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All reference priors for the parameter K when b is known and for ðb, KÞ when b is unknown with corresponding triplets ðl 1 , l 2 , l 3 Þ:
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The assumption of independence of causes for modeling a competing risks scenario is not presumable always. In this paper, cause dependent competing risks model has been analyzed under Marshall-Olkin set-up. A Marshall-Olkin generalized lifetime distribution has been established to address a competing risks model. The essential statistical propertie...
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... reference prior 'nð:Þ' defined in (4.1) and (4.3) can take different form of reference priors according to triplet ðl 1 , l 2 , l 3 Þ where, l i ¼ 0 or 1; i ¼ 1, 2, 3: All considered reference priors with corresponding triplet indicators are given in Table 5 for both the case when the parameter b is known and when it is unknown. ...Context 2
... posterior analysis of the parameters can be done by using the conditional posterior distributions for different reference priors based on values of triplet ðl 1 , l 2 , l 3 Þ: We use Gibbs sampling method, a Markov chain Monte Carlo (MCMC) technique Chen, Shao and Ibrahim (2000), to obtain the Bayesian estimator of the parameters. A step-by-step procedure is defined for obtaining the samples (estimates) for all parameters in both the cases and for all considered priors in Table 5. ...Context 3
... As in Table 5, we have three reference priors, x 11 , x 12 and x 13 with l 3 ¼ 0: So, in all three priors, we get exact gamma distribution to generate a sample for k 1 , ...Context 4
... is easy to see that all three reference priors x 11 , x 12 and x 13 in Table 5 have different set of l 1 and l 2 . So, in respect to each reference prior a unique procedure needed to apply. ...Context 5
... To obtain a posterior sample for b from (4.7), we use the Metropolis-Hastings (M-H) algorithm by considering normal distribution as a proposal density, such as the candidate point b c $ N ð ^ b, Varð ^ bÞÞ: As in Table 5, we consider four different reference priors, x 21 , x 22 , x 23 and x 24 , for unknown b. As for reference priors x 21 and x 24 , we have l 3 ¼ 0 and hence a sample can be generated directly for k 1 from Gðn, P n j¼1 gðt j Þ Â Ã b Þ: But for the reference prior x 22 and x 23 , where l 3 ¼ 1, we use M-H algorithm using gamma distribution as a proposal density such as candidate point k c 1 $ Gðn, ...Citations
... In analysis, dependent competing risks may be described using various methods, with a common approach being the application of bivariate and multivariate models. For more insight, one can see the work of Marshall and Olkin [11], Basu and Ghosh [12], Samanta and Kundu [14], and Barnwal and Panwar [13]. In most conventional analyses, the failure times X of items are recorded as the first instance of the different cause of failures, i.e. ...
In this article, inference for a competing risk model is discussed when the cause of failures of units is dependent on using the maximum ranked set sampling (MaxRSS) approach. When the latent failure times follow the bivariate generalized Rayleigh distributions, inference for unknown parameters is obtained using frequentist and Bayesian approaches. Maximum likelihood estimators of the unknown parameter and its existence and uniqueness are also provided. Subsequently, approximate confidence intervals have also been constructed based on the observed Fisher information matrix. In competing risk, it is obvious that one risk is at a higher risk than the other, the MLE of the parameter has been discussed under re-parameterization of the parameter. Further, Bayes estimates have been discussed using informative and non-informative (Jeffrey's and reference) prior under the squared error loss function. Moreover, the associated highest posterior density credible interval is also developed. The performance of various estimators is evaluated based on extensive simulation study, and comments are obtained. Finally, two different real-life applications are also provided for illustrative purposes.
... Marshall and Olkin [24] introduced a popular bivariate exponential lifetime distribution from the perspective of the shock model, and such Marshall-Olikin bivariate model was later extended by many authors with different baseline distributions such as Weibull, Kumaraswamy, Gompertz, generalized distributions among others (e.g. [7,18,35,36], etc). ...
... From Samanta and Kundu [28], it can be seen that out of these 71 data points, there are 28 failures occurred due to cause 1, 33 failures occurred due to cause 2, and about 10 failures, the cause of failure couldn't be observed. Barnwal and Panwar [7] also analyzed this data using a generalized Marshall-Olikin bivariate distribution. ...
... The listed probability models indicate that GLD family can have variety of hazard rate functions such as increasing, decreasing, constant and bathtubshaped behavior. Barnwal and Panwar (2022) discuss important properties of this family which are useful in different inference problems. Remark 1 Here, we considered different cases, like when β and α are unknown and when β is unknown but α is known. ...
... So, this data set can be used to obtain inferences related to a competing risk problem with two failure causes. Barnwal and Panwar (2022) also analyzed this data set using Marshall-Olikin bivariate generalized lifetime distribution and obtained useful estimates for unknown quantities. We also study this data in the context of the considered problem. ...
We make inference for a competing risks model under the assumption that observations are left-truncated and right-censored and failure causes are partially observed. When the latent failure times follow a generalized family of distributions, inference for unknown parameters is provided using classical and Bayesian approaches. Particularly existence-uniqueness properties of maximum likelihood estimators are established. Subsequently interval estimators are constructed based on observed Fisher information matrix. Bayes estimates and associated highest posterior density intervals are developed using gamma-beta prior distributions by considering squared error loss function. We also study estimation problem when parameters are order restricted. The performance of all estimators is evaluated based on an extensive simulation study and comments are obtained. A real data set is also analyzed for illustration purposes.
... D. Proof of Theorem 5 When n 1 < n 2 , the conditional MLEs of β k , k = 1, 2, 3 for given λ can be obtained as the ones given in Theorem 3 under the order restriction β 1 < β 2 . Whereas when n 1 ≥ n 2 , the conditional MLEs from Theorem 3 do not satisfy the order restriction condition, and the likelihood function ℓ(β 1 , β 2 , β 3 , α) given in (9) is maximized in the line β 1 = β 2 = β and β 3 > 0 under order restriction β 1 < β 2 . In this manner, for given α > 0, the log-likelihood function of β 3 and β can be expressed as ℓ(β 3 , β, α) ∝ n ln α + n 12 ln β + n 3 ln β 3 − α ...
Ranked set sampling (RSS) has been proved to be an efficient sampling design for parametric and nonparametric inference. This paper explores inference for a maximum RSS procedure with unequal samples (MRSSU) with multiple dependent failure causes. When the lifetimes of units are characterized by a proposed complementary competing risks model, classical likelihood and Bayesian approaches are discussed for parameters and reliability estimation. Maximum likelihood estimators of model parameters and associated existence and uniqueness are established, and approximate confidence intervals are constructed using asymptotic theory and delta methods. With respect to general flexible priors, Bayesian estimates of interested quantities are also performed and a Monte Carlo sampling algorithm is proposed for complex posterior computation. Additionally, when extra historical information between the competing risks parameters is available, likelihood and Bayesian estimation are also studied under an order restriction case. Different methods are compared based on extensive simulation studies, and another real data example is demonstrated for application propose.
... While Zhang et al. (2022) obtained the system reliability under the Bayesian paradigm for a multicomponent stressstrength model assuming Marshall-Olkin bivariate Weibull distribution. Recently, Barnwal and Panwar (2022) considered the Marshall-Olkin bivariate generalized lifetime family of distribution to model the dependency between competing risks and performed the classical and Bayesian inference with reference priors derived for different ordered grouping of the parameters of interest. Also, Kundu and Gupta (2013) performed the Bayes estimation of the parameters of Marshall-Olkin bivariate Weibull distribution considering the Gamma prior for scale parameters. ...
In competing risks problem, a subset of risks is needed more attention for inferential purposes. In the objective Bayesian paradigm, reference priors enable to achieve such inferential objectives. In this article, the Marshall-Olkin bivariate Weibull distribution is considered to model the competing risks data. In the availability of partial information for some of the parameters, the reference priors are derived as per the importance of the parameters. The Dirichlet prior is taken as a conditional subjective prior and the marginal reference prior has been derived. Also, the propriety of the resulting posterior density has been proved. The Bayesian estimates of the parameters are obtained under squared error and linear-exponential loss functions. Further, the derived reference prior is used for the computation of Bayes factors or posterior odds in testing the hypothesis that the competing risks are identical. The performance of established Bayesian estimators is illustrated using the Diabetic Retinopathy Study (DRS) and Prostate Cancer data sets. Finally, the model compatibility is done for the considered datasets under Bayesian Paradigm.
In this paper, we consider the problem of dependent censoring models with a positive probability that the times of failure are equal. In this context, we propose to consider the Marshall-Olkin type model and studied some properties of the associated survival copula in its application to censored data. We also introduce estimators for the marginal distributions and the joint survival probabilities under different schemes and showed their asymptotic normality under appropriate conditions. Finally, we evaluate the finite-sample performance of our approach relying on a small simulation study on synthetic data, and an application to real data.
The statistical inference under competing risks model is of great significance in reliability analysis and it is more practical to assume that they have dependent competing causes of failure in actual situations. In this article, we make inference for unknown parameters of a Marshall-Olkin bivariate Kumaraswamy distribution under adaptive progressive hybrid censoring mechanism. The maximum likelihood estimations of the unknown parameters are derived, and the Fisher information matrix is then employed to construct asymptotic confidence intervals. Bayes estimates are evaluated against squared error and linex loss functions assuming ordered Gamma-Dirichlet and Gamma-Dirichlet prior distributions for order restriction and without order restriction cases respectively. The Metropolis-Hasting and Lindley techniques are applied to acquire the estimates of all unknown parameters. A thorough simulation analysis is demonstrated to assess the performance of the supplied approaches across various sample sizes. The usefulness of the techniques is illustrated using real engineering data to prove their versatility in practical applications.
