Reliability modeling using the left-truncated logistic distribution

Abstract

This research applies the left-truncated logistic distribution to reliability modeling. The resulting models are applicable when the system components are subject to wearout and their individual time-to-failures can be described by the left-truncated logistic density function. The left-truncated logistic probability density function is the logistic density function truncated to the left of zero. The logistic probability density function closely parallels the normal density function, but its cumulative distribution can be represented as a closed form expression. The logistic distribution graphical representation is virtually indistinguishable from the normal distribution. Reliability functions are derived for various systems where the components are in either series, active series-parallel, active parallel-series, and partial stand-by redundant configurations. The generalized reliability models can be used as prediction tools in system reliability analyses, to aid the system designer. The statistical properties needed for reliability applications are developed. The probability density function, cumulative distribution, reliability function, hazard function, moment generating function, mean, and variance of the left-truncated logistic probability density function are determined.Two procedures to estimate the parameters of the left truncated density function are developed. The maximum likelihood estimates are determined by an iterative procedure. A modified Newton-Ralphson method is used to determine the roots of the non-linear maximum likelihood equations. Another method is developed which yields additional point estimates of the parameters. The second procedure parallels the method of moments technique. Sometimes for modeling purposes, it is desired that the density function have a certain mean and standard deviation. The second method has been modified to determine the parameter values which yield the desired mean and standard deviation values. Tests of hypotheses are developed to make inferences about the population parameters. The sample median is employed to construct a hypothesis test for one of the parameters. The sample range is used as the test statistic in the other hypothesis test for the second parameter.