A Generalized Continuous Bernoulli Distribution: Statistical Properties, Methods of Estimation, and Applications

This paper introduces a generalized continuous Bernoulli distribution based on the Marshall–Olkin technique for generating new distributions. We refer to the proposed distribution as the Marshall–Olkin continuous Bernoulli (MOCB) distribution. Useful statistical properties and mathematical expressions of the seven methods of parameter estimation for the unknown parameters of the MOCB distribution, including the maximum likelihood, ordinary least squares, weighted least squares, maximum product spacing, percentile, Anderson–Darling, and Cramér–von Mises estimators, are derived. The asymptotic behavior of the unknown parameter estimates of the MOCB distribution using these methods revealed that the maximum product spacing method provides a better estimate of the parameters of the MOCB distribution. Furthermore, the applicability of the MOCB distribution in practical data fitting is illustrated using two real-life datasets. Results obtained from the fitting of the two datasets suggest that the proposed generalized continuous Bernoulli distribution based on the Marshall–Olkin scheme offers a better fit than those based on the power and transmuted transformations. In particular, the likelihood ratio test (LRT) results for the two datasets indicate a significant improvement in the continuous Bernoulli distribution via the Marshall–Olkin scheme.