Asymptotic properties of the minimum contrast estimators for projections of inhomogeneous space-time shot-noise Cox processes

We consider a flexible class of space-time point process models—inhomogeneous shot-noise Cox point processes. They are suitable for modelling clustering phenomena, e.g. in epidemiology, seismology, etc. The particular structure of the model enables the use of projections to the spatial and temporal domain. They are used to formulate a stepwise estimation method to estimate different parts of the model separately. In the first step, the Poisson likelihood approach is used to estimate the inhomogeneity parameters. In the second and third steps, the minimum contrast estimation based on K-functions of the projected processes is used to estimate the interaction parameters. We study the asymptotic properties of the resulting estimators and formulate a set of conditions sufficient for establishing consistency and asymptotic normality of the estimators under the increasing domain asymptotics.     

Rates of convergence for minimum contrast estimators

Summary We shall present here a general study of minimum contrast estimators in a nonparametric setting (although our results are also valid in the classical parametric case) for independent observations. These estimators include many of the most popular estimators in various situations such as maximum likelihood estimators, least squares and other estimators of the regression function, estimators for mixture models or deconvolution... The main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters. Optimal rates depending on entropy conditions are already known, at least for some of the models involved, and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large. But, under some circumstances (large entropies or changes in the entropy structure due to local perturbations), the resulting the rates are only suboptimal. Counterexamples are constructed which show that the phenomenon is real for non-parametric maximum likelihood or regression. This proves that, under purely metric assumptions, our theorem is optimal and that minimum contrast estimators happen to be suboptimal.     

Asymptotic bounds for estimators without limit distribution

Let be a general family of probability measures,κ : a functional, and the optimal limit distribution for regular estimator sequences of κ. On intervals symmetric about 0, the concentration of this optimal limit distribution can be surpassed by the asymptotic concentration of an arbitrary estimator sequence only forP in a “small” subset of . For asymptotically median unbiased estimator sequences the same is true for arbitrary intervals containing 0. The emphasis of the paper is on “pointwise” conditions for , as opposed to conditions on shrinking neighbourhoods, and on “general” rather than parametric families.     

Asymptotic expansions for the distribution functions of Pickands-type estimators

The asymptotic expansions for the distribution functions of Pickands-type estimators in extreme statistics are obtained. In addition, several useful results on regular variation and intermediate order statistics are presented. Project supported by the National Natural Science Foundation of China (Grant No. 19601007) and Doctoral Program Foundation of Higher Education of China.     

Validity of Edgeworth expansions of minimum contrast estimators for Gaussian ARMA processes

Let {X_t} be a Gaussian ARMA process with spectral density f_θ(λ), where θ is an unknown parameter. To estimate θ we propose a minimum contrast estimation method which includes the maximum likelihood method and the quasi-maximum likelihood method as special cases. Let θ̂_τ be the minimum contrast estimator of θ. Then we derive the Edgewroth expansion of the distribution of θ̂_τ up to third order, and prove its valldity. By this Edgeworth expansion we can see that this minimum contrast estimator is always second-order asymptotically efficient in the class of second-order asymptotically median unbiased estimators. Also the third-order asymptotic comparisons among minimum contrast estimators will be discussed.     

Asymptotic properties of the estimators of the Weibull distribution parameters

The properties of the maximum likelihood and moment estimators are investigated for the three-dimensional Weibull distribution in the case of arbitrary values of the shape parameter.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 9–16, 1988.     

Asymptotic behavior of polynomial Pitman estimators

The asymptotic normality of polynomial Pitman estimators for the location parameter is proved, the principal term of the formula for the variance of these estimators is obtained, and certain characterization problems are considered that occur in the study of polynomial Pitman estimators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 43, pp. 30–39, 1974.     

Asymptotic distribution theory of statistical functionals: The compact derivative approach for robust estimators

Summary Derivatives of statistical functionals have been used to derive the asymptotic distributions ofL-,M- andR-estimators. This approach is often heuristic because the types of derivatives chosen have serious limitations. The Gateaux derivative is too weak and the Fréchet derivative is too strong. In between lies the compact derivative. This paper obtains strong results in a rigorous manner using the compact derivative onC ₀(R). This choice of space allows results for a broader class of functionals than previous choices, and the fact that is often tight provides the compact set required. A major result is the derivation of the compact derivative of the inverse c.d.f. when the range space is endowed with the uniform norm. It has applications to the asymptotic theory ofL-,M- andR-estimators. We illustrate the power of this result by applications toL-estimators in settings including the one sample problem, data grouped by quantiles, and censored survival time data.     

Asymptotic optimality of estimators in non-regular cases

Summary The bound of the asymptotic distributions of for all asymptotically median unbiased (AMU) estimators is given in non-regular cases. It provides us with a powerful criterion for an AMU estimator to be two-sided asymptotically efficient and also useful in the cases when there may not exist a two-sided asymptotically efficient estimator since we may find an AMU estimator whose asymptotic distribution attains at least at a point, or an AMU estimator whose asymptotic distribution is uniformly “close” to it. Some examples are given. The results of this paper have been presented at the Meeting on Statistical Theory of Model Analysis at Tsukuba University in Japan, October 1979.     

Asymptotic variance of grey-scale surface area estimators

Grey-scale local algorithms have been suggested as a fast way of estimating surface area from grey-scale digital images. Their asymptotic mean has already been described. In this paper, the asymptotic behavior of the variance is studied in isotropic and sufficiently smooth settings, resulting in a general asymptotic bound. For compact convex sets with nowhere vanishing Gaussian curvature, the asymptotics can be described more explicitly. As in the case of volume estimators, the variance is decomposed into a lattice sum and an oscillating term of at most the same magnitude.     

Asymptotic minimum norm quadrature formulae

The asymptotic limit of minimum norm quadrature formulae for some Hilbert spaces of functions regular and analytic in a domainB is studied, asB expands.     

Asymptotic theory for estimators under random censorship

Summary The product limit estimator of an unknown distributionF is represented as aU-statistic plus an error of the ordero(1/n). Using this, the maximum likelihood estimator of the specific risk rate in the time interval [0,M], is shown to admit a two term Edgeworth expansion. This risk rate for a specific cause of death is defined as the ratio of the probability of death, due to that particular cause, in the time interval [0,M], to the mean life time of an individual up to that time pointM. Similar expansions for the bootstrapped statistics are used to show that the bootstrap distribution, of the studentized estimator of the risk rate, approximates the sampling distribution better than the corresponding normal distribution.Research supported in part by NSA Grant MDA 904-90-H-1001 and by NSF Grant DMS-9007717     

Asymptotic normality for wavelet deconvolution density estimators

This current paper shows the asymptotic normality for wavelet deconvolution density estimators, when a density function belongs to some $L^p(\mathbb{R})$ $(p>2)$ and the noises are moderately ill-posed with the index β. The estimators include both the linear and non-linear wavelet ones. It turns out that the situation for $0<\beta \le 1$ is more complicated than that for $\beta >1$.     

Asymptotic properties of parallel Bayesian kernel density estimators

Annals of the Institute of Statistical Mathematics - In this article, we perform an asymptotic analysis of Bayesian parallel kernel density estimators introduced by Neiswanger et al. (in:...     

Asymptotic approximation of kernel-type estimators with its application

Sufficient conditions are given under which a generalized class of kernel-type estimators allows asymptotic approximation on the modulus of continuity. This generalized class includes sample distribution function, kernel-type estimator of density function, and an estimator that may apply to the censored case. In addition, an application is given to asymptotic normality of recursive density estimators of density function at an unknown point.     

Asymptotic expansion of Bayes estimators for small diffusions

Summary Using the Malliavin calculus we derived asymptotic expansion of the distributions of the Bayes estimators for small diffusions. The second order efficiency of the Bayes estimator is proved.     

Asymptotic normality of p-norm estimators in multiple regression

Summary The consistency and asymptotic normality of p-norm estimators (1<p<2) is established by applying some of the ideas of Huber (1973), where asymptotic normality of the so-called M-estimators for regression parameters is shown. A central role is played by a weight function . Huber assumed that , and are bounded. This is, however, not the case for p-norm estimators with 1<p<2, but some of his ideas can still be applied.     

Asymptotic normality of some kernel-type estimators of probability density

Central limit theorems are proved for some kernel-type estimators of probability density in the case where the observations form a strictly random sequence satisfying the ?-mixing condition with a certain logarithmic mixing rate.     

Asymptotic properties of nonparametric estimators of the characteristic function

One investigates the asymptotic properties of nonparametric estimators of the characteristic function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 136, pp. 97–112, 1984.