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

For a sample \(X_1, X_2,\ldots X_N\) of 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 \(x_L>0\) is introduced. Due to scaling properties it is sufficient to investigate the case \(x_L=1\). Here the corresponding maximum likelihood equations for a normalised sample (i.e. a sample divided by \(x_L\)) do not always possess a solution. A simple criterion guarantees the existence of a solution: Let \(\mathbb {E}(\cdot )\) denote the expectation induced by the normalised sample and denote by \(\beta _0=\mathbb {E}(\ln {X})^{-1}\), the inverse value of expectation of the logarithm of the sampled random variable X (which is greater than \(x_L=1\)). If this value \(\beta _0\) is bigger than a certain positive number \(\beta _C\) then a solution of the maximum likelihood equation exists. Here the number \(\beta _C\) is the unique solution of a moment equation,\(\mathbb {E}(X^{-\beta _C})=\frac{1}{2}\). 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 \(L^1\)-limit of the degenerated left-truncated log-logistic distribution, where \(L^1(\mathbb {R}^+)\) 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.