Max-Erlang and Min-Erlang power series distributions as two new families of lifetime distribution

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1 BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 275, 2014, Pages ISSN Max-Erlang Min-Erlang power series distributions as two new families of lifetime distribution Alexei Leahu, Bogdan Gheorghe Munteanu, Sergiu Cataranciuc Abstract. The distribution of the minimum maximum of a rom number of independent, identically Erlang distributed rom variables are studied. Some particular cases of such kind of lifetime distributions are discussed too. Mathematics subject classification: 62K10, 60N05. Keywords phrases: Power series distribution, distribution of the maximum, distribution of the minimum, Erlang distribution. 1 Preliminary results In this paper we present two new families of distribution, namely the Max- Erlang power series MaxErlPS distribution, respectively the Min-Erlang power series MinErlPS distribution. These are obtained by mixing the distribution of the maximum or the minimum of a fixed number of independent Erlang distributed rom variables, where the combination is obtained by the techniques that have been treated by Adamidis Loukas 1998, 1 more generally by Chahki Ganjali 2009, 6 or Baretto-Souza Cribari 2009, 3. Recently, the new distributions that model the reliability systems were obtained by exponential distribution with several discrete distributions the families of the power series distributions. For example, the distribution of the minimum of a sample of rom size with the exponential distribution was obtained. In this connection, the geometric distribution, the Poisson distribution the logarithmic distribution were considered by Adamidis Loukas 1998, 1, Kus 2007, 10, Tahmasbi Rezaei 2008, 19. Then, the previous results have been generalized by Chahki Ganjali 2009, 6 using the compounding exponential distribution with the power series distribution, thus obtaining the exponential power series distribution EPS type. Later, Morais Baretto-Souza 2011, 16 replaced the exponential distribution with the Weibull power series distribution WPS of the minimum of a sequence of the independent identically distributed rom variables i. i. d. r. v. in a rom number, studying the distribution of the strength of 1.5 cm glass fibers. The case of the maximum has been discussed analyzed by Munteanu 2013, 17, introducing the Max Weibull power series MaxWPS distribution that particularizes the complementary exponential geometric CEG distribution introduced by c A. Leahu, B. Gh. Munteanu, S. Cataranciuc,

2 MAX-ERLANG AND MIN-ERLANG POWER SERIES DISTRIBUTIONS Louzada, Roman Cancho 2011, 14, the complementary exponential Poisson CEP distribution introduced by Cancho, Louzada Barriga 2011, 5 the complementary exponential logaritmic CEL distribution introduced by Flores, Borges, Cancho Louzada 2013, 8. The geometric distribution which contains a power parameter was considered by Adamidis, Dimitrakopoulou Loukas 2005, 2 later generalized by Silva, Baretto-Souza Cordeiro 2010, 18. The Poisson distribution being treated by Cancho, Louzada Barriga 2011, 5 which then was generalized by Cordeiro, Rodriques Castro 2011, 7 as the COMPoisson distribution, this contains a parameter power, the power series distribution type is not considered because the maximum number will being one deterministic. The methodology techniques used in this article are shown in the paper of Leahu, Munteanu Cataranciuc 2013, 12, general framework illustrating the particular cases treated in the works of Adamidis Loukas 1998, 1, Kus 2007, 10, Tahmasbi Rezaei 2008, 19, Leahu Lupu 2010, 11, Baretto- Souza, Morais Cordeiro 2011, 4, Morais Baretto-Souza 2011, 16, Cancho, Louzada Barriga 2011, 5, Louzada, Roman Cancho 2011, 14, Flores, Borges, Cancho Louzada 2013, 8. Let s consider r.v. Z such that P Z {1,2,...} = 1. Definition 1 see 9. We say that r.v. Z has a power series distribution if: P Z = z = a zθ z, z = 1,2,... ; Θ 0,τ, 1 AΘ where a 1,a 2,... are nonnegative real numbers, τ is a positive number bounded by the convergence radius of power series series function AΘ = z 1a z Θ z, Θ 0,τ, Θ is power parameter of the distribution Table 1. PSD denotes the power series distribution functions families. If the r. v. Z has the distribution from relationship 1, then we write that Z PSD. Table 1. The representative elements of the PSD families for various truncated distributions Distribution a z Θ AΘ τ Binom n,p n p z 1 p 1 + Θ n 1 Poisson 1 α z! α e Θ 1 1 Logp z p ln1 Θ 1 Geom Θ p 1 1 p 1 Θ 1 P ascalk, p z 1 k k 1 1 p Θ 1 Θ 1 Bineg k,p p 1 Θ k 1 1 z+k 1 z

3 62 A. LEAHU, B. GH. MUNTEANU, S. CATARANCIUC 2 On the properties of the Max-Erlang the Min-Erlang power series distributions We consider that X i Erlangk,λ, k N, k 1, λ > 0, where X i i 1 i.i. d.r.v. with the distribution function F Xi x F Erl x = 1 e λx, x > 0 the pdf f Xi x f Erl x = λk x k 1 e λx k 1! We note that U Erl = max {X 1, X 2,...,X Z } V Erl = min {X 1, X 2,...,X Z }. The results in this section are obtained using the general framework of the work 12, for which reason some proofs are not presented. Proposition 1. If r.v. U Erl = max {X 1, X 2,...,X Z } V Erl = min{x 1, X 2,...,X Z }, where X i i 1 are nonnegative i.i.d.r.v., X i Erlangk,λ, k N, k 1, λ > 0 Z PSD with P Z = z = azθz AΘ, z = 1,2,... ; Θ 0,τ, τ > 0, r.v. X i i 1 Z being independent, then the distribution functions of the r.v. U Erl, respectively V Erl are the following: U Erl x = V Erl x = 1 A Θ 1 e λx AΘ A Θe λx AΘ 2 3 We denote a r.v. U Erl following Max-Erlang power series MaxErlPS distribution with parameters k,λ Θ by U Erl MaxErlPSk,λ,Θ, respectively a r.v. V Erl following Min-Erlang power series MinErlPS distribution with parameters k,λ Θ by V Erl MinErlPSk,λ,Θ. The following results characterize the survival functions the probability density functions pdf for the maximum, respectively minimum of a sequence of independent Erlang distributed rom variables in a rom number. Consequence 1. If r.v. U Erl MaxErlPSk,λ,Θ V Erl MinErlPSk,λ,Θ, then the survival functions of the r.v. U Erl, respectively V Erl are the following: S UErl x = 1 S VErl x = A Θ 1 e λx AΘ A Θe λx AΘ 4 5

4 MAX-ERLANG AND MIN-ERLANG POWER SERIES DISTRIBUTIONS Consequence 2. The pdf s of the r.v. U Erl, respectively V Erl are the following: Θλ k x k 1 e λx A Θ 1 e λx u Erl x =, x > 0 6 AΘ h UErl x = v Erl x = Θλ k x k 1 e λx A Θe λx AΘ u Erlx 1 U Erl x = 7 Proposition 2. The hazard rates for the r.v. U Erl, respectively V Erl are characterized by the following relations: Θλ k x k 1 e λx A Θ 1 e λx h VErl x = v Erlx 1 V Erl x = AΘ A Θ 1 e λx Θλ k x k 1 e λx A Θe λx A Θe λx. The next result shows a characteristic of the MaxErlPS MinErlPS distributions. Proposition 3. If X i i 1 is a sequence of independent, identically Erlang distributed r.v., with parameters λ > 0, k {1,2,...} Z PSD with P Z = z = azθz AΘ, where a z z 1 is a sequence of nonnegative real numbers, AΘ = z 1a z Θ z, Θ 0,τ, then lim U Erlx = considering m = min {n N, a n > 0}. lim U Erlx = k 1 m 1 e λx Proof. By applying the l Hospital rule m-time, we have: 1 e λx = A Θ m lim m!a m 1 e λx m = min {n N, a n > 0}. m A m Θ 1 e λx m k 1 m = 1 e λx, x > 0 m!a m

5 64 A. LEAHU, B. GH. MUNTEANU, S. CATARANCIUC Applying the same method of proof of Proposition 3, we obtain: Proposition 4. Under the conditions of the Proposition 3, when Θ 0 +, we have that lim V Erlx = 1 where l = min {n N, a n > 0}. l e λx Consequence 3. The r th moments, r N, r 1 of the r.v. U Erl MaxErlPSk,λ,Θ V Erl MinErlPSk,λ,Θ are given by EU r Erl = z 1 EV r Erl = z 1 a z Θ z AΘ E max {X 1, X 2,...,X z } r 8 a z Θ z AΘ E min {X 1, X 2,...,X z } r, 9 where pdf s fmax{x 1, X 2,...,X z}x = zf Erl xf Erl x z 1 f min{x1, X 2,...,X z} x = zf Erl x1 F Erl x z 1. The distribution functions pdf s of the r.v. U Erl MaxErlPSk,λ,Θ for different combinations of the r.v. Z PSD Table 1, are the following: Z Binom n,p; U ErlB MaxErlBk,λ,n,p, λ > 0; k,n {1,2...}; p 0,1: k 1 n 1 pe λx 1 U ErlB x = 1 1 p n u ErlB x = npλ k x k 1 e λx 1 pe λx n p n Z Poisson α; U ErlP MaxErlPk,λ,α, λ,α > 0; k {1,2...}: e αe λx e α U ErlP x = 1 e α u ErlP x = αλk x k 1 e λx αe λx 1 e α

6 MAX-ERLANG AND MIN-ERLANG POWER SERIES DISTRIBUTIONS Z Logp; U ErlLog MaxErlLogk,λ,p, λ > 0; k {1,2...}; p 0,1: k 1 a U ErlLog x = ln 1 p + p e λx u ErlLog x = where a = 1/ln1 p. apλ k x k 1 e λx 1 p + p e λx Z Geom p; U ErlG MaxErlGk,λ,p, λ > 0; k {1,2...}; p 0,1 : p 1 e λx U ErlG x = p + 1 pe λx u ErlG x = U ErlPas x = pλ k x k 1 e λx k 1 2 p + 1 pe λx Z Pascalk,p; U ErlPas MaxErlPask,λ,k,p, λ > 0; k, k {1,2...}; p 0,1 : k 1 k p 1 e λx u ErlPas x = p + 1 pe λx k p k λ k x k 1 e λx 1 e λx p + 1 pe λx k 1 k +1 Z Bineg k,p; U ErlBineg MaxErlBinegk,λ,k,p, λ > 0; k, k {1,2...}; p 0,1 : U ErlBineg x = u ErlBineg x = 1 p + pe λx 1 p k 1 k k pλ k x k 1 e λx 1 p + pe λx 1 p k 1 1 k 1

7 66 A. LEAHU, B. GH. MUNTEANU, S. CATARANCIUC The above results shows that the following result is valid: Proposition 5. If X i i 1, X i Erlangk,λ, k {1,2,...}, λ > 0 Y j j 1 are i.i.d.r.v., Y j MaxErlGk,λ,p, p 0,1, then the r.v. max {Y 1,Y 2,...,Y k } has the same distribution as the r.v. max {X 1,X 2,...,X Z }, where Z Pascalk,p, k {1,2,...}, p 0,1. Proof. Indeed, it is known that if Y j j 1 are independent r.v. Max-Erlang- Geometric MaxErlG distributed, with the distribution function U ErlG x we have: Fmax{Y 1,Y 2,...,Y k }x = U ErlG x k = U ErlPas x, x > 0, where U ErlPas x represents the distribution function of the Max-Erlang-Pascal MaxErlPas distribution. The distribution functions pdf s of the r.v. V Erl MinErlPSk,λ,Θ for different combinations of the r.v. Z PSD Table 1, are the following: Z Binom n,p; V ErlB MinErlBk,λ,n,p λ > 0; k,n {1,2...}; p 0,1: k 1 n 1 1 p + pe λx V ErlB x = 1 1 p n v ErlB x = npλ k x k 1 e λx 1 p + pe λx n p n Z Poisson α; V ErlP MinErlPk,λ,α, λ,α > 0; k {1,2...}: V ErlP x = 1 e α 1 e λx v ErlP x = αλk x k 1 e λx α+αe λx 1 e α 1 e α Z Logp; V ErlLog MinErlLogk,λ,p, λ > 0; k {1,2...}; p 0,1: k 1 a V ErlLog x = 1 + ln 1 p e λx v ErlLog x = where a = 1/ln1 p. apλ k x k 1 e λx 1 p e λx

8 MAX-ERLANG AND MIN-ERLANG POWER SERIES DISTRIBUTIONS Z Geom p; V ErlG MinErlGk,λ,p, λ > 0; k {1,2...}; p 0,1 : V ErlG x = 1 e λx k pe λx v ErlG x = pλ k x k 1 e λx k pe λx Z Pascalk,p; V ErlPas MinErlPask,λ,k,p λ > 0; k, k {1,2...}; p 0,1 : V ErlPas x = 1 e λx k pe λx k v ErlPas x = k 1 k p k λ k x k 1 e k λx 1 1 pe λx k 1 k +1 Z Bineg k,p; V ErlBineg MinErlBinegk,λ,k,p, λ > 0; k, k {1,2...}; p 0,1 : V ErlBineg x = 1 p k 1 pe λx 1 p k 1 k v ErlBineg x = k pλ k x k 1 e λx 1 pe λx 1 p k 1 k 1 Figure 1 shows several representations of pdf s of some particular MaxErlPS distributions MaxErlBk, λ, n, p, MaxErlPk, λ, α, for different values of their parameters: k = 2,λ = 3.5,α = 7,n = 21,p = 1/4. Figure 2 shows the behavior of the pdf s of the MinErlBk,λ,n,p, MinErlPk,λ,α for some values of the parameters: k = 2,λ = 0.5,α = 5,n = 25,p = 1/6.

9 68 A. LEAHU, B. GH. MUNTEANU, S. CATARANCIUC Figure 1. Pdf s for Max-Erlang-Binomial Max-Erlang-Poisson distributions Figure 2. Pdf s for Min-Erlang-Binomial Min-Erlang-Poisson distributions

10 MAX-ERLANG AND MIN-ERLANG POWER SERIES DISTRIBUTIONS Special cases In this section, we shall illustrate the characteristics of four distributions: the Max-Erlang-Binomial MaxErB distribution, the Min-Erlang-Binomial MinErlB distribution, the Max-Erlang-Poisson MaxErlP distribution, respectively the Min- Erlang-Poisson MinErlP distribution, so that later we can formulate a Poisson limit theorem. 3.1 The MaxErlB MinErlB distributions The MaxErlB MinErlB distributions is defined by the distribution functions 2 3, with AΘ = Θ + 1 n 1, namely: U ErlB x = V ErlB x = k 1 n 1 + Θ Θe λx Θ n, x > Θ n 1 + Θe λx 1 + Θ n 1 n 11 where n is integer pozitive. The survival functions, defined by the relationships 4, 5, for the r.v. U ErlB, respectively V ErlB are the following: S UErlB x = S VErlB x = 1 + Θ n 1 + Θ Θe λx 1 + Θ n Θe λx 1 + Θ n 1 n 1 n, x > 0 By using the relationships 6, 7 Proposition 2, the pdf s hazard rates are given by: u ErlB x = v ErlB x = nθλ k x k 1 e λx 1 + Θ Θe λx 1 + Θ n 1 nθλ k x k 1 e λx 1 + Θe λx 1 + Θ n 1 n 1 n 1, x > 0

11 70 A. LEAHU, B. GH. MUNTEANU, S. CATARANCIUC respectively h UErlB x = nθλ k x k 1 e λx 1 + Θ Θe λx 1 + Θ n 1 + Θ Θe λx n 1 n h VErlB x = nθλ k x k 1 e λx 1 + Θe λx 1 + Θe λx n 1 n 3.2 The MaxErlP MinErlP distributions The MaxErlP MinErlP distributions is defined by the distribution functions 2 3 with AΘ = e Θ 1, Θ = α > 0, namely: e Θe λx e Θ U ErlP x = 1 e Θ, x > 0 12 V ErlP x = 1 e Θ 1 e λx 1 e Θ 13 By using Consequences 1, the survival functions for the r.v. U ErlP, respectively V ErlP are the following: S UErlP x = 1 e Θe λx S VErlP x = e Θ 1 e Θ, x > 0 e Θ 1 e Θ 1 e λx With the formulas 6, 7 Proposition 2, the pdf s the hazard rates are given by: u ErlP x = Θλk x k 1 e λx Θe λx 1 e Θ

12 MAX-ERLANG AND MIN-ERLANG POWER SERIES DISTRIBUTIONS respectively v ErlP x = Θλk x k 1 e λx Θ+Θe λx h UErlP x = Θλk x k 1 e λx Θe λx 1 e Θ, x > 0 1 e Θe λx h VErlP x = Θλk x k 1 e Θe λx e Θe λx 3.3 On the Poisson limit theorem The next proposition shows that the MaxErlP MinErlP distributions approximate the MaxErlB, respectively MinErlB distributions under certain conditions. Proposition 6. Poisson limit theorem. The MaxErlP MinErlP distributions can be obtained as limiting of the MaxErlB, respectively MinErlB distributions with ditribution functions given by if nθ α > 0 when n Θ 0 +. Proof. We shall study the convergence in terms of the distributions U ErlB x, V ErlB x, U ErlP x V ErlP x, x > 0 of the two types of distributions. By calculating separately three elementary limits: lim 1 + Θ n = lim 1 + Θ 1/Θ nθ = e α, lim 1 + ΘAk,λ,x n = { = lim 1 + ΘAk,λ,x = e αak,λ,x lim 1 + Θ 1 Ak,λ,x n = } 1 nθak,λ,x ΘAk,λ,x

13 72 A. LEAHU, B. GH. MUNTEANU, S. CATARANCIUC we obtain: { 1 = lim 1 + Θ 1 Ak,λ,x Θ1 Ak,λ,x k 1 = e α1 Ak,λ,x,where Ak,λ,x = e λx, lim U ErlB x = lim V ErlB x = = e α lim 1 + Θ Θe λx 1 + Θ n 1 1 e λx e α 1 } nθ1 Ak,λ,x n 1 1 = U ErlP x 1 + Θ n 1 + Θe lim 1 + Θ n 1 = eα e αe λx e α 1 λx k 1 = V ErlP x. n References 1 Adamidis K., Loukas S. A Lifetime Distribution with Decreasing Failure Rate. Statistics Probability Letters, 1998, 39, No. 1, Adamidis K., Dimitrakopoulou T., Loukas S. On an extension of the exponentialgeometric distribution. Statistics Probability Letters, 2005, 73, No. 3, Barreto-Souza W., Cribari-Neto F. A generalization of the exponential-poisson distribution. Statistics Probability Letters, 2009, 79, Barreto-Souza W., Morais A. L., Cordeiro G. M. The Weibull-geometric distribution, Journal of Statistical Computation Simulation, 2011, 81, No. 5, Cancho V., Louzada F., Barriga G. The Poisson-Exponential Lifetime Distribution. Computational Statistics Data Analysis, 2011, 55, No. 8, Chahki M., Ganjali M. On some lifetime distributions with decreasing failure rate. Computational Statistics Data Analysis, 2009, 53, No. 12, Cordeiro G., Rodriques J., Castro M. The exponential com-poisson distribution. Statistical papers, 2011, Flores J., Borges P., Cancho V.G. The complementary exponential power series distribution. Braz. J. Probab. Stat., 2013, 27, No. 4, Johnson N. L., Kemp A. W., Kotz S. Univariate Discrete Distribution. New Jersey, 2005.

14 MAX-ERLANG AND MIN-ERLANG POWER SERIES DISTRIBUTIONS Kus C. A New Lifetime Distribution Distributions. Computational Statistics Data Analysis, 2007, 51, No. 9, Leahu A., Lupu C. E. On the Binomially Mixed Exponential Lifetime Distributions. Proceedings of the Seventh Workshop on Mathematical Modelling of Environmental Life Sciences Problems, Bucharest, 2010, Leahu A., Munteanu B.Gh., Cataranciuc S. On the lifetime as the maximum or minimum of the sample with power series distributed size. Romai J., 2013, 9, No. 2, Lehmann E. L. Testing Statistical Hypotheses second edition. New York, Wiley, Louzada F., Roman M., Cancho V. The Complementary Exponential Geometric Distribution: Model, Properties a Comparison with Its Counterpart. Computational Statistics Data Analysis, 2011, 55, No. 8, Louzada F., Bereta M. P. E., Franco M. A. P. On the Distribution of the Minimum or Maximum of a Rom Number of i. i. d. Lifetime Rom Variables. Applied Mathematics, 2012, 3, No. 4, Morais A. L., Barreto-Souza W. A Compound Class of Weibull Power Series Distributions, Computational Statistics Data Analysis, 2011, 55, No. 3, Munteanu B. Gh. The Max-Weibull Power Series Distribution. Annals of Oradea University, Fasc. Math., 2014, XXI, No. 2, Silva R. B., Barreto-Souza W., Cordeiro G. M. A new distribution with decreasing, increasing upside-down bathtub failure rate. Computational Statistics Data Analysis, 2010, 54, No. 4, Tahmasbi R., Rezaei S. A Two-Parameter Life-Time Distribution with Decreasing Failure Rate. Computational Statistics Data Analysis, 2008, 52, No. 8, Alexei Leahu Technical University of Moldova, Chisinau alexe.leahu@gmail.com Received December 28, 2013 Bogdan Gheorghe Munteanu Henri Coa Air Forces Academy, Brasov munteanu.b@afahc.ro Sergiu Cataranciuc Moldova State University, Chisinau s.cataranciuc@gmail.com