Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns

The likelihood method is developed for the analysis of socalled regular point patterns. Approximating the normalizing factor of Gibbs canonical distribution, we simultaneously estimate two parameters, one for the scale and the other which measures the softness (or hardness), of repulsive interactions between points. The approximations are useful up to a considerably high density. Some real data are analyzed to illustrate the utility of the parameters for characterizing the regular point pattern.     

面板数据公共变点的非参数最大似然估计

Gao Pengli;Xia Zhiming(School of Mathematics,Northwest University,Xi'an 710127,China)     

Penalized maximum likelihood estimation for a function of the intensity of a Poisson point process

Let be an unknown 2 times differentiable function and consider M to be an α- homogeneous Poisson process on Graf(f). The goal is to estimate f having a sample of the inhomogeneous Poisson process N constructed by dislocating each point of M perpendicularly to Graf(f) by a normal random variable with zero mean and constant variance σ². The exact formulas for the mean measure and the intensity function of N are obtained. Then, the function f is estimated directly using a hybrid spline approach to penalized maximum likelihood. Simulation results indicate the procedure to be consistent as and .      

Bayesian modeling of continuously marked spatial point patterns

Many analyses of continuously marked spatial point patterns assume that the density of points, with differing marks, is identical. However, as noted in the seminal paper of Goulard et al. (Scand J Stat 23:365–379, 1996), such an assumption is not realistic in many situations. For example, a stand of forest may have many more small trees than large, hence the model should allow for a higher density of points with small marks. In addition, as suggested by Ogata and Tanemura (Biometrics 41:421–433, 1985), the interaction between points should be a function of their mark, allowing, for example, the range of interaction for large trees to exceed that of smaller trees. The aforementioned articles use frequentist inferential techniques, but interval estimation presents difficulties due to the extremely complex distributional properties of the estimates; it might be possible, however, to use parametric bootstrap methodology for such inferences (Baddeley et al. in J Roy Stat Soc Ser B 67:617–666, 2005). We suggest the use of Bayesian inferential techniques. Although a Bayesian approach requires a complex, computational implementation of (reversible jump) Markov Chain Monte Carlo methodology, it enables a wide variety of inferences (including interval estimation). We demonstrate our approach by analyzing the well known Norway spruce dataset.     

Clustering estimates for spatial point distributions with unstable potentials

Summary Several authors have tried to model highly clustered point patterns by using Gibbs distributions with attractive potentials. Some of these potentials violate a stability condition well known in statistical mechanics. We show that such potentials produce patterns which are much more tightly clustered than those considered by the authors. More generally, our estimates provide a useful test for rejecting unsuitable potentials in models for given patterns. We also use instability arguments to reject related approximations and simulations. Csiro     

Efficient simulated maximum likelihood estimation through explicitly parameter dependent importance sampling

There exists an overall negative assessment of the performance of the simulated maximum likelihood algorithm in the statistics literature, founded on both theoretical and empirical results. At the same time, there also exist a number of highly successful applications. This paper explains the negative assessment by the coupling of the algorithm with “simple importance samplers”, samplers that are not explicitly parameter dependent. The successful applications in the literature are based on explicitly parameter dependent importance samplers. Simple importance samplers may efficiently simulate the likelihood function value, but fail to efficiently simulate the score function, which is the key to efficient simulated maximum likelihood. The theoretical points are illustrated by applying Laplace importance sampling in both variants to the classic salamander mating model.     

Likelihood analysis of spatial inhomogeneity for marked point patterns

An objective method is developed for estimations of both spatial intensity of the point locations and spatial variation of a characteristic parameter of the distributions for the attached marks. Its utility is demonstrated by means of analyses of seismological and ecological data sets.     

On the consistency of conditional maximum likelihood estimators

Let {P _, : , H} be a family of probability measures admitting a sufficient statistic for the nuisance parameter . The paper presents conditions for consistency of (asymptotic) conditional maximum likelihood estimators for . An application to the Rasch-model (a stochastic model for psychological tests) yields a condition on the sequence of nuisance parameters which is sufficient for strong consistency of conditional maximum likelihood estimators, and necessary for the existence of any weakly consistent estimator-sequence.     

On properties of the iterative maximum likelihood reconstruction method

In this paper, we continue our investigations⁶ on the iterative maximum likelihood reconstruction method applied to a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Equations of this type often occur in connection with the determination of density functions from measured data. There are certain relations between the directed Kullback–Leibler divergence and the iterative maximum likelihood reconstruction method some of which were already observed by other authors. Using these relations, further properties of the iterative scheme are shown and, in particular, a new short and elementary proof of convergence of the iterative method is given for the discrete case. Numerical examples have already been given in References 6. Here, an example is considered which can be worked out analytically and which demonstrates fundamental properties of the algorithm.     

A third-order optimum property of the maximum likelihood estimator

Let θ⁽ⁿ⁾ denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T⁽ⁿ⁾ there exists q such that the risk of θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾) exceeds the risk of T⁽ⁿ⁾ by an amount of order o(n^?1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If $\text{q}{}^1$ is chosen such that $\text{θ}{}^{\mathrm{(n)}}\text{+ n}{}^{\mathrm{?1}}\text{q}{}^1\text{(θ}{}^{\mathrm{(n)}}\text{)}$ is unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$, then this estimator minimizes the risk up to an amount of order o(n^?1) in the class of all estimators which are unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$.The results are obtained under the assumption that T⁽ⁿ⁾ admits a stochastic expansion, and that either the distributions have—roughly speaking—densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth.     

On a fundamental optimality of the maximum likelihood estimator

It is well-known that the rate of exponential convergence for any consistent estimator is less than or equal to the Bahadur bound. In this paper we have proven, for the one-dimensional case, that the rate of exponential convergence for the maximum likelihood estimator (m.l.e.) attains the Bahadur bound if and only if the underlying distribution is a member of the exponential family of distributions.     

Uniform LAN condition of planar Gibbsian point processes and optimality of maximum likelihood estimators of soft-core potential functions

Summary A parametric model of planar point patterns in a bounded region is constructed using grand canonical Gibbsian point processes with soft-core potential functions. A simple and explicit condition that this model becomes a uniform locally asymptotic normal (ULAN) family will be given. From this result we can conclude that the maximum likelihood estimator of the potential function is asymptotically efficient for a wide class of loss functions.     

Computation of the maximum likelihood estimate of a noncentrality parameter

An algorithm and error analysis are presented for finding the maximum likelihood estimator of the noncentrality parameter of the χ² and F distributions.     

Finding maximum likelihood estimators for the three-parameter Weibull distribution

Much work has been devoted to the problem of finding maximum likelihood estimators for the three-parameter Weibull distribution. This problem has not been clearly recognized as a global optimization one and most methods from the literature occasionally fail to find a global optimum. We develop a global optimization algorithm which uses first order conditions and projection to reduce the problem to a univariate optimization one. Bounds on the resulting function and its first order derivative are obtained and used in a branch-and-bound scheme. Computational experience is reported. It is also shown that the solution method we propose can be extended to the case of right censored samples.