MODIFIED ESTIMATORS OF RENYI ENTROPY WITH APPLICATION IN TESTING EXPONENTIALITY BASED ON TRANSFORMED DATA

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1 İSTATİSTİK: JOURNAL OF THE TURKISH STATISTICAL ASSOCIATION Vol. 9, No. 2, July 2016, pp iss İSTATİSTİK MODIFIED ESTIMATORS OF RENYI ENTROPY WITH APPLICATION IN TESTING EXPONENTIALITY BASED ON TRANSFORMED DATA Maliheh Abbasejad* Departet of Statistics Haki Sabzevari Uiversity, Sabzevar, Ira. Abstract: This paper deals with the estiatio proble of Reyi etropy of a cotiuous rado variable. Three estiators are obtaied by correctig the coefficiets of the Vasicek-type estiator of Reyi etropy. We perfor a siulatio study to copare these estiators with the Vasicek-type estiator. The results show that our estiators are better tha the existig oe i ters of bias ad root ea squared error. We also itroduce goodess of fit tests for testig expoetiality based o the estiated Reyi etropy of trasfored data. By Mote Carlo siulatio, the powers of the proposed tests uder various alteratives are copared with the faous existig tests. Key words : Reyi iforatio; Etropy estiator; Goodess of fit test; Uifority History : Subitted: 17 October 2015; Revised: 12 May 2016; Accepted: 22 Jue Itroductio Let X be a absolutely cotiuous rado variable with distributio fuctio (cdf), F (x), ad cotiuous desity fuctio (pdf), f(x). The basic easure of ucertaity cotaied i rado variable X is defied by Shao (1948) as H(f) = + f(x) l f(x)dx = El f(x)]. (1.1) Etropy has bee used i a wide rage of probles, icludig characterizatio of probability distributios ad goodess of fit tests, aalysis of cesored data ad reliability theory. By icreasig applicatios of etropy easures, it sees ecessary to fid accurate ad robust oparaetric estiatios of these easures. The estiatio of (1.1) has bee discussed by ay authors icludig Ahad ad Li (1976), Vasicek (1976), Dudewicz ad Va der Meule (1981), Bowa (1992), Va Es (1992), Ebrahii et al. (1994), Correa (1995), Yousefzadeh ad Arghai (2008), Alizadeh (2010), Alizadeh ad Arghai (2010) ad Zaazade ad Arghai (2011). May researchers are also iterested i developig etropy based goodess of fit tests icludig Vasicek (1976), Arizoo ad Ohta (1989), Bowa (1992), Ebrahii et al. (1992), Choi et al. (2004), ad Abbasejad (2011a, 2011b). After the Shao work, iterest has also icreased i the applicatios of other easures of ucertaity, such as Reyi etropy, cuulative residual etropy ad survival etropy (See Rao et al. (2004) ad Abbasejad et al. (2010)). Reyi (1961) has geeralized the etropy as H r (f) = 1 r 1 l * E-ail address:.abbasejad@hsu.ac.ir + f r (x)dx = 1 r 1 l Ef r 1 (X)]. (1.2) 42

2 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik 43 It is to be oted that, as r 1, (1.2) reduces to H(f). Wachowiak et al. (2005) have proposed a estiator of Reyi etropy which was based o the fact that usig p = F (x), (1.2) ca be expressed as H r (f) = 1 r 1 l 1 0 ] df 1 1 r (p) dp. Replacig the distributio fuctio F by the epirical distributio fuctio F, ad usig a differece operator istead of the differetial operator, the estiator is costructed as { } HV r,, = 1 r 1 l 1 ] 1 r 2 (X i+: X i : ), (1.3) where X 1:... X :, are order statistics of rado saple X 1,..., X, is a positive iteger, 2, ad X i: = X 1: if i < 1, X i: = X : if i >. I Sectio 2, we propose three odified estiators of Reyi etropy. I Sectio 3, we copare the proposed estiators with the existig estiators based o their ea squared errors (MSEs). I Sectio 4, a goodess of fit test for expoetiality is proposed based o Reyi distace. Trasforatios of the observatios are used to tur the test of expoetiality ito oe of uifority ad a correspodig test based o Reyi etropy is give. 2. Three odified estiators By rewritig equatio (1.3) as HV r,, = 1 r 1 l { 1 dp ] r 1 } F (X i+: ) F (X i : ), (2.1) X i+: X i : we ca see that iside the brackets i the equatio (2.1) is the slope of the straight lie that jois poits (F (X i+: ), X i+: ) ad (F (X i : ), X i : ). It is clear that (X 2 i+: X i : ) is ot a good approxiatio for the slope whe i or i + 1. The ai idea is to prepare the better approxiatio for the slope ad propose odified estiators of Reyi etropy. If i equatio (2.1) we replace F (X i : ) by F (X 1: ) = 1 for i ad F (X i+: ) by F (X : ) = 1 for i + 1, we obtai the first odified estiator of Reyi etropy as The first odified estiator { HE r,, = 1 r 1 l 1 ] 1 r } c i (X i+: X i : ), where 1 + i 1 1 i c i = i 1 + i + 1 i. Siulatio results show that the bias of HE r,, is egative i ost cases, that is, the first odified estiator uderestiates the value of Reyi etropy. It ay be because of the fact that we put too uch weight o the extree observatios (X i: s, 1 i ad + 1 i ). So replacig c i s for 1 i ad + 1 i by saller ubers could decrease the bias without affectig the stadard deviatio of the estiator. Therefore, we propose the secod ad third odified estiators as The secod odified estiator { HA r,, = 1 r 1 l 1 ] 1 r } a i (X i+: X i : ),

3 44 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik where The third odified estiator { HZ r,, = 1 r 1 l 1 where 1 1 i a i = i c i = i i i. b i = i ] 1 r } b i (X i+: X i : ), 1 i i i i. We choose the otatios HE, HA ad HZ because these estiators were obtaied siilar to odified estiators of Shao etropy proposed by Ebrahii et al. (1994), Alizadeh ad Arghai (2010) ad Zaazade ad Arghai (2011), respectively. Mote Carlo studies show that these odificatios reduce the bias ad thus RMSE. I followig theore we show that the order of estiators of Reyi etropy. Theore 1. Let X 1:,..., X : be a ordered rado saple fro a ukow cotiuous distributio F with pdf f. The The secod ad third equalities hold if ad oly if = 1. HV r,, HE r,, HA r,, HZ r,,. (2.2) Proof. For 1 i, we have c i 2, the ] 1 r ] 1 r 2 (X i+: X i : ) ( ) c i (X i+: X i : ), r < 1 (r > 1). Siilar result holds for + 1 i ad so the first iequality of (2.2) follows by defiitios of HV r,, ad HE r,,. Also, we ca siilarly prove the secod ad third iequalities. I the ext theore, it is show that the bias ad ea squared error of HE r,,, HA r,,, HZ r,, ad HV r,, are scale ivariat. Theore 2. Let H r (X) ad H r (Y ) deote Reyi etropies of cotiuous rado variables X ad Y, respectively ad Y = bx, where b > 0. The: i) E ] Hir,,] Y = E Hi X r,, + l b, ii) V ar ] Hir,,] Y = V ar Hi X r,,, iii) MSE ] Hir,,] Y = MSE Hi X r,,, for i = 1, 2, 3, 4, where the superscripts X ad Y refer to the correspodig distributio ad H1 = HE r,,, H2 = HA r,,, H3 = HZ r,,, H4 = HV r,,. Proof. It is easy to see that { HE Y r,, = 1 r 1 l 1 { = 1 r 1 l 1 ] 1 r } c i (Y i+: Y i : ) b 1 r c i (X i+: X i : ) ] 1 r } = HE X r,, + l b,

4 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik 45 ad (i), (ii) ad (iii) are proved easily. The proof for H2, H3 ad H4 is siilar to H1. To study the behavior of the proposed odified estiators of Reyi etropy, a siulatio study was perfored. We cosider four estiators, Vasicek-type estiator HV r,,, the first odified estiator HE r,,, the secod odified estiator HA r,, ad the third odified estiator HZ r,,. For our siulatio we have used oral, expoetial ad uifor distributios which are the sae distributios cosidered i Correa (1995), Alizadeh (2010), Ebrahii et al. (1994), Alizadeh ad Arghai (2010) ad Zaazade ad Arghai (2011). Table 1. RMSE ad bias of the estiators of Reyi etropy of the uifor distributio. RMSE Bias r HV r HE r HA r HZ r HV r HE r HA r HZ r

5 46 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik Table 2. RMSE ad bias of the estiators of Reyi etropy of the stadard expoetial distributio RMSE Bias r HV r HE r HA r HZ r HV r HE r HA r HZ r Table 3. RMSE ad bias of the estiators of Reyi etropy of the stadard oral distributio. RMSE Bias r HV r HE r HA r HZ r HV r HE r HA r HZ r

6 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik 47 Tables 1-3 cotai the results of siulatios (of saples of size 5,10,20 ad 30) per each case, to obtai the RMSE (Root of Mea Squared Error) values of the four estiators at differet saple sizes for each of the three cosidered distributios. The optial choice of (the value which results the iiu value of RMSE for give ) is still a ope proble. Grzegorzewski ad Wieczorkowski (1999) suggested the heuristics forula for choosig for estiatio of Shao etropy as = + 0.5]. We perfor the siulatio study for all values of /2 ad choose the value of which gives the iiu RMSE for ost of the values of r for three distributios. Our siulatio results show that the optial choice of for proposed odified estiators are as Vasicek-type estiator { + 0.5] 1 r < 1 = + 0.5] + 1 r > 1. The first odified estiator The secod odified estiator = /2]. { + 0.5] r < 1 = + 0.5] + 1 r > 1. The third odified estiator { + 0.5] < 10 = + 0.5] Fro Tables 1-3, we observe that ostly, the ai copetitio is betwee HA r,, ad HZ r,,. The third odified estiator has less RMSE ad absolute of bias tha the secod odified estiator for sall values of. However by icreasig, HA r,, behaves better tha HZ r,,. O the other had, if we copare the estiators for sall ad large values of r, it is see that geerally for sall values of r, (r = 0.2, 0.5), HZ r,, behaves better tha HA r,,. Also, for r = 5, HE r,, copete favorably with HA r,, ad HZ r,,. 3. Testig expoetiality 3.1. Test statistics I ay situatios such as life testig, whe X is the life of a product, it is assued that X has a expoetial distributio. Suppose X 1,..., X be a rado saple fro a cotiuous oegative probability distributio fuctio F with a desity fuctio f. we are iterested i testig the hypothesis H 0 : f(x) = f 0 (x; θ), where f 0 (x; θ) = θe θx, x > 0 ad θ is ukow, agaist the H 1 : f(x) f 0 (x; θ). I order to obtai a test statistic, we use the followig trasforatios. Let X 1 ad X 2 be two idepedet observatios fro a cotiuous distributio fuctio F. It is show by Kotz ad Stetel (1983), that W = X 1 X 1 +X 2 is distributed as U(0, 1) if ad oly if F is expoetial. Also, Alizadeh ad Arghai (2011) proved that Y = X 1 X 2 X 1 +X 2 is distributed as U(0, 1) if ad oly if F is expoetial. Let X 1:,..., X : be a ordered rado saple of size. Cosider the followig trasforatios of the saple data as W ij = X i: X i: + X j:, i j, j = 1,...,,

7 48 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik V ij = X i: X j: X i: + X j:, i > j, j = 1,...,. Uder the ull hypothesis, each W ij ad V ij have uifor distributio o (0, 1). Hece, to test H 0, we ca trasfor X i to W ij or V ij ad test H 0 : g(u) = g 0 (u), where g 0 (u) = 1, 0 < u < 1. The asyetric Reyi distace of g fro g 0 is: D r (g, g 0 ) = 1 r 1 l = 1 r 1 l ] r 1 g(u) g(u)du g 0 (u) g r (u)du = H r (g), (3.1) where the H r (g) is the Reyi etropy of order r of distributio with desity g. D r (g, g 0 ) 0 ad the equality holds if ad oly if H 0 holds. If a saple coes fro a expoetial distributio, D r (g, g 0 ) should be close to zero value ad thus, large values of D r (g, g 0 ) lead us to reject the ull hypothesis H 0 i favor of the alterative hypothesis H 1. To derive a test statistic by evaluatig the iforatio fuctio (3.1), the desity g ust be copletely specified, which is ot operatioal. Thus, it is ecessary to estiate the iforatio fuctio (3.1) fro a saple. Toward this ed, we use four estiators of Reyi etropy HV r,,, HE r,,, HA r,, ad HZ r,, based o the trasfored data W i for i = 1, 2,...,, where = ( 1) ad V i for i = 1, 2,...,, where = ( 1)/2. Sice the perforace of the first ad secod trasforatios were siilar i ters of power, we just cosider the secod trasforatio, that is, we have four test statistics as: T V = HV r,, = 1 r 1 l 1 ] 1 r 2 (V i+: Y i : ), T E = HE r,, = 1 r 1 l 1 ] 1 r c i (V i+: V i : ), T A = HA r,, = 1 r 1 l 1 ] 1 r a i (V i+: V i : ), T Z = HZ r,, = 1 r 1 l 1 ] 1 r b i (V i+: V i : ). We reject the ull hypothesis for sall values of the test statistics. Critical poits are deteried by the quatiles of the distributios of the test statistics uder hypothesis H 0. Sice the saplig distributios of test statistics are itractable, we deterie the critical poits usig Mote Carlo ethod. Ufortuately, there is o choice criterio of r ad (the optiu values of r ad ). I geeral they depeds o the type of the alteratives. We suggest to choose r = 1.5 based o siulatio results. The results are give i Tables 4-8 for r = 1.5 ad differet values of.

8 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik 49 Table 4. Critical values of the T V, α = Table 5. Critical values of the T E, α = Table 6. Critical values of the T A, α = Table 7. Critical values of the T Z, α = Power Study For power coparisos, we choose the copetitor tests fro the existig expoetiality tests discussed i Heze ad Meitais (2005). The test statistics of copetitor tests are as follows. We

9 50 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik will deote the scaled observatios x i / x by y i. (1) The Kologorov-Sirov test statistic (D Agostio ad Stephes, 1986): where z i = (1 exp( y i )). KS = sup x 0 = ax F (x) (1 exp( x)) { ] i ax 1 i z (i), ax 1 i z (i) i 1 ]}, (2) The Craer-vo Mises test statistic (D Agostio ad Stephes, 1986): CV = 1 ( 2i z (i) ) 2. (3) The Watso test statistic: W = T ( z 1 2 )2. (4) The Aderso-Darlig test statistic: AD = 2i 1 l z(i) + l(1 z ( i+1) ) ]. (5) The test statistic of D Agostio ad Stephes (1986) which is based o the oralized spacigs D i = ( + 1 i)(x i: X i 1: ), i = 1,...,, X 0: = 0: S = 2 2 iy (i:). (6) The test statistic of Cox ad Oakes (1984): CO = + (1 y i ) l(y i ). (7) The test statistic of Epps ad Pulley (1986): EP = ( ) 1 48 exp( y i ) 1. 2 (8) The test statistic of Barighaus ad Heze (1991) which is based o the epirical Laplace trasfor: BH = 0 (1 + t)ψ (t) + ψ (t)] 2 exp( at)dt, where ψ (t) = 1/ exp( ty i) is the epirical Laplace trasfor of the expoetial distributio. Their suggested value of a is 2.5. (9) The test statistic of Ebrahii et al. (1992) which is based o Kullback-Leibler iforatio: KL = exp(h ) exp(l x + 1),

10 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik 51 where H = 1 l 2 (X i+: X i : ) ] is Vasicek s (1976) etropy estiator of Shao etropy. We have take = 3, followig the recoedatio of Ebrahii et al. (1992). (10) The test statistic of Heze (1993) which is based o the epirical Laplace trasfor: H = 0 ψ (t) 1 ] 2 exp( at)dt, 1 + t where ψ (t) = 1/ exp( ty i) is the epirical Laplace trasfor of the expoetial distributio. His suggested value of a is 2.5. (11) The Kologorov-Sirov ad Craer-vo Mises-type statistics proposed by Barighaus ad Heze (2000): KS = 1 sup i(y t 0 i, t) 1 I {yi t}, 1 CV = i(y i, t) 1 2 I {yi t}] exp( t)dt, 0 where I {A} is the idicator of the evet A. (12) The test statistics proposed by Heze ad Meitais (2005) which are based o the epirical characteristic fuctio: T (1),a = a ] a 2 + y 2 j,k=1 jk a 2 + yjk+ 2 2a ] 1 2 a 2 + (y jk y l ) a 2 + (y jk + y l ) 2 j,k=1 l=1 + a ] 1 3 a 2 + (y jk y l ) + 1, 2 a 2 + (y jk + y l ) 2 j,k=1 l=1 T (2),a = 1 π ( ) ( )] exp y2 jk + exp y2 jk+ 2 a 4a 4a j,k=1 1 π exp ( (y ) jk y l ) 2 + exp ( (y )] jk + y l ) 2 2 a 4a 4a j,k=1 l=1 + 1 π exp ( (y ) jk y l ) 2 + exp ( (y )] jk + y l ) 2, 2 3 a 4a 4a j,k=1 l=1 where y jk = y j y k, y jk+ = y j +y k ad a > 0. We have take a = 2.5, followig the recoedatio of Heze ad Meitais (2005). (13) The test statistics proposed by Alizadeh ad Arghai (2011) which are based o the Shao etropy of trasfored data: T 1 = = 1 ] l 2 (w i+: w i : ),

11 52 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik T 2 = = 1 T 3 = = 1 l ] 2 (t i+: t i : ) + 2 l(1 + t i ), ] l 2 (z i+: z i : ) + l 2, where = ( 1), w ij = x i x i +x j, t ij = x i x j ad z ij = x i x j x i +x j for i j = 1,...,. For power coparisos, we cosider the sae alteratives listed i Heze ad Meitais (2005) ad Alizadeh ad Arghai (2011) ad their choices of paraeters: the Weibull distributio with desity θx θ 1 exp( x θ ), deoted by W (θ), the gaa distributio with desity Γ(θ) 1 x θ 1 exp( x), deoted by Γ(θ), the logoral distributio with desity (θx) 1 (2π) 1/2 exp( (l x) 2 /(2θ 2 )), deoted by LN(θ), the half-oral distributio with desity Γ(2/π) 1/2 exp( x 2 /2), deoted by HN, the uifor distributio with desity 1, 0 x 1, deoted by U(0, 1), the odified extree value distributio with distributio fuctio 1 exp(θ 1 (1 e x )), deoted by EV (θ), the liear icreasig failure rate law with desity (1 + θx) exp( x θx 2 /2), deoted by LF (θ), Dhillo s (1981) law with distributio fuctio 1 exp( (l(x + 1)) θ+1 ), deoted by DL(θ), Che s (2000) distributio with distributio fuctio 1 exp(2(1 e xθ )), deoted by CH(θ). These distributios coprise of widely used alteratives to the expoetial odel ad iclude desities f with decreasig hazard rates, icreasig hazard rates as well as odels with oootoe hazard fuctios. Tables 8 ad 9 shows the estiated powers of the tests T V, T E, T A ad T Z ad those of the copetig tests, at the sigificace level α = 0.05, based o the results of siulatios of saple size 20. The axiu power are give i bold type. For these alteratives, we have take = 50 for = 20 ( = 190), based o siulatio results. It is evidet fro Tables 8 ad 9 that, the test statistic T Z has the greatest power for alost all alteratives aog the test statistics based o etropy ad aog other test statistics T (1),a has the best perforace i ters of the powers. I geeral, we ca ot say a sigle test perfor the best for all alteratives. We observe that the proposed test statistics based o Reyi etropy perfor very well copared with the other tests for Weibull(0.8), gaa(2), logoral(0.8) ad Dhillo, DL(θ) alteratives. However, the test of Heze ad Meitais (2005), T (1),a, has the greatest powers for alost all other alteratives.

12 Abbasejad: Modified Estiators of Reyi Etropy with Applicatio i Testig Expoetiality Based o Trasfored Data İSTATİSTİK: Joural of the Turkish Statistical Associatio 9(2), pp , c 2016 İstatistik 53 Table 8. Power copariso of various expoetiality tests for = 20 with α = Alter. KS CV W AD EP KS CV S CO BH HE T (1),a T (2),a W (0.8) W (1.4) Γ(0.4) Γ(1) Γ(2) LN(0.8) LN(1.5) HN U CH(0.5) CH(1) CH(1.5) LF (2) LF (4) EV (0.5) EV (1.5) DL(1) DL(1.5) Table 9. Power copariso of expoetiality tests based o etropy for = 20 with α = Alter. KL T 1 T 2 T 3 T V T E T A T Z W (0.8) W (1.4) Γ(0.4) Γ(1) Γ(2) LN(0.8) LN(1.5) HN U CH(0.5) CH(1) CH(1.5) LF (2) LF (4) EV (0.5) EV (1.5) DL(1) DL(1.5) Coclusio I this paper, we proposed three odified estiators of Reyi etropy ad copared the with the existig estiator based o their RMSEs. It is observed that oe of these estiators has better perforace tha other estiators especially for sall saple sizes. Also we itroduced a goodess of fit test for expoetiality based o Reyi distace. At first we use trasforatios of data which tur the test of expoetiality ito oe of uifority ad the we use a correspodig test based o estiators of Reyi etropy. We ca see that the proposed tests perfors well for soe alteratives. Refereces 1] Abbasejad, M. (2011a). Soe goodess of fit tests based o Reyi iforatio. Applied Matheatical Scieces, 5, ] Abbasejad, M. (2011b). Testig expoetiality based o Reyi etropy of trasfored data. Joural of Statistical Research of Ira, 8, ] Abbasejad, M., Arghai, N. R., Morgethaler, S. ad Mohtashai Borzadara, G. R. (2010). O the dyaic survival etropy. Statistics ad Probability letters, 80,

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