This book relies on maximum likelihood (ML) estimation of parameters. Asymptotic theory assumes regularity conditions hold when the ML estimator is consistent. Typically an additional third derivative condition is assumed to ensure that the ML estimator is also asymptotically normally distributed. Standard asymptotic results that then hold are summarized in this chapter; for example, the asymptotic variance of the ML estimator is then given by the Fisher information formula, and the log-likelihood ratio, the Wald and the score statistics for testing the statistical significance of parameter estimates are all asymptotically equivalent. Also, the useful profile log-likelihood then behaves exactly as a standard log-likelihood only in a parameter space of just one dimension. Further, the model can be reparametrized to make it locally orthogonal in the neighbourhood of the true parameter value. The large exponential family of models is briefly reviewed where a unified set of regular conditions can be obtained.
We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l
n
between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l
n
/ n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.
The choice of a probability distribution function and confidence interval of estimated design values have long been of interest in flood frequency analysis. Although the four-parameter exponential gamma (FPEG) distribution has been developed for application in hydrology, its maximum likelihood estimation (MLE)-based parameter estimation method and asymptotic variance of its quantiles have not been well documented. In this study, the MLE method was used to estimate the parameters and confidence intervals of quantiles of the FPEG distribution. This method entails parameter estimation and asymptotic variances of quantile estimators. The parameter estimation consisted of a set of four equations which, after algebraic simplification, were solved using a three dimensional Levenberg-Marquardt algorithm. Based on sample information matrix and Fisher’s expected information matrix, derivatives of the design quantile with respect to the parameters were derived. The method of estimation was applied to annual precipitation data from the Weihe watershed, China and confidence intervals for quantiles were determined. Results showed that the FPEG was a good candidate to model annual precipitation data and can provide guidance for estimating design values
We study optimal estimation of a signal in parametric neuronal models on the basis of interspike interval data. Fisher information is the inverse asymptotic variance of the best estimator. Its dependence on the parameter value indicates accuracy of estimation. Our models assume that the input signal is estimated from neuronal output interspike interval data where the frequency transfer function is sigmoidal. If the coefficient of variation of the interspike interval is constant with respect to the signal, the Fisher information is unimodal, and its maximum for the most estimable signal can be found. We obtain a general result and compare the signal producing maximal Fisher information with the inflection point of the sigmoidal transfer function in several basic neuronal models.
On the Markov Transition Kernels for First Passage Percolation on the Ladder
