The lognormal and inverse Gaussian distributions as lifetime models

Abstract

The purpose of this dissertation is to investigate the statistical and distributional properties of the Inverse Gaussian and the Lognonnal distributions which are both appropriate when analyzing right skewed data. Moreover, it investigates how the Inverse Gaussian distribution can be applied to lifetime problems, describing failure rate and mean residual lifetimes and derive maximum likelihood estimates for these measures. The thesis explains why the Inverse Gaussian distribution is more appropriate than the Lognonnal distribution when the failure rate decreases to a non-zero asymptotic value. The dissertation also provides a description of the theoretical framework of generalized linear models (GLMs) and explains in detail maximum likelihood estimation, which is equivalent to reweighted least squares estimation. GLMs accommodate any distribution that is a member of the exponential family including the Inverse Gaussian and the Lognormal distributions. Furthermore, a number of diagnostic measures for GLMs are described and an explanation is provided how they identify outliers, influential observations and model oddities. This dissertation presents two models for analyzing right skewed data. The responses are repair times of airborne communication transceivers which comprise four components (the transmitter-receiver block, the control box, the audio switch box and the AF block). The four predictors are categorical variables identifying which components are defective and which are not. The EasyFitXL software is used to identify the best distributions for the repair times and two contender models are identified. The first model assumes an Inverse Gaussian distribution and an inverse quadratic link function; while the second model is a Lognormal regression model, which assumes a Normal distribution and an identity link function to log-transformed repair times. By using a forward procedure the two parsimonious models are identified and compared. Diagnostic tools are also used to detect anomalous data points and other model misspecifications.