Songklanakarin Journal of Science and Technology SJST R1 Thongchan. A Modified Hyperbolic Secant Distribution

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1 A Modified Hyperbolic Secnt Distribution Journl: Songklnkrin Journl of Science nd Technology Mnuscript ID SJST-0-0.R Mnuscript Type: Originl Article Dte Submitted by the Author: 0-Mr-0 Complete List of Authors: Thongchn, Pnu; Ksetsrt University, Sttistics; Bodhisuwn, Wini; Ksetsrt University, Deprtment of Sttistics Keyword: hyperbolic function, hyperbolic secnt distribution, secnt function

2 Pge of Abstrct Originl Article A Modified Hyperbolic Secnt Distribution Pnu Thongchn nd * Wini Bodhisuwn Deprtment of Sttistics, Fculty of Science, Ksetsrt University, Chtuchk, Bngkok, 000 Thilnd * Corresponding uthor. Emil: fsciwnb@ku.c.th The ims of this pper re to introduce nd vlidte new distribution, which is relted to the hyperbolic function. The differentil eqution is pplied to obtin the survivl nd probbilistic functions. The moment generting function is provided by using the integrl trnsform. The proposed distribution re pplied to rel dt sets. It is illustrted tht it cn be used s n lterntive model in vrious disciplines such s electronics, finncil, wether, nd rrivl times. Keywords: hyperbolic function, hyperbolic secnt distribution, secnt function. Introduction The hyperbolic function is n importnt mthemticl function in reltion to trigonometric functions, exponentil functions, nd complex numbers. The hyperbolic secnt is prt of set of the hyperbolic functions, which is defined s sech( x) =. In, Bten introduced study relted to the hyperbolic secnt ( x x e + e ) function for usge with the probbility distribution, the hyperbolic secnt distribution

3 Pge of (HSD) (Bten, ). This work ws then expnded by Tlcko () who proposed the distribution for finncil return models. Consequently, the HSD provides n optiml fit nd exhibits more leptokurtosis thn both the norml nd logistic distributions. However, it is limited in its utility; it cnnot tke on vrious shpes due to lck of flexibility in its prmeters, thus nturl behviors cnnot be sufficiently explined with this model. Mny brnches of nturl sciences emphsize the study of phenomen such s the spreding of disese or growth of popultion by looking t rtes of chnge. Differentil eqution is n importnt technique used to solve these types of problems. Differentil eqution is useful techniqe in explining sttisticl properties nd vitl tools in proving or disproving sttistic-bsed issues. This model cn be pplied to the survivl nlysis, life time dt nlysis, nd relibility nlysis. This pper introduces techniques used to generte new distribution tht is constructed by modifying the hyperbolic functions into more flexible model using differentil equtions. We implements the technique with set ( S( t )) be stte t time t, defined ( S( t0) = S( t)),, ( S( tn) = S( t+ t)) t t 0 = t, t = t+ τ, t = t+ τ,, t n = t+ nτ where n belongs to {,,,,T}, τ = T / n, nd S( tk ) = S( tk ) S( tk ). The process cn be expressed ss( t0) = S( t), S( t) S( t0) = S, S( t) S( t) = S,, S( tk ) S( tk ) = Sk. Under the ssumption tht τ is very smll, we obtin S( tk ) S( tk ) = Sk = ds( t). Some properties of the survivl function, S( ), re right continuous in t. Nevertheless, for continuous survivl time T, S( ) is continuous with non-incresing function, ( S( t 0) = nd lim S( t ) = 0 ). n n

4 Pge of The rest of the pper is orgnized s follows. In Section, new fmily of distribution is proposed from the hyperbolic functions using differentil equtions to develop this new probbility function. Also we will pply rel dt sets to the proposed model in Section before strting our conclusions in Section.. A New Distribution This section presents n lterntive wys in which the new probbility function could be developed. We pply the differentil eqution to derive probbility density function by setting up the first order differentil eqution, nd solve it to get survivl function. We then tke derivtive of the survivl function. Consequently, probbility density function ssocited with survivl function will be obtined. Definition : Let T be rndom vrible on probbility spce ( Ω, F, P) with probbility density function ( f ( t; θ )), nd hs distribution function ( F ( t; θ )) nd survivl function ( S( t; θ ) = F ( t; θ )) nd ( f ( t; θ )) defined by ( F ( t; θ )) or ( S ( t; θ )) Proposition : Let T be rndom vrible on probbility spce ( Ω, F, P) with probbility density function f ( t; k, ) nd T [0, ), which produces the eqution below: + t e k t t e (π k log[ e ] + k log[ e + e ]) t ( e + e ) π π f ( t) = + π where is n initil vlue nd 0 k. e ( ) + log() log( e ), ()

5 Pge of Proof: We propose the new distribution by finding the solution of the following differentil eqution when ki S ( t) + S( t) =, S( ), ( t ) ( t ) π ( e + e ) = I [0, ) = 0 otherwise. Tking the first derivtive, we will obtin the probbility functions s seen in Eq. (). As for the solution function of the differentil eqution, we set U ( t ) s n rbitrry function, then multiplying Eq. () by U ( t ) we obtin U ( t) ki U ( t) S ( t) + U ( t) S( t) =, ( t ) ( t ) π ( e + e ) by derivtive product rule, it is then set to the following form where U ( t) = e dt. U ( t) ki ( U ( t) S( t)) =, ( t ) ( t ) π ( e + e ) Therefore, the result of differentil eqution Eq. () is which is the survivl function. U ( t) ki S( t) = [ + ] U ( t) π ( e + e ) dt C, ( t ) ( t ) t In this cse U ( t) = e nd the solution of the differentil eqution with initil vlue problem S( ) = is ( t ) t e (π k log( e ) + k log(e + e )) S( t) =. π () ()

6 Pge 0 of Now, tking the first derivtive, the pdf of T is Corollry : t ( t ) t + t e ( e (π k log( e ) + k log(e + e ))) e k t π ( ) f ( t) =. e + e π Let T be rndom vrible with the pdf f ( t; b ), which could be expressed s t b e f ( t; b) =, t b ( + e ) π with the loction prmeter b nd initil vlue c b. Proof: t b e Similr to Eq. (), f ( t) = obtined by solving the differentil eqution t b ( + e ) π expressed s ( ) =, S( c) =, ( ) ( ) ( t b t b π e + e ) ' S t by integrting both sides, we obtin the survivl function, which is the result from solving Eq.() c b t b π + rctn( e ) rctn( e ) S( t) =, π then by tking first derivtive, the pdf is obtined in the form where T (, ). t b e f ( t; b) =, t b ( + e ) π It is lso possible to present the generl differentil eqution s () ()

7 Pge of AkI ( A) ( ) + ( ) =, ( t ) ( t ) ( t b) ( t b) π ( e + e ) π ( e + e ) ' S t AS t by imposing A= 0 or. The Eq.() cn be reduced to Eq. () nd Eq.(), respectively. Following this logic, S ( ) = nd S ( ) c = re therefore the initil vlues of Eq. () nd Eq. (), respectively. () Moreover, the pdf derived from the survivl function is probbility function, which is sstified the following properties;. Setting S ( ) = nd S ( c ) = re initil vlue functions.the survivl function is monotonic decresing. lim S( t) = 0 t.if the pdf correspond to prmeter spce, then it will be greter thn zero. Some Properties of the New Distribution There re mny methods to solve the differentil equtions. An importnt method use the Lplce trnsform, which is relted to the moment generting functions. The Lplce trnsform of Eq. () is given by st L( s) = e f ( t) dt ; s> 0 + t t t st e k e (π k log[ e ] k log[ e e ]) = e ( ) d( t) t ( e + e ) π π ( s ) t ( s+ ) t ( s+ ) t e k e π ke log[ e ] = [ ( ) + ( ) ( ) t ( + e ) π π π ( s+ ) t t ke log[ e + e ] ( )] dt π + ( s ) t ( s+ ) t ( s+ ) t e k e (log[ e ] k) e = ( ) dt+ ( ) ( ) t ( + e ) π ( s+ ) π ( s+ )

8 Pge of ( s+ ) t t ke log[ e + e ] ( ) dt π + ( s ) t s s e k e (log[ e ] k) e = ( ) dt+ ( ) ( ) t ( e ) π ( s ) π ( s ) ( s+ ) t t ke log[ e + e ] + ( ) dt. () π Focusing on the lst term nd using the by prt integrtion technique, we hve ( s+ ) t t t ( s+ ) t ( s+ ) t ke log[ e + e ] k log[ e + e ] e e k dt= + t ( s+ ) ( s+ ) ( + e ) () ( ) [ ( ) dt], π π π substitue the lst term of Eq. () with Eq. (), L( S ) becomes ( s+ ) t s s e k e (log[ e ] k) e L( s) = ( ) dt+ ( ) ( ) t ( + e ) π ( s+ ) π ( s+ ) t ( s+ ) t ( s+ ) t k log[ e + e ] e e k t + [ ] + ( ) dt π ( s+ ) ( s+ ) ( + e ) π ( s+ ) t s s e k e (log[ e ] k) e dt t ( ) s+ ( + e ) π ( s+ ) π ( s+ ) = ( ) ( ) + ( ) ( ) s e k log[ e ] + ( ). (( s+ ))π The convergence of the integrl prt of Eq. () will be shown in Appendix. Under the condition of 0 t with power series, we rerrnge Eq. () to obtin the following Lplce trnsformtion k L S e e dt s ( s+ ) t n ( t) n ( ) = ( ) (( ) ) ( ). π ( s ) + + n= 0 ( s+ ) In ddition, the moment generting function (mgf) of t is given by st M ( t) = E( e ) = L( s), for ll moments if the Lplce trnsform exists (s shown in r r r Appendix), nd where the rth moment is E( t ) = ( ) L (0). In prticulr, e ()

9 Pge of n k ( ) Et ( ) = [ ] + ( + ). π (n+ ) n= 0 The moment generting function of Eq. () is given by = st M ( s) e f ( t) dt = e dt + e π st ( t b) ( t b) ( ) e ( s+ ) t b ( s ) t+ b = b e e dt dt ; s 0 ( t b) ( t b) ( ( e )) π b + ( ( e )) π > n sb ( ) e = (0) π ( s+ + n) n= 0 In the cse of Eq. () where symmetric properties define the set of domin s (, ), then M ( s ) cn be expressed s Eq.(0) Some plots of the survivl nd relibility functions relted to the proposed distribution re presented in Figure. Figure Some plots of the survivl function of the new distribution where k = 0,,,, from the bottom line respectively The survivl function of Eq. () re exhibited in Figure. Figure Some plots of the survivl function for the new distribution with some prmeter vlues of Eq. () In survivl nlysis, the hzrd function is defined s f ( t) ht ( ) = where f ( t ) S( t) nd S( t ) re the pdf nd survivl function, respectively. The hzrd function of the proposed distribution (Eq. ()) cn be written s

10 Pge of t e k t t ( e + e )( k log[ e ] + k log[ e + e ])) ht ( ) = +. π Some shpes of the hzrd function of the proposed distribution (Eq. ()) with some prmeters re shown in Figure Figure : Some plots of hzrd functions for the proposed distribution with prmeter =0 of Eq.() The hzrd function of Eq. () cn be written s t b e ht ( ) = ), ( t b) ( c b) ( t b) ( + e )( π+ rctn( e )) rctn( e ) where severl function with some prmeters re illustrted below. Figure Some plots of the hzrd functions for the proposed distribution with some prmeter vlues of Eq.() Figure shows some shpes of the pdf bsed on some selected prmeter vlues. It shows tht the new proposed distribution consists of vrious shpes. Figure Some pdf plots of new distribution (Eq. ()) with some prmeter vlues The symmetric behviors of Eq. () re illustrted in Figure. Figure Some pdf plots of the new distribution (Eq. ()) with some prmeter vlues

11 Pge of Applictions Four dt sets re fitted with the proposed distribution. The first exmple dels with the rte of chnge of filure times of electronic devices reported by Domm (0). The second exmple is the rte of chnge of soyben prices t Chicgo Bord of Trde (CBOT) from June 0 June fitted with the proposed distribution. The third nd fourth dt sets re US july precipittion (Top soybens production stte) from -0 nd the inter-rrivl times(0 minutes) dt set for crs (Lw, 0), respectively. The estimted prmeters could be crried out by mximum likelihood estimtion (MLE). In this study, we use bbmle (Bolker nd R Tem, 0) pckge of R progrmming lnguge (R Core Tem, 0) to obtin the prmeter estimtes. The fitting distribution for four exmples re verified by Figure. In ddition, the estimted prmeters using MLE re shown in Tble Figure Some fitted distributions with = min( t) Tble The estimte prmeters using MLE. Conclusions The im of this pper is to propose n lterntive method to generte new survivl function. We hve done so by the reltionship between differentil eqution nd hyperbolic function, nd obtin the solution of the method s non-incresing function; we obtin the survivl nd the probbility density functions from these techniques. Moreover, we orgnize the generl form of the model, which produces two

12 Pge of survivl nd the probbilistic functions. The vrious grphicl styles of the result cn be used to informed its flexibility in nlyzing behvior in rel dt. Acknowledgements The uthors would like to thnk to Deprtment of Sttistics, Fculty of Science nd the grdute school of Ksetsrt University. Also we thnk to Chroen Pokphnd Foods (CPF) for supporting to the first uthor. References Bten, W.D.. The probbility lw for the sum of n independent vribles. Bulletin of Americn Mthemticl Society. 0, - 0. Bten, W.D A new model with bthtub-shped filure rte using n dditive burr xii distribution. F.K wng. 0(). 0. Bten, W.D. 0. A new bthtub curve model with finite support. Relibility Engineering nd System Sfety., -. Bolker, B. nd Tem, R.D.C. 0. bbmle: Tools for generl mximum likelihood estimtion. R pckge version.0.. Domm, F. nd Condino, F. 0. A new clss of distribution functions for lifetime dt. Relibility Engineering nd System Sfety.,. Fisher, M.J. 0. Generlized Hyperbolic Secnt Distribution. Springer. Jeffrey, A. nd Zwillinger, D. 00. Tble of integrl, series, nd product. New York: Acdemic Press.

13 Pge of Klbfleisch, J.D. nd Prentice, R.L. 0. The Sttisticl Anlysis of Filure Dt. John Wiley nd son. Lw, A. 0. Simultion Modeling nd Anlysis (Mcgrw-Hill Series in Industril Engineering nd Mngement). Lwless, J.F.. Sttisticl Models nd Methods for Lifetime Dt. John Wiley nd son. Mnoukin, E.B. nd Ndeu, P.. A note on the hyperbolic secnt distribution. The Americn Sttisticin.,. R Core Tem. 0. R: A Lnguge nd Environment for Sttisticl Computing. R Foundtion for Sttisticl Computing, Vienn, Austri. Tlcko, J.. Perk distributions nd their role in the theory of weiner s stochstic vribles. Trbjos de Esttistic., -.

14 Pge of A convergence will be proved APPENDIX ( s+ ) t s s e k e (log[ e ] k) e dt+ t ( ) + ( + ) ( + ) ( + ) ( ) ( ) ( ) ( ) s. e π s π s The integrl is improper integrl, we use the comprison test to prove convergent s follow theorem A comprison test theorem then Suppose tht f nd g re continuous functions with 0 f ( t) g(t) for t.if.if g(t) dt is convergent, then f (t) dt is convergent g(t) dt is divergent, then f (t) dt is divergent. Espectilly, this improper integrl s shown ( s+ ) t e k t ( + ) ( ) dt. e π We set ( s+ ) t g( t) dt= e k, is greter thn f ( t ) for ll t, s> 0, nd ( s+ ) t g( t) dt= e k = ek ( s+ ) ( s+ ) t e k, wheres g( t ) t ( + ) f ( t) dt= ( ) dt e π ( s+ ) [ e ] ; t, s> 0.

15 Pge of e k It is verified tht g(t) dt converge to e ( s+ ) ( s+ ) t e k is convergent too. t ( + ) f ( t) dt= ( ) dt e π ( s+ ) [ ], thererfore,

16 Pge 0 of () prmeters = 0 (b) prmeters = Figure Some plots of the survivl function of the new distribution where k = 0,,,, from the bottom line respectively () prmeters b = 0; =,,,, Figure Some plots of the survivl function for the new distribution with some prmeter vlues of Eq. ()

17 Pge of () prmeters k = 0 (b) prmeters k = 0 (c) prmeters k = (d) prmeters k = π Figure : Some plots of hzrd functions for the proposed distribution with prmeter =0 of Eq.()

18 Pge of () prmeters = 0; b = (b) prmeters = 0; b = 0 Figure Some plots of the hzrd functions for the proposed distribution with some prmeter vlues Eq.()

19 Pge of () prmeters = 0; k = 0 (b) prmeters = 0; k = (c) prmeters = 0; k = (d) prmeters = 0; k = (e) prmeters = 0; k = (f) prmeters = 0; k = π (g) prmeters = ;k = Figure Some pdf plots of new distribution with some prmeter vlues of Eq. ()

20 Pge of () prmeters b = (b) prmeters b = (c) prmeters b = Figure : Some pdf plots of the new distribution with some prmeter vlues of Eq. ()

21 Pge of Density Density () prmeters k =. 0 0 (b) prmeters k = t t. 0

22 Pge of Density Density (c) prmeters k =. t t (d) prmeters k = Figure Some fitted distributions with = min( t)

23 Pge of Tble The estimte prmeters using MLE Dt set Estimted prmeters.the rte of chnge of filure times of electronic devices 0..The rte of chnge of soyben prices t Chicgo Bord of Trde (CBOT) from June 0 June July precipittion from the inter-rrivl times(0 minutes) dt set for crs 0 k