Maximum likelihood estimation for left-truncated log-logistic distributions with a given truncation point

For a sample X1,X2,…XNof independent identically distributed copies of a log-logistically distributed random variable X the maximum likelihood estimation is analysed in detail if a left-truncation point xL>0is introduced. Due to scaling properties it is sufficient to investigate the case xL=1. Here the corresponding maximum likelihood equations for a normalised sample (i.e. a sample divided by xL) do not always possess a solution. A simple criterion guarantees the existence of a solution: Let E(·)denote the expectation induced by the normalised sample and denote by β0=E(lnX)-1, the inverse value of expectation of the logarithm of the sampled random variable X (which is greater than xL=1). If this value β0is bigger than a certain positive number βCthen a solution of the maximum likelihood equation exists. Here the number βCis the unique solution of a moment equation, E(X-βC)=12. In the case of existence a profile likelihood function can be constructed and the optimisation problem is reduced to one dimension leading to a robust numerical algorithm. When the maximum likelihood equations do not admit a solution for certain data samples, it is shown that the Pareto distribution is the L1-limit of the degenerated left-truncated log-logistic distribution, where L1(R+)is the usual Banach space of functions whose absolute value is Lebesgue-integrable. A large sample analysis showing consistency and asymptotic normality complements our analysis. Finally, two applications to real world data are presented.