Research Article Generalized Residual Entropy and Upper Record Values

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1 Joural of Probability ad Statistics Volume 2015, Article ID , 5 pages Research Article Geeralized Residual Etropy ad Upper Record Values Suchada Kayal Departmet of Mathematics, Natioal Istitute of Techology Rourkela, Rourkela , Idia Correspodece should be addressed to Suchada Kayal; suchada.kayal@gmail.com Received 1 Jue 2015; Accepted 26 July 2015 Academic Editor: RamóM. Rodríguez-Dagio Copyright 2015 Suchada Kayal. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. I this commuicatio, we deal with a geeralized residual etropy of record values ad weighted distributios. Some results o mootoe behaviour of geeralized residual etropy i record values are obtaied. Upper ad lower bouds are preseted. Further, based o this measure, we study some compariso results betwee a radom variable ad its weighted versio. Fially, we describe some estimatio techiques to estimate the geeralized residual etropy of a lifetime distributio. 1. Itroductio There have bee several attempts made by various researchers to geeralize the Shao etropy (see Shao [1]) sice its appearace i Bell System Techical Joural. For various properties ad applicatios of the geeralized etropy measures, we refer to Kapur [2], Reyi [3], Tsallis [4], ad Varma [5]. I this paper, we cosider geeralized residual etropy due to Varma [5]. Let X be a oegative radom variable represetig the lifetime of a system with a absolutely cotiuous cumulative distributio fuctio F(x), probability desity fuctio f(x), survival fuctio F(x)(= 1 F(x)), ad hazard rate λ F (x). The geeralized etropy of X is give by (see Varma [5]) V α,β (X) =( 1 ) l f α+β 1 (x) dx, 0 (1) 0 β 1 <α<β. Measure (1) reduces to Reyi etropy (see Reyi [3]) whe β = 1 ad reduces to Shao etropy (see Shao [1]) whe β = 1adα 1. We ofte fid some situatios i practice where the measure defied by (1) is ot a appropriate tool to deal with ucertaity. For example, i reliability ad life testig studies, sometimes it is required to modify the curret age of a system. Here, oe may be iterested to study the ucertaity of the radom variable X t = [X t X t].the radom variable X t is dubbed as the residual lifetime of a system which has survived up to time t 0 ad is still workig. Aalogous to Ebrahimi [6], the geeralized etropy of the residual lifetime X t is give by V α,β (X; t) =( 1 ) l f α+β 1 (x) dx, t F α+β 1 (t) 0 β 1 <α<β, t 0, which is also kow as the geeralized residual etropy. It reduces to (1) whe t = 0. Also (2) reduces to Reyi s residual etropy (see Asadi et al. [7]) whe β = 1ad reduces to residual etropy (see Ebrahimi [6]) whe β=1 ad α 1. Basedothegeeralizedetropymeasures give i (1) ad (2), several authors obtaied various results i the literature. I this directio, we refer to Kayal [8 11], Kayal ad Vellaisamy [12], Kumar ad Taeja [13], ad Sati ad Gupta [14]. I this paper, we study some properties ad characterizatios of the geeralized residual etropy give by (2) based o the upper record values. Let {X : = 1, 2,...} be a sequece of idetically ad idepedetly distributed oegative radom variables havig a absolutely cotiuous cumulative distributio fuctio F(x), probability desity fuctio f(x),adsurvival fuctio F(x). AobservatioX j i a ifiite sequece X 1,X 2,... is said to be a upper record value if its value is greater tha that of all the previous observatios. For coveiece, we deote U 1 = 1ad,fori 2, U i = mi{j; U i 1 < j ad X j > X Ui 1 }.The,X U1,X U2,... is called a sequece of upper record values. The probability (2)

2 2 Joural of Probability ad Statistics desity fuctio ad the survival fuctio of the th upper record value X U are give by f U (x) = H 1 (x) f (x), (3) ( 1)! F U (x) = 1 j=0 H j (x) F (x) = j! Γ (; H (x)), (4) Γ () respectively, where H(x) = l F(x) ad Γ(a; x) = x e u u a 1 du, a>0, x 0. Note that Γ(a; x) is kow as icomplete gamma fuctio. Also the hazard rate of X U is λ (x) = H 1 (x) / ( 1)! FU 1 λ F (x). (5) j=0 H (x) /j! Record values have wide spread applicatios i real life. For the applicatios of record values i destructive testig of woode beams ad idustrial stress testig, oe may refer to Glick [15] adahmadiadarghami[16]. Record values are also useful i meteorological aalysis ad hydrology. For a extesive study of record values ad applicatios, we refer to Arold et al. [17]. The paper is arraged as follows. I Sectio 2, we obtai various properties o the geeralized residual etropy. It is show that the measure give by (2) of the th upper record value of ay distributio cabeexpresseditermsofthatoftheth upper record valuefromu(0, 1) distributio. Upper ad lower bouds are obtaied. Mootoe behaviour of (2) based o the upper record values is ivestigated. I Sectio 3, based o(2), we study comparisos betwee a radom variable ad its weighted versio. We describe some estimatio techiques to estimate the geeralized residual etropy of a life distributio i Sectio 4. Some cocludig remarks have bee added i Sectio 5. Throughout the paper, we assume that the radom variables are oegative. The terms icreasig ad decreasig stad for odecreasig ad oicreasig, respectively. 2. Mai Results I this sectio, we study several properties of the geeralized residual etropy give by (2) based o the upper record values. First, we state the followig lemma. The proof is straightforward hece omitted. Lemma 1. Let X U deote the th upper record value from a sequece of idepedet observatios from U(0, 1).The, V α,β (X U =( 1 Γ(γ( 1) + 1, l (1 t)) ) l Γ γ. (; l (1 t)) I the followig theorem, we show that the geeralized residual etropy of the upper record value X U ca be (6) expressed i terms of that of X U.LetX be a radom variable havig trucated gamma distributio with desity fuctio f (x a,b) = b a Γ (a, t) e bx x a 1, For coveiece, we deote X Γ t (a, b). x>t 0, a > 0, b > 0. Theorem 2. The geeralized residual etropy of X U ca be expressed as V α,β (X U =V α,β (X U ;F(t)) +( 1 ) l [E (fγ 1 (F 1 (1 e V )))], where V Γ lf(t) (γ( 1)+1, 1). Proof. From (2), (3), ad(4) ad usig the trasformatio u= l F(x),weobtai V α,β (X U =( 1 ) f γ 1 (F 1 (1 e u )) e u u ( 1)γ l lf(t) [Γ (; l F (t))] γ du =V α,β (X U ;F(t))+( 1 ) l [E (f γ 1 (F 1 (1 e V )))], sice V α,β (X U ; F(t)) = Γ(γ( 1)+1, lf(t)).thiscompletes the proof. As a cosequece of Theorem 2, wegetthefollowig remark. Remark 3. Let X U E deote the th upper record value from a sequece of idepedet observatios from a stadard expoetial distributio. The, V α,β (X U E =( 1 Γ(γ( 1) + 1,t) ) l Γ γ (, t) +( 1 ) l E(e (γ 1)Z ), where Z Γ t (γ( 1)+1, 1). (7) (8) (9) (10) I Table 1, we obtai expressios of V α,β (X U for Weibull ad Pareto distributios. It is easy to show that λ FU (t)/λ F (t) ad f U (t)/f(t) are icreasig fuctios i t 0(see Li ad Zhag[18]). Therefore, we have the followig theorem whose proof follows alog the lies similar to those i Theorem 8 of Kayal [11].

3 Joural of Probability ad Statistics 3 Table 1: Expressios of V α,β (X U. Probability desity fuctios (PDF) V α,β (X U f (x θ) =θx θ 1 e xθ, x>0, θ>0 ( 1 ) l Γ(γ ((γ 1)/θ); tθ ) +( γ 1 (1/θ)(γ 1) ) l θ (γ ) l γ Γ γ (; t θ ) f (x θ,δ) = θδθ 1, x δ>0, θ>0 ( xθ+1 ) l ( (θ/δ) γ 1 Γ(γ( 1)+1; θl(δ/t)) ((1 +(1/θ))(γ 1)+1) γ( 1)+1 Γ γ (; θ l(δ/t)) ) Table 2: Bouds of V α,β (X U. PDF Bouds 1 f (x θ,δ) = θδθ x, x δ>0, θ>0 B r (t) +( ) l [ θ γ δ θ(γ 1) (θ+1)(γ 1)t ] (θ+1)(γ 1) θ+1 ( ) V α,β (X U +( γ 1 ) l ( θ δ ),forα+β>(<)2 f (x θ) = 1 θ exp ( x θ ), x>0, θ>0 B r (t) + 1 θ1 γ l ( γ+1 t (γ + 1) exp ( )) θ ( ) V α,β (X U γ 1 l(θ),forα+β>(<)2 Theorem 4. The th upper record value X U is icreasig (decreasig) geeralized residual etropy (IGRE (DGRE)) if X is IGRE (DGRE). Note that for th ad ( + 1)th upper record values, λ FU+1 (t)/λ FU (t) is icreasig i t 0(seeKochar[19]) lr ad hece X U XU+1. Therefore, the followig corollary immediately follows from Theorem 4. Corollary 5. The ( + 1)th upper record value X U+1 is IGRE (DGRE) if th upper record value X U is IGRE (DGRE). The followig theorem provides bouds for the geeralized residual etropy of the th upper record X U.Weomit theproofasitfollowsiaaalogousapproachsimilarto that of Theorem 11 give i Kayal [11]. Theorem 6. (a) Let V α,β (X U be fiite. If m r = max{γ( 1), l F(t)} is the mode of Γ lf(t) (γ( 1)+1, 1),the V α,β (X U B r (t) +( 1 ) l λ F (x) f γ 1 (x) dx, t (11) where B r (t) = (γ/(β α)) l Γ(; l F(t)) + (γ( 1)/(β α)) l m r (1/(β α))m r. (b) Let M = f(m) <,wherem is the mode of the distributio with desity fuctio f(x). AlsoX U deote the th upper record value from a sequece of observatio from U(0, 1).The,forα + β > (<)2, V α,β (X U ( ) V α,β (X U +( γ 1 ) l M. (12) As a applicatio of Theorem 6, we obtai bouds of Pareto ad expoetial distributios preseted i Table 2. The followig theorem gives the mootoe behaviour of the geeralized residual etropy of upper record values i terms of. Defiitio 7. Let X ad Y be two oegative radom variables with survival fuctios F(x) ad G(x),respectively. The, X is said to be smaller tha Y i the usual stochastic orderig, deoted by X st Y,ifF(x) G(x),forallx 0. Theorem 8. Let X U1,X U2,... be a sequece of upper record values from a distributio with cumulative distributio fuctio F(x) ad probability desity fuctio f(x).alsoletf(x) be a icreasig fuctio. The, V α,β (X U is icreasig (decreasig) i for α + β > (<)2. Proof. From (8),wehave V α,β (X U+1 V α,β (X U =V α,β (X U +1 ;F(t)) V α,β (X U ;F(t)) +( 1 E(f γ 1 (F 1 (1 e V +1 ))) ) l. E(f γ 1 (F 1 (1 e V ))) Moreover, for α + β > (<)2, lf(t) e x x γ( 1) dx lf(t) e x x ( 1) dx (13) (14) is a icreasig (decreasig) fuctio i t. Therefore, for u(x) = x ad V γ (x) = e x x γ st,wehavew γ ( )W st 1 for α + β > (<)2. Hece, from Theorem 12 of Kayal [11], it ca be proved that V α,β (X U ; F(t)) is icreasig (decreasig) for α+ β > (<)2. Now, alog the lies of the proof of Theorem 3.7 of Zarezadeh ad Asadi [20], the proof follows. This completes the proof of the theorem.

4 4 Joural of Probability ad Statistics 3. Weighted Distributios To overcome the difficulty to model oradomized data set i evirometal ad ecological studies, Rao [21]itroduced the cocept of weighted distributios. Let the probability desity fuctio of X be f(x), adletw(x) be the oegative fuctio with μ w = E(w(X)) <. Alsoletf w (x) ad F w (x), respectively, be the probability desity fuctio ad survival fuctio of the weighted radom variable X w,which are give by f w (x) = F w (x) = The hazard rate of X w is λ Fw (x) = w (x) f (x) μ w, (15) E (w (X) X x) F (x) μ w. (16) w (x) E (w (X) X x) λ F (x). (17) Forsomeresultsadapplicatiosoweighteddistributios, oe may refer to Di Crescezo ad Logobardi [22], Gupta ad Kirmai [23], Kayal [9], Maya ad Suoj [24], Navarro et al. [25], ad Patil [26]. I the preset sectio, we obtai some compariso results based o the geeralized residual etropy betwee a radom variable ad its weighted versio. We eed the followig defiitio i this directio. Defiitio 9. A radom variable X with hazard rate λ F (x) is said to have a decreasig (icreasig) failure rate (DFR (IFR)), if λ F (t) is decreasig (icreasig) i t 0. Theorem 10. Let X ad Y be two radom variables with cumulative distributio fuctios F(x) ad G(x), probability desity fuctios f(x) ad g(x), survival fuctios F(x) ad G(x), ad hazard rates λ F (x) ad λ G (x), respectively.if λ F (t) λ G (t), forallt 0, ad either F(x) or G(x) is DFR, the V α,β (X; t) ( )V α,β (Y; t),forα + β > (<)2. Proof. The proof follows alog the lies of that of Theorem 4 of Asadi et al. [7]. Theorem 11. UdertheassumptiosofTheorem10,ifλ F (t) λ G (t), forallt 0, ad either F(x) or G(x) is DFR, the V α,β (X; t) ( )V α,β (Y; t),forα + β > (<)2. Proof. Proof follows from that of Theorem 4 of Asadi et al. [7]. Theorem 12. (a) Suppose E(w(X) X t), or w(t), is decreasig. If X or X w is DFR, the, for all t 0, V α,β (X; t) ( )V α,β (X w,forα + β > (<)2. (b) Suppose E(w(X) X t), or w(t), is icreasig. If X or X w is DFR, the, for all t 0, V α,β (X; t) ( )V α,β (X w, for α + β > (<)2. Proof. It is ot difficult to see that λ F (t) λ Fw (t),forallt 0, whe either E(w(X) X t) or w(t) is decreasig. Now, the proof of part (a) follows from Theorem 10. Part(b)ca be proved similarly. This completes the proof of the theorem. Let X be a radom variable with desity fuctio f(x) ad cumulative distributio fuctio F(x). Alsoletμ x = E(X) > 0 be fiite. Deote the legth biased versio of X by X L. The, the probability desity fuctio of X L is give by f L (x) = xf (x) μ x. (18) The radom variable X L arises i the study of lifetime aalysis ad various probability proportioal-to-size samplig properties. Associated with a radom variable X, oe ca defie aother radom variable X E with desity fuctio f E (x) = F (x) μ x. (19) ThisdistributioiskowasequilibriumdistributioofX. The radom variables X L ad X E are weighted versios of X with weight fuctio w L (x) = x ad w E (x) = 1/λ F (x), respectively. The followig corollary is a cosequece of Theorem 12. Corollary 13. Let X be DFR. The, for all t 0, (a) V α,β (X; t) ( )V α,β (X L for α + β > (<)2; (b) V α,β (X; t) ( )V α,β (X E for α + β > (<)2. 4. Estimatio I this sectio, we discuss the problem of estimatio of the geeralized residual etropy of a statistical distributio based o upper record values. Here, we cosider expoetial distributio. It has various applicatios i practice. Let X follow expoetial distributio with mea λ.the,from(2), we obtai V α,β (X; t) =( α+β 1 ) l λ ( 1 ) l (α+β 1). (20) Based o the upper record values, the maximum likelihood estimator (mle) of λ ca be obtaied as δ = X U /, where X U is the th upper record value. Now, applyig ivariace property, we obtai the mle of V α,β (X; t) as V ml α,β =(α+β 1 ) l (X U ) ( 1 ) l (α+β 1). (21) Also the uiformly miimum variace ubiased estimator (umvue) of V α,β (X; t) ca be obtaied as

5 Joural of Probability ad Statistics 5 V mv α,β =(α+β 1 ) l ( X U exp (ψ ()) ) ( 1 ) l (α+β 1), (22) where ψ() is a digamma fuctio. To illustrate the estimatio techiques developed i this sectio, we cosider simulated data from expoetial distributio with mea 1. I this purpose, we use Mote-Carlo simulatio. Example 14. I this example, we cosider a simulated sample of size =5 from the expoetial distributio with mea 0.5. The simulated upper records are as follows: , , , , (23) Basedotheseupperrecordvalues,wehave V ml α,β = ad V mv α,β = whe α=1.2adβ= Cocludig Remarks I this paper, we cosider geeralized residual etropy due to Varma [5] of record values ad weighted distributios. We obtai some results o mootoe behaviour of this measure i upper record values. Some bouds are obtaied. Further, some compariso results betwee a radom variable ad its weighted versio based o the geeralized residual etropy are studied. Fially, two estimators of the geeralized residual etropy of expoetial distributio have bee described. Coflict of Iterests The author declares that there is o coflict of iterests regardig the publicatio of this paper. Refereces [1] C. E. Shao, A mathematical theory of commuicatio, The Bell System Techical Joural,vol.27,o.3,pp ,1948. [2] J. N. Kapur, Geeralized etropy of order alpha ad type beta, i Proceedigs of the Mathematical Semiar, pp.78 94,New Delhi, Idia, [3] A. Reyi, O measures of etropy ad iformatio, i Proceedigs of the 4th Berkeley Symposium o Mathematics, Statistics ad Probability 1960,vol.1,pp ,Uiversityof Califoria Press, Berkeley, Calif, USA, [4] C. Tsallis, Possible geeralizatio of Boltzma-Gibbs statistics, Joural of Statistical Physics, vol.52,o.1-2,pp , [5] R. S. Varma, Geeralizatio of Reyi s etropy of order α, Joural of Mathematical Scieces,vol.1,pp.34 48,1966. [6] N. Ebrahimi, How to measure ucertaity i the residual life time distributio, Sakhyā,vol.58,o.1,pp.48 57,1996. [7] M. Asadi, N. Ebrahimi, ad E. S. Soofi, Dyamic geeralized iformatio measures, Statistics & Probability Letters, vol. 71, o. 1, pp , [8] S. Kayal, O geeralized dyamic survival ad failure etropies of order (α, β), Statistics ad Probability Letters,vol.96,pp , [9] S. Kayal, Some results o dyamic discrimiatio measures of order (α, β), Hacettepe Joural of Mathematics ad Statistics, vol. 44, pp , [10] S. Kayal, Characterizatio based o geeralized etropy of order statistics, Commuicatios i Statistics Theory ad Methods.Ipress. [11] S. Kayal, Some results o a geeralized residual etropy based o order statistics, Statistica.Ipress. [12] S. Kayal ad P. Vellaisamy, Geeralized etropy properties of records, Joural of Aalysis,vol.19, pp.25 40,2011. [13] V. Kumar ad H. C. Taeja, Some characterizatio results o geeralized cumulative residual etropy measure, Statistics & Probability Letters, vol. 81, o. 8, pp , [14] M. M. Sati ad N. Gupta, O partial mootoic behaviour of Varma etropy ad its applicatio i codig theory, Joural of the Idia Statistical Associatio.Ipress. [15] N.Glick, Breakigrecordsadbreakigboards, The America Mathematical Mothly, vol. 85, o. 1, pp. 2 26, [16] J. Ahmadi ad N. R. Arghami, Comparig the Fisher iformatio i record values ad iid observatios, Statistics,vol.37,o. 5,pp ,2003. [17] B. C. Arold, N. Balakrisha, ad H. N. Nagaraja, Records, Joh Wiley & Sos, New York, NY, USA, [18] X. Li ad S. Zhag, Some ew results o Réyi etropy of residual life ad iactivity time, Probability i the Egieerig ad Iformatioal Scieces,vol.25,o.2,pp ,2011. [19] S. C. Kochar, Some partial orderig results o record values, Commuicatios i Statistics Theory ad Methods,vol.19,o. 1, pp , [20] S. Zarezadeh ad M. Asadi, Results o residual Reyi etropy of order statistics ad record values, Iformatio Scieces, vol. 180,o.21,pp ,2010. [21] C. R. Rao, O discrete distributios arisig out of methods of ascertaimet, Sakhyā,vol.27,pp ,1965. [22] A. Di Crescezo ad M. Logobardi, O weighted residual ad past etropies, Scietiae Mathematicae Japoicae,vol.64,o.2, pp , [23]R.C.GuptaadS.N.U.A.Kirmai, Theroleofweighted distributios i stochastic modelig, Commuicatios i Statistics Theory ad Methods, vol.19,o.9,pp , [24] S. S. Maya ad S. M. Suoj, Some dyamic geeralized iformatio measures i the cotext of weighted models, Statistica,vol.68,o.1,pp.71 84(2009),2008. [25] J. Navarro, Y. del Aguila, ad J. M. Ruiz, Characterizatios through reliability measures from weighted distributios, Statistical Papers,vol.42,o.3,pp ,2001. [26] G. P. Patil, Weighted distributios, i Ecyclopedia of Evirometrics, A.H.El-ShaarawiadW.W.Piegorsch,Eds.,Joh Wiley & Sos, Chichester, UK, 2002.

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