Tag Archives: Weibull Distribution

Estimation and Forecasting Age Groups wise Survival of CABG Patients (Kalman Filter Smoothing Approach) (Published)

In this paper we present a new approach (Kalman Filter Smoothing) to estimate and forecast, Age Groups wise, survival of Coronary Artery Bypass Graft Surgery (CABG) patients. Survival proportions of the patients are obtained from a lifetime representing parametric model (Weibull distribution with Kalman Filter approach). Moreover, an approach of complete population (CP) from its incomplete population (IP) of the patients with 12 years observations/ follow-up is used for their survival analysis [23]. The survival proportions of the CP obtained from Kaplan Meier method are used as observed values at time t (input) for Kalman Filter Smoothing process to update time varying parameters. In case of CP, the term representing censored observations may be dropped from likelihood function of the distribution. Maximum likelihood method, in-conjunction with Davidon-Fletcher-Powell (DFP) optimization method [8] and Cubic Interpolation method is used in estimation of the survivor’s proportions. The estimated and forecasted, Age Groups wise survival proportions of CP of the CABG patients from the Kalman Filter Smoothing approach are presented in terms of statistics, survival curves, discussion and conclusion.

Keywords: CABG Patients, Complete and Incomplete populations, DFP method, Estimation and Forecasting of Survivor’s Proportions., Kalman Filter, Maximum Likelihood method, Weibull Distribution

Bayesian Inferences for Two Parameter Weibull Distribution (Published)

In this paper, Bayesian estimation using diffuse (vague) priors is carried out for the parameters of a two parameter Weibull distribution. Expressions for the marginal posterior densities in this case are not available in closed form. Approximate Bayesian methods based on Lindley (1980) formula and Tierney and Kadane (1986) Laplace approach are used to obtain expressions for posterior densities. A comparison based on posterior and asymptotic variances is done using simulated data. The results obtained indicate that, the posterior variances for scale parameter  obtained by Laplace method are smaller than both the Lindley approximation and asymptotic variances of their MLE counterparts. 

Keywords: Laplace Approximation, Lindley Approximation, Maximum Likelihood Estimates, Weibull Distribution

Impact Assessment of the Logarithm Transformation on the Error Component of the Multiplicative Error Model (Published)

In this study we examine the implication of logarithm transformation on the two most popular distributions (Gamma and Weibull) of the error component of the multiplicative error model. The kth moment ( k =1, 2, 3, …) of the logarithm transformed Gamma distribution was established while that of the log-transformed Weibull distribution was found not to be solvable in its closed form hence further investigations were limited to the Gamma distributed error component. The mean of the log-transformed Gamma distribution as required in statistical modeling was found to exist for while its variance exits for . However using simulations the region for successful application of log-transformed distribution was found to be . Furthermore, it was discovered that the log-transform led to a significant reduction of the variance of the distribution, however the expected zero-mean assumption after linearing a multiplicative model with a logarithm transformation is not met even though there were decreases in the mean values after the transformation.

Finally as a result of the findings of this study, we recommend in statistical modeling, that logarithm transformation is not appropriate in a multiplicative error model (with a unit mean error component) for either linearizing or stabilizing the variance of the model or both since it leads to a distribution whose kth moment ( k = 1, 2, 3, . .) is not solvable in a closed form (for the Weibull distrib


Keywords: Gamma Distribution, Logarithm Transformation, Moments, Multiplicative Error Model, Weibull Distribution