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 THE EXPONENTIATED EXPONENTIAL-GEOMETRIC DISTRIBUTION: A DISTRIBUTION WITH DECREASING, INCREASING AND UNIMODAL HAZARD FUNCTION Francisco Louzada-Neto Vitor Marchi Mari Roman RELATÓRIO TÉCNICO DEPARTAMENTO DE ESTATÍSTICA TEORIA E MÉTODO SÉRIE A Janeiro/211 nº 235

2 The Exponentiated Exponential-Geometric Distribution: A distribution with decreasing, increasing and unimodal hazard function Francisco Louzada, Vitor Marchi and Mari Roman Department of Statistics, Universidade Federal de São Carlos, Brazil Abstract In this paper we proposed a new family of distributions namely Exponentiated Exponential- Geometric (E2G) distribution. The E2G distribution is a straightforwardly generalization of the EG distribution proposed by [1], which accommodates increasing, decreasing and unimodal hazard functions. It arises on a latent competing risk scenarios, where the lifetime associated with a particular risk is not observable but only the minimum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal prove of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments, rth moment of the ith order statistic, mean residual lifetime and modal value. Maximum likelihood inference is implemented straightforwardly. From a misspecification simulation study performed in order to assess the extent of the misspecification errors when testing the EG distribution against the E2G one we observed that it is usually possible to discriminate between both distributions even for moderate samples in presence of censoring. The practical importance of the new distribution was demonstrated in three applications where we compare it with several former lifetime distributions. Key-words: Exponentiated Exponential Distribution, Geometric Distribution, Latent Competing Risks, Survival Analysis, Censured data, unimodal failure rate function 1 Introduction In recent years, several new classes of models were introduced grounded in the simple exponential distribution, which is a wide used lifetime distribution for modeling many survival problems. The main idea is to propose lifetime distributions which can accommodate practical applications where the underlying hazard functions are non-constant, presenting monotone shapes. The exponential distribution however does not provide a reasonable parametric fit for such practical applications. For instance, we can cite [1], which proposed a variation of the exponential distribution, the exponential geometric (EG) distribution, with decreasing hazard function, [2], which proposed a generalized exponential distribution, which can accommodate data with increasing and decreasing hazard functions, [3], which proposed another modification of the exponential distribution with decreasing hazard function, and [4], which generalizes the distribution proposed by [3] by including a power parameter in his distribution, which can accommodate increasing, decreasing and unimodal hazard functions. [5] proposed the complementary exponential power distribution by exponentiating the exponential power distribution proposed by [6]. In this paper, we propose a new lifetime distribution family, an direct extension of the EG distribution [1], obtained by compounding the exponentiated exponential distribution [7] with the 1

3 usual geometric distribution. Hereafter we shall refer to the new distribution as the exponentiated exponential geometric (E2G) distribution. Besides being able to accommodate increasing, decreasing and unimodal hazard functions, the E2G distribution can be seen as a mechanistic framework of easy practical interpretation. Its genesis is based on a latent competing risk problem [8], in the sense that there is no information about which factor was responsible for the component failure and only the minimum lifetime value among all risks is observed. On many occasions this information is not available or it is impossible that the true cause of failure is specified by an expert. Further, the true cause of failure can be masked from our view. The paper is organized as follows. In Section 2, we introduce the new E2G distribution and present some of its properties. Furthermore, we derive the expressions for the probability density function and survival function, r-th raw moments of the E2G distribution and modal value. Also, in this section we present the inferential procedure. In Section 3 we discuss the relationship between the Exponential, EG, and E2G distributions based on the hazard function and report the results of a misspecification simulation study performed in order to verify whether we can distinguish between the EG and E2G distributions in the light of the data based on some usual distribution comparison criterion. In Section 4 we fit the E2G distribution to three real datasets and compare it with the fits of several usual lifetime distributions, pointing out its relative superiority. Some final comments in Section 5 conclude the paper. 2 The E2G model Let Y a nonnegative random variable denoting the lifetime of a component in some population. The random variable Y is said to have a E2G distribution with parameters λ >, α > and < θ < 1 if its probability density function is given by, αλθe λy (1 e λy ) α 1 f(y) = [1 (1 θ)(1 (1 e λy ) α )] 2, (1) where λ is a scale parameter of the distribution, and α and θ are shape parameters. For α = 1 the E2G is reduced to the EG distribution [1]. The Figure 1 (left panel) shows the E2G probability density function for θ =.1,.5,.99 and α =.3, 1, 1. The survival function of a E2G distributed random variable is given by, S(y) = θ(1 (1 e λy ) α ) 1 (1 θ)(1 (1 e λy ) α ), (2) where, α >, θ (, 1) and λ >. From (2), the hazard function, according to the relationship h(y) = f(y)/s(y), is given by, h(y) = αλe λy (1 e λy ) α 1 (1 (1 e λy ) α ) [1 + (1 θ) (1 (1 e λy ) α )]. (3) Its initial value is not finite if α < 1. Otherwise it is given by h() = λ/θ if α = 1 and h() = if α > 1. The long-term hazard function value is h( ) = λ. The hazard function (3) can be increasing, decreasing or unimodal as shown in the Figure 2 (right panel), which shows some hazard function shapes for θ =.1,.5,.99 and α =.3, 1, 1. The pth quantile of the E2G distribution, the inverse of the distribution function F (x p ) = p, is given by, Q(u) = F 1 (u) = ln(1 ( uθ 1 u+uθ )1/α ), (4) λ where u has the uniform U(, 1) distribution and F (y) = 1 S(y) is cumulative distribution function of Y. 2

4 λ=1., θ=.1 λ=1., θ=.5 λ=1., θ=.99 Density α=.3 α=1. α=1. Density α=.3 α=1. α=1. Density α=.3 α=1. α= λ=1., θ=.1 λ=1., θ=.5 λ=1., θ=.99 Hazard Function α=.3 α=1. α=1. Hazard Function α=.3 α=1. α=1. Hazard Function α=.3 α=1. α= Figure 1: Left panel: Probability density function of the E2G distribution. Right panel: hazard function of the E2G distribution. We fixed λ = Genesis In the classical competing risks scenarios the lifetime associated with a particular risk is not observable, rather we observe only the minimum lifetime value among all risks. Simplistically, in reliability, we observe only the minimum component lifetime of a serial system. That is, the observable quantities for each component are the minimum lifetime value to failure among all risks, and the cause of failure. Competing risks problems arise in several areas and full statistical procedures and extensive literature are available. Interested readers can refer to [9], [1] and [11]. A difficulty however 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 competing risks data [8]. 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 medical studies, a patient can die and the true cause can be attributed to multiple unknown risks. Then, in this context, our model can be derived as follows. Let M be a random variable denoting the number of failure causes, m = 1, 2,... and considering M with geometrical distribution of probability given by, P (M = m) = θ(1 θ) m 1, (5) where < θ < 1 and M = 1, 2,... Also consider t i, i = 1, 2, 3,... realizations of a random variable denoting the failure times, ie, the time-to-event due to the j th latent competing risk and T i has an exponentiated exponential distribution with probability density function, indexed by λ and α, given by, f(t i ; λ, α) = αλ exp{ λt i }(1 exp{ λt i }) α 1. (6) 3

5 The exponentiated exponential distribution is an alternative to the Weibull and the gamma distributions, firstly proposed by [2] and [7]. In the latent competing risks scenario, the number of causes M and the lifetime t j associated with a particular cause are not observable (latent variables), but only the minimum lifetime Y among all causes is usually observed. So, we only observe the random varible given by, Y = min (t 1, t 2..., t M ). (7) The following result shows that the random variable Y have probability density function given by (1). Proposition 2.1. If the random variable Y is defined as 7, then, considering (5) and (6), Y is distributed according to a E2G distribution, with probability density function given by (1). Proof 2.1. The conditional density function of (7) given M = m is given by f(y M = m, λ) = mαλe λy (1 e λy ) α 1 ( 1 (1 e λy ) α) m 1 ; t >, m = 1,... Them, the marginal probability density function of Y is given by f(y) = mαλe λy (1 e λy ) α 1 ( 1 (1 e λy ) α) m 1 θ(1 θ) m 1 m=1 This completes the proof. ( =θαλe λy (1 e λy ) α (1 (1 e λy ) α )(1 θ) m=1 ( ) 2 =θαλe λy (1 e λy ) α (1 θ)(1 (1 e λy ) α ) ) m Some Properties Many of the interesting characteristics and features of a distribution can be studied through its moments, such mean, variance. Expressions for mathematical expectation, variance and the rth moment on the origin of X can be obtained using the well-known formula E[X r ] = r x r 1 S(x)dx. (8) A general expression for r-th ordinary moment µ r = E(Y r ) of the Y variable, with density function given by (1) can be obtained analytically, if we consider the binomial series expansion given by, (1 x) r = k= (r) k x k, (9) k! where (r) k is a Pochhammer symbol, given (r) k = r(r + 1) (r k + 1) and if x < 1 the series converge, and ( r) k = ( 1) k (r k + 1) k. (1) Proposition 2.2. For the random variable Y with E2G distribution, we have that, rth moment function is given by µ r = θr! ( 1) l+m (j l + 2) l (αl + k m + 1) m (1 θ) j λ r l!j!(m + 1) r. l= m= k= j= 4

6 Proof 2.2. From (2) and (8), and using (9), we have that µ r =r =rθ =rθ 1 = rθ( 1)r 1 λ r = rθ( 1)r 1 λ r = rθ( 1)r 1 λ r = rθ λ r = rθ λ r y r 1 S(y)dy y r 1 (1 (1 e λy ) α ) 1 (1 θ)(1 (1 e λy ) α ) dy ( ) r 1 ln(1 z) (1 z α ) λ [1 (1 θ)(1 z α )](1 z)λ dz (1) 1 k k! k= k= j= k= j= l= (1) j j! k= j= l= k= j= l= m= ( (j + 1)) l l! (ln(1 z)) r 1 z k (1 z α ) 1 (1 θ)(1 z α ) dz 1 (1 θ) j (ln(1 z)) r 1 z k (1 z α ) j+1 dz ( (j + 1)) l l! ( (j + 1)) l l! 1 (1 θ) j (ln(1 z)) r 1 z αl+k dz (1 θ) j u r 1 (1 e u ) αl+k e u du ( (αl + k)) m m! (1 θ) j u r 1 e u(m+1) du where the last equality follows from the integrate of a gamma distribution and the property (1), completing complete the proof. Order statistics are among the most fundamental tools in non-parametric statistics and inference. Let Y 1,..., Y n be a random sample taken from the E2G distribution and Y 1:n,..., Y n:n denote the corresponding order statistics. Then, the probability density function f i:n (y) of the ith order statistics Y i:n is given by f i:n (x) = n! (k 1)!(n k)! F (y)k 1 (1 F (y)) n k f(y) The rth moment of the ith order statistic X i:n, according with [12], can be represented as E [Y r i:n] = r n p=n i+1 ( )( ) p 1 n ( 1) p n+i 1 x r 1 [S(y)] p dy. (11) n i p Proposition 2.3. For the random variable Y with E2G distribution the rth moment of the ith order statistic is given by E [Y r i:n] = r! λ r n p=n i+1 l= m= k= j= ( n p ( ) p 1 ( 1) p n+l+m+i+2r n i ) (p)j (p + j l + 1) l (1 θ) j (αl + k m + 1) m j!l!(m + 1) r. Proof 2.3. From (11), using (2) and (9), and proceeding in a similar way as in the proof 2.2 the result follows. θ m 5

7 Given that there was no failure prior to time t, the residual lifetime distribution of a random variable X, distributed as E2G distribution, has the survival function given by ( 1 (1 e λ(x+t) ) α ) ( 1 (1 θ)(1 e λx ) α ) S t (x) = Pr[X > x + t X > t] = 1 (1 e λx ) α. 1 (1 θ)(1 e λ(x+t) ) α The mean residual lifetime of a continuous distribution with survival function F (x) is given by µ(t) = E(X t X > t) = 1 S(t) t S(u)du. (12) Proposition 2.4. For the random variable Y with E2G distribution the mean residual lifetime is given by µ(t) = θ ( 1 (1 θ)(1 e λt ) ) (1 θ) i ( 1) j ( (i j + 2) j 1 (1 e λt ) k+αj+1 ) λ 1 (1 e λt ) α. j! k + αj + 1 k= i= j= Proof 2.4. From (12) and using S(y) given by (2) we have that 1 S(t) t S(u)du = θ 1 (1 θ)(1 e λt ) 1 (1 e λt ) α = θ 1 (1 θ)(1 e λt ) λ 1 (1 e λt ) α t 1 1 (1 e λu ) α 1 (1 θ)(1 e λu ) du 1 e λt 1 x α (1 (1 θ)(1 x α ))(1 x) dx. Now using (9) and proceeding in a similar way as in the proof 2.2 the result follows. Proposition 2.5. The modal value for the variable Y with density given by (1) is given by, Ỹ = ln(root[θ(αx 1) + (1 x) α ( 1 + θ + αx(θ 1))])/λ if α > 1, (13) where Root[f(x)] is the x root of f(x) and its is evaluated by numerical methods. Proof 2.5. This proof is straightly obtained by solving, from (1), the equation df(y)/dy =. 2.3 Inference Assuming the lifetimes are independently distributed and are independent from the censoring mechanism, the maximum likelihood estimates (MLEs) of the parameters are obtained by direct maximization of the log-likelihood function given by, n n n l(θ, λ, α) = n ln(θ) + ln(αλ) c i λ c i y i + (α 1) c i ln(1 e λyi ) + (14) + i=1 i=1 n (1 c i ) ln ( 1 (1 e λyi ) α) i=1 i=1 n (1 + c i ) ln ( 1 (1 θ)(1 (1 e λyi ) α ) ), where c i is a censoring indicator, which is equal to or 1, respectively, if the data is censured or observed, respectively. The advantage of this procedure is that it runs immediately using existing statistical packages. We have considered the optim routine of the R [13]. Large-sample inference for the parameters are based on the MLEs and their estimated standard errors. In order to compare distributions we consider the max l(.) values and the Akaike information criterion (AIC), which are defined, respectively, by 2l(.) + 2q where l(.) is the log-likehood evaluated in the MLE vector of parameters of the respective distribution and q is the number of estimated parameters. The best distribution corresponds to a lower max l(.), AIC and BIC values. i=1 6

8 3 On the relationship between the EG and E2G distributions In this section, we discuss some relationship between the EG distribution [1], who motivated this article, and E2G distribution proposed here. The hazard function of the EG distribution is given by h(y) = λ/(1 (1 θ)e λy ). (15) Proposition 3.1. The hazard functions, (15) and (3), diverge for y and converge for the same point λ when y. λ Proof 3.1. (a) For y, for the EG model, lim y h(y) = lim y 1 θe λy = λ 1 θ while for the E2G distribution, if α = 1, lim y h(y) = λθ if α < 1, lim y h(y) = and if α > 1, lim y h(y) = concluding the proof of the initial divergence between the EG and E2G hazard functions. λ (b) For y, for the EG model, lim y h(y) = lim y = λ and for the E2G 1 θe λy distribution lim y h(y) = λ, concluding the proof of the converge. The Figure 2 shows the behavior of hazard functions of EG model for θ =.1,.9 and for rate function of E2G with θ =.1,.9 and α =.25, 4. The EG hazard function is decreasing while the E2G hazard function can be decreasing, increasing and unimodal, but both hazards converge to λ for y corroborating with the Proposition 3.1. EG, with λ=1. E2G, with λ=1. Hazard Function θ=.1 θ=.9 Hazard Function (θ,α)=(.1,.25) (θ,α)=(.9,.25) (θ,α)=(.1,4.) (θ,α)=(.9,4.) Figure 2: Comparing the hazard function of the EG and E2G distributions for same fixed θ values. A misspecification study was performed in order to verify if we can distinguish between the EG and E2G distributions, in the light of a dataset, based on an usual comparison criterion. We consider here the max l(.) values. The preferred distribution is the one with the smaller max l(.) value. We generate 1. samples of each one of the EG and E2G distributions. We consider different sample sizes, n, equal to 1, 2, 3, 5 and 1, and different censoring percentages, p, equal to.1, 7

9 Table 1: Percentage of times that the ( EG / E2G) distribution, which originated the sample, was the best fitted distribution. θ =.9 n/p / / /.94.76/ / / / / / / / / / / /.826 θ =.1 n/p / / / / / / / / / / / / / / /.999.2,.3. We fixed λ =.5 and θ =.1 and θ =.9 for both distributions, and α = 3 to the E2G distribution. The two distributions were fitted to each sample and their max l(.) values were calculated. For θ =.1, and α = 3, the data have a unimodal hazard function and for θ =.9, and α = 3 have a increasing hazard function. In principle, the case θ =.1, θ =.9 combined with α = 3 is better adjusted by a E2G distribution. Table 1 shows the percentage of time that the distribution, which originated the sample, was the best fitted distribution according to the max l(.) values. We observe that it is usually possible to discriminate between the distributions even for moderate samples in presence of censoring. We also performed the study by considering the AIC instead of the max l(.) values. However the results leads to similar conclusions and are omitted here. 4 Aplications In this section, we compare the E2G distribution fit with several usual lifetime distributions on three datasets extracted from the literature. one with increasing hazard function, one with unimodal hazard function and one with decreasing hazard function. The following lifetime distributions were considered.the Exponential distribution with probability density function given by f(x) = λe λx, the Weibull distribution with probability density function given by f(x) = θ x ( ) θ 1 λ λ e (x/λ) θ, the gamma distribution with probability density function given by f(x) = 1 λ θ Γ(θ) xθ 1 e x/λ, the EG distribution [1] with probability density function given by f(x) = λ(1 (1 θ)e λx ) 1, the Modified Weibull (MW) distribution [14] with probability density function given by f(x) = αx θ 1 (θ + λx)e λx e αxθ exp{λx}, and the generalized exponential-poisson (GEP) distribution [4] with probability density function given by f(x) = αβλ (1 e λ+λ exp ( βx) ) α 1 e λ βx+λ exp ( βx). (1 e λ ) α The first set, hereafter T 1, are observed survival times for 65 breast cancer patients treated over the period , quoted by [15]. The second set, hereafter T 2, are survival times for patients with bile duct cancer, which took part in a study to determine whether a combination of a radiation treatment (RRx) and the drug 5-fluorouracil (5-FU) prolonged survival, extracted from [16]. Survival times, in days, are given for a control group. The third set, hereafter T 3, consists of the number of successive failure for the air conditioning system of each member in a fleet of 13 Boeing 72 jet airplanes. The pooled data with 214 observations was considered by [1]. It was first analyzed by [17]and discussed further by [18],[19], [3] and [2]. 8

10 Firstly, in order to identify the shape the hazard function we shall consider a graphical method based on the TTT plot [21]. In its empirical version the TTT plot is given by G(r/n) = [( r i=1 Y i:n) + (n r)y r:n ]/( n i=1 Y i:n), where r = 1,..., n and Y i:n represent the order statistics of the sample. It has been shown that the hazard function is increasing (decreasing) if the TTT plot is concave (convex). The left panels of Figure 4 shows concave TTT plots for T 1, concave/convex for T 2 and convex for T 3, indicating increasing, unimodal and decreasing hazard functions, respectively. Table 2 provides the max l(.) and the AIC values for all fitted distributions. Both criteria provide evidence in favor of our E2G distribution for T 2 and T 3, and give similar values for the E2G and gamma distributions for T 1, corroborating the fact that the E2B distribution can be seen as a competitive distribution of practical interest for the analysis of survival data. Figure 4 (right panels) shows the fitted survival superimposed to the empirical Kaplan-Meier survival function. The the MLEs (and their corresponding standard errors in parentheses) of the parameters α, θ and λ( 1) of the E2G distribution are given, respectively, by 2.12(.719),.582(.551) and 41.11(9.98) for T 1, by 1.978(2.377),.21(.489) and.85(.188) for T 2, and by 1.238(.273),.314(.141) and 6.9(.72) for T 3. T1 T2 T3 Table 2: Values of the max l(.) and AIC for all fitted distributions. E EG Weibull Gamma E2G MW GEP max l(.) AIC max l(.) AIC max l(.) AIC Concluding remarks In this paper we propose a new lifetime distribution. The E2G distribution is a straightforwardly generalization of the EG distribution proposed by [1], which accommodates increasing, decreasing and unimodal hazard functions. It arises on a latent competing risk scenarios, where the lifetime associated with a particular risk is not observable but only the minimun lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal prove of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments, rth moment of the ith order statistic, mean residual lifetime and modal value. Maximum likelihood inference is implemented straightforwardly. From a misspecification simulation study performed in order to assess the extent of the misspecification errors when testing the EG distribution against the E2G one we observed that it is usually possible to discriminate between both distributions even for moderate samples in presence of censoring. The practical importance of the new distribution was demonstrated in three applications where the E2G distribution provided the better fitting in comparison with the EG one and several other former lifetime distributions. Acknowledgments: The research of Francisco Louzada is supported by the Brazilian organization CNPq. 9

11 TTT Plot G(n/r) S(t) estimated E EG Weibull Gamma E2G MW GEP n/r Time TTT Plot G(n/r) S(t) estimated E EG Weibull Gamma E2G MW GEP n/r Time TTT Plot G(n/r) S(t) estimated E EG Weibull Gamma E2G MW GEP n/r Time Figure 3: Left Panels: Empirical TTT-Plot. Right Panels: Kaplan-Meier and survival fitted curves. Upper panels are for dataset T 1, Middle panels are for dataset T 2 and lower panels are for dataset T 3. 1

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14 PUBLICAÇÕES 21. RODRIGUES, J.; CASTRO, M.; BLAKRISHNAN, N.; GARIBAY, V.; Destructive weighted Poisson cure rate models Novembro/29 Nº 21. 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/21 Nº 211. DINIZ, C. A. R. ; LOUZADA-NETO, F.; MORITA, L. H. M.; The Multiplicative Heteroscedastic Von Bertalan_y Model Fevereiro/21 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/21 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/21 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/21 Nº CORDEIRO, G. M.; RODRIGUES, J.; CASTRO, M. The exponential COM-Poisson distribution, Abril/21 Nº 216. CANCHO, V. G.; LOUZADA-NETO, F.; BARRIGA, G. D. C. The Poisson-Exponential Survival Distribution For Lifetime Data, Maio/21 Nº 217. DIINIZ, C. A. R.; FURLAN, C. P. R.; LEITE, J. G. A Bayesian Estimation of Lag Length in Distributed Lag Models, Julho/21 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/21 Nº 219. LOUZADA-NETO, F.; CANCHO, V. G.; BARRIGA, G. D. C.; A Bayesian Analysis For The Poisson-Exponential Distribution Agosto/21 Nº 22. 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/21 Nº 221. TOMAZELLA, V. L., BERNARDO, J. M.; Testing for Hardy-Weinberg Equilibrium in a Biological Population: An Objective Bayesian Analysis Setembro/21 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/21 Nº 223. ARA-SOUZA, A. L.; LOUZADA-NETO, F.; Caracterização dos Docentes e Necessidade de Doutores dentro das Graduações de Estatística do Brasil Setembro/21 Nº 224. LOUZADA-NEDTO, F.; ROMAN, M.; CANCHO, V. G.; The Complementary Exponential Geometric Distribution: Model, Properties and a Comparison With Its Counterpart Setembro/21 Nº 225. LOUZADA-NETO,F.; BORGES, P.; The Exponential Negative Binomial Distribuition Setembro/21 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/21 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/21 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/21 Nº 229. TOJEIRO, C.; A. V. LOUZADA-NETO, F.; A General Threshold Stress Hybrid Hazard Model for Lifetime Data Outubro/21 Nº 23. TOMAZELLA, V. L.D.; CANCHO, V. G.; LOUZADA-NETO, F.; Objective Bayesian Reference Analysis for the Poissonexponential lifetime distribution Outubro/21 Nº 231. DELGADO, J. F.; BORGES, P.; CANCHO, V.G.; LOUZADA-NETO, F.; A compound class of exponential and power series distributions with increasing failure rate Novembro/21 Nº 232. DELGADO. J. J. F., CANCHO V. G., LOUZADA-NETO F.; The power series cure rate model: an application to a cutaneous melanoma data Novembro/21 Nº 233 Os recentes relatórios poderão ser obtidos pelo endereço Mais informações sobre publicações anteriores ao ano de 21 podem ser obtidas via dfln@ufscar.br