Statistical Inference for Middle Censored Data with Applications

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Statistical Inference for Middle Censored Data with Applications

Introduction to Middle Censoring

Middle censoring is a unique censoring scheme where the actual failure data of an observation becomes unobservable if it falls within a random interval. This type of censoring is particularly relevant in various fields such as medical research and reliability engineering, where certain data points may be obscured due to practical constraints.

Maximum Likelihood Estimation (MLE) for Middle Censored Data

Dependent Middle Censoring

In scenarios where middle censoring is dependent, the lifetime and the lower bound of the censoring interval are variables that follow a Marshall-Olkin bivariate exponential distribution. The maximum likelihood estimates (MLEs) of the unknown parameters can be derived using iterative methods. This approach is crucial for accurately estimating parameters in dependent setups.

Competing Risks Model

For dependent competing risks models, the Marshall-Olkin bivariate Weibull (MOBW) distribution is used. The MLEs, midpoint approximation (MPA) estimations, and approximate confidence intervals (ACIs) of the unknown parameters are obtained. This method provides a comprehensive framework for handling middle-censored data in competing risks scenarios.

Discrete Middle Censoring

When dealing with discrete middle-censored data, where the lifetime, lower bound, and length of the censoring interval follow a geometric distribution, the likelihood function of the observed data is derived. The MLE of the unknown parameter is obtained using the Expectation-Maximization (EM) algorithm, which is effective for discrete data scenarios.

Bayesian Estimation for Middle Censored Data

Gamma Priors and Squared Error Loss

Bayesian estimation methods are also applied to middle-censored data. For dependent middle censoring, Bayes estimates of the parameters are proposed under gamma priors and the squared error loss function. Monte Carlo simulations are used to compare these estimators, providing a robust framework for Bayesian inference.

Gamma-Dirichlet Prior

In the context of dependent competing risks models, the Bayes approach is considered based on the Gamma-Dirichlet prior of the scale parameters, with a given shape parameter. The acceptance-rejection sampling method is used to obtain Bayes estimations and construct credible intervals (CIs), offering a reliable Bayesian inference method.

Exponential Lifetime Distributions

For middle-censored data with exponential lifetime distributions, the Bayes estimate of the exponential parameter is derived under a gamma prior. Gibbs sampling is proposed to construct credible intervals, which is particularly useful when theoretical construction becomes difficult.

Nonparametric and Self-Consistent Estimators

Self-Consistent Estimator (SCE)

Nonparametric methods provide the self-consistent estimator (SCE) and the nonparametric maximum likelihood estimator (NPMLE) for middle-censored data. An algorithm is provided to find the SCE, and sufficient conditions for its consistency are established. This method is validated through simulation results and real data examples, demonstrating its practical applicability.

Applications and Practical Implications

Real Data Analysis

The methods discussed are applied to real data sets to illustrate their practical applications. For instance, the analysis of Danish melanoma data set using nonparametric methods showcases the effectiveness of these techniques in real-world scenarios. Similarly, the analysis of a real data set with exponential lifetime distributions highlights the practical utility of the proposed Bayesian methods.

Conclusion

Statistical inference for middle-censored data involves a variety of methods, including maximum likelihood estimation, Bayesian estimation, and nonparametric approaches. These methods are essential for accurately analyzing data where certain observations are obscured within random intervals. The practical applications of these methods in real data scenarios underscore their importance in fields such as medical research and reliability engineering.