Analyzing lifetime data with long-tailed skewed distribution: the logistic-sinh family

A new two-parameter family of distribution is presented. It is derived to model the highly negatively skewed data with extreme observations. The new family of distribution is referred to as the logistic-sinh distribution, as it is derived from the logistic distribution by appropriately replacing an exponential term with a hyperbolic sine term. The resulting family provides not only negatively skewed densities with thick tails but also variety of monotonic density shapes. The space of shape parameter, lambda greater than zero is divided by boundary line of lambda equals one, into two regions over which the hazard function is, respectively, increasing and bathtub shaped. The maximum likelihood parameter estimation techniques are discussed by providing approximate coverage probabilities for uncensored samples. The advantages of using the new family are demonstrated and compared by illustrating well known examples.