Non-Additive Entropy Measure and Record Values

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1 Appl. Math. If. Sci. 9, No. 3, ) 54 Applied Mathematics & Iformatio Scieces A Iteratioal Joural No-Additive Etropy Measure ad Record Values Vikas Kumar, ad H. C. Taeja 2 Departmet of Applied Sciece, UIET, M. D. Uiversity, Rohtak-24, Idia 2 Departmet of Applied Mathematics, Delhi Techological Uiversity, Delhi - 42, Idia Received: 2 Aug. 24, Revised: 2 Nov. 24, Accepted: 2 Nov. 24 Published olie: May 25 Abstract: No-additive etropy measures are importat for may applicatios. We study Havrda ad Charvat etropy for record values ad have show that this characterizes the uderlyig distributio fuctio uiquely. Also the o-additive etropy of record values has bee derived i case of some specific distributios. Further we propose a geeralized residual etropy measure for record value. Keywords: Geeralized etropy, Record value, Probability itegral trasformatio, Residual etropy. Itroductio Suppose X, X 2,, X be a sequece of idepedet ad idetically distributed i.i.d.) radom variables with a commo absolutely cotiuous distributio fuctio cdf) F, probability desity fuctio pdf) f, ad survival fuctio F = F. A observatio X j will be called a upper record value if its value exceeds that of all previous observatios. Thus X j is a upper record if X j > X i for every i < j. A aalogous defiitio ca be give for a lower record value. Let R j) deote the time idex) at which the jth record value is observed. Sice the first observatio is always a record value, we have R)=,,RJ+ )= mi i : X i > X R j), where R) is defied to be. The sequece of upper record values ca thus be defied by U j = X R j), j =,2,3, Let D j) = R j + ) R j) deote the iter-record time betwee the jth record value ad j+ )th record value, ad let the jth record value X R j) be deoted by X j for simplicity. The the probability desity fuctio of the jth record value X j is give by g Xj x)= l Fx)) j fx), x> ) Γj) The survival fuctio is Ḡ Xj x)= l Fx)) j Fx) = Γ ; log Fx)), j= Γ j) Γ) 2) where Γa; x), the icomplete gamma fuctio, is defied as Γa;x)= x a e x dx, x, a>, x refer to, David ad Nagaraja 23, p.32). Records ca be viewed as order statistics from a sample whose size is determied by the values ad the order of occurrece of the observatios. I reliability theory, order statistics ad record values are used for statistical modelig. The m+)th order statistics i a sample of size represets the life legth of a m out of system. Record values are used i shock models ad miimal repair systems, refer to Kamps 994). Record values arise aturally i problems such as idustrial stress testig, meteorological aalysis, hydrology, sportig ad athletic evets, ad ecoomics, refer to Arold et al. 998), Nevzorov 2) ad Ahsaullah 24). Several authors have studied the characterizatio of distributio fuctio F based o the properties of order statistics ad record values, refer to Nagaraja ad Nevzorov 997), Raqab ad Awad 2), ad Balakrisha ad Stepaov 24). The idea of iformatio-theoretic etropy was first itroduced by Shao 948) ad later by Weier 949) i Cyberetics. Let X be a absolutely cotiuous radom variable which deotes the lifetime of a device or, a system with probability desity fuctio fx). The the average amout of ucertaity associated with the radom Correspodig author vikas iitr82@yahoo.co.i c 25 NSP

2 542 V. Kumar, H. C. Taeja: No-Additive Etropy Measure ad Record Values variable X, as give by Shao etropy 948) is HX)= fx)log fx)dx. 3) The measure 3) is additive i ature i the sese that for two idepedet radom variables X ad Y HX Y)=HX)+HY), where X Y deotes the joit radom variable. I literature may authors have geeralized Shao etropy 3) i differet ways. A well-kow parametric extesio of the Shao etropy measure was defied by Havrda ad Charvat 967) as H α X)= f α x)dx, α, α >. 4) α Although etropy measure 4) was first itroduced by Havrda ad Charvat 967) i the cotext of cyberetics theory, it was Tsallis 988) who exploited its o-extesive features ad placed it i a physical settig. Hece etropy measure 4) is also kow as Tsallis etropy 988). Clearly as α, 4) reduces to 3). This etropy is similar to Shao etropy except for its o-additive ature, that is HX Y)=HX)+HY)+ α)hx)hy). I geeral, the o-additive measures of etropy fid justificatios i may biological ad chemical pheomea. Some properties ad applicatios of o-additive etropy measure 4) have bee studied by Tsallis 998, 22) ad, Tsallis ad Brigatti 24). Baratpour et al. 27, 28) obtaied results for the Shao etropy ad Reyi etropy of the order statistics ad record values. I this commuicatio we study results for the record values based o the o-additive etropy measure 4). The paper is orgaized as follows. A geeral expressio for the etropy measure 4) of a record value distributio is derived i Sectio 2. The etropies of record values associated with the uiform, expoetial, weibull, pareto, fiite rage ad gamma are preseted i Sectio 3. I a attempt to establish a coherece betwee the etropies of the paret ad the correspodig record value distributios, Sectio 4 is devoted to a characterizatio result. Geeralized residual etropy of record values has bee studied i Sectio 5. 2 Geeralized Etropy for Differet Uivariate distributio I this sectio, we derive geeralized etropy measure 4) for some specific uivariate cotiuous distributios. 2. Expoetiated Expoetial Distributio A radom variable X is said to have the expoetiated expoetial distributio deoted by X EEγ, θ), if its probability desity fuctio pdf) ad cumulative distributio fuctio cdf) are give by ad fx)=γθ exp θx) exp θx) γ, x> 5) Fx)= exp θx) γ, γ >, θ >, 6) respectively. I particular for γ =, 5) is the expoetial distributio. The expoetiated expoetial distributio itroduced by Gupta ad Kudu 999) has some iterestig physical iterpretatios. Cosider a parallel system cosistig of γ compoets, the system works, oly whe at least oe of the γ-compoets works. If the lifetime distributios of the compoets are idepedet idetically distributed i.i.d.) expoetial radom variables, the the lifetime distributio of the system is defied as 6). Etropy of a radom variable X is a measure of its ucertaity. Let X EEγ, θ); we derive explicit form for the geeralized etropy measure 4) for X. For the pdf give by 5), f α x)dx=γθ) α exp θαx) exp θx)αγ α dx. 7) O substitutig y = exp θ x), 7) reduces to γ α θ α yα y αγ α dy=γ α θ α Bα;αγ α+ ). So 4) takes the form H α X)= γ α θ α Bα;αγ α+ ). 8) α If γ =, the 8) reduces to H α X) = α θ α α, the etropy measure 4) for expoetial distributio. Table gives the o-additive Havrda ad Charvat etropy measure 967) for some specific probability distributios. 3 Geeralized Etropy of Record Value Obtaied for Specific Distributios We will use the probability itegral trasformatio of the radom variable U = Fx), where the distributio of U is the stadard uiform distributio. The probability itegral trasformatio provides the followig useful c 25 NSP

3 Appl. Math. If. Sci. 9, No. 3, ) / Table : yyyyyy Distributio Desity fuctio fx) Etropy H α X) Uiform b a) ; x [a,b] α) b a) α Expoetial θe θx ; x>, θ > θ α α) [ α ] θ Pareto θ θ ; x >, θ > α x θ+) α) [αθ+α ] α Fiite Rage a b b x)a ; x b, a> a α b α α) [ +a )α) ] x Beta a x) b αa )+,αb )+) a,b) ; x, a, b> α) α a,b) Levy 2π σ )/2 e σ x µ) 3/2 2x µ) ; x> µ, µ > α) 2 σ ) 3 α) 2 α 3α 2 )Γ 3α π /2 2 ) Folded Cramer θ +θx) 2 ; x, θ > α) θ α 2α ) represetatio of the etropy measure 4) for the radom variable X H α X)= f α ) F u)du. 9) α Next we prove the followig result. Lemma 3The etropy measure 4) of the j th record value X j ca be expressed as Γ[ j )α+] Γj) α α) E[ f α F e v )] α), ) where v Γ j )α+ ad E is the expectatio. Proof Geeralized etropy 4) of the j th record value is defied as g Xj x) α dx, α, α >. α Usig ), this ca be rewritte as α)γj) α l Fx) j )α f α x)dx Γj) α Substitutig l Fx) = u, ad hece, x = F e u ), we have α)γj) α u j )α e u [ f α F e u ) ] du Γj) α It ca be rewritte as So, the result follows. j )α+ ] Γ[ α) Γj) α E[ f α F e v )]. ) 3. Uiform Distributio If a radom variable X is uiformly distributed over a, b), a < b, the its desity ad distributio fuctios are give respectively by We have fx)= b a ad Fx)= x a b a, a<x<b. α)γj) α u j )α e u f α x)du Γj) α. Thus geeralized etropy 4) of the j th record value for uiform distributio is give as u j )α e u du α)γj) α b a) α Γj) α, which gives α)γj) α Γ j )α+ b a) α Γj) α. The geeralized etropy of the first record, that is, X, is H α X )= b a) α α α. 2) This is the etropy of the paret distributio for uiform variate as idicated i Table 2.. The etropy of the otrivial record, that is, X 2, is give as H α X 2 )= α) 3.2 Expoetial Distributio Γα+ ) b a) α. Let X be a radom variable havig the expoetial distributio with pdf fx) = θexp θx). Substitutig c 25 NSP

4 544 V. Kumar, H. C. Taeja: No-Additive Etropy Measure ad Record Values l Fx)=u, hece, u θ = F e u ) we have [ u )] α f α F e u )= f = θ α e uα ). θ 3) Thus geeralized etropy 4) of the j th record value for expoetial distributio is give as which gives α)γj) α u j )α e αu du θ α Γj) α, j )α+ θ α α)γj) αγ α j )α+ Γj) α. 4) For j =, that is, the etropy of the first record, we have H α X )= θ α α α) α. This is the etropy of the paret distributio. The etropy of the o-trivial record, that is, X 2, is give as Γα+ )θ α H α X 2 )= α) α α Pareto Distributio Let X be a radom variable havig Pareto distributio with pdf fx)= θθ xθ+, x >, θ >. 5) Substitutig l Fx) = u, we observe that x = F e u )= e u θ ad for computig H α X j ), we have ) θ α f α F e u ) = e uα )+ θ ). Thus geeralized etropy 4) of the j th record value for Pareto distributio is give as which gives, θ α)γj) α ) α u j )α α α+ e θ )u du Γ j) α, Γ j )α+ θ jα α)γj) α α [αθ + α ] j )α+ Γj) α. 6) Therefore, H α X )= α θ α [αθ + α ] α the etropy for the paret distributio. The etropy for o trivial record X 2 is give as H α X 2 )= Γα+ )θ 2α α α [αθ + α ] α+. 7) 3.4 Fiite Rage Distributio The pdf of the fiite rage distributio is give by fx)= a b The survival fuctio is x b) a, a >, x b. 8) Fx)= Fx)= x b) a. Substitutig l Fx) = u, we observe that x = F e u )=b e u a) ad for computig H α X j ), we have a ) α f α F e u )= e uα ) a ) 9) b Lemma 3. gives Γ j )α+ a jα α)γj) α b α [aα α+ ] j )α+ Γj) α. 2) For j =, the etropy for paret distributio is H α X )= θ α α [αθ + α ] α The etropy for o trivial record X 2 is give as H α X 2 )= Γα+ )a 2α α b α [aα α+ ] α+. 2) 3.5 Weibull Distributio A o-egative radom variable X is Weibull distributed, if its pdf is fx)=λ x exp λ x, λ, >, x> where λ ad are scale ad shape parameters respectively. The survival fuctio is Fx)= Fx)=e λ x., c 25 NSP

5 Appl. Math. If. Sci. 9, No. 3, ) / Substitutig l Fx) = u, we observe that x = F e u )= u λ ad for computig H α X j ), we have ) α u f α F e u )= λ α ) ) e uα ) Lemma 3. gives α)γj) α λ ) α Γjα α jα α α Γj) α. 22) For j = the etropy for paret distributio is λ α ) Γα α H α X )= α α) α α. For =, 22) reduces to 4), the etropy for j th record of expoetial distributio. 4 Characterizatio Problem I this sectio, we show that the distributio fuctio F ca be uiquely specified up to a locatio chage by the equality of Tsallis etropy of record values. First we state the followig lemma, due to Goffma ad Pedrick 965). Lemma 4A complete orthogoal system for the space L 2, ) is give by sequece of Laguerre fuctio φ x)=! e x 2 L x),. where L x) is the Laguerre polyomial, defied as the sum of coefficiets of e x i the th derivative of x e x, that is L x)=e x d dx x e x )= ) k ) k+)x k. k= 23) The completeess of Laguerre fuctios i L 2, ) meas that if f L 2, ) ad fx)e 2 x L x)dx=,, the f is zero almost everywhere. Theorem 4Let X ad Y be two radom variables with pdfs fx) ad gx) ad absolutely cotiuous cdfs Fx) ad Gx) respectively, with E[log fx)] 2 < ad E[loggx)] 2 <. The F ad G belog to the same locatio family of distributio, if, ad oly if H α Y j ), j, 24) where X j ad Y j are the j th upper records of X ad Y respectively. Proof The ecessary part is obvious. We oly eed to prove the sufficiecy part. Let We Kow that H α Y j ), j. α)γj) α l Fx) j )α f α x)dx Γj) α. 25) Substitutig l Fx) ) α = u, ad hece, x=f exp u α, i 25) we get α)γj) α u j u α e u) α Similarly, we get f α F e u) α )du Γ j) α. H α Y j )= α)γj) α u j u α e u) α g α G e u) α )du Γ j) α. If for two cdfs F ad G, these differeces coicide, we ca coclude that u α e u) α [ f α F e u) α ) 26) g α G e u) α )]u L u)du=, for all, By 26), we ca coclude that u α e u 2 u) α [ f α F e u) α ) g α G e u) α )]e u 2 u L u)du=, 27) for all, where L u) is Laguerre polyomial give i Lemma 4.. Usig the assumptio E[log fx)] 2 < ad E[loggx)] 2 <, ad Mikowski iequality, we ca coclude that u α e u 2 u) α f α F e u) α ) g α G e u) α ) L 2, ). Hece, by the completeess property of Lemma 4., we coclude that ff ν))=gg ν)), ν,). As, df ν)) dν =. Therefore, we have ff ν)) F ν) = G ν), ν,) or, F ν) = G ν)+c, where c is a costat. This cocludes the proof. c 25 NSP

6 546 V. Kumar, H. C. Taeja: No-Additive Etropy Measure ad Record Values 5 Geeralized Residual Etropy of Record Values I reliability theory ad survival aalysis, X usually deotes a duratio such as the lifetime of a compoet. The residual lifetime of the system whe it is still operatig at time t is X t = X t X > t) which has the probability desity, x t >, where Ft) = Ft) >. Ebrahimi 996) proposed the etropy of the residual lifetime X t as HX;t)= t fx;t) = fx) Ft) fx) Ft) fx) log dt,t >. 28) Ft) This measures the ucertaity of the residual lifetime of the system whe it is still operatig at time t. The role of residual etropy as a measure of ucertaity i order statistics ad record values has bee studied by may researchers, refer to, Zarezadeh ad Asadi 2), Bartapour et al. 27, 28). The geeralized residual etropy of order α is defied as H α X;t)= α) [ t f α x)dx F α t) ] ; α >, α. 29) For more details ad applicatio of this dyamic iformatio measure refer to Nada ad Paul 25), ad Kumar ad Taeja 2). Obviously, whe t =, 28) ad 29) reduce to iformatio measures 3) ad 4) respectively. Let X,X 2, X are i.i.d. radom variables with a absolutely cotiuous distributio F ad desity fuctio f, deotig the lifetime of compoets. The Z = mix,x 2,...,X ) represets the lifetime of the system, whose compoets are coected i series. The residual etropy measure 29) of the series system is idepedet of t, whe X i s are expoetially variate. I this cotext, we prove the followig theorem. Theorem 5If X,X 2,...,X are idepedet radom variables havig a expoetial distributio with parameters θ i,i =,2,...,, the the residual etropy 29), of the radom variable Z = mix,x 2,...,X ) is idepedet of the parameters t. Proof Sice Z = mix,x 2,...,X ), therefore the cumulative distributio fuctio c.d.f.) Z is F Z z)=pz z)= i= exp θ iz))= exp z i= θ i). Survival fuctio of Z is F Z z)= F Z z)=exp The its p.d.f. is give by f Z z)= d dz F Zz)= i= θ i [exp z i= z θ i ). i= θ i )]. Substitutig these values i 29) ad simplify, we obtai H α Z;t)= i= θ i α ), α α which is idepedet of t. Corollary 5If X,X 2,...,X are idepedet ad idetically distributed i.i.d.) radom variables, the : H α Z;t)= i= θ α ). α α The role of residual etropy as a measure of ucertaity i order statistics ad record value has bee studied by Zarezadeh ad Asadi 2). Next, we derive geeralized residual etropy of order α for the jth upper record value. Before the mai result we state the followig two lemmas which are easy to prove. Lemma 5Let Uj be the j th upper record value for a sequece of observatios from uiform distributio o,). The H α U j ;t)= Γ[ j )α+ ; log t)] α Γ α. ; log t)) 3) Proof For uiform distributio, usig 29) we have [ H α U j ;t)= t g α ] X j x)dx α) ḠX α ; α >, α. j t) 3) Puttig values from ) ad 2) i 3), we get the desired result 3). Lemma 52Let Ū j be the j th upper record values for a sequece of observatios from stadard expoetial distributio. The H α Ū j ;t)= α Γ[ j )α+ ;t] Γ α ;t) where U t Γ j )α+ ;t Ee α )U t, The proof follows o the same lies as i Lemma 5.. Now we state the mai result. 32) Theorem 52Let X, > be a sequece of i.i.d. cotiuous radom variable from the distributio Fx) with desity fuctio fx) ad the quatile fuctio F.). Let U j deote the j th upper record. The the dyamic geeralized etropy 29) of j th upper record value ca be expressed as H α Ū j ;t)= α Γ[ j )α+ ; log Ft)] Γ α ; log Ft)) E f α F e V z ), 33) where z= log Ft) ad V z Γ j )α+; log Ft) ad E is the expectatio. c 25 NSP

7 Appl. Math. If. Sci. 9, No. 3, ) / Proof Dyamic geeralized etropy 29) of the j th upper record value is defied as [ t g α ] X j x)dx H α U j ;t)= α) ḠX α ; α >, α j t) = α) t l Fx) j )α f α x)dx Γ α ; log Ft). 34) Substitutig log Fx) = u ad x = F e u ), we have H α U j ;t)= It ca be rewritte as u j )α e u log Ft) [ f α F e u ) ] du α)γ α ; log Ft) ) H α Ū j ;t)= α Γ[ j )α+ ; log Ft)] Γ α ; log Ft)) So, the result follows. E f α F e V z ). Example Let X have Weibull distributio with desity fx)=λ x exp λ x, λ, >, x> Here, x=f e u )= u λ.the we have ) α u f α F e u )= λ α ) ) e uα ) Therefore H α Ū j ;t)= α [ α ) λ α Γ jα α ;λt] Γ α ;λt jα α [α]. 35) Remark For b=2, 35) reduces to 2 λ α Γjα α 2 H α Ū j ;t)= ;λt2 ; α) Γ α ;λt 2 α jα α 2 36) the residual etropy of the jth record value from a Rayleigh distributio, that is, X Rayleighλ > ). Remark 2 For b=, 35) reduces to H α Ū j ;t)= α) λ α Γjα α+ ;λt Γ α ;λtα jα α+ 37) the residual etropy of the jth record value from a expoetial distributio, that is, X expλ). ; 6 Coclusio Iformatio theoretic measures of Shao ad Reyi which are additive i ature have bee studied by may researchers for record values. It is of iterest to study o-additive etropy measures, which fid applicatios i may pheomea, for record values. We have see that Havrda ad Charvat etropy measure for record values characterize the uderlyig distributio fuctio uiquely except for the locatio. Also the cocept geeralized residual etropy for record values has bee studied which ca be explored further both theoretical iterest ad applicatio poit of view. Refereces [] Ahsaullah, M. Record Values-Theory ad Applicatios, Uiversity Press of America Ic., New York, 24. [2] Arold, B. C. Balakrisha, N. Nagaraja, H. N. Records, Wiley, New York, 998. [3] Baratpour, S. Ahmadi, J. Arghami, N. R. 27). Etropy properties of record statistics, Statistical Papers, 48, [4] Baratpour, S. Ahmadi, J. Arghami, N. R. 28). Characterizatios based o Reyi etropy of order statistics ad record values, Joural of Statistical Plaig ad Iferece, 38, [5] David, H. A. Nagaraja, H. N. 23) Order Statistics. Wiley, Hoboke, New Jersey. [6] Ebrahimi, N. 996). How to measure ucertaity i the residual lifetime distributios, Sakhya A, 58, [7] Goffma, C. Pedrick, G. First course i fuctioal aalysis. Pretice Hall Ic. 965). [8] Havrda, J. ad Charvat, F. 967). Quatificatio method of classificatio process:cocept of structural α-etropy, Kyberetika, 3, [9] Kumar, V. Taeja, H. C. 2). A geeralized etropybased residual lifetime distributios, Iteratioal Joural of Biomathematics, 42), [] Kamps, U. 994). Reliability properties of record values from o-idetically distributed radom variables. Commuicatios i Statistics Theory ad Methods, 23, [] Nada, A. K. Paul, P. 26). Some results o geeralized residual etropy, Iformatio Scieces, 76, [2] Nevzorov, V. Records: Mathematical Theory, Traslatio of Mathematical Moographs, vol. 94, America Mathematical Society Providece, RI, USA, 2. [3] Raqab, M. Z., Awad, A. M. 2). Characterizatios of the Pareto ad related distributios. Metrika, 52, [4] Shao, C. E. 948). A mathematical theory of commuicatio, Bell System Techical Joural, 27, [5] Tsallis, C. 988). Possible geeralizatio of Boltzma- Gibbs statistics, Joural Statistical Physics, 52, [6] Tsallis, C. ad Brigatti, E. 24). No-extesive statistical mechaics: a brief itroductio, Cotiuum Mechaics ad Thermodyamics, 6, c 25 NSP

Results o residual Reyi etropy of order statistics ad record values, Iformatio Scieces, 8, 495-426. Vikas Kumar has obtaied his M.Sc ad M.

8 548 V. Kumar, H. C. Taeja: No-Additive Etropy Measure ad Record Values [7] Zahedi, H. ad Shakil, M. 26). Properties of etropies of record values i reliability ad life testig cotext, Commuicatios i Statistics - Theory ad Methods, 35, [8] Zarezadeh, S. ad Asadi, M. 2). Results o residual Reyi etropy of order statistics ad record values, Iformatio Scieces, 8, Vikas Kumar has obtaied his M.Sc ad M.Phil degree i Applied Mathematics from IIT Roorkee ad ISM Uiversity, Dhabad i 25 ad 27 respectively. He received the Ph.D. degree i Mathematics from Uiversity of Delhi, Delhi. Curretly, he is a Assistat Professor i Mathematics, UIET, M. D. Uiversity, Rohtak, Idia. His research iterests are iformatio theory ad its applicatios ad mathematical modelig. He has published research articles i reputed iteratioal jourals of Mathematics ad Statistical scieces. H. C. Taeja presetly Professor i Applied Mathematics ad Dea Alumui & Iteratioal Affairs) at Delhi Techological Uiversity, Delhi got his Ph.D. i the field of Iformatio Theory i the year 985. He has vast experiece of about 35 years of teachig i mathematics ad statistics both at UG ad PG levels at differet uiversities ad techical istitutios. Dr. Taeja has published more tha sixty research papers i the field of Iformatio Theory & Codig i jourals of iteratioal repute ad is a member of various atioal ad iteratioal societies. He has authored two books, oe?advaced Egieerig Mathematics? ad the secod?statistical Methods for Egieerig ad Scieces?. c 25 NSP

Confidence interval for the two-parameter exponentiated Gumbel distribution based on record values

Confidence 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