Statistical analysis of dependent competing risks model from Gompertz distribution under progressively hybrid censoring

Previous studies have mostly considered the competing risks to be independent even when the interpretation of the failure modes implies dependency. This paper studies the dependent competing risks model from Gompertz distribution under Type-I progressively hybrid censoring scheme. We derive the maximum likelihood estimations of the model parameters, and then the asymptotic likelihood theory and Bootstrap method are used to obtain the confidence intervals. The simulation results are provided to investigate the effects of different dependence structures on the estimations of parameters. Finally, one data set was used for illustrative purpose.

number of corresponding works have been devoted to the dependent competing risks model. Zheng and Klein (1995) considered the dependence structure between failure modes is represented by an assumed Archimedean copula. Other works see Escarela and Carriere (2003); Kaishev et al. (2007).
In this paper, we present a dependent competing risks model from Gompertz distribution under Type-I progressively hybrid censoring scheme (PHCS). The Gompertz distribution is one of classical mathematical models and was first introduced by Gompertz (1825), which is a commonly used growth model in actuarial and reliability and life testing, and plays an important role in modeling human mortality and fitting actuarial tables and tumor growth. This distribution has been widely used, see, Ali (2010); Ghitany et al. (2014).
The Type-I PHCS was first proposed by Kundu and Joarder (2006) [see also Childs et al. (2008)]. This censoring scheme has been widely used in reliability analysis, see, Chien et al. (2011); Cramer and Balakrishnan (2013). It can be defined as follows: suppose n identical units are put to life test with progressive censoring scheme (r 1 , r 2 , . . . , r m ), 1 ≤ m ≤ n, the experiment is terminated at time τ, where τ ∈ (0, ∞), r i (i = 1, · · · , m) and m are fixed in advance. At the time of the first failure t 1 , r 1 of the remaining units are randomly removed, at the time of the second failure t 2 , r 2 of the remaining units are randomly removed and so on. If the mth failure time t m occurs before time τ, all the remaining units R * m = n − m − (r 1 + · · · + r m−1 ) are removed and the terminal time of the experiment is t m . On the other hand, if the mth failure time t m does not occur before time τ and only J failures occur before time τ, where 0 ≤ J ≤ m. Then all the remaining units R * J = n − J − (r 1 + · · · + r J ) are removed and the terminal time of the experiment is τ. We denote the two cases as Case I t 1 < t 2 < · · · < t m , if t m < τ Case II t 1 < t 2 < · · · < t J < τ < t J +1 < · · · < t m , if t m > τ The rest of the paper is organized as follows. "Model description" section provides the model description, "Maximum likelihood estimations (MLEs)" section presents the maximum likelihood estimations of the model parameters. The confidence intervals are provided in "Confidence intervals" section. "Simulation and data analysis" section presents the simulation and data analysis. Conclusion appears in "Conclusion" section.

Model description
It is assumed that the Gompertz distribution with shape parameter λ and scale parameter θ has the following probability density function (PDF), cumulative distribution function (CDF) and survival function respectively, where t > 0, > 0, θ > 0. We denote the Gompertz distribution by GP( , θ).

Theorem 1
The joint survival function of (T 1 , T 2 ) is Proof Corollary 1 The joint PDF of (T 1 , T 2 ) can be written as Proof For the cases t 1 > t 2 and t 1 < t 2 , f 1 (t 1 , t 2 ), f 2 (t 1 , t 2 ) can be easily obtained by − ∂ 2 S T 1 , T 2 (t 1 ,t 2 ) ∂t 1 ∂t 2 . For the case t 1 = t 2 = t, by the full probability formula, we have the fact that

Competing risks model
Consider two competing failure modes with latent lifetimes T 1 , T 2 in the experiment under Type-I PHCS, the failure of an individual is caused by any single one of the two failure modes, obviously, the actual lifetime span is X = min(T 1 , T 2 ). Let r denotes the number of failures that occur before time τ, τ* denotes the terminal time. Then, at time all the remaining R * r = n − r − r l=1 r l units are removed and the experiment is terminated, where r = m, τ * = t r , r m = 0 in Case I and r = J, τ * = τ in Case II.

Maximum likelihood estimations (MLEs)
The likelihood function for the two competing risks model under Type-I PHCS can be written as where So the likelihood function can be written as �� .
By setting the first partial derivative of log L about θ 0 , θ 1 , θ 2 , to zero, we get From (7), (8) and (9), the estimates of θ j , j = 0, 1, 2 are given by Substituting θ j ( ) into log L and ignoring the constant, we obtain the profile log-likelihood function of λ as , which implies that the second derivative of g( ) is negative, so g( ) is concave. □ From Lemma 1, we know that g( ) is unimodal and it has a unique maximum. Since g( ) is unimodal, most of the standard iterative procedure can be used to find the MLE.
So we propose to use the following simple algorithm. Substituting θ j ( ) into (10), the MLE ˆ of satisfies the following equation, Using the method of a simple iterative scheme proposed in the literature by Kundu (2007), we can solve the shape parameter from (13). Start with an initial guess of , say (0) , then obtain (1) = h( (0) ) and proceed in this way to obtain (n+1) = h( (n) ). Stop the iterative procedure when (n+1) − (n) < ε, some pre-assigned tolerance limit. Once we obtain ˆ , the MLEs of θ j , j = 0, 1, 2 can be obtained from (11) as θ j , j = 0, 1, 2.

Observed fisher information
In this section, we will construct the asymptotic confidence intervals (ACIs) for the parameters θ 0 , θ 1 , θ 2 , using the asymptotic likelihood theory. The observed Fisher information matrix is denoted by where the elements of which are negative second partial derivatives of log L.
a2. We obtain the failures r before time τ and the terminal time τ * .

Simulation
In this section, we presented some simulation results to evaluate the performance of all the methods proposed in the previous sections for different sample size n, different effective sample size m and different dependence structure θ 0 .
Repeat 10,000 times for each given n, m, θ 0 and censoring scheme, the average mean squared errors (MSEs) and the average absolute relative bias (RABias) and the coverage percentage of the ACIs and Boot-P CIs are shown in Tables 1, 2 and 3. From Tables 1, 2 and 3, the observations can be made. For fixed sampling scheme, sample size n and dependence structure θ 0 , the MSEs and RABias decrease as the effective sample size m increase.
For fixed sampling scheme, sample size n and effective sample size m, as the dependence structure of competing failure modes become stronger, the MSEs and RABias get smaller, while the MSEs and RABias with θ 0 = 0 are bigger, which shows that the performance of the MLEs depends on the strength of dependence. This also shows that the dependence structure is very important in the competing risks model. For fixed sampling scheme, n, m and dependence structure θ 0 , the ACIs are stable than the Boot-P CIs, they can maintain their coverage percentages at the pre-fixed normal level.

Data analysis
Using the procedures above, we generate the Type-I PHC samples when (n, m, τ ) = (30, 10, 1) with initial value for parameters (θ 1 , θ 2 , ) as (1.2, 1, 0.6), and the dependence structure θ 0 = 0.8, the censoring scheme as r 1 = r 2 = · · · = r m = 2. The simulated data is listed in Table 4. The MLEs and 95 % ACIs and Boot-P CIs are shown in Table 5. The trace plot of the MLE for parameter using the iterative procedure is shown in Fig. 3, which shows that the estimate of converges to a value after about 1000 iterations.

Conclusion
This paper proposed the dependent competing risks model from Gompertz distribution under Type-I PHCS. We obtained the MLEs and ACIs and Boot-P CIs for the parameters. Simulations showed that the ACIs are more stable than the Boot-P CIs and that the dependence structure is important in the competing risks model. For a given sample size, the performance of the MLEs declined with increasing dependence, which suggests that greater dependence will require a larger sample size to achieve a particular level of precision in estimation.    Fig. 3 Trace plot of MLE for λ using the iterative procedure