UNIVERSIDADE FEDERAL DE SÃO CARLOS CENTRO DE CIÊNCIAS EXATAS E TECNOLÓGICAS

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1 ISSN UNIVERSIDADE FEDERAL DE SÃO CARLOS CENTRO DE CIÊNCIAS EXATAS E TECNOLÓGICAS A COMPOUND CLASS OF EXPONENTIAL AND POWER SERIES DISTRIBUTIONS WITH INCREASING FAILURE RATE José Flores Delgado Patrick Borges Vicente G. Cancho Francisco Louzada-Neto RELATÓRIO TÉCNICO DEPARTAMENTO DE ESTATÍSTICA TEORIA E MÉTODO SÉRIE A Novembro/2010 nº 232

2 A compound class of exponential and power series distributions with increasing failure rate José Flores Delgado a,c, Patrick Borges b, Vicente G. Cancho a, Francisco Louzada-Neto b a Department of Applied Mathematics and Statistics, Universidade de São Paulo, Brazil b Department of Statistics, Universidade Federal de São Carlos, Brazil c Department of Science, Pontificia Universidad Católica del Perú Abstract In this paper, we introduce the class exponential power series distributions with increasing failure rate, which is complementary to the exponential power series model proposed by Chahkandi & Ganjali The new class of distribution arises on a latent complementary risks scenarios, where the lifetime associated with a particular risk is not observable, rather we observe only the maximum lifetime value among all risks. This new class contains several distributions as particular case. The properties of the proposed distribution class are discussed such as quantiles, moments and order statistics. Estimation is carried out via maximum likelihood. Simulation results on maximum likelihood estimation are presented. An real application illustrate the usefulness of the new distribution class. Key words: Complementary risks, Exponential Distribution, Increasing failure rate, Power series distribution, Exponential power series distribution. 1. Introduction The exponential distribution is widely used for modeling many problems in lifetime testing and reliability studies. However, the exponential distribution does not provide a reasonable parametric fit for some practical applications where the underlying failure rates are nonconstant, presenting monotone shapes. In recent years, in order to overcame such problem, new classes of models were introduced grounded in its simple elegant and close form, and interested readers can refer to Gupta & Kundu 1999, which proposed a generalized exponential distribution, which can accommodate data with increasing and decreasing and decreasing failure rates, Kus 2007, which proposed another modification of the exponential distribution with decreasing failure rate, and Barreto-Souza & Cribari-Neto 2009, which generalizes the distribution proposed by Kus 2007 by including a power parameter in his distribution. We focus on Chahkandi & Ganjali 2009, which proposed a variation of the exponential distribution, the exponential power series EPS distribution, with decreasing hazard function. Its genesis is based on a competing risk problem in presence of latent risks Louzada-Neto, 1999, in the sense that there is no Corresponding author: Francisco Louzada-Neto. Departamento de Estatística, Universidade Federal de São Carlos, Caixa Postal 676, CEP , São Carlos, São Paulo, Brasil. dfln@ufscar.br

3 information about which factor was responsible for the component failure and only the minimum lifetime value among all risks is observed. In this paper, in an opposite way to Chahkandi & Ganjali 2009, we propose a new distribution family based on a complementary risk problem Basu & J., 1982 in presence of latent risks. We assume that there is no information about which factor was responsible for the component failure but only the maximum lifetime value among all risks is observed instead of the minimum lifetime value among all risks as in Chahkandi & Ganjali The distribution is a counterpart of the EPS distribution and then, hereafter it shall be called complementary exponential power series CEPS distribution. The paper is organized as follows. In Section 2, we define the CEPS distribution. In Section 3 the survival and failure rate function, the quantiles, moments, order statistics and moments of the order statistics are provided. In Section 4 some special cases are studied in details. In Section 5 we discuss the relationship between the EPS and CEPS distributions. Estimation of the parameters by maximum likelihood is given in the section 6. In Section 7 presents the results of an simulation study. In Section 8 an application to one real data set is provided. Final remarks in the Section 9 concludes the paper. 2. The CEPS distribution In the classical complementary risks scenarios Basu & J., 1982 the lifetime associated with a particular risk is not observable, rather we observe only the maximum lifetime value among all risks. simplistically, in reliability, we observe only the maximum component lifetime of a parallel system. That is, the observable quantities for each component are maximum lifetime value to failure among all risks, and the cause of failure. Complementary risks problems arise in several areas and available with this and interested readers can refer to Lawless 2003, Crowder et al and Cox & Oakes A difficulty arises if the risks are latent in the sense that there is no information about which factor was responsible for the component failure, which can be often observed in field data. We call these latent complementary risks data. On many occasions this information is not available or it is impossible that the true cause of failure is specified by an expert. In reliability, the components can be totally destroyed in the experiment. Further, the true cause of failure can be masked from our view. In modular systems, the need to keep a system running means that a module that contains many components can be replaced without the identification of the exact failing component. Goetghebeur & Ryan 1995 addressed the problem of assessing covariate effects based on a semi-parametric proportional hazards structure for each failure type when the failure type is unknown for some individuals. Reiser et al considered statistical procedures for analyzing masked data, but their procedure can not be applied when all observations have an unknown cause of failure. Lu & Tsiatis 2001 presents a multiple imputation method for estimating regression coefficients for risk modeling with missing cause of failure. A comparison of two partial likelihood approaches for risk modeling with missing cause of failure is presented in Lu & Tsiatis The proposed distribution can be derived as follows. Let Z be a random variable denoting the number of failure causes, z = 1, 2,... and considering Z following a power series distribution truncated at zero with probability function given by P [Z = z; θ] = a zθ z, z = 1, 2,... 1 Aθ 2

4 where a z depends only on z, Aθ = a zθ z and θ 0, s s can be is such that Aθ is finite. For more details on the power series class of distributions, see Johnson et al Table 2.1 shows useful quantities of some power series distributions truncated at zero such as Poisson, logarithmic, geometric and binomial with m being the number of replicas distributions. The quantities A θ and A θ are the derivation of Aθ and A θ with respect to θ, respectively. The quantile A 1 θ is the inverse function of Aθ. Table 2.1: Useful quantities of some power series distributions. Distribution a z Aθ A θ A θ A 1 θ s Poisson z! 1 e θ 1 e θ e θ log1 + θ Logarithmic z 1 log1 θ 1 θ 1 1 θ 2 1 e θ 1 Geometric 1 θ1 θ 1 1 θ 2 21 θ 3 θ1 + θ 1 1 Binomial m z 1 + θ m 1 m1 + θ m 1 mm 1 1+θ θ 1 1/m 1 2 m Let s also consider y i, i = 1, 2, 3,... realizations of a random variable denoting the failure times, i.e., the time-to-event due to the i th CR and Y j has an exponential distribution with probability density function pdf with parameter β > 0, given by fy i ; β = β exp{ βy i }. 2 In the latent complementary risks scenario, the number of causes Z and the lifetime y i associated with a particular cause are not observable latent variables, but only the maximum lifetime X among all causes is usually observed. So, we only observe the random variable given by Then, fx z; β = zβe βx 1 e βx z 1 and the marginal pdf X is but So fx; θ, β = = = X = max{y 1,..., Y Z }. 3 fx z, βp Z = z; θ zβe βx 1 e βx z 1 a z θ z Aθ [ ] θ 1 e a z zθβe βx βx z 1 Aθ a z zθβe βx [ θ 1 e βx] z 1 fx; θ, β = x A θ 1 e βx Aθ = x [ a z θ 1 e βx ] z = x A θ 1 e βx. = θβe βx A θ 1 e βx, x > 0, θ, β > 0, 5 Aθ where A θ 1 e βx is the derivative of A evaluated at θ1 e βx. We denote a random variable X following CEPS distribution with parameters θ, β by X CEP Sθ, β. 4 3

5 3. Some properties of the CEPS distribution 3.1. The distribution, survivor, failure rate functions Let X a nonnegative random variable denoting the lifetime of a component in some population with CEPS distribution with parameters θ and β, i.e., X CEPSθ, β. The distribution function is given by and survival function is F x; θ, β = A θ 1 e βx, x > 0, 6 Aθ Sx; θ, β = 1 A θ 1 e βx, x > 0. 7 Aθ The following proposition shows that our distribution has exponential distribution as limiting distribution. Proposition 3.1. The exponential distribution with parameter β is a limiting special case of the CEPS distribution when θ 0 +. Proof. Using Aθ = a zθ z in 6, we have that lim F x = lim θ 0 + θ 0 + and using L Hospital s rule, it follows that a z θ 1 e βx z a zθ z a 1 lim F x = lim θ 0 + θ e βx + z=2 za zθ z 1 1 e βx z a 1 + z=2 za zθ z 1 = 1 e βx. The pdf of the CEPS distribution have an interesting representation. For ω < 1 and γ > 0, we have that 1 ω γ 1 1 j Γγ = Γγ jj! ωj. 8 If γ is an integer, the sum in 8 stops at γ 1. Plugging 8 into 4, we obtain j=0 j=0 z 1 a z θ z Γz j fx; θ, β = Aθ j + 1!Γz j f Expx; βj + 1, 9 where f Exp, βj + 1 is the pdf of a random variable with exponential distribution with parameter βj + 1. We then have that the CEPS pdf can be written as a linear combination of exponentials. The hazard function is given by fx; θ, β hx; θ, β = Sx; θ, β = θβe βx A θ 1 e βx Aθ A θ 1 e βx, x > The following result shows that the hazard function 10 is increasing for all θ,β > 0. Proposition 3.2. The hazard function given in 10 is increasing for all θ,β > 0. 4

6 Proof. Define the function ηx = f x fx, where f denotes the first derivative of f. Therefore, Hence, the first derivative of η is given by ηx = βa θ 1 e βx + θβe βx A θ 1 e βx A θ 1 e βx. η x = θβ2 e βx A θ 1 e βx A θ 1 e βx A θ 1 e βx 2 + θβe βx 2 A θ 1 e βx 2 A θ 1 e βx A θ 1 e βx A θ 1 e βx Therefore, η x > 0, x > 0. Hence, using theorem b of Glaser 1980 paper, we conclude that the hazard function is increasing Quantiles, moments, mean residual and order statistics The quantile γ x γ = F 1 γ; θ, β of the CEPS distribution follows from 6 as x γ = F 1 γ; θ, β = β 1 log { 1 θ 1 A 1 γaθ }, 12 where A 1 is the inverse function of A. An expression for the moments of a CEPS distribution can be derived as following. Proposition 3.3. The r th moment of CEP Sθ, β distribution is given by E[X r ] = Γr + 1 β r z 1 a z θ z Γz j 1 Aθ j + 1!Γz j j + 1 r. 13 j=0 Proof. A random variable X following exponential distribution with pdf in the form 2 has the r th moment given by E[X r ] = Γr+1 β. Using this in 9, we obtain the result. r An expression for the mean residual of a CEPS distribution can be derived as following. Proposition 3.4. The mean residual, given the survival to time x 0, until the time of failure, of the CEPS distribution can be obtained as follows: mx 0 = EX x 0 X x 0 = 1 β Sx 0 z 1 a z θ z Γz j Aθ j + 1!Γz j j=0 e βj+1x0 j Also, by the propositions 3.2 and 5.1, mx 0 decreases monotonically from EX down to its limit at infinity 1/β, therefore the mean lifetime exceeds the mean residual lifetime for all x 0 > 0. Proof. EX x 0 X x 0 = 1 Sx 0 0 xf x + x 0 dx, then the proof is similar to the above proposition. X 5

7 An explicit expression for the density of the i th order statistic X i:n, say f i:n x, in a random sample of size n from the CEPS distribution is derived in the sequel. It is well-known that f i:n x = 1 Bi, n i + 1 fxf i 1 x1 F x n i, for i = 1,..., n, where B, is the beta function. Using the binomial expansion in the last equation, f i:n x becomes f i:n x = n i k=0 1 k Bi, n i + 1 n i k fxf i+k 1 x, 15 where f and F are pdf and cdf given by 5 and 6, respectively. By inserting these equations in 15 we obtaing f i:n x; θ, β = θβe βx A θ 1 e βx AθBi, n i + 1 n i [ n i 1 k A θ 1 e βx ] i+k 1, x > k Aθ k=0 The moments of the CEPS distribution order statistics are obtained by using a result due to Barakat & Abdelkader 2004 applied to the independent and identically distributed iid case, leading to E[X r i:n; θ, β] = r = r n k=n i+1 n k=n i+1 k 1 n 1 k n+i 1 x r 1 S k x; β, pdx n i k 0 [ k 1 n 1 k n+i 1 x r 1 1 A θ 1 e βx ] k dx. 17 n i k Aθ 0 4. Special cases In this Section we present some special cases of the CEPS distribution. Expressions for mean, variance and mean residual are presented. Also, in order to illustrate the flexibility of the distributions, plots of the pdf and the failure rate function for some values of the parameters are provided Complementary exponential binomial distribution The complementary exponential binomial CEB distribution is defined from the cdf 5 with Aθ = 1 + θ m 1, which is given by 1 + θ 1 e βx m 1 F x; θ, β = 1 + θ m, x > 0, 18 1 where m is integer positive. The associated pdf and failure rate function are given, respectively, by fx; θ, β = θβe βx m 1 + θ 1 e βx m θ m 1 for x > 0. and hx; θ, β = θβe βx m 1 + θ 1 e βx m θ m 1 + θ 1 e βx m, 6

8 The mean, variance and mean residual of the CEB distribution are given, respectively, by V ar[x] = and E[X] = 1 + θ m 2 β θ m 1 mx = 1 + θ m β 1 + θ m 1 m 1 z+1 z 2 m 1 z+1 z m θ, z z 1 + θ z m θ z 1 + θ 1 + θ m β[ 1 + θ m 1 + θ1 e βx m ] 4.2. Complementary exponential Poisson distribution 1 + θm m 1 + θ m 1 m 1 z z m z 1 z+1 z θe βx x. 1 + θ z 2 m θ z 1 + θ The pdf and survival function of the complementary exponential Poisson CEP distribution are given by θβe βx θe βx fx; θ, β = 1 e θ and Sx; θ, β = 1 e θe βx 1 e θ, for x > 0, respectively. The next Proposition gives us another characterization of the CEP distribution. Proposition 4.1. The CEP distribution can be obtained as limiting of CEB distribution with cdf given by 18, if mθ β > 0, when m. Proof. We shall show that the survival function of the CEB distribution converges to the survival function of the CEP distribution under conditions of the Proposition. It is straightforward to show that if mθ λ > 0, when m, we obtain lim 1 + θ 1 e βx m = lim 1 + mθ 1 e βx = e λ1 e βx m m m and Hence, it is easy to see that lim 1 + m θm = lim 1 + mθ m = e λ. m m lim 1 m 1 + θ 1 e βx m θ m 1 which is the survival function of a CEP distribution. βx = 1 eλ1 e 1 e λ = 1 e λe βx 1 1 e λ, The failure rate function of the CEP distribution is given by θβe βx hx; θ, β = e θe βx 1, x > 0. The mean, variance and mean residual of the CEP distribution are given, respectively, by E[X] = θ β 1 e θ F 2,2 [1, 1], [2, 2], θ, 7

9 and [ ] 2θ θ V ar[x] = β 2 1 e θ 2F 3,3 [1, 1, 1], [2, 2, 2], θ 1 e θ F 2,2 2 [1, 1], [2, 2], θ mx = θe βx λ1 e θe βx F 2,2 [1, 1], [2, 2], θe βx, where F p,q n, d, λ is the generalized hypergeometric function. This function is also known as Barnes s extended hypergeometric function. The definition of F p,q n, d, λ is F p,q n, d, λ = λ k p i=1 Γn i + kγ 1 n i Γk + 1 q i=1 Γd i + kγ 1 d i, k=0 where n = [n 1, n 2,..., n p ], p is the number of operands of n, d = [d 1, d 2,..., d q ] and q is the number of operands of d. The generalized hypergeometric function is quickly evaluated and readily available in standard software such as Maple Complementary exponential geometric distribution The complementary exponential geometric CEG distribution is defined by the cdf 5 with Aθ = θ1 θ 1, which is given by F x; θ, β = 1 θ1 e βx 1 θ 1 e βx, x > 0 θ 0, 1. The associated pdf and failure rate function are given respectively by 1 θβe βx 1 θβ fx; θ, β = 1 θ 1 e βx 2 and hx; θ, β = 1 θ1 e βx, for x > 0 and θ 0, 1. The mean, variance and mean residual of the CEG distribution are given, respectively, by and E[X] = 1 log1 θ, θβ V ar[x] = 1 [ 2 β 2 1 θ F 3,2 [1, 1, 1], [2, 2], θ 1 ] 1 θ θ 2 log2 1 θ mx = 1 θ1 e βx βθe βx [ log1 θ log1 θ1 e βx ] Complementary exponential logarithmic distribution The cdf of the complementary exponential logarithmic CEL distribution is defined by 5 with Aθ = log1 θ, 0 < θ < 1. The associated pdf and failure rate function are θβe βx fx; θ, β = log1 θ 1 θ 1 e βx for x > 0, respectively. and hx; θ, β = log 1 θ 1 θ1 e βx θβe βx 1 θ1 e βx, 8

10 The mean, variance and mean residual of the CEL distribution are given, respectively, by 2θ E[X] = β 2 1 θ log1 θ F 3,2 [1, 1, 1], [2, 2], θ, 1 θ θ1 θ V ar[x] = β 2 log1 θ and [2F 4,3 [1, 1, 1, 1], [2, 2, 2], θ 1 θ θe βx mx = β1 θ[log1 θ log1 θ1 e βx ] F 3,2 θ1 θ log1 θ F 3,2 [1, 1, 1], [2, 2], θ ] 1 θ [1, 1, 1], [2, 2], θe βx. 1 θ 5. On the relationship between the EPS and CEPS distributions In this section, we discuss some relationship between the EPS distribution Chahkandi & Ganjali, 2009 and CEPS distribution proposed here. The failure rate function of the EPS distribution is given by hx; β, p = θβe βx A θe βx A θe βx. 19 Proposition 5.1. The failure rate functions, 10 and 19, of both distributions diverge for x 0 but converge to β when x. Proof. For x 0, for the EPS model, while, for CEPS model, lim hx = lim x 0 + lim hx = lim x 0 + concluding the proof of the initial divergence. For x, for the EPS model βθe βx A θe βx x 0 + A θe βx βθe βx A A θ 1 e βx x 0 + Aθ Aθ 1 e βx = βθ A θ Aθ, = βθ A 0 Aθ, βθe βx A θe βx βθe βx 2 A θe βx + β 2 θe βx A θe βx lim x A θe βx = lim x βθe βx A θe βx = β; concluding the proof. θβe βx A θ 1 e βx θβe βx 2 A lim x Aθ A θ 1 e βx = lim θ 1 e βx x θβe βx A θ 1 e βx + θβ 2 e βx A θ 1 e βx θβe βx A θ 1 e βx = β, 9

11 Hazard EB distribution CEB distribution θ=0.01 θ=0.1 θ=0.5 θ=0.99 Hazard EP distribution CEP distribution θ=0.01 θ=0.1 θ=0.5 θ= x x Hazard EG distribution CEG distribution θ=0.01 θ=0.1 θ=0.5 θ=0.99 Hazard EL distribution CEL distribution θ=0.01 θ=0.1 θ=0.5 θ= x x Figure 5.1: Comparing the failure rate function of the EPS and CEPS distributions for same fixed θ values. β = 1.0. We fixed The Figure 5.1 shows the behavior of failure rate functions of both class of distributions for some values of the parameters. The CEPS failure rate function 10 increases while the EPS failure rate function 15 decreases with x, but both converge to β when x corroborating the Proposition Inference 6.1. Maximum likelihood estimation Let x = x 1,..., x n be a random sample of the CEPS distribution with unknown parameter vector ξ = θ, β. The log-likelihood l = lξ; x is given by l = n logθβ β n x i + i=1 n log A θ1 e βxi n logaθ. 20 i=1 10

12 The maximum likelihood estimations MLEs of θ and β can be derived directly either from the log-likelihood 20 or by solving the following nor-linear system: lξ; x θ lξ; x β = n θ na θ Aθ + = n n β x i + θ i=1 n 1 e βxi A θ1 e βxi A = 0, θ1 e βxi 21 n x i e βxi A θ1 e βxi A = 0, θ1 e βxi 22 i=1 i=1 where A θ1 e βxi is the second derivative of A evaluated at θ1 e βxi. Large sample inference for the parameters can be based, in principle, on the MLEs and their estimated standard errors. Following Cox & Hinkley, 1974, under suitable regularity conditions, it can be shown that n θ θ, β β N2 0, I 1 θ, β, where Iθ, β is Fisher information matrix, i.e, Iθ, β = E E 2 Lθ, β; X θ 2 2 Lθ, β; X θ β E E 2 Lθ, β; X θ β 2 Lθ, β; X β 2, where Lθ, β ; x = logfx; θ, β is the logarithm of density CEP Sθ, β distribution given by equation 5, then the second derivatives are given by 2 Lθ, β ; x θ 2 = 1 θ 2 A θaθ A 2 θ A 2 + b2 x[ A θbxa θbx A 2 θbx ] θ A 2 θbx 2 Lθ, β ; x = xe βx { A θbx [ A θbx + θbxa θbx ] θbxa 2 θbx } θ β A 2 θbx 2 Lθ, β ; x β 2 = 1 β 2 θx2 e βx { A θbx [ A θbx θe βx A θbx ] + θe βx A 2 θbx }, A 2 θbx where where A θbx is the third derivative of A evaluated at bx and bx = θ1 e βx. 7. Simulation study This section presents the results of a simulation study carried out to assess the accuracy of the approximation of the variance and covariance of the MLEs determined from the Fisher information matrix. Five thousand samples of sizes n = 20, 30, 50, 100 and 200 were generated from a Binomial-Exponential distribution with m = 13 for each combination of the parameter values θ; β = 0, 1; 2, 0, 3; 2, 0, 5; 2, 0, 7; 2, 0, 9; 2, 0, 1; 6, 0, 3; 6, 0, 5; 6, 0, 7; 6 and 0, 9; 6. Overall, 50 different set ups were considered. The simulated values of varˆθ, var ˆβ and covˆθ, ˆβ as well as their approximate values obtained by averaging the corresponding values obtained from the expected and observed information matrices are presented in Table 7.1. It is observed that the approximate values determined from the expected and observed information matrices are close to the simulated values for large values of n. Furthermore, it is noted that the approximation becomes quite accurate as n increases. Additionally, variances and covariances of MLEs obtained from the observed information matrix are quite close from 11

13 the variances and covariances obtained from the expected information matrix for large value of n. Also, the simulation study include the mean square error mse of the MLEs, as well as the empirical coverage probabilities of the 95% confidence intervals for the parameters θ and β, which are closer to the nominal coverage as the sample size increases. Table 7.1: Mean of the variances and covariances of the MLEs, mean square error mse of the MLEs and coverage probabilities of the 95% confidence intervals for the parameters. Simulated Expected Information Observed Information Coverage n θ; β ˆθ ˆβ varˆθvar ˆβ mseˆθ mse ˆβ covˆθ, ˆβ varˆθ var ˆβ covˆθ, ˆβ varˆθvar ˆβ covˆθ, ˆβ θ β 20 0,1; ,3; ,5; ,7; ,9; ,1; ,3; ,5; ,7; ,9; ,1; ,3; ,5; ,7; ,9; ,1; ,3; ,5; ,7; ,9; ; ,3; ,5; ,7; ,9; ,1; ,3; ,5; ,7; ,9; ,1; ,3; ,5; ,7; ,9; ,1; ,3; ,5; ,7; ,9; ,1; ,3; ,5; ,7; ,9; ,1; ,3; ,5; ,7; ,9; Application In this section we reanalyze the data set extracted from Lawless, The lifetimes are the number of million revolutions before failure for each one of the 23 ball bearings on an endurance test of deep groove ball bearings. From the practical point of view, even though the risks for deep groove ball bearing failure are unobserved, one may speculate on some possible competing risks. For instance, we can consider the risk of contamination from dirt from the casting of the casing, the wear particles from hardened steel gear wheels and the harsh working environments, amongst others, which can leads to the deep groove ball bearing failure Cancho et al., The shape of the failure rate function for the dataset can be determined from a TTT plot Aarset This plot is built from the points r n, G r n, where G r n = [ r Y i + n ry r ]/ r Y i, r = 1,..., n, and Y i is the i th order statistic of the sample. i=1 i=1 12

14 As pointed out in the literature, it is shown that the rate of failure is increasing decreasing if the TTT plot is concave convex. However, since the TTT plot is a sufficient condition, but not necessary, to indicate the rate of failure, it is taken here only as a reference to determine the shape of the rate of failure. As we can see in Cancho et al the TTT plot is concave, indicating an increasing failure rate function. Therefore, the dataset may be fitted by the CEPS distribution. Thus, the Poisson, geometric, logarithmic and binomial distributions can be used for this purpose. The maximum likelihood estimation is obtained by direct maximization of 20 via the optim function of the R program R Development Core Team Also, for sake of comparison, we fit more one usual lifetime distribution generally used for fitting dataset with increasing failure rate function: the Weibull distribution. As well known, the Weibull distribution is indexing by two parameters as it is the case for the CEPS distribution. The density functions of the Weibull distributions, with parameters β and α is given by βα β x β 1 exp x/α β, respectively. We compare the fitting of the CEPS particular distributions and the Weibull one by considering the AIC Akaike s information criterion, 2lˆθ, ˆβ+2p and BIC Schawartz s Bayesian information criterion, 2lˆθ, ˆβ + 2 logn, where p is the number of parameters in the model and n is size sample. Both criterion penalize overfitting and the preferred model is the one with the smaller value on each criterion. The table 8.1 presents the maximum values of the log-likelihood function l and the estimated AIC and BIC criteria considering the all fitted distributions. Table 8.1: Parameter estimates, l value, and AIC and BIC values for the fitted distributions. Distribution l AIC BIC CEB CEP CEG CEL Weibull The CEB distribution outperforms its concurrent distributions in all considered criteria. The parameter estimates of the CEB distribution of θ and β and their standard errors are and , respectively. Figure 9.1 shows the fitted density functions of the all fitted distributions superimposed to the histogram and the fitted survival superimposed to the empirical survival function. 9. Concluding remarks In this paper we propose the CEPS distribution, which is complementary to the EPS distribution proposed by Chahkandi & Ganjali, 2009, and accommodates increasing failure rate functions. It arises on a latent complementary risks scenarios, where the lifetime associated with a particular risk is not observable but only the maximum lifetime value among all risks. We provide a mathematical treatment of the new distribution including expansions for its density and cumulative distributions, survival and failure rate functions. We derive expansions for the moments and quantile function, obtain the density of the order statistics and provide expansions for moments of the order statistics. Maximum likelihood inference is implemented straightforwardly. Finally, we fit the CEPS distribution to a real data set in order 13

15 to show its flexibility and potentially as a lifetime distribution compared with an usual two-parameters lifetime distributions. Density function CEB CEP CEG CEL Weibull Surviving function CEB CEP CEG CEL Weibull Time Time Figure 9.1: Left panel: The plots of the fitted CEB, CEP, CEG, CEL and Weibull densities. Right panel: Kaplan-Meir curve together with the fitted survival functions. References Aarset, M. V The null distribution for a test of constant versus bathtub failure rate. Scandinavian Journal of Statistics, 121, Barakat, H. M. & Abdelkader, Y. H Computing the moments of order statistics from nonidentical random variables. Statistical Methods and Applications, 13, Barreto-Souza, W. & Cribari-Neto, F A generalization of the exponential-poisson distribution. Statistics and Probability Letters, 79, Basu, A. & J., K Some recent development in competing risks theory. Survival Analysis, Edited by Crowley, J. and Johnson, R. A, Hayvard:IMS, 1, Cancho, V. G., Louzada-Neto, F. & Barriga, G. D The poisson-exponential lifetime distribution. Computational Statistics & Data Analysis, In Press, Corrected Proof. Chahkandi, M. & Ganjali, M On some lifetime distributions with decreasing failure rate. Computational Statistics and Data Analysis, 53, Cox, D. R. & Hinkley, D. V Theoretical statistics. Chapman and Hall, London. Cox, D. R. & Oakes, D Analysis of Survival Data. Chapman & Hall. Crowder, M., Kimber, A., Smith, R. & Sweeting, T Chapman & Hall. Statistical Analysis of Reliability Data. 14

16 Glaser, R. E Bathtub and related failure rate characterizations. Journal of the American Statistical Association, 75, Goetghebeur, E. & Ryan, L A modified log rank test for competing risks with missing failure type. Biometrics, 77, Gupta, R. & Kundu, D Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41, Johnson, N. L., Kemp, A. W. & Kotz, S UnivariateDiscreteDistribution. New Jersey. Kus, C A new lifetime distribution. Computational Statistics and Data Analysis, 51, Lawless, J. F Statistical Models and Methods for Lifetime Data, volume second edition. Wiley. Louzada-Neto, F Poly-hazard regression models for lifetime data. Biometrics, 55, Lu, K. & Tsiatis, A. A Multiple imputation methods for estimating regression coefficients in the competing risks model with missing cause of failure. Biometrics, 54, Lu, K. & Tsiatis, A. A Comparision between two partial likelihood approaches for the competing risks model with missing cause of failure. Lifetime Data Analysis, 11, Reiser, B., Guttman, I., Lin, D., Guess, M. & Usher, J Bayesian inference for masked system lifetime data. Applied Statistics, 44, R Development Core Team R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. 15

17 PUBLICAÇÕES RODRIGUES, J.; CASTRO, M.; BLAKRISHNAN, N.; GARIBAY, V.; Destructive weighted Poisson cure rate models Novembro/2009 Nº 210. COBRE, J.; LOUZADA-NETO, F., PERDONÁ, G.; A Bayesian Analysis for the Generalized Negative Binomial Weibull Cure Fraction Survival Model: Estimating the Lymph Nodes Metastasis Rates Janeiro/2010 Nº 211. DINIZ, C. A. R. ; LOUZADA-NETO, F.; MORITA, L. H. M.; The Multiplicative Heteroscedastic Von Bertalan_y Model Fevereiro/2010 Nº 212. DINIZ, C. A. R. ; MORITA, L. H. M, LOUZADA-NETO, F.; Heteroscedastic Von Bertalanffy Growth Model and an Application to a Kubbard female chicken corporeal weight growth data Fevereiro/2010 Nº 213. FURLAN, C. P. R.; DINIZ, C. A. R.; FRANCO, M. A. P.; Estimation of Lag Length in Distributed Lag Models: A Comparative Study Março/2010 Nº 214. RODRIGUES, J., CANCHO, V. G., CASTRO, M., BALAKRISHNAN, N., A Bayesian destructive weighted Poisson cure rate model and an application to a cutaneous melanoma data, Março/2010 Nº CORDEIRO, G. M.; RODRIGUES, J.; CASTRO, M. The exponential COM-Poisson distribution, Abril/2010 Nº 216. CANCHO, V. G.; LOUZADA-NETO, F.; BARRIGA, G. D. C. The Poisson-Exponential Survival Distribution For Lifetime Data, Maio/2010 Nº 217. DIINIZ, C. A. R.; FURLAN, C. P. R.; LEITE, J. G. A Bayesian Estimation of Lag Length in Distributed Lag Models, Julho/2010 Nº 218. CONCEIÇÃO, K. S.; PIRES, R. M.; LOUZADA-NETO, F.; ANDRADE, M. G.; DINIZ, C. A. R. A Generalized Species-Area Relationship for Estimating Species Diversity: The Poisson Distribution Case Julho/2010 Nº 219. LOUZADA-NETO, F.; CANCHO, V. G.; BARRIGA, G. D. C.; A Bayesian Analysis For The Poisson-Exponential Distribution Agosto/2010 Nº 220. SCACABAROZI, F.N., DINIZ C. A. R., FRANCO M. A. P., A Comparative Study of Credibility an Confidence Intervals for the Parameter of a Poisson Distribuition Setembro/2010 Nº 221. TOMAZELLA, V. L., BERNARDO, J. M.; Testing for Hardy-Weinberg Equilibrium in a Biological Population: An Objective Bayesian Analysis Setembro/2010 Nº 222. LOUZADA-NETO F., DINIZ C. A. R., COSTA, C. C., SILVA P. H. F., DESTEFANI, C. R., TEMPONI A. P. O. Procedimentos Estatísticos para Segmentação de Base de Dados Setembro/2010 Nº 223. ARA-SOUZA, A. L.; LOUZADA-NETO, F.; Caracterização dos Docentes e Necessidade de Doutores dentro das Graduações de Estatística do Brasil Setembro/2010 Nº 224. LOUZADA-NEDTO, F.; ROMAN, M.; CANCHO, V. G.; The Complementary Exponential Geometric Distribution: Model, Properties and a Comparison With Its Counterpart Setembro/2010 Nº 225. LOUZADA-NETO,F.; BORGES, P.; The Exponential Negative Binomial Distribuition Setembro/2010 Nº 226. PEREIRA, G.A; LOUZADA-NETO, F.; MORAES-SOUZA, H. ; FERREIRA-SILVA, M. M.; BARBOSA, V. F.; General Bayesian latent class model for the evaluation of the performance of L diagnostic tests for Chagas disease in the absence of a gold standard considering M covariates and V different disease prevalences Setembro/2010 Nº 227. SARAIVA, E. F.; MILAN, L. A.; LOUZADA-NETO, F.; A Posterior Split-Merge MCMC Algorithm for Mixture Models with an unknown number of components Setembro/2010 Nº 228. BORGES, P.; ROMAN, M.; TOJEIRO, C. A. V.; LOUZADA-NETO, F.; The complementary exponential logarithmic distribution: A two-parameter lifetime distribution with increasing failure rate Outubro/2010 Nº 229. TOJEIRO, C.; A. V. LOUZADA-NETO, F.; A General Threshold Stress Hybrid Hazard Model for Lifetime Data Outubro/2010 Nº 230. TOMAZELLA, V. L.D.; CANCHO, V. G.; LOUZADA-NETO, F.; Objective Bayesian Reference Analysis for the Poissonexponential lifetime distribution Outubro/2010 Nº 231 Os recentes relatórios poderão ser obtidos pelo endereço Mais informações sobre publicações anteriores ao ano de 2010 podem ser obtidas via dfln@ufscar.br