We study maximum likelihood estimation in the three-parameter $r$-largest generalized extreme-value ($r$GEV) model with a fixed number $r$ of upper order statistics per block. Because the support depends on all three parameters, the classical regular theory does not apply. We show that the maximum likelihood estimator is strongly consistent on compact parameter sets contained in $k<1$, where $k=-\gamma$ denotes the shape parameter. For $k_0<1/2$, the range in which the model is differentiable in quadratic mean, the estimator is root-$n$ consistent, asymptotically normal and asymptotically efficient. The proof verifies the conditions of the moving-support likelihood theorem of B\"ucher and Segers for the ordered $r$-dimensional likelihood; the main new ingredients are a sequential modified likelihood criterion and a support-loss bound showing that the regularity threshold $k_0<1/2$ does not depend on $r$. In simulated heavy-tailed cases with only $50$ blocks, using five to seven order statistics instead of the block maximum reduces the relative bias of the 100-year return level from $6$--$13\%$ to below $5\%$.
The $r$ largest observations of a block form a dependent random vector whose support moves with all three parameters of the $r$-largest generalized extreme-value ($r$GEV) model. For a fixed number $r$ of order statistics, we show that the model is differentiable in quadratic mean if and only if $k_0<1/2$, where $k=-\ga...
Jihong Park, Yi-Re Shin, Jeong-Soo Park· 1 citation
We study the maximum likelihood estimation of the coefficient {\beta} in well-specified Poisson regression. Using tools from empirical process theory and random conic geometry, we show that the probability of existence of the maximum likelihood estimator (MLE) exhibits a sharp phase transition at the threshold n>d. We...
This work addresses a longstanding gap in the statistical foundations of marginal maximum likelihood estimation for high-dimensional latent variable models. Marginal maximum likelihood estimation is widely used to fit latent variable models across the social sciences, ecology, and machine learning. Despite its broad us...
We extend the Ibragimov-Khasminskii likelihood ratio analysis method, in the one-dimensional parameter case, to a framework based on the Skorokhod $M\_1$ topology. The proposed framework applies to a broad class of statistical models, including those for which the normalized likelihood ratio processes are continuous wh...
We study the asymptotic behaviour of the solutions to the unconstrained generalized variational inference problem regularized by an $\alpha$-divergence. Under mild conditions, the minimizer is explicit up to a scalar normalizing constant, and we show that it admits a rescaling under which it converges in $L^1$ to a fix...
Hermite and Laguerre $\beta$-ensembles are important and well studied models in random matrix theory, with the special cases $\beta=1,2,4$ corresponding to classical matrix ensembles. Although sharp moderate-deviation estimates have been established for the largest eigenvalues, substantially less is known for other edg...
Jnaneshwar Baslingker· 0 citations
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