Local likelihood smoothing of sample extremes

Trends in sample extremes are of interest in many contexts, an example being environmental statistics. Parametric models are often used to model trends in such data, but they may not be suitable for exploratory data analysis. This paper outlines a semiparametric approach to smoothing example extremes, based on local polynomial fitting of the generalized extreme value distribution and related models. The uncertainty of fits is assessed by using resampling methods. The methods are applied to data on extreme temperatures and on record times for the womens 3000m race.

Stochastic comparability and dual q-functions

In this paper we discuss the relationship and characterization of stochastic comparability, duality, and Feller–Reuter–Riley transition functions which are closely linked with each other for continuous time Markov chains. A necessary and sufficient condition for two Feller minimal transition functions to be stochastically comparable is given in terms of their density q-matrices only. Moreover, a necessary and sufficient condition under which a transition function is a dual for some stochastically monotone q-function is given in terms of, again, its density q-matrix. Finally, for a class of q-matrices, the necessary and sufficient condition for a transition function to be a Feller–Reuter–Riley transition function is also given.

Maximum curves and isolated points of entire functions

Given M(r; f) =maxjzj=r (jf(z)j) , curves belonging to the set of points M = fz : jf(z)j = M(jzj; f)g were de�ned by Hardy to be maximum curves. Clunie asked the question as to whether the set M could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions f1(z) and f2(z), if the maximum curve of f1(z) is the real axis, conditions are found so that the real axis is also a maximum curve for the product function f1(z)f2(z). By means of these results an entire function of in�nite order is constructed for which the set M has an in�nite number of isolated points. A polynomial is also constructed with an isolated point.

Applications of Feller-Reuter-Riley transition functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.

Strong ergodicity of monotone transition functions

By revealing close links among strong ergodicity, monotone, and the Feller–Reuter–Riley (FRR) transition functions, we prove that a monotone ergodic transition function is strongly ergodic if and only if it is not FRR. An easy to check criterion for a Feller minimal monotone chain to be strongly ergodic is then obtained. We further prove that a non-minimal ergodic monotone chain is always strongly ergodic. The applications of our results are illustrated using birth-and-death processes and branching processes.

Feller transition functions, Resolvent Decomposition Theorems, and their application in unstable denumerable Markov processes

This paper surveys the recent progresses made in the field of unstable denumerable Markov processes. Emphases are laid upon methodology and applications. The important tools of Feller transition functions and Resolvent Decomposition Theorems are highlighted. Their applications particularly in unstable denumerable Markov processes with a single instantaneous state and Markov branching processes are illustrated.

Likelihood, entropy and species diversity; some comparisons in a Sumatran forest

In attempts to conserve the species diversity of trees in tropical forests, monitoring of diversity in inventories is essential. For effective monitoring it is crucial to be able to make meaningful comparisons between different regions, or comparisons of the diversity of a region at different times. Many species diversity measures have been defined, including the well-known abundance and entropy measures. All such measures share a number of problems in their effective practical use. However, probably the most problematic is that they cannot be used to meaningfully assess changes, since thay are only concerned with the number of species or the proportions of the population/sample which they constitute. A natural (though simplistic) model of a species frequency distribution is the multinomial distribution. It is shown that the likelihood analysis of samples from such a distribution are closely related to a number of entropy-type measures of diversity. Hence a comparison of the species distribution on two plots, using the multinomial model and likelihood methods, leads to generalised cross-entropy as the LRT test statistic of the null that the species distributions are the same. Data from 30 contiguous plots in a forest in Sumatra are analysed using these methods. Significance tests between all pairs of plots yield extremely low p-values, indicating strongly that it ought to been "Obvious" that the observed species distributions are different on different plots. In terms of how different the plots are, and how these differences vary over the whole study site, a display of the degrees of freedom of the test, (equivalent to the number of shared species) seems to be the most revealing indicator, as well as the simplest.

Maximum-likelihood estimator for coarse carrier frequency offset estimation in OFDM systems

Orthogonal frequency division multiplexing (OFDM) systems are more sensitive to carrier frequency offset (CFO) compared to the conventional single carrier systems. CFO destroys the orthogonality among subcarriers, resulting in inter-carrier interference (ICI) and degrading system performance. To mitigate the effect of the CFO, it has to be estimated and compensated before the demodulation. The CFO can be divided into an integer part and a fractional part. In this paper, we investigate a maximum-likelihood estimator (MLE) for estimating the integer part of the CFO in OFDM systems, which requires only one OFDM block as the pilot symbols. To reduce the computational complexity of the MLE and improve the bandwidth efficiency, a suboptimum estimator (Sub MLE) is studied. Based on the hypothesis testing method, a threshold Sub MLE (T-Sub MLE) is proposed to further reduce the computational complexity. The performance analysis of the proposed T-Sub MLE is obtained and the analytical results match the simulation results well. Numerical results show that the proposed estimators are effective and reliable in both additive white Gaussian noise (AWGN) and frequency-selective fading channels in OFDM systems.