m j j m The the log-lielihood fuctio is The, we have! ( = ( ( ( ( ij( ( l L r m l m m l x l l r m j= i= d r d j= i= ( ij( m ( ij( m ( ( m l x ( l x rl r ( m l x ( l x m j j= m j( ( m xij( m l( xij( m x ( m j r d r m L rm = m l ( x ij( l x ( j= i= j i= d xij( m m x l x m m r j j( m l x m x j( = 0 j= m j (43 ( ij m

9 Cofidece iterval estimatio 4635 The solutio of (53 with respect to is the MLE of, ie MLE, e which caot be obtaied i a closed form The Fisher iformatio of uder ERSS i this case which ca be foud as follows: r m l ( xij( m ( l ( x L rm ij( m IERSS ( = E ( m E = j i= d ( x ij( m x l x m m r j j( m E, j= x m j m3 ( 5 / 5 3 ( v m m ( s m rm v ( m m! s = ( m, 3 3 v= 0 ( v ( m s= 0 ( m s! Therefore, the ACI for is as i (5 5 ACI based o a ad hoc estimator We suggest the followig ad hoc estimator: (a I the case whe m is eve, rq Adhoc =, eve r m/ ( l ( x(: mi l ( x( mmi : / j= i= m m m where Q= m m ( = 0 ( Note that the distributio of ( X(: m i Yi = l is the same as the distributio of the miimum order statistics based o simple radom sample of size m from a expoetial distributio with mea Therefore, it is easy to show EY ( i = Also, ote that the m distributio of Zi = l ( X( m: m i is the same as the distributio of the maximum order statistics of a simple radom sample of size m from a expoetial distributio with mea = ad V, V, K, V / are iid with mea Let V i [l ( x (: m i l ( x ( m: m i /

10 4636 A Omar, K Ibrahim ad A M Razali m m Q m m ( = = 0 ( ad variace σ(: m σ( m: m m where σ ad σ are the variaces of the miimum ad maximum order (: m ( mm : statistics of a simple radom sample of size m from a expoetial distributio with mea, respectively The by CLT m/ Q d r V N 0, ( σ(: mi σ( mmi : m/, i= ad by usig the delta method we have m / Q d Q r N 0, g' σ(: mi σ( mmi : m/, V V i= 4 m / = N 0, ( σ(: mi σ( mmi : m/ rq i= Thus a 00( 0% cofidece iterval for is (, ( Z SE Z SE Adhoce, ( 0/ Adhoce, Adhoce, ( 0/ Adhoce, (44 4 m / where SE ( Adhoc, e = σ (: m i σ ( m: m i m/ rq i= (b I the case whe m is odd ru Adhoc =, eve r ( m/ l ( x(: mij l ( x( mm : ( i / j l ( x(( m /: m j j= i= where m m m m U = m m ( 0 ( = ( m/m 4 m! s ( m s= 0 ( m s! s Followig the same procedures i the case where m is eve we get a 00( % cofidece iterval for as (55 where SE 0 σ σ σ 4 d m Adhoc, e = (: m i ( m: m i (( m /: m ru i= i= d ( ad σ is the (( m / : m

11 Cofidece iterval estimatio 4637 variace of the media of order statistics of a simple radom sample of size m from a expoetial distributio with mea 6 Compariso of the Cofidece Itervals We ow compare the asymptotic cofidece itervals via their legths ad their coverage probabilities Sice all the cofidece itervals above are of the form ( σ Z, σ Z, it follows that the legths are give by ( 0/ ( 0/ L= σ Z( 0 / The coverage probability of the cofidece iterval ( σ Z( 0/, σ Z( 0/ is defied as P ( σz( 0/ < < σ Z( 0/ The smaller the legth ad the larger the coverage probability is the better the cofidece iterval A simulatio coducted to compute the legths ad coverage probabilities of the cofidece itervals for =3, 4, 5 ad 6, m =,3, 4, r =3,4,6,8,, where = rm ad 0 = 005 The results are summarized i Table ad below Note that all cofidece itervals obtaied are depedet o which is uow, therefore, we replaced by their correspodig estimators From tables ad we may coclude the followig: The estimator mle, e has the closer coverage probability to the omial value 0 The estimator mle, m has the shorter expected legth The expected legth of ACIs for based o RSS ad ERSS is smaller tha the expected legth of their couterparts estimators based o SRS The coverage probability of ACIs based o RSS ad ERSS is more close to the omial value i the case of ACIs based o SRS

12 4638 A Omar, K Ibrahim ad A M Razali Table The coverage probabilities of the cofidece itervals mom mle umvue mome ( mle ( Adhoc, m mle, m Adhoc, e mle, e m=, r= m=3, r= m=4, r= m=,r= m=3, r= m=4, r= m=, r= m=3, r= m=4, r= m=,r= m=3, r= m=4, r= m=, r= m=3, r= m=4, r= m=,r= m=3, r= m=4, r= m=, r= m=3, r= m=4, r= m=,r= m=3, r= m=4, r=

13 Cofidece iterval estimatio 4639 Table The expected legth of the cofidece itervals mom mle umvue mome ( mle ( Adhoc, m mle, m Adhoc, e mle, e m=, r= m=3, r= m=4, r= m=,r= m=3, r= m=4, r= m=, r= m=3, r= m=4, r= m=,r= m=3, r= m=4, r= m=, r= m=3, r= m=4, r= m=,r= m=3, r= m=4, r= m=, r= m=3, r= m=4, r= m=,r= m=3, r= m=4, r=

14 4640 A Omar, K Ibrahim ad A M Razali Acowledgmets We tha Uiversiti Kebagsaa Malaysia for providig partial support for this research uder the research grats UKM-Gup-0-3 Refereces [] W Abu-Dayyeh, A Assrhai, ad K Ibrahim, Estimatio of the shape ad scale parameters of Pareto distributio usig raed set samplig, Statistical Papers, (0, -9 [] M F Al-Saleh ad M A Al-Kadiri, Double-raed set samplig, Statistics & Probability Letters, 48 (000, 05- [3] M F Al-Saleh ad A I Al-Omari, Multistage raed set samplig, Joural of Statistical Plaig ad Iferece, 0 (00, [4] B C Arold, Pareto Distributios, Iteratioal Co-operative Publishig House, Fairlad MD, 983 [5] C M Harris, The Pareto Distributio as a Queue Service Disciplie, Operatios Research, 6 (968, [6] A Helu, M Abu-Salih, ad O Alam, Bayes Estimatio of Weibull Distributio Parameters Usig Raed Set Samplig, Commuicatios i Statistics - Theory ad Methods, 39 (00, [7] R V Hogg ad S A Klugma, Loss Distributios, Wiley, New Yor, 984 [8] E L Lehma, Theory of poit estimatio, J Wiley, New Yor, 983 [9] G A McItyre, A method for ubiased selective samplig, usig, raed sets, Australia Joural of Agricultural Research 3 (95, [0] H A Muttla, Media raed set samplig, Joural of Applied Statistical Scieces, 6 (997, [] S Nadarajah ad M Ali, Pareto Radom Variables for Hydrological Modelig, Water Resources Maagemet, (008, [] A Omar, K Ibrahim, ad A F Shahabuddi, Estimatio of the Shape ad Scale Parameters of the Pareto Distributio usig Extreme Raed Set Samplig "Mauscript submitted for publicatio", (0 [3] M Rytgaard, Estimatio i the Pareto Distributio, ASTIN BULLLETIN a Joural of Iteratioal Actuarial Associatio, 0 (990, 0-5 [4] H M Samawi, M S Ahmed, ad W Abu-Dayyeh, Estimatig the Populatio Mea Usig Extreme Raed Set Samplig, Biometrical Joural, 38 (996, Received: April, 0