On Energy and Expected Uncertainty Measures in Weighted Distributions

Transcription

1 Inernaional Mahemaical Forum, 2, 27, no. 2, On Energy and Expeced Uncerainy Measures in Weighed Disribuions Broderick O. Oluyede Deparmen of Mahemaical Sciences Georgia Souhern Universiy, Saesboro, GA 346 Mekki Terbeche Deparmen of Mahemaics Faculy of Sciences Universiy of Oran, Es-Senia, Oran Absrac In his noe, bounds and inequaliies for he comparisons of weighed energy funcions, enropy, and discriminaion informaion measures and heir unweighed counerpars are presened. Inequaliies for weighed expeced uncerainy, cross-enropy or discriminaion informaion measures are also presened. A useful resul on he convergence of he weighed kernel densiy informaional energy esimaes is given and some informaional energy applicaions presened. Mahemaics Subjec Classificaion: 62B1, 62N5 Keywords: Sochasic inequaliies, Weighed disribuion funcions, Bounds. 1 Inroducion The use of weighed disribuions in research relaed o reliabiliy, bio-medicine, ecology and several oher areas is of remendous pracical imporance in mahemaics, probabiliy and saisics. These disribuions arise naurally as a resul of observaions generaed from a sochasic process and recorded wih some weigh funcion. Several auhors have presened imporan resuls on lengh-biased disribuions and on weighed disribuions in general. Rao [7] unified he concep of weighed disribuions. Bhaacharyya e al [1] sudied

2 948 Broderick O. Oluyede and Mekki Terbeche and compared nonparameric unweighed and lengh-biased densiy esimaes of fibers. Vardi [1] derived he nonparameric maximum likelihood esimae (NPMLE) of a lifeime disribuion in he presence of lengh bias and esablished convergence o a pinned Gaussian process wih a simple covariance funcion under mild condiions. For addiional and imporan resuls on weighed disribuions see Pail and Rao [6], Gupa and Keaing [4] among ohers. Le {P θ : θ Θ} be a family of probabiliy disribuions associaed wih a σ finie measure λ. Assume ha he funcion P θ is coninuous and he mapping θ p θ, where p θ = dp θ /dλ is almos surely (a.s.) upper semiconinuous, separable random process, and he energy funcion e(p θ )=E[p θ ] exiss and is finie on he parameer space. Energy funcions are measures of dispersion of disribuions ha varies monoonically wih dispersive order. The purpose of his noe is o compare energy, enropy and discriminaion informaion measures involving lengh-biased disribuion funcions and weighed disribuion funcions in general. We also presen some inequaliies involving cerain cross-enropy or discriminaion measures for he comparisons of weighed disribuions wih he paren disribuions. Comparisons beween weighed and he paren disribuions ha involves he informaional energy funcion and enropy are also presened. Secion 2 conains some basic definiions and imporan uiliy noions. In secion 3, comparisons and inequaliies involving uncerainy and cross-enropy or discriminaion measures are presened. In secion 4, basic convergence resul for he energy funcion is presened via he applicaion of Schuser s lemma for kernel lengh-biased disribuions. In secion 5, some applicaions are presened. 2 Some Basic Resuls In his secion, we presen some definiions and useful uiliy noions. Le F be he se of absoluely coninuous disribuion funcions saisfying F () =, x lim F (x) =1, sup{x : F (x) < 1} =. (1) Noe ha if he mean of a random variable wih a disribuion funcion in F is finie, i is posiive. Le X be a non-negaive random variable wih disribuion funcion F, survival funcion F and probabiliy densiy funcion (pdf) f. In a weighed disribuion problem, a realizaion x of X eners ino he invesigaors record wih probabiliy proporional o a weigh funcion W (x). Obviously, he recorded x is no an observaion on X, bu raher an observaion on a weighed random variable X W. The weighed random variable X W has a survival or reliabiliy funcion given by G W (x) = E F [W (X) X >x] F (x). (2) E F (W (X))

3 Expeced Uncerainy Measures 949 This survival or reliabiliy funcion can also be expressed as G W (x) =F (x)(w (x)+t F (x))/e F (W (X)), (3) where T F (x) = x (F (u)w (u)du)/f (x), and W (u) =dw (u)/du, assuming ha W (x)f (x) asx. The probabiliy densiy funcion corresponding o he reliabiliy funcion given in (2) is referred o as a weighed probabiliy densiy funcion (wpdf) wih weigh funcion W (x). The wpdf is given by (x) =W (x)f(x)/δ, (4) where <δ < is a normalizing consan. When W (x) =x, he corresponding pdf is called he lengh-biased probabiliy densiy funcion and is given by g l (x) =xf(x)/μ F, (5) where <μ F is given by: = F (x)dx <. The lengh-biased survival funcion G l (x) G l (x) =F (x)v F (x)/μ F, (6) where V F (x) =E F (X X >x) is he vialiy funcion. Noe ha G l (x) and F (x) are sochasically ordered, ha is G l (x) F (x) for all x. Nex we presen some well known and useful definiions on he hazard rae funcion, mean residual life funcion, informaional energy funcion and enropy measure. Definiion 2.1. A disribuion funcion F is said o have increasing (decreasing) hazard rae or failure rae on [, ), denoed by IHR (DHR) or IFR (DFR), if F ( ) =,F() < 1 and P (X >x+ X >)=F (x + )/F () is decreasing (increasing) in for each x>. Definiion 2.2. A disribuion funcion F is said o have decreasing (increasing) mean residual life if F ( ) =,F() < 1 and F (x)dx/f () is decreasing (increasing) in. Noe ha if F has DHR and μ F mean residual life (IMRL). = F (x)dx <, hen F has increasing Definiion 2.3. The informaional energy associaed wih a probabiliy densiy funcion (pdf) f in F is given by e(f) = f 2 (x)dx, (7) where f(x) =df (x)/dx and F is he corresponding disribuion funcion.

4 95 Broderick O. Oluyede and Mekki Terbeche Definiion 2.4.LeX be a non-negaive random variable wih finie variance and differeniable probabiliy densiy funcion (pdf) f. The uncerainy measure associaed wih a disribuion funcion F in F is he differenial enropy given by H(f(X)) = E f (logf(x)) = f(x)logf(x)dx, (8) where f(x) =df (x)/dx. The cross-enropy (Guiasu [3]) is H(f 1,f 2 )= E f1 (log(f 1 (X)/f 2 (X)) = f 1 (x)ln(f 1 (x)/f 2 (x))dx. (9) H(f(X)) is commonly referred o as he Shannon enropy measure (Shannon [9]). I is he expeced uncerainy conained in f(x) abou he predicabiliy of an oucome X. The following resul is well known. Le X and Y be random variables and suppose ha he densiy of Y has bounded derivaive and if U is any random variable independen of X and Y, hen convoluion decreases Fisher informaion and increases enropy. Tha is, I(Y +U) I(Y ) and H(X+ U) H(X). These inequaliies follow from he convexiy of he funcions x 2 and xlogx. Noe ha equaliy hold if and only if U is a consan, almos surely. The following definiion is due o Ebrahimi and Pellerey [2]. Definiion 2.5. The uncerainy of residual life disribuion H(X; ), of a componen a ime, is he enropy of he residual life random variable (X X >), and is given by f(x) H(X; ) = F () logf(x) F () dx = logf () {F ()} 1 f(x)logf(x)dx = 1 {F ()} 1 f(x)log{λ F (x)}dx, (1) where λ F (x) =f(x)/f (x) is he hazard or failure rae funcion. The enropy of he weighed residual life random variable (X W X W >) is given by f W (x) H(X W ; ) = F W () log f W (x) F W () dx = logf W () {F W ()} 1 f W (x)logf W (x)dx (11)

5 Expeced Uncerainy Measures 951 Under lengh biased sampling H(X W ; ) reduces o H(X l ; ) = 1 {F ()V F ()} 1 xf(x)log(xλ F (x)/v F (x))dx. (12) where V F () =E(X X >)+, is he vialiy funcion and E F (X X >) is he mean residual life funcion. H(X; ) is he expeced uncerainy in he condiional disribuion of X given X > abou he predicabiliy of he remaining lifeime of he componen ha has survived for ime. We define cerain weighed cross-enropy or discriminaion informaion measures as follows: and Similarly, define H (f, ; ) = H (f, ; ) = H (,f; ) = and H (,f; ). Also, define H(f, ; ) = f(x)log(f(x)/ (x))dx (13) f(x)log(f(x)/ (x))dx. (14) (x)log( (x)/f(x))dx (15) f(x) F () log{ f(x)/f () }dx. (16) (x)/g W () Noe ha if he weigh funcion W (x) is non-decreasing in x>, hen e( ) e(f). 3 Inequaliies for Uncerainy Measures In his secion, we presen some resuls including inequaliies and comparisons of cross-enropy and uncerainy measures for weighed and unweighed or paren disribuions. Le X and Y be wo random variables, and assume ha he densiy of Y has a bounded derivaive. If U is any random variable independen of X and Y, hen i is well known ha convoluion increases enropy. This leads o he following quesion: If X W and X are he weighed and unweighed random variables respecively, can X W be expressed as X W = X + U, where U is independen of X and X W? In he nex resul, we show ha he measure of uncerainy presen wih respec o he value of a random variable X is greaer han or equal o he measure of uncerainy presen wih respec o he value of he corresponding weighed random variable X W. Theorem 3.1. If he weigh funcion W (x) is non-decreasing for all x, hen H( (X)) H(f(X)).

6 952 Broderick O. Oluyede and Mekki Terbeche Proof: Noe ha (x)/f(x) =W (x)/δ is non-decreasing in x. Now, for <β<f(x)/ (x), and using he expansion of log() abou = 1, we have (x)log(f(x)/ (x)) = +. (x){( f(x) (x) f(x)dx (x)dx (f(x) (x)) 2 (x) 2 1 2β 2 dx 1) ( f(x) (x) 1)2 } 1 2β 2 dx (17) Noe also ha, since W (x) is non-decreasing, we have (x)log( (x)) (x)logf(x) f(x)logf(x), (18) for all x>. Therefore, (x)log( (x))dx (x)log(f(x))dx f(x)log(f(x))dx. (19) Equivalenly, E gw ( log( (X))) E f ( logf(x)). (2) Consequenly, H( (X)) H(f(X)). Theorem 3.2.LeW (x) be a non-decreasing weigh funcion wih W (x) >. Then (i) H(f, ; ) H (f, ; ), for all, (ii) H (f, ;)=E F [log(w (X))] + C, where C = log(δ ). Proof: (i) Noe ha G W (x) F (x), so ha G W (x)/f (x) 1 and log(g W (x)/f (x)), for all x. Now, H(f, ; ) = (F ()) 1 f(x)log(f(x)/ (x))dx + log G (x) W F (x) = (F ()) 1 H (f, ; )+log G (x) W F (x) (F ()) 1 H (f, ; ) H (f, ; ), (21) for all.

7 Expeced Uncerainy Measures 953 (ii) Le W (x) be a non-decreasing weigh funcion wih W (x) >. Then where C = log(δ ). H (f, ;) = = = = f(x) f(x)/f () log F () (x)/g W () dx f(x)log f(x) (x) dx f(x) f(x)log W (x)f(x)/δ dx f(x)log(w (x))dx + log(δ ) = E F (logw(x)) + C, (22) 4 Kernel Lengh-Biased Esimaes In his secion, kernel lengh-biased densiy esimaes are presened and convergence resul on he energy funcions is esablished in Theorem 4.2. Le f n (x) be he kernel esimae based on n independen observaions. The corresponding lengh-biased esimae is given by where f n (x) =(nh n ) 1 g n (x) =xf n (x)/μ n, (23) n i=1 K((x X i )/h n ), (24) μ n is an esimae of μ F, and h n is a sequence of posiive consan ending o zero, and K is a bounded densiy funcion saisfying lim u uk(u) =. Parzen (1962) showed ha E(f n (x)) f(x) asn,e(f n (x) f(x)) 2 as n, fn(x) E(fn(x)) σ(f n(x)) converges in disribuion o he sandard normal disribuion and nh n σ 2 (f n (x)) f(x) K 2 (u)du as n. Using he esimae μ n = n/ n i=1 Xi 1, we have g n (x) =x(n 2 h n ) 1 n i=1 K((x X i )/h n )/ n i=1 X 1 i. (25) The lengh-biased pdf can also be esimaed by using he fac ha μ n μ n = ni=1 X i /n, o ge g n (x) =x(nh n μ n ) 1 n i=1 K((x X i )/h n ). (26)

8 954 Broderick O. Oluyede and Mekki Terbeche Lemma 4.1. (Schuser (1969)). If f and is firs r +1 derivaives are bounded and {δ n } is a sequence of posiive numbers such ha b n = o(δ n ), hen here exiss posiive consans C 1 and C 2, such ha P {Sup x f n (r) (x) f (r) (x) >δ n } C 1 exp{ C 2 nδn 2 b2r+2 n }, (27) for sufficienly large n. Theorem 4.2. Le g n (x) be given by equaions (23) or (25). Then for δ n >, P { gn 2 (x)dx g 2 (x)dx δ n }, (28) as n. Proof: We noe ha gn 2 (x)dx g 2 (x)dx g 2 n (x) g2 (x) dx Sup x g 2 n (x) g2 (x). An applicaion of Schuser s Lemma leads o P { gn(x)dx 2 g 2 (x)dx δ n } P{ Sup x gn(x) 2 g 2 (x) δ n 2 } C 1 exp{ C 2 nδn 2 b2 n }. Applying Borel-Canelli Lemma, i follows ha C 1 exp{ C 2 nδ 2 n b2 n } as n, by an appropriae choice of he sequence b n, and he fac ha he series n=1 exp{ C 2 nδ 2 nb 2 n} <. Consequenly, P { g 2 n (x)dx g 2 (x)dx δ n }, as n. 5 Applicaions In his secion, we presen examples on energy funcions involving he normal and Rayleigh disribuions respecively. 1. Normal Disribuion. Le f(x; μ, σ) and (x; μ, σ) denoe he pdf for he normal and weighed normal disribuions respecively. Le W (x) be a non-decreasing weigh funcion. In paricular, we le W (x) = x. The energy funcion associaed wih he normal pdf f(x; μ, σ) is given by e(f(x; μ, σ)) = f 2 (x; μ, σ)dx = π 1/2 (2σ) 1. (29) The energy funcion e(f(x; μ, σ)) is a bijecive funcion of σ. If σ f and σ gw are he sandard deviaions of he normal disribuion funcions F and G W respecively, hen σ gw σ f if and only if e(f(x; μ, σ)) e( (x; μ, σ)).

9 Expeced Uncerainy Measures Rayleigh Disribuion. Le f(x; β) =2β 1 xe βx2, if x> and β>. (3) Then μ F = β 1/2 π 1/2 /2 and he corresponding weighed pdf wih W (x) =x is given by (x; β) =4π 1/2 β 3/2 x 2 e βx2, if x> and β>. (31) Noe ha ( (x; β)) 2 dx = = 16π 1 x 4 β 3 e βx2 dx 4π 1/2 x 2 β 3/2 e βx2 dx f 2 (x; β)dx. (32) Consequenly e( ) e(f) for all x>. ACKNOWLEDGEMENTS. The auhors are graeful o he edior and he referee for aking he ime o review his paper. References [1] Bhaacharyya, B. B., Franklin, L. A., and Richardson, G. D., A Comparison of Nonparameric Unweighed and Lengh-Biased Densiy Esimaion of Fibers, Communicaions in Saisics-Theory and Mehods, 17(11) (1988), [2] Ebrahimi, N. and Pellerey, F., New Parial Ordering of Survival Funcions Based on he Noion of Uncerainy, Journal of Applied Probabiliy, 32 (1995), [3] Guiasu, S., Informaion Theory wih Applicaions, New York, McGraw- Hill, [4] Gupa, R.C., and Keaing, J.P., Relaions for Reliabiliy Measures Under Lengh-Biased Sampling, Scandinavian Journal of Saisics, 13 (1985), [5] Parzen, E., On Esimaion of a Probabiliy Densiy Funcion and Mode, Annals of Mahemaical Saisics, 33 (1962),

10 956 Broderick O. Oluyede and Mekki Terbeche [6] Pail, G. P., and Rao, C. R., Weighed Disribuions and Size-Biased Sampling wih Applicaions o Wildlife and Human Families, Biomerics, 34 (1978), [7] Rao, C. R., On Discree Disribuions Arising Ou of Mehods of Ascerainmen, In Classical and Conagious Discree Disribuions, G. P. Pail (ed.), Calcua, India, Pergamon Press and Saisical Publishing Sociey, (1965), [8] Schuser, E. F., Esimaion of a Probabiliy Densiy Funcion and is Derivaives, Annals of Saisics, 4 (1969), [9] Shannon, C. E., Mahemaical Theory of Communicaion, Bell Sysem Technical Journal, 27 (1948), and [1] Vardi, Y., Nonparameric Esimaion in he Presence of Bias, Annals of Saisics, 1(2) (1982), Received: June 7, 26