THE COMPOSED ZERO TRUNCATED LINDLEY-POISSON DISTRIBUTION

ABSTRACT

In this paper, a new compounding distribution, named zero truncated Lindley-Poisson distribution is introduced. The probability density function, cumulative distribution function, survival function, failure rate function and quantiles expressions of it are provided. The parameters estimatives were obtained by six methods: maximum likelihood (MLE), ordinary least-squares (OLS), weighted least-squares (WLS), maximum product of spacings (MPS), Crame´r-von-Mises (CM) and Anderson-Darling (AD), and intensive simulation studies are conducted to evaluate the performance of parameter estimation. Some generalizations are also proposed. Application in a real data set is given and shows that the composed zero truncated Lindley-Poisson distribution provides better fit than the Lindley distribution and three of its generalizations. The paper is motivated by application in real data set and we hope this model may be able to attract wider applicability in survival and reliability.

Keywords:
compounding; estimation methods; Lindley distribution; survival analysis; zero truncated Poisson distribution

1 INTRODUCTION

The one parameter Lindley distribution was introduced by Lindley (see, ^{Lindley 1958} and ¹⁹⁶⁵) as a new distribution useful to analyze lifetime data, especially in stress-strength reliability modeling. Suppose that T ₁ , ... , T_M are independent and identically distributed random variables following the one parameter Lindley distribution with probability density function and distribution function written, respectively, as:

(1)

(2)

where t > 0 and θ > 0.

For a random variable with the one parameter Lindley distribution, the probability density function, (1), is unimodal for 0 < θ < 1 and decreasing when θ > 1 (see Fig. 1-a). The hazard rate function is an increasing function in t and θ (see Fig. 1-b) and given by:

(3)

Figure 1
Probability density function and hazard rate function behavior for different values of θ .

^{Ghitany et al. (2008b}) studied the properties of the one parameter Lindley distribution under a careful mathematical treatment. They also showed, in a numerical example, that the Lindley distribution is a better model than the Exponential distribution. A generalized Lindley distribution, which includes as special cases the Exponential and Gamma distributions was proposed by ^{Zakerzadeh & Dolati (2009}), and ^{Nadarajah et al. (2011}) introduced the exponentiated Lindley distribution. ^{Ghitany & Al-Mutari (2008}) considered a size-biased Poisson-Lindley distribution and ^{Sankaran (1970}) proposed the Poisson-Lindley distribution to model count data. Some properties of Poisson-Lindley distribution and its derived distributions were considered in ^{Borah & Begum (2002}) while ^{Borah & Deka (2001a}) considered the Poisson-Lindley and some of its mixture distributions. The zero-truncated Poisson-Lindley distribution and the generalized Poisson-Lindley distribution were considered in ^{Ghitany et al. (2008a}) and ^{Mahmoudi & Zakerzadeh (2010}), respectively. A study on the inflated Poisson-Lindley distribution was presented in ^{Borah & Deka (2001b}) and ^{Zamani & Ismail (2010}) considering the Negative BinomialLindley distribution. The weighted and extended Lindley distribution were considered by ^{Ghitany et al. (2011}) and ^{Bakouch et al. (2012}), respectively. The one parameter Lindley distribution in the competing risks scenario was considered in ^{Mazucheli & Achcar (2011}). The exponential Poisson Lindley distribution was presented in ^{Barreto-Souza & Bakouch (2013}). ^{Ghitany et al. (2013}) introduced the power Lindley distribution. ^{Ali (2015}) investigated various properties of the weighted Lindley distribution which main focus was the Bayesian analysis. A new four-parameter class of generalized Lindley distribution called the beta-generalized Lindley distribution is proposed by ^{Oluyede & Yang (2015}).

Aim to offers more flexible distributions for modeling lifetime data set, in this paper, is proposed an extension of the Lindley distribution. We consider that T_j, j = 1, . . ., M is a random sample from the one parameter Lindley distribution and that our variable of interest is defined as:

(i) Y = min (T ₁, . . ., T_M ) and (ii) Y = max (T ₁ , .. ., T_M )

representing, respectively, the first and the last failure time of a certain device subject to the presence of an unknown number M of causes of failures. Furthermore, we consider that M has a zero truncated Poisson distribution, M ∼ PoissonT runc (λ), λ > 0, and that T_j , j = 1, .. ., M, and M are independent random variables, leading to the composed zero truncated Lindley-Poisson distribution. The process of composition using the zero truncated Poisson distribution has been fairly used in the literature. In ^{Kus¸ (2007}) was considered the zero truncated Exponential-Poisson distribution in the competing risks scenario. ^{Hemmati et al. (2011}) developed the zero truncated Weibull-Poisson distribution. Also in 2011, the same distribution was studied by ^{Ristic´ & Nadarajah (2012}) and ^{Lu & Shi (2012}). The zero truncated Exponential-Poisson distribution in the complementary risks scenario was introduced by ^{Rezaei & Tahmasbi (2012}).

The paper is organized as follows: in Section 2 the zero truncated Lindley-Poisson distribution is formulated. In Section 3 six estimation methods are presented. A simulation study is introduced in Section 4. The Section 5 brings a real data application. And finally, conclusions are presented in Section 6.

2 MODEL FORMULATION

In the theory of competing risks and complementary risks the number of risk factors (or causes) that may lead to the event of interest, is known and denoted as M. However, in models of distributions composition is assumed that M is unknown. Therefore, there is a number M of latent risk factors competing to cause the event of interest. In what follows, let us consider the situation where an individual or unit is exposed to M possible causes of death or failure, such that the exact cause is fully known (^{David & Moeschberger, 1978}). The model for lifetime in the presence of suchcompeting risks structure or complementary risks structure is known as model of composition distributions. If T_j , j = 1, ..., M denote the latent failure times of a individual subject to M risks, which are independent of M, what is observed is the time to failure Y = min (T ₁ , ..., T_M ). Given M = m, under the assumption that the latent failure times T_j , j = 1, ... , M are independent and identically distributed random variables with the distribution function (2), the probability density function and the cumulative distribution function are written, respectively, as:

(4)

(5)

It is important to note that (4) and (5) are uniquely determined by the distribution function of the minimum, that is, P(Y ≤ y) = 1 − [1 − F1(y | θ )]m , (^{Arnold et al., 2008}).

Now, assuming the number of causes of death or failure, M, is a zero truncated Poisson random variable with probability mass function given by:

(6)

where m = 1, 2, ... and λ> 0, ^{Rezaei & Tahmasbi (2012}).

The marginal probability density function, f _min (y | θ, λ), the marginal cumulative distribution function, F _min (y | θ, λ), and the marginal hazard rate function, h _min (y | θ, λ) of Y = min (T ₁ , .. ., T_M ) are given, respectively, by:

(7)

(8)

(9)

where θ > 0,λ > 0 and y > 0, which defines the zero truncated Lindley-Poisson distribution in the competing risks scenario. Taking the λ = 0 we have the one parameter Lindley distribution as a particular case. Note that f _min (0 | θ, λ) = λθ ²/(θ + 1)(1 - e ^{- λ} ) and f _min (∞ | θ, λ) = 0. For all θ > 0 and λ > 0, the probability density function, (7), is decreasing or unimodal (see Fig. 2). For values of λ close to 1, the curve resembles the one parameter Lindley distribution, while when λ → 0 the curve tends to be symmetric.

Figure 2
The zero truncated Lindley-Poisson probability density function for different values of the λ and θ = 0.5 if Y = min(T ₁ , . . . , T_M ).

The hazard rate, (9), is increasing, increasing-decreasing-increasing and decreasing (see Fig. 3). Is easy to see that h _min (0 | θ, λ) = f _min (0 | θ, λ) = λθ ²/(θ + 1)(1 - e ^{- λ} ) and h _min (∞ | θ, λ) = θ.

Figure 3
The zero truncated Lindley-Poisson hazard rate function for different values of the λ and θ = 2.0 if Y = min(T ₁ , . . . , T_M ).

Now, under the same assumptions and considering a complementary risks scenario, (^{Basu, 1981}), where Y = max (T ₁ , . . ., T_M ) is observed, the marginal probability density function, f _max (y | θ, λ), the cumulative distribution function, F _max (y | θ, λ), and the hazard rate function, h _max (y | θ, λ), are given, respectively, by:

(10)

(11)

(12)

where θ > 0, λ > 0 and y > 0.

Note that f _max (0 | θ, λ) = λθ ² e ^{- λ} /(θ + 1)(1 - e ^{- λ} ) and f _max (∞ | θ, λ) = 0. For all θ > 0 and λ > 0 the probability density function (10), is decreasing or unimodal (see Fig. 4). For values of λ close to 1, the curve resembles the one parameter Lindley distribution, while when λ → ∞ the curve tends to be symmetric.

Figure 4
The zero truncated Lindley-Poisson probability density function for different values of the λ and θ = 2.0 if Y = max(T ₁ , . . . , T_M ).

For all θ > 0 and λ > 0, the hazard rate function, (12), is increasing (see Fig. 5). Is easy to see that h _max (0 | θ, λ) = f _max (0 | θ, λ) = λθ ² e ^{- λ} /(θ + 1)(1 - e ^{- λ} ) and h _max (∞ | θ, λ) = θ. Note that h _min (∞ | θ, λ) = h _max (∞ | θ, λ) = θ.

Figure 5
The zero truncated Lindley-Poisson hazard rate function for different values of the λ and θ = 2.0 if Y = max(T ₁ ,. .. , T_M ).

^{Glaser (1980}) and ^{Chechile (2003}) studied the hazard rate function behavior by the η (y) = -f’(y|θ, λ)/f(y|θ, λ) function and its derivative η’(y). Because of the complexity of such studies, this work only presents the functions η (y) and η’(y). Considering the hazard rate functions (9) and (12), we have:

(13)

(14)

and its first derivatives are:

Therefore, the hazard rate function behavior properties of the zero truncated Lindley-Poisson distribution follows from the results in ^{Glaser (1980}) and ^{Chechile (2003}).

2.1 Quantile function

The quantile function of the zero truncated Lindley-Poisson distribution is given by:

if Y = min (T ₁ , . . ., T_M), where 0 < u < 1 and W−1 (·) denotes the negative branch of the Lambert W function (i.e., the solution of the equation W(z)e^{W ( z )} = z) because (1 + θ + θ y) > 1 and e, 0). And, the quantile function of the zero truncated Lindley- Poisson distribution is given by: ∈ (-1/

if Y = max (T ₁ , . . ., T_M), where 0 < u < 1 and W−1 (·) denotes the negative branch of the Lambert W function because (1 + θ + θ y) > 1 and e, 0) (^{Jodrá, 2010}; ^{Ghitany et al., 2012}). ∈ (-1/

Our approach may be generalized in some different ways, for instance, it is important to note that for any probability density function f ₁ (y | θ), θ = θ ₁ , . . . , θp), and M ∼ PoissonT runc (λ) as the discrete distribution, the general marginal probability density function can be written as:

(15)

where F_p (y | θ) = F ₁ (y | θ) when Y = min (T ₁ , . . ., T_M) and F_p (y | θ) = 1−F ₁ (y | θ) when Y = max (T ₁ , . . ., T_M ).

From (15), the cumulative distribution, survival and hazard functions for Y = min (T ₁ , . . ., T_M ) and Y = max (T ₁ , . . ., T_M ) can be generically written as:

3 ESTIMATION METHODS

In this section, considering the distribution obtained by the composition of distributions we describe six methods used to estimate λ and θ. For all methods we consider the case when both λ and θ are unknown. This is also considered in the simulation study presented in Section 4. Note that the methods were presented for a general baseline function f ₁ (y | θ).

3.1 Maximum Likelihood

Let y = (y ₁ , ... , y_n ) be a random sample of n size from the distribution obtained by the composition of distributions with parameters λ and θ , the likelihood and log-likelihood function are, respectively:

(16)

(17)

where:

1. i) F_p (y_i | θ) = F ₁ (y_i | θ) if Y = min (T ₁ , . . ., T_M )

2. ii) F_p (y_i | θ) = 1 − F ₁ (y_i | θ) if Y = max (T ₁ , . . ., T_M ).

The maximum likelihood estimates of θ and λ, MLE and MLE respectively, can be obtained numerically by maximizing the log-likelihood function (17). In this case, the log-likelihood function is maximized by solving numerically ∂/∂θ l (θ,λ | y) = 0 and ∂/∂λ l (θ,λ | y) = 0 in θ and λ, respectively, where:

(18)

(19)

where f ‘ ₁ (y_i | θ) = ∂/∂θ f ₁ (y_i | θ) and F’_p (y_i | θ) = ∂/∂θ F_p (y_i | θ).

3.2 Ordinary Least-Squares

Let y _{1: n} < y _{2: n} ··· < y_{n : n} be the order statistics of a random sample of n size from a distribution with cumulative distribution function F (y). It’s well known that:

(20)

For the distribution obtained by the composition process, the least square estimates OL S and OL S of the parameters θ and λ, respectively, are obtained by minimizing the function:

(21)

when Y = min (T ₁ , .. ., T_M), and minimizing

(22)

when Y = max (T ₁, . . ., T_M ).

Therefore, if Y = min (T ₁ , .. ., T_M), these estimates can also be obtained by solving the nonlinear equations:

(23)

(24)

where:

(25)

(26)

But, if Y = max (T ₁ , .. ., T_M ), these estimates can also be obtained by solving the nonlinear equations:

(27)

(28)

where:

(29)

(30)

Note that ∆₁ and ∆₂ are derivative from first order distribution function for parameters θ and λ, respectively.

3.3 Weighted Least-Squares

The weighted least-squares estimates WL S of the parameters θ and λ, respectively, are obtained by minimizing the function:WL S and

(31)

if Y = min (T ₁ , .. ., T_M ), and minimizing

(32)

if Y = max (T ₁ , .. ., T_M ).

The correction factor w_i is given by:

(33)

Therefore, if Y = min (T ₁ , .. ., T_M), these estimates can also be obtained by solving the nonlinear equations:

(34)

(35)

where ∆₁ (y_{i : n} |θ, λ) and ∆₂ (y_{i : n} |θ, α) are given by (25) and (26), respectively.

Thus, if Y = max (T ₁ , . . ., T_M ), these estimates can also be obtained by solving the nonlinear equations:

(36)

(37)

where ∆₁ (y_{i : n} |θ, λ) and ∆₂ (y_{i : n} |θ, α) are given by (29) and (30), respectively.

3.4 Maximum Product of Spacings

^{Cheng & Amin (1979}, ¹⁹⁸³) introduced the maximum product of spacings (MPS) method as alternative to MLE for the estimation of parameters of continuous univariate distributions. ^{Ranneby (1984}) independently developed the same method as an approximation of Kullback-Leibler measure of information. In what follows, let y_1:n < y_2:n < ··· < y_{n : n} be an ordered random sample drawn from the general model of composition distribution. Are defined as the uniform spacings of the sample the quantities: D ₁ = F (y _{1: n} | θ, λ), D_{n +1} = 1 − F (t_{n : n} | θ, λ) and D_i = F (t_{i : n} | θ, λ) − F t _{( i −1): n} | θ, λ , i = 2, ... , n. There are (n + 1) spacings of the first order.

Following ^{Cheng & Amin (1983}), the maximum product of spacings method consists in finding the values of θ and λ which maximize the geometric mean of the spacings, the MPS statistics, is given by:

(38)

or, equivalently, its logarithm H = log(G). Considering 0 = F (t _{0: n} | θ, λ) < F (y _{1: n} | θ, λ) < ··· < F (y_{n : n} | θ, λ) < F y _{( n +1 ) : n} | θ, λ = 1 the quantitie H = log(G) can be calculated as:

(39)

The estimates for θ and λ can be found solving, respectively in θ and λ, the nonlinear equations:

(40)

(41)

where ∆ is the first order difference operator.

^{Cheng & Amin (1983}) showed that maximizing H as a method of parameter estimation is as efficient as MLE estimation and the MPS estimators are consistent under more general conditions than the MLE estimators.

Therefore, if Y = min (T ₁ , .. ., T_M ), the estimates MP S and MP S can be obtained by solving the nonlinear equations:

(42)

(43)

Thus, if Y = max (T ₁ , .. ., T_M ), the estimates MP S can be obtained by solving the nonlinear equations:MP S and

(44)

(45)

3.5 Minimum distance methods

In this subsection we present two estimation methods for θ and λ based on the minimization of the goodness-of-fit statistics. This class of statistics is based on the difference between the estimate of the cumulative distribution function and the empirical distribution function (^{Luceño, 2006}).

3.5.1 Crame´r-von-Mises

The Crame´r-von-Mises estimates of the parameters CM and CM, respectively, are obtained by minimizing, in θ and λ, the function:

(46)

These estimates can also be obtained by solving the nonlinear equations:

(47)

(48)

where ∆₁ (·|θ, λ) and ∆₂ (·|θ, λ) are given, respectively, by (25) and (26) if Y = min(T ₁ ,... , T_M ) and, respectively, by (29) and (30) if Y = max(T ₁ , ... , T_M ).

3.5.2 Anderson-Darling

The Anderson-Darling estimates of the parameters AD , respectively, are obtained by minimizing, with respect to θ and λ, the function:AD and

(49)

These estimates can also be obtained by solving the nonlinear equations:

(50)

(51)

where ∆₁ (·|θ, λ) and ∆₂ (·|θ, λ) are given, respectively, by (25) and (26) if Y = min(T ₁ ,... , T_M ) and, respectively, by (29) and (30) if Y = max(T ₁ , ... , T_M ).

4 SIMULATION STUDY

In this section we present results of some numerical experiments to compare the performance of the different estimation methods discussed in the previous section. We have taken sample sizes n = 20, 50, 100 and 200, θ = 1.0 and λ = 0.5, 1.0, 2.0, 3.0 and 5.0. For each combination (n,θ, λ) we have generated B = 500, 000 pseudo random samples from the zero truncated Lindley-Poisson distribution.

The estimates were obtained in Ox version 6.20 (^{Doornik, 2007}) using MaxBFGS function in MLE, OLS, WLS, MPS, CM and AD methods. For each estimate we computed the bias, the root mean-squared error, the average absolute difference between the true and estimate distributions functions and the maximum absolute difference between the true and estimate distributions functions, respectively, as:

(52)

(53)

(54)

(55)

In Tables 1, 2, 3, 4 and 5 we show the calculated values of (52)-(55). The superscript values indicate the rank obtained by each of the methods considered, and the total line shows the global rank for each method based on measures (52)-(55).

For the simulations, the MLE method proved to be the most efficient for estimate the parameters of zero truncated Lindley-Poisson distribution to Y = min(T₁, . . . , T_M ) when λ= 0.5 and λ = 1.0. For λ = 2.0, 3.0 and 5.0, the OLS, in general, proved to be better.

Table 1
Simulations results for θ = 1.0 and λ = 0.5.

n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.12701	-0.27046	-0.20964	-0.22305	-0.16922	-0.19783
	RMSE(θ)	0.27912	0.35496	0.28423	0.30195	0.26881	0.29754
	Bias(λ)	1.00861	1.64176	1.23243	1.34805	1.10702	1.31524
20	RMSE(λ)	1.60543	2.09246	1.48742	1.66204	1.42621	1.75735
	Dabs	0.04722	0.04691	0.05016	0.04894	0.04975	0.04833
	D max	0.07392	0.07101	0.07765	0.07604	0.07866	0.07573
	Total	111	265	234	276	172	223
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.11991	-0.24326	-0.17715	-0.17714	-0.14972	-0.15673
	RMSE(θ)	0.24532	0.33366	0.24753	0.25805	0.23371	0.24874
	Bias(λ)	0.84751	1.52886	0.98104	1.03015	0.87422	0.95703
50	RMSE(λ)	1.54365	2.16276	1.31022	1.45564	1.24331	1.43543
	Dabs	0.03092	0.03061	0.03236	0.03164	0.03225	0.03153
	D max	0.04832	0.04741	0.05085	0.04984	0.05106	0.04963
	Total	131	265	254	265	172	193
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.08971	-0.19266	-0.14125	-0.12634	-0.12083	-0.11492
	RMSE(θ)	0.20633	0.29416	0.21635	0.21294	0.20461	0.20532
	Bias(λ)	0.60621	1.21996	0.75765	0.71384	0.67223	0.66682
100	RMSE(λ)	1.30385	1.97126	1.13112	1.17144	1.06951	1.14183
	Dabs	0.02232	0.02231	0.02336	0.02274	0.02325	0.02273
	D max	0.03511	0.03522	0.03726	0.03624	0.03715	0.03623
	Total	131	275	296	244	183	152
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.05321	-0.12556	-0.09895	-0.07703	-0.08494	-0.07182
	RMSE(θ)	0.15691	0.22906	0.17865	0.16173	0.17014	0.15802
	Bias(λ)	0.34691	0.77376	0.51745	0.42193	0.45764	0.40022
200	RMSE(λ)	0.95705	1.53326	0.90794	0.85082	0.86063	0.83231
	Dabs	0.01611	0.01622	0.01686	0.01643	0.01675	0.01644
	D max	0.02571	0.02612	0.02736	0.02654	0.02725	0.02643
	Total	101	285	316	183	254	142

• Qtd: measures (52)-(55)

Table 2
Simulations results for θ = 1.0 and λ = 1.0.

n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.05391	-0.20786	-0.13144	-0.15385	-0.08802	-0.13003
	RMSE(θ)	0.28645	0.32746	0.25491	0.27873	0.25512	0.28374
	Bias(λ)	0.63711	1.25636	0.83973	0.99615	0.71652	0.97104
20	RMSE(λ)	1.41013	1.77816	1.18412	1.41464	1.16251	1.53885
	Dabs	0.04711	0.04742	0.05036	0.04934	0.04985	0.04833
	D max	0.07322	0.07121	0.07695	0.07554	0.07786	0.07493
	Total	131	276	213	255	182	224
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.05961	-0.19536	-0.10494	-0.12185	-0.07482	-0.09803
	RMSE(θ)	0.24965	0.31286	0.22082	0.24264	0.21801	0.24083
	Bias(λ)	0.55232	1.23466	0.63223	0.76595	0.52521	0.67704
50	RMSE(λ)	1.42195	1.91286	1.06982	1.32484	1.03721	1.29553
	Dabs	0.03072	0.03071	0.03226	0.03164	0.03225	0.03143
	D max	0.04782	0.04701	0.04975	0.04904	0.05016	0.04873
	Total	172	265	224	265	161	193
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.04391	-0.16286	-0.07814	-0.07955	-0.05552	-0.06703
	RMSE(θ)	0.22235	0.28966	0.20312	0.21504	0.19981	0.21193
	Bias(λ)	0.40632	1.05436	0.47284	0.50865	0.38521	0.46173
100	RMSE(λ)	1.29285	1.83266	0.98412	1.12134	0.95361	1.11113
	Dabs	0.02232	0.02221	0.02316	0.02274	0.02315	0.02263
	D max	0.03512	0.03481	0.03625	0.03564	0.03646	0.03563
	Total	172	265	234	265	161	183
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.02451	-0.11946	-0.05085	-0.04444	-0.03402	-0.03833
	RMSE(θ)	0.18945	0.25136	0.18484	0.18473	0.18231	0.18242
	Bias(λ)	0.24881	0.77836	0.32085	0.30124	0.25292	0.27703
200	RMSE(λ)	1.08765	1.59696	0.88122	0.93084	0.85601	0.92333
	Dabs	0.01622	0.01621	0.01675	0.01644	0.01676	0.01643
	D max	0.02592	0.02581	0.02675	0.02624	0.02686	0.02623
	Total	161	265	265	234	183	172

• Qtd: measures (52)-(55)

Table 3
Simulations results for θ = 1.0 and λ = 2.0.

n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	0.11666	-0.07615	0.01502	-0.02713	0.05464	-0.00851
	RMSE(θ)	0.39546	0.32984	0.27841	0.31963	0.31572	0.33525
	Bias(λ)	-0.01911	0.65556	0.25553	0.57894	0.19112	0.58015
20	RMSE(λ)	1.49873	1.63394	1.07381	1.63895	1.21542	1.70526
	Dabs	0.04771	0.04873	0.05056	0.05055	0.04994	0.04862
	D max	0.07462	0.07421	0.07764	0.07775	0.07826	0.07593
	Total	192	235	171	256	203	224
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	0.07345	-0.09026	0.00682	-0.04744	0.03463	0.00421
	RMSE(θ)	0.33166	0.30465	0.24651	0.27813	0.27222	0.29204
	Bias(λ)	0.05581	0.73656	0.23473	0.62285	0.16772	0.37024
50	RMSE(λ)	1.51823	1.66445	1.17541	1.73686	1.25642	1.54714
	Dabs	0.03131	0.03152	0.03256	0.03204	0.03245	0.03163
	D max	0.04932	0.04901	0.05014	0.05025	0.05066	0.04963
	Total	182	255	171	276	204	193
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	0.05725	-0.08796	0.00101	-0.03354	0.02062	0.02373
	RMSE(θ)	0.29736	0.28825	0.24441	0.27814	0.26332	0.27453
	Bias(λ)	0.05251	0.71276	0.26384	0.52255	0.21533	0.19332
100	RMSE(λ)	1.41094	1.61665	1.26921	1.66196	1.33382	1.35353
	Dabs	0.02271	0.02272	0.02325	0.02324	0.02336	0.02303
	D max	0.03612	0.03601	0.03633	0.03686	0.03675	0.03654
	Total	193	255	151	296	204	182
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	0.03935	-0.08136	-0.00121	-0.00842	0.01253	0.03144
	RMSE(θ)	0.25844	0.26245	0.23931	0.26276	0.25383	0.24632
	Bias(λ)	0.05861	0.63696	0.25704	0.32635	0.22433	0.07982
200	RMSE(λ)	1.25363	1.48626	1.25082	1.41705	1.30384	1.12801
	Dabs	0.01642	0.01641	0.01675	0.01673	0.01686	0.01674
	D max	0.02621	0.02632	0.02663	0.02695	0.02706	0.02674
	Total	161	265	161	265	254	173

• Qtd: measures (52)-(55)

Table 4
Simulations results for θ = 1.0 and λ = 3.0.

n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	0.10375	-0.08234	0.04903	-0.03582	0.02261	-0.11516
	RMSE(θ)	0.51686	0.35395	0.31521	0.32662	0.33893	0.34734
	Bias(λ)	0.77984	0.91085	0.04561	0.63623	0.34932	1.28836
20	RMSE(λ)	4.11806	2.90455	1.55501	2.28613	1.88622	2.78334
	Dabs	0.04903	0.05226	0.04965	0.04954	0.04831	0.04892
	D max	0.07545	0.07776	0.07452	0.07504	0.07441	0.07493
	Total	295	316	132	183	101	254
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.02001	-0.14664	-0.04772	-0.18566	-0.07193	-0.18455
	RMSE(θ)	0.44296	0.33815	0.28931	0.33373	0.30762	0.33464
	Bias(λ)	1.33683	1.34004	0.58801	1.66535	0.81702	1.69556
50	RMSE(λ)	4.10316	3.00495	1.93321	2.99054	2.13632	2.96613
	Dabs	0.03424	0.03566	0.03332	0.03455	0.03301	0.03343
	D max	0.05354	0.05436	0.05091	0.05385	0.05132	0.05233
	Total	243	306	81	285	122	243
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.08251	-0.17674	-0.13672	-0.26566	-0.16123	-0.23585
	RMSE(θ)	0.40276	0.32753	0.30441	0.35535	0.31952	0.34094
	Bias(λ)	1.41362	1.41633	1.19631	2.22296	1.42214	1.98915
100	RMSE(λ)	3.58526	2.67683	2.35281	3.25135	2.53632	3.05074
	Dabs	0.02724	0.02785	0.02562	0.02806	0.02551	0.02663
	D max	0.04314	0.04325	0.04021	0.04486	0.04062	0.04263
	Total	233	233	81	346	142	245
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.12451	-0.19662	-0.20863	-0.32726	-0.22704	-0.28165
	RMSE(θ)	0.35425	0.30971	0.31752	0.37456	0.32903	0.34824
	Bias(λ)	1.26251	1.38402	1.65173	2.62376	1.82554	2.20345
200	RMSE(λ)	2.72054	2.22941	2.55452	3.34556	2.69413	3.00075
	Dabs	0.02223	0.02264	0.02091	0.02456	0.02092	0.02335
	D max	0.03533	0.03554	0.03361	0.03986	0.03402	0.03775
	Total	173	142	121	366	184	295

• Qtd: measures (52)-(55)

Table 5
Simulations results for θ = 1.0 and λ = 5.0.

n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.18026	-0.10654	0.03062	-0.06613	-0.01101	-0.13635
	RMSE(θ)	0.47686	0.36991	0.38372	0.38573	0.40024	0.41395
	Bias(λ)	3.03506	0.75184	-0.47893	0.35192	-0.03351	1.13225
20	RMSE(λ)	6.65076	3.87665	2.40451	2.99533	2.54652	3.38294
	Dabs	0.07455	0.07906	0.07022	0.07344	0.06681	0.07313
	D max	0.10785	0.11086	0.10262	0.10764	0.10041	0.10593
	Total	346	265	122	193	101	254
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.33016	-0.25893	-0.19451	-0.28174	-0.22432	-0.30435
	RMSE(θ)	0.44316	0.35782	0.35111	0.38494	0.36443	0.39475
	Bias(λ)	4.16776	2.13973	1.18581	2.25194	1.53662	2.50265
50	RMSE(λ)	6.79316	4.42025	2.81211	3.78934	3.01802	3.78733
	Dabs	0.06843	0.07006	0.06632	0.06935	0.06501	0.06884
	D max	0.09813	0.09834	0.09772	0.10256	0.09721	0.10005
	Total	306	233	81	274	112	274
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.40124	-0.33712	-0.33551	-0.40155	-0.34883	-0.41536
	RMSE(θ)	0.45066	0.38231	0.38522	0.43364	0.39393	0.44505
	Bias(λ)	4.85246	3.00863	2.58841	3.73474	2.79012	4.02875
100	RMSE(λ)	6.97206	4.83514	3.48911	4.71153	3.64232	4.90555
	Dabs	0.06793	0.06824	0.06672	0.07105	0.06621	0.07196
	D max	0.09742	0.09621	0.09903	0.10556	0.09904	0.10485
	Total	274	152	101	274	152	326
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.43113	-0.37831	-0.43102	-0.47436	-0.43704	-0.46495
	RMSE(θ)	0.45424	0.40001	0.44412	0.48486	0.44923	0.47475
	Bias(λ)	4.87495	3.35451	4.03762	5.06606	4.17463	4.83984
200	RMSE(λ)	6.45266	4.62852	4.51211	5.72155	4.63153	5.33584
	Dabs	0.06804	0.06753	0.06692	0.07276	0.06651	0.07075
	D max	0.09742	0.09571	0.10054	0.10866	0.10033	0.10355
	Total	244	91	132	356	173	285

• Qtd: measures (52)-(55)

In Tables 6, 7, 8, 9 and 10 we show the calculated values of (52)-(55). The superscript values indicate the rank obtained by each of the methods considered, and the total line shows the global rank for each method based on measures (52)-(55).

Table 6
Simulations results for θ = 1.0 and λ = 0.5.

n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.24895	0.12321	0.20894	0.20132	0.27676	0.20253
	RMSE(θ)	0.32152	0.27761	0.37915	0.36444	0.44016	0.34703
	Bias(λ)	0.81892	0.76081	1.09005	1.06064	1.34726	1.04693
20	RMSE(λ)	1.53443	1.21231	1.67375	1.61904	1.98926	1.53112
	Dabs	0.15106	0.05751	0.06044	0.05913	0.06115	0.05822
	D max	0.26456	0.08941	0.09944	0.09703	0.10535	0.09582
	Total	244	61	275	203	346	152
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.22296	0.05991	0.11184	0.10683	0.14125	0.10682
	RMSE(θ)	0.28106	0.16731	0.22004	0.20933	0.24285	0.20362
	Bias(λ)	0.51272	0.39251	0.57455	0.55604	0.67866	0.55113
50	RMSE(λ)	1.37746	0.74041	0.93404	0.90373	1.03375	0.89142
	Dabs	0.11906	0.03521	0.03694	0.03613	0.03745	0.03582
	D max	0.20466	0.05571	0.06104	0.05943	0.06345	0.05892
	Total	326	61	254	193	315	132
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.18066	0.03051	0.06674	0.06233	0.08275	0.06212
	RMSE(θ)	0.22266	0.11751	0.15144	0.14223	0.16245	0.13942
	Bias(λ)	0.14511	0.21062	0.34125	0.32444	0.39906	0.32073
100	RMSE(λ)	0.98126	0.52231	0.64624	0.62163	0.69465	061402
	Dabs	0.07986	0.02431	0.02574	0.02503	0.02595	0.02492
	D max	0.13356	0.03901	0.04254	0.04123	0.04365	0.04092
	Total	315	71	254	193	315	132
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.14346	0.01151	0.03604	0.03283	0.04505	0.03242
	RMSE(θ)	0.16426	0.08481	0.10644	0.09883	0.11155	0.09742
	Bias(λ)	-0.15002	0.09001	0.18365	0.17084	0.21726	0.16703
200	RMSE(λ)	0.54556	0.38201	0.45974	0.43973	0.48275	0.43512
	Dabs	0.04636	0.01701	0.01804	0.01753	0.01815	0.01742
	D max	0.07576	0.02761	0.02984	0.02893	0.03035	0.02882
	Total	326	61	254	193	315	132

• Qtd: measures (52)-(55)

Table 7
Simulations results for θ = 1.0 and λ = 1.0.

n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.30306	0.05231	0.12253	0.11952	0.19055	0.12844
	RMSE(θ)	0.35875	0.23171	0.30774	0.29643	0.36036	0.28602
	Bias(λ)	0.30351	0.43942	0.77785	0.75593	1.09026	0.76924
20	RMSE(λ)	1.45003	1.12491	1.59845	1.53544	1.95886	1.44862
	Dabs	0.14196	0.05561	0.05854	0.05713	0.05965	0.05672
	D max	0.25886	0.08631	0.09534	0.09313	0.10165	0.09272
	Total	275	71	254	183	336	162
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.27206	0.00591	0.04812	0.04843	0.07895	0.05204
	RMSE(θ)	0.31566	0.14791	0.18204	0.17403	0.19905	0.17072
	Bias(λ)	-0.05621	0.11322	0.29754	0.29343	0.42896	0.30395
50	RMSE(λ)	1.36706	0.71811	0.86984	0.84493	0.96425	0.84012
	Dabs	0.10396	0.03441	0.03614	0.03533	0.03675	0.03522
	D max	0.19006	0.05521	0.05974	0.05833	0.06205	0.05812
	Total	315	71	224	183	315	172
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.22816	-0.01071	0.01782	0.01953	0.03555	0.02094
	RMSE(θ)	0.25446	0.11211	0.13124	0.12403	0.13765	0.12232
	Bias(λ)	-0.47766	-0.01621	0.11192	0.11613	0.18965	0.12004
100	RMSE(λ)	1.05056	0.56021	0.63374	0.61443	0.66905	0.61032
	Dabs	0.06236	0.02451	0.02564	0.02513	0.02595	0.02502
	D max	0.11326	0.04011	0.04274	0.04173	0.04365	0.04162
	Total	366	61	204	183	305	162
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.19646	-0.01585	0.00191	0.00492	0.01234	0.00513
	RMSE(θ)	0.20616	0.08651	0.09834	0.09133	0.10005	0.09062
	Bias(λ)	-0.75456	-0.06635	0.01641	0.02813	0.06334	0.02782
200	RMSE(λ)	0.86986	0.44781	0.48634	0.46543	0.49565	0.46322
	Dabs	0.03406	0.01781	0.01844	0.01803	0.01855	0.01802
	D max	0.06326	0.02941	0.03104	0.03013	0.03135	0.03012
	Total	366	142	184	173	285	131

• Qtd: measures (52)-(55)

Table 8
Simulations results for θ = 1.0 and λ = 2.0.

n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.55616	-0.03063	0.02461	0.03052	0.09505	0.04504
	RMSE(θ)	0.60786	0.20181	0.24524	0.23743	0.28015	0.22782
	Bias(λ)	2.65296	-0.04831	0.34882	0.37453	0.82255	0.41294
20	RMSE(λ)	5.70006	1.25981	1.81604	1.78273	2.29375	1.57612
	Dabs	0.26666	0.05531	0.05764	0.05622	0.05895	0.05623
	D max	0.54056	0.08721	0.09404	0.09192	0.09985	0.09213
	Total	366	81	194	152	305	183
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.54416	-0.04005	-0.00702	0.00151	0.02614	0.00743
	RMSE(θ)	0.59566	0.14291	0.15834	0.14963	0.16425	0.14652
	Bias(λ)	2.46536	-0.19474	0.00871	0.05022	0.20885	0.08063
50	RMSE(λ)	5.52206	0.88871	0.99914	0.95193	1.06615	0.94132
	Dabs	0.23976	0.03601	0.03724	0.03633	0.03765	0.03632
	D max	0.51106	0.05881	0.06194	0.06033	0.06335	0.06032
	Total	366	131	194	153	295	142
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.66816	-0.02975	-0.00873	-0.00122	0.00944	0.00071
	RMSE(θ)	0.70916	0.10622	0.11574	0.10723	0.11645	0.10571
	Bias(λ)	5.76226	-0.15765	-0.03283	0.00771	0.07574	0.01812
100	RMSE(λ)	8.05316	0.67141	0.72384	0.68093	0.73545	0.67442
	Dabs	0.33026	0.02601	0.02684	0.02603	0.02685	0.02602
	D max	0.71446	0.04291	0.04484	0.04343	0.04525	0.04332
	Total	366	152	224	152	285	101
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.78716	-0.01765	-0.00514	-0.00022	0.00413	0.00001
	RMSE(θ)	0.80306	0.07451	0.08195	0.07523	0.08184	0.07472
	Bias(λ)	9.16526	-0.09385	-0.02093	0.00631	0.03384	0.00702
200	RMSE(λ)	10.26636	0.47291	0.50934	0.47573	0.51155	0.47342
	Dabs	0.42146	0.01841	0.01905	0.01843	0.01904	0.01842
	D max	0.90896	0.03051	0.03184	0.03073	0.03195	0.03072
	Total	366	142	254	153	254	111

• Qtd: measures (52)-(55)

Table 9
Simulations results for θ = 1.0 and λ = 3.0.

n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.82476	-0.05894	-0.01042	0.00241	0.06245	0.01793
	RMSE(θ)	0.83386	0.19341	0.22614	0.21563	0.24945	0.20452
	Bias(λ)	9.22016	-0.30153	0.19331	0.27682	0.88755	0.33404
20	RMSE(λ)	11.06006	1.55841	2.33133	2.35134	3.00185	1.95322
	Dabs	0.42866	0.05721	0.05914	0.05773	0.05985	0.05742
	D max	0.91996	0.09071	0.09654	0.09423	0.10105	0.09372
	Total	366	111	184	163	305	152
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.87206	-0.04265	-0.01253	-0.00271	0.01884	0.00302
	RMSE(θ)	0.87396	0.12922	0.14244	0.13233	0.14525	0.12841
	Bias(λ)	11.46986	-0.26745	-0.01951	0.04352	0.23704	0.08253
50	RMSE(λ)	12.40366	0.99861	1.17544	1.08983	1.26365	1.05992
	Dabs	0.48246	0.03682	0.03804	0.03683	0.03815	0.03671
	D max	0.98526	0.05991	0.06304	0.06083	0.06385	0.06052
	Total	366	163	204	152	285	111
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.89326	-0.02565	-0.00673	-0.00021	0.00894	0.00112
	RMSE(θ)	0.89346	0.09052	0.09904	0.09143	0.09975	0.09011
	Bias(λ)	14.34616	-0.16295	-0.01531	0.02702	0.10824	0.03713
100	RMSE(λ)	15.00656	0.70381	0.79084	0.73343	0.81505	0.72402
	Dabs	0.50006	0.02602	0.02695	0.02603	0.02694	0.02601
	D max	0.99856	0.04261	0.04474	0.04313	0.04505	0.04292
	Total	366	163	214	152	275	111
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.90426	-0.01405	-0.00323	0.00072	0.00454	0.00071
	RMSE(θ)	0.90426	0.06271	0.06934	0.06383	0.06965	0.06342
	Bias(λ)	17.45926	-0.08805	-0.00731	0.01862	0.05324	0.01903
200	RMSE(λ)	17.93896	0.49131	0.54734	0.50743	0.55535	0.50402
	Dabs	0.50166	0.01831	0.01905	0.01843	0.01904	0.01842
	D max	0.99976	0.03011	0.03174	0.03053	0.03185	0.03042
	Total	366	142	214	163	275	121

• Qtd: measures (52)-(55)

Table 10
Simulations results for θ = 1.0 and λ = 5.0.

n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.89966	-0.06155	-0.01543	-0.00301	0.05434	0.01232
	RMSE(θ)	0.90016	0.17561	0.20614	0.19733	0.22405	0.18262
	Bias(λ)	12.04836	-0.45182	0.41991	0.58223	1.63825	0.63024
20	RMSE(λ)	13.68936	2.46221	3.87553	4.41874	5.13225	3.32432
	Dabs	0.45726	0.05862	0.06075	0.05913	0.06054	0.05791
	D max	0.99206	0.09251	0.09844	0.09583	0.10115	0.09362
	Total	366	121	204	173	285	132
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.91766	-0.03805	-0.00703	0.00071	0.02034	0.00382
	RMSE(θ)	0.91786	0.10851	0.12404	0.11423	0.12865	0.11002
	Bias(λ)	17.17036	-0.35984	0.09561	0.15942	0.49495	0.18633
50	RMSE(λ)	18.47696	1.29171	1.78764	1.61263	2.03725	1.50732
	Dabs	0.47716	0.03672	0.03835	0.03693	0.03824	0.03661
	D max	0.99926	0.05891	0.06284	0.06023	0.06355	0.05952
	Total	366	142	214	153	285	121
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.92806	-0.02285	-0.00403	0.00111	0.00944	0.00152
	RMSE(θ)	0.92806	0.07501	0.08584	0.07823	0.08725	0.07672
	Bias(λ)	22.52636	-0.22745	0.02851	0.07592	0.21314	0.08033
100	RMSE(λ)	23.46306	0.89131	1.13144	1.01833	1.20095	0.98932
	Dabs	0.47816	0.02571	0.02705	0.02603	0.02704	0.02592
	D max	0.99996	0.04151	0.04444	0.04243	0.04475	0.04212
	Total	366	142	214	153	275	131
n	Qtd	MLE	MPS	OLS	WLS	CM	AD
	Bias(θ)	-0.93966	-0.01315	-0.00223	0.00082	0.00444	0.00051
	RMSE(θ)	0.93966	0.05231	0.06004	0.05443	0.06045	0.05392
	Bias(λ)	31.18416	-0.13675	0.00511	0.03643	0.09434	0.03312
200	RMSE(λ)	31.38096	0.62691	0.76604	0.68663	0.78755	0.67662
	Dabs	0.48626	0.01811	0.01915	0.01833	0.01914	0.01832
	D max	1.00006	0.02931	0.03144	0.02993	0.03155	0.02982
	Total	366	142	214	173	275	111

• Qtd: measures (52)-(55)

In general, the MPS method proved to be the best method to estimate the parameters of the zero truncated Lindley-Poisson distribution to Y = max(T ₁ , ..., T_M ). The MLE method showed the worst results even for the large sample size. For future work, further study the zero truncated Lindley-Poisson distribution to Y = max(T ₁ , ... , T_M ) to understand why the MLE method was not as good would be very relevant.

For λ = 0.5 and 1.0 the MPS method had the highest rank and AD method the second. For λ = 3.0 and 5.0 the AD method was the best, the MPS rank was the second one, only when n = 20 the MPS was better, and for λ = 2.0, the MPS was better for n = 20 and 50 while the AD was better for n = 100 and 200.

5 REAL DATA APPLICATION

In this section we fit the zero truncated Lindley-Poisson distribution (LP) to a real data set. For comparison, we also have considered four alternative models: the one parameter Lindley f (y| θ) = θ ²/1+θ (1 + y) e^{−θ y} , the weighted Lindley distribution (WL)

the exponentiated or generalized Lindley distribution (EL)

and the power Lindley distribution (PL)

The data set was extracted from ^{Lee & Wang (2003}) and refers to remission times (in months) of a randomly censored of 137 bladder cancer patients. Out of 137 data points, 9 observations are right censored. We considered (y ₁ , y ₂ , ..., y_n ) the observed values from Y = min (T ₁ , ... , T_M ).

In Table 11 we present, for all models, the maximum likelihood, maximum product of spacings, ordinary least-squares, weighted least-squares, Crame´r-von-Mises and Anderson-Darling estimates for θ and λ and its respectivally standard errors estimates. The maximum likelihood estimates were obtained in SAS/SEVERITY procedure (^{SAS, 2011}) and others estimates were obtained in R version 2.15, using the “fitdist”, “max.Lik” and “nls” functions. The dotted in Table 11 indicates is not possible to calculate standard errors estimates to the Crame´r-von-Mises and Anderson-Darling methods.

Table 11
Maximum likelihood, Maximum product of spacings, Ordinary least-squares, Weighted least-squares, Crame´r-von-Mises and Anderson-Darling estimates and (standard errors) estimates.

Model	MLE		MPS		OLS		WLS		CM		AD
	θ	λ	θ	λ	θ	λ	θ	λ	θ	λ	θ	λ
L	0.1964		0.1861		0.2290		0.2253		0.2291		0.2315
	(0.0119)		(0.0117)		(0.0014)		(0.0019)		-		-
LP	0.1115	3.1053	0.1020	3.3173	0.1445	2.0188	0.1108	3.6056	0.1435	2.0522	0.1361	2.3210
	(0.0201)	(0.9803)	(0.0199)	(1.1061)	(0.0160)	(0.5222)	(0.0016)	(0.0660)	-	-	-	-
WL	0.1641	0.7200	0.1525	0.6932	0.1942	0.7889	0.1939	0.8115	0.1980	0.8114	0.2265	0.9739
	(0.0172)	(0.1139)	(0.0149)	(0.0978)	(0.0039)	(0.0230)	(0.0042)	(0.0244)	-	-	-	-
EL	0.1688	0.7617	0.1570	0.7404	0.2013	0.8337	0.1993	0.8461	0.2045	0.8512	0.2268	0.9756
	(0.0163)	(0.0923)	(0.0159)	(0.0919)	(0.0032)	(0.0173)	(0.0035)	(0.0186)	-	-	-	-
PL	0.2872	0.8410	0.2841	0.8259	0.2690	0.9055	0.2657	0.9028	0.2651	0.9145	0.2400	0.9745
	(0.0352)	(0.0460)	(0.0360)	(0.0470)	(0.0040)	(0.0082)	(0.0044)	(0.0088)	-	-	-	-

From Table 11, it is observed that all estimation methods were effective to estimate the parameters θ and λ, in addition, the standard errors there were small.

The SAS/SEVERITY procedure can fit multiple distributions at the same time and choose the best distribution according to a specified selection criterion. Seven different statistics of fit can be used as selection criteria. They are log likelihood, Akaike’s information criterion (AIC), corrected Akaike’s information criterion (AICC), Schwarz Bayesian information criterion (BIC), Kolmogorov-Smirnov statistic (KS), Anderson-Darling statistic (AD) and Crame´r-von-Mises statistic (CvM). The calculed values of theses statistics are report in Table 12. In Figure 6 is possible to see similar fit for the five models applied to the data set.

Table 12
2log-likelihood values and goodness of fit measures.

Model	−2 × log lik	AIC	AICC	BIC	KS	AD	CvM
L	895.7115	897.7115	897.7411	900.6314	1.4358	3.0061	0.5537
LP	879.3024	883.3024	883.3920	889.1424	1.3375	332.3061	0.5713
WL	890.9094	894.9094	894.9990	900.7494	11.7211	84085.45	59.2112
EL	890.4974	894.4774	894.5669	900.3173	1.1176	1.5698	0.02796
PL	884.2719	888.2719	888.3615	894.1119	0.8391	0.9585	0.1384

Figure 6
Fitted survival curves.

A close examination of Table 12 reveals that the zero truncated Lindley-Poisson model is the best choice among the competing models, since it has the lowest AIC, AICC and others statistics. This is also supported by the survival curves in Figure 6.

6 CONCLUDING REMARKS

In this paper we proposed the composed zero truncated Lindley-Poisson distribution, which was obtained by compounding an one parameter Lindley distribution with a zero truncated Poisson under the first and last failure time when a device is subjected to the presence of an unknown number M of causes of failures. Both alternative distributions have the one parameter Lindley distribution as a particular case. For the first distribution we assume we have a series system and observe the time to the first failure, Y = min(T ₁ , ... , T_M ) while for the second distribution we assume we have a parallel system and observe the time to the last failure of the device,Y = max(T ₁ , ... , T_M ).

We compared, via intensive simulation experiments, the estimation of parameters of the zero truncated Lindley-Poisson distribution using six known estimation methods, namely: the maximum likelihood, maximum product of spacings, ordinal and weighted least-squares, Crame´r-von Mises and Anderson-Darling.

In general, the methods of estimation showed to be efficient to estimate the parameters of the zero truncated Lindley-Poisson distribution. Motivated by application in real data set, we hope this model may be able to attract wider applicability in survival and reliability. For possible future works, there the interest of authors in studies of the Fisher information matrix, Confidence intervals, Hypothesis test and Bayesian estimates.

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Publication Dates

• Publication in this collection
  Sep-Dec 2016

History

• Received
  23 Jan 2015
• Accepted
  20 Oct 2016