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

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