The logistic–exponential survival distribution

Abstract

For various parameter combinations, the logistic-exponential survival distribution belongs to four common classes of survival distributions: increasing failure rate, decreasing failure rate, bathtub-shaped failure rate, and upside-down bathtub-shaped failure rate. Graphical comparison of this new distribution with other common survival distributions is seen in a plot of the skewness versus the coefficient of variation. The distribution can be used as a survival model or as a device to determine the distribution class from which a particular data set is drawn. As the three-parameter version is less mathematically tractable, our major results concern the two-parameter version. Boundaries for the maximum likelihood estimators of the parameters are derived in this article. Also, a fixed-point method to find the maximum likelihood estimators for complete and censored data sets has been developed. The two-parameter and the three-parameter versions of the logistic-exponential distribution are applied to two real-life data sets.

Kurtosis of the logistic-exponential survival distribution

“…Of course, it is well known that 3 = 2 and 4 = 9 for = 1 (exponential distribution). It was furthermore proved by Lan and Leemis (2008) that 3 # 2:1126 and 4 # 8:6876 as # 0 and that 3 # 0 and 4 # 4:2 as ! 1 (note that the logistic distribution has 3 = 0 and 4 = 4:2).…”

“…As discussed by Lan and Leemis (2008), it is the only curve that is bounded and hence it is the only curve for which maximum values for 3 and 4 are achieved. In the next two sections the kurtosis properties of the logistic-exponential distribution are studied.…”

“…The logistic-exponential distribution, introduced by Lan and Leemis (2008), is a useful model in survival analysis, since it encompasses failure rates that are increasing, decreasing, bathtub-shaped and upside-down bathtub-shaped, and because its cumulative distribution function, F (x), probability density function, f (x), and quantile function, Q(u), all have closed-form expressions. This survival distribution has a two-parameter version with scale parameter > 0 and shape parameter > 0, and a three-parameter version which includes a location parameter, > 0.…”

“…Likewise the logistic-exponential distribution and the loglogistic distribution tend to the logistic distribution as the values of their shape parameters tend to in…nity -see Lan and Leemis (2008) and Tadikamalla and Johnson (1982) respectively for details.…”

Inverse Rayleigh Software Reliability Growth Model

“…These assumptions determine the form of the model and the meaning of the model's parameters. Some recent works in this regard are by Akaike(1974) [12]. With this backdrop, we study the modeling of software reliability as a Non Homogenous Poisson Process (NHPP) with mean value function based on inverse Rayleigh distribution.…”