Souther Illiois Uiversity Carbodale OpeSIUC Publicatios Couselig, Quatitative Methods, ad Special Educatio 2017 Derivatio of the Expoetial Distributio through a Ifiite Sie Series Todd C. Headrick Souther Illiois Uiversity Carbodale, headrick@siu.edu Michael E. May Souther Illiois Uiversity Carbodale Follow this ad additioal works at: http://opesiuc.lib.siu.edu/cqmse_pubs Copyright c 2017 Todd Christopher Headrick ad Michael E. May. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Recommeded Citatio Headrick, Todd C. ad May, Michael E. "Derivatio of the Expoetial Distributio through a Ifiite Sie Series." Applied Mathematical Scieces 11 ( Ja 2017): 2023-2030. doi:10.12988/ams.2017.76197. This Article is brought to you for free ad ope access by the Couselig, Quatitative Methods, ad Special Educatio at OpeSIUC. It has bee accepted for iclusio i Publicatios by a authorized admiistrator of OpeSIUC. For more iformatio, please cotact opesiuc@lib.siu.edu.
Applied Mathematical Scieces, Vol. 11, 2017, o. 41, 2023-2030 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.76197 Derivatio of the Expoetial Distributio through a Ifiite Sie Series Todd Christopher Headrick Departmets of CQMSE ad Psychology (Quatitative Methods-Statistics) Wham Bldg., Mail Code 4618 Souther Illiois Uiversity Carbodale Carbodale, IL, USA 62901-4618 Michael E. May Departmet of CQMSE ad the Rehabilitatio Istitute (Behavior Aalysis ad Therapy) Wham Bldg., Mail Code 4618 Souther Illiois Uiversity Carbodale Carbodale, IL, USA 62901-4618 Copyright c 2017 Todd Christopher Headrick ad Michael E. May. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract This commuicatio provides a derivatio of the ofte-used oeparameter expoetial family of distributios based o a ifiite sie series. The mai results associated with this derivatio are the fiite approximatios of the probability desity fuctio (pdf) ad the cumulative distributio fuctio (cdf) of the exact oe-parameter expoetial family of distributios. The limit of these fiite fuctios are the exact pdf ad cdf. Numerical examples are provided to compare ad cotrast the fiite approximatios of distributios i terms of error usig the fiite-based cdf ad the exact stadard expoetial cdf. We would also ote that the fiite approximatios of the pdf ad cdf vary i terms of both shape ad percetage poits. I view of this, the fiite pdf ad cdf offer a user the flexibility to potetially provide more accurate approximatios i the cotext of fittig distributios to data rather tha approximatios that are based solely o the exact expoetial pdf or cdf most otably whe distributios are heavy-(right)tailed.
2024 Todd Christopher Headrick ad Michael E. May Keywords: Statistical Distributio Theory; Fiite Approximatios; Probability Desity Fuctio; Distributio Fuctio 1 Itroductio The cotiuous oe-parameter expoetial distributio is oe of the most ofte used distributios to make iferetial or probability statemets with respect to evets that occur ad reoccur at radom over time e.g. life time evets ad survival aalysis. Specifically, Epstei ad Sobel [5] discussed the use of the oe-parameter expoetial distributio i terms of life testig, fatigue testig, ad i other cotexts of destructive test situatios such as the amout of (i) curret eeded to blow out a fuse, (ii) voltage eeded to break dow a codeser, or (iii) force to rupture some physical material. Further, the expoetial distributio is closely related to both the Poisso ad geometric distributios. For example, the expoetial distributio has bee demostrated to be related to the Poisso distributio i terms of the umber of successive chages of discrete observatios that occur durig a give cotiuous time iterval e.g. the waitig times betwee successive chages; which are also radom variables. Furthermore, the expoetial distributio ca also be derived based o the Poisso or geometric distributios see e.g. Johso, Kotz, ad Kemp [13]; Johso, Kotz, ad Balakrisha [12]; Marsaglia [15]; Leemis ad McQuestio [14]. The expoetial or approximate expoetial distributios have also bee demostrated to be useful i may Mote Carlo studies i terms of comparig ad cotrastig the Type I error rates ad (or) power properties associated with parametric ad (or) oparametric statistics. See, for example, Algia, Oshima ad Lie [1]; Boeau [2]; Coover ad Ima [3]; Doaldso [4]; Harwell ad Serli [6]; Headrick ad Sawilowsky [9, 10]; Headrick [7]; Headrick ( [8], p. 106). I view of the importace of the oe-parameter expoetial distributio, the purpose of this commuicatio is to derive this statistical distributio through a ifiite sie series; which is, as far as we are aware, wholly ew. It is demostrated that the fiite derivatios of the pdf ad cdf provided below ca yield good approximatios to the exact pdf ad cdf. Further, the derivatio provided herei of the (smaller values of the atural umber idex set) fiite approximatios to the oe-parameter expoetial distributio may vary cosiderably i terms of shape ad percetage poits. As such, this may provide a user with more accurate approximatios i the cotext of fittig distributios to real-world data.
Derivatio of the expoetial distributio through a ifiite sie series 2025 2 Mathematical Developmet The exact forms of the oe-parameter expoetial probability desity fuctio (pdf) ad cumulative distributio fuctio (cdf) cosidered herei are respectively as follows (e.g. Hogg ad Tais, 2001, pp. 178-179): where 0 x < ad the parameter θ > 0. f(x) = 1 θ e x/θ, (1) F (x) = 1 e x/θ (2) The derivatio of equatios (1) ad (2) begis by makig use of the followig lemma. LEMMA: If is the idex set, such that N, the the followig ifiite sie series si x = 1 (π x), 0 < x < 2π. (3) 2 PROOF: The proof begis by first settig As such, we will let m It follows that for y < 1 yields y 1 (cos x+i si x) = Cosequetly, we have z = cos x + i si x. (4) y 1 z = z{1 (yz)m }. (5) 1 yz cos x + i si x 1 y cos x yi si x y 1 (si x) = = (cos x y) + i si x 1 2y cos x + y 2. (6) si x 1 2y cos x + y 2. (7) The ifiite series i equatio (7) is uiformly coverget for all values of y ad for y p < 1. It follows that itegratig with respect to y, where 0 < y < 1, yields
2026 Todd Christopher Headrick ad Michael E. May ( ) si x y y du = si x 1 2u cos x + u 2 = si x = 0 y 0 du (u cos x) 2 + (si x) 2 [ ( u cos x ta 1 si x ) = ta 1 ( y cos x si x )] u=y u=0 ta 1 ( cot x) ad thus we have ( ) si x y = ( ) y cos x ( π ) ta 1 + si x 2 x, 0 < x < π ( ) ( ) y cos x 3π ta 1 + si x 2 x, π < x < 2π. (8) Let us suppose that x is either zero or a multiple of 2π. As such, it ecessarily follows that the ifiite sie series is coverget for 0 y 1 ad y ad is positive, mootoic, decreasig, ad bouded. Hece, if we let y 1 i equatio (8), ad (agai) if x is either zero or a multiple of 2π, yields ( si x ta 1 ta 1 ) ( π ) 2 x + 2 x, 0 < x < π = ( ta 1 ta 1 ) ( ). (9) 3π 2 x + 2 x, π < x < 2π Whece, based o equatio (9), we the have equatio (3) as give above i.e. si x = 1 (π x), 0 < x < 2π. 2 REMARK: We would ote that the fial step, to obtai equatio (3), is justified by the fact that if the series a (x) is uiformly coverget for a x b to the sum a(x), ad if, for each value of, a (x) teds to the limit (s ) as x x 0, where x 0 is some poit i the rage of (a, b), the, as x x 0, a(x) teds to the limit δ, where δ is the sum of the ifiite series s.
Derivatio of the expoetial distributio through a ifiite sie series 2027 3 Mai Result Expoetiatio of equatio (3) ad usig a geeral scaler of 1/θ (i lieu of 1/2 i equatio 3) for values of θ > 0 ad subsequetly itegratig with respect to x yields: 0 e (1/θ)(π x) dx = θe π/θ. (10) Solvig equatio (10) for a costat to esure that this equatio itegrates to oe yields a costat of (e π/θ )/θ for θ > 0. As such, it ecessarily follows that the fuctios provided below i equatios (11) ad (12) are the fiite approximatios of the expoetial pdf ad cdf i equatios (1) ad (2). Specifically, for fiite values of k we have { f(x) = 1 ( (e π/θ ) 1 + π x ) } k, (11) θ kθ ad { ( ) k F (x) 1 π + kθ x = (e π/θ ) ( π kθ + (π + kθ) 1+k (π + kθ + x) k ) + x}, θ(1 + k) kθ (12) where x > 0, 0 < k <, θ > 0, ad π +kθ > x. Takig the limit as k i equatios (11) ad (12) will result i these approximatig fuctios to coverge to the exact oe-parameter expoetial pdf ad cdf give i equatios (1) ad (2), respectively. 4 Numerical Examples ad Coclusio Figure 1 below gives the graphs of the approximate pdfs associated with equatio (11) for a expoetial distributio with θ = 1, ad k = 10, 10 2, 10 3, 10 5. Further, preseted below i Table 1 are approximate cumulative proportios, based o equatio (12), ad are associated with the exact 90-th percetile value for the stadard expoetial pdf (x = 2.302585...), as well as the error i these proportios. Ispectio of Table 1 idicates that as the value of k icreases the the approximatio becomes closer to the exact proportio (0.90) of the stadard expoetial distributio. If we were to icrease the value of k to k = 10 6 (or k = 10 7 ) i Table 1, the the error would be less tha 2.7 10 6 (or 2.3 10 7 ). I coclusio, it is worthy to poit out that there is o real-world set of data that will exactly follow a theoretical statistical distributio. I view of this, the derivatio preseted i this paper offers a variety of approximate expoetial
2028 Todd Christopher Headrick ad Michael E. May distributios with varyig shapes ad percetage poits. Thus, this allows a user to alter the shape of a approximate expoetial distributio i such a maer that may create a more suitable approximatio to a set of data i terms of distributio fittig. Specifically, whe empirical distributios are cotamiated with outliers i the right tail of the distributio. Coflict of Iterests The authors declare that there are o coflict of iterests regardig the publicatio of this paper. Refereces [1] J. Algia, T.C. Oshima ad W. Li, Type I error for Welch s test ad James s secod-order test uder o-ormality ad iequality of variace whe there are two groups, Joural of Educatioal ad Behavioral Statistics, 19 (1994), 275-291. https://doi.org/10.3102/10769986019003275 [2] C.A. Boeau, The effects of violatios of assumptios uderlyig the t-test, Psychological Bulleti, 57 (1960), 49-64. https://doi.org/10.1037/h0041412 [3] W.J. Coover ad R.L. Ima, Aalysis of covariace usig the rak trasformatio, Biometrics, 38 (1982), 715-724. https://doi.org/10.2307/2530051 [4] T.S. Doaldso, Robustess of the F-test to errors of both kids ad the correlatio betwee the umerator ad deomiator of the F-ratio, Joural of the America Statistical Associatio, 63 (1968), 660-676. https://doi.org/10.2307/2284037 [5] B. Epstei ad M. Sobel, Life testig, Joural of the America Statistical Associatio, 48 (1953), 486-502. https://doi.org/10.1080/01621459.1953.10483488 [6] M.R. Harwell ad R.C. Serli, A empirical study of a proposed test of oparmetric aalysis of covariace, Psychological Bulleti, 104 (1988), 268-281. https://doi.org/10.1037/0033-2909.104.2.268 [7] T.C. Headrick, Fast fifth-order polyomial trasforms for geeratig uivariate ad multivariate o-ormal distributios, Computatioal Statistics & Data Aalysis, 40 (2002), 685-711. https://doi.org/10.1016/s0167-9473(02)00072-5
Derivatio of the expoetial distributio through a ifiite sie series 2029 [8] T.C. Headrick, Statistical Simulatio: Power Method Polyomials ad other Trasformatios, Chapma & Hall/CRC, 2010. [9] T.C. Headrick ad S.S. Sawilowsky, Properties of the rak trasformatio i factorial aalysis of covariace, Commuicatios i Statistics: Simulatio ad Computatio, 29 (2000), 1059-1087. https://doi.org/10.1080/03610910008813654 [10] T.C. Headrick ad S.S. Sawilowsky, Weighted simplex procedures for determiig boudary poits ad costats for the uivariate ad multivariate power methods, Joural of Educatioal ad Behavioral Statistics, 25 (2000), 417-436. https://doi.org/10.3102/10769986025004417 [11] R.V. Hogg ad E.A. Tais, Probability ad Statistical Iferece, Pretice Hall, 2001. [12] N.L. Johso, S. Kotz ad N. Balakrisha, Cotiuous Uivariate Distributios, Vol. 1, Joh Wiley ad Sos, 2001. [13] N.L. Johso, S. Kotz ad A.W. Kemp, Uivariate Discrete Distributios, Vol. 1, Joh Wiley ad Sos, 1993. [14] L.M. Leemis ad J.T. McQuestio, Uivariate distributio relatioships, America Statisticia, 62 (2008), 45-53. https://doi.org/10.1198/000313008x270448 [15] G. Marsaglia, Geeratig expoetial radom variables, Aals of Mathematical Statistics, 32 (1961), 899-900. https://doi.org/10.1214/aoms/1177704984 Received: Jue 30, 2017; Published: August 11, 2017 Value of k Cumulative Proportio Error k = 10 0.697852... 0.202148... k = 10 2 0.873523... 0.026476... k = 10 3 0.897268... 0.002732... k = 10 5 0.899973... 0.000027... Table 1: Cumulative proportios usig the exact value (x = 2.302585...) associated with the stadard expoetial distributio for the 90-th percetile. The proportios are based o equatio (12) ad the values of k = 10, 10 2, 10 3, 10 5 that are provided i Figure 1.
2030 Todd Christopher Headrick ad Michael E. May (a) k = 10 (b) k = 10 2 (c) k = 10 3 (d) k = 10 5 Figure 1: Approximatios of the stadard expoetial distributio based o equatio (11)