Nonparametric estimation of intensities of nonhomogeneous Poisson processes

The problem of nonparametric estimation of the intensity of a nonhomogeneous Poisson process is considered. A kernel estimator of the intensity is introduced with data driven bandwidth. The bandwidth is obtained from an L₂ cross validation procedure. Results on almost sure convergence of the estimator are obtained, provided the number of observed realizations n tends to infinity. The limiting distribution of the estimator is presented for n→∞.     

Consistent parametric estimation of the intensity of a spatial–temporal point process

We consider conditions under which parametric estimates of the intensity of a spatial–temporal point process are consistent. Although the actual point process being estimated may not be Poisson, an estimate involving maximizing a function that corresponds exactly to the log-likelihood if the process is Poisson is consistent under certain simple conditions. A second estimate based on weighted least squares is also shown to be consistent under quite similar assumptions. The conditions for consistency are simple and easily verified, and examples are provided to illustrate the extent to which consistent estimation may be achieved. An important special case is when the point processes being estimated are in fact Poisson, though other important examples are explored as well.     

Consistent Smooth Bootstrap Kernel Intensity Estimation for Inhomogeneous Spatial Poisson Point Processes

Non‐parametric estimation and bootstrap techniques play an important role in many areas of Statistics. In the point process context, kernel intensity estimation has been limited to exploratory analysis because of its inconsistency, and some consistent alternatives have been proposed. Furthermore, most authors have considered kernel intensity estimators with scalar bandwidths, which can be very restrictive. This work focuses on a consistent kernel intensity estimator with unconstrained bandwidth matrix. We propose a smooth bootstrap for inhomogeneous spatial point processes. The consistency of the bootstrap mean integrated squared error (MISE) as an estimator of the MISE of the consistent kernel intensity estimator proves the validity of the resampling procedure. Finally, we propose a plug‐in bandwidth selection procedure based on the bootstrap MISE and compare its performance with several methods currently used through both as a simulation study and an application to the spatial pattern of wildfires registered in Galicia (Spain) during 2006.     

Isotonic estimation of the intensity of a nonhomogeneous Poisson process: The multiple realization setup

In the 1950s Brunk and Van Eeden each obtained maximum-likelihood estimators of a finite product of probability density functions under partial or complete ordering of their parameters. Their results play an important role in the general theory of inference under order restrictions and lead to an isotonic estimator of the intensity of a nonhomogeneous Poisson process. Here an elementary derivation of the maximum likelihood estimator (m.l.e.) for the intensity of a nonhomogeneous Poisson process is given when several (possibly censored) realizations are available. Boswell obtained the m.l.e. based on a single realization as well as a conditional m.l.e. under the same conditions. An example is given to show that in the multirealization setup a conditional m.l.e. may not exist; the proofs are, we believe, new and elementary. An illustrative application is given.     

Re-colouring the Intensity-Based Bootstrap for Point Processes

Abstract

A method for obtaining bootstrapping replicates for one-dimensional point processes is presented. The method involves estimating the conditional intensity of the process and computing residuals. The residuals are bootstrapped using a block bootstrap and used, together with the conditional intensity, to define the bootstrap realizations. The method is applied to the estimation of the cross-intensity function for data arising from a reaction time experiment.     

Iterated Bootstrap-t Confidence Intervals for Density Functions

Abstract.  Conventional bootstrap- t intervals for density functions based on kernel density estimators exhibit poor coverages due to failure of the bootstrap to estimate the bias correctly. The problem can be resolved by either estimating the bias explicitly or undersmoothing the kernel density estimate to undermine its bias asymptotically. The resulting bias-corrected intervals have an optimal coverage error of order arbitrarily close to second order for a sufficiently smooth density function. We investigated the effects on coverage error of both bias-corrected intervals when the nominal coverage level is calibrated by the iterated bootstrap. In either case, an asymptotic reduction of coverage error is possible provided that the bias terms are handled using an extra round of smoothed bootstrapping. Under appropriate smoothness conditions, the optimal coverage error of the iterated bootstrap- t intervals has order arbitrarily close to third order. Examples of both simulated and real data are reported to illustrate the iterated bootstrap procedures.     

Kernel Smoothing to Improve Bootstrap Confidence Intervals

Some studies of the bootstrap have assessed the effect of smoothing the estimated distribution that is resampled, a process usually known as the smoothed bootstrap. Generally, the smoothed distribution for resampling is a kernel estimate and is often rescaled to retain certain characteristics of the empirical distribution. Typically the effect of such smoothing has been measured in terms of the mean-squared error of bootstrap point estimates. The reports of these previous investigations have not been encouraging about the efficacy of smoothing. In this paper the effect of resampling a kernel-smoothed distribution is evaluated through expansions for the coverage of bootstrap percentile confidence intervals. It is shown that, under the smooth function model, proper bandwidth selection can accomplish a first-order correction for the one-sided percentile method. With the objective of reducing the coverage error the appropriate bandwidth for one-sided intervals converges at a rate of n ^−1/4, rather than the familiar n ^−1/5 for kernel density estimation. Applications of this same approach to bootstrap t and two-sided intervals yield optimal bandwidths of order n ^−1/2. These bandwidths depend on moments of the smooth function model and not on derivatives of the underlying density of the data. The relationship of this smoothing method to both the accelerated bias correction and the bootstrap t methods provides some insight into the connections between three quite distinct approximate confidence intervals.     

Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.     

Uncertainty estimation in heterogeneous capture–recapture count data

The Conway–Maxwell–Poisson estimator is considered in this paper as the population size estimator. The benefit of using the Conway–Maxwell–Poisson distribution is that it includes the Bernoulli, the Geometric and the Poisson distributions as special cases and, furthermore, allows for heterogeneity. Little emphasis is often placed on the variability associated with the population size estimate. This paper provides a deep and extensive comparison of bootstrap methods in the capture–recapture setting. It deals with the classical bootstrap approach using the true population size, the true bootstrap, and the classical bootstrap using the observed sample size, the reduced bootstrap. Furthermore, the imputed bootstrap, as well as approximating forms in terms of standard errors and confidence intervals for the population size, under the Conway–Maxwell–Poisson distribution, have been investigated and discussed. These methods are illustrated in a simulation study and in benchmark real data examples.     

Invariance principles and almost sure central limit theorem for the error variance estimator in linear models

A number of score statistics are derived for a heterogeneous spatial Poisson process which has a composite intensity. The intensity consists of a 'background' process which is estimated

from a control point process by kernel density estimation. The parametric form of the composite intensity yields score tests for particular spatial effects. A numerical example concerning respiratory cancer mortality is given.     

Bayesian inference for power law processes with applications in repairable systems

Statistical models for recurrent events are of great interest in repairable systems reliability and maintenance. The adopted model under minimal repair maintenance is frequently a nonhomogeneous Poisson process with the power law process (PLP) intensity function. Although inference for the PLP is generally based on maximum likelihood theory, some advantages of the Bayesian approach have been reported in the literature. In this paper it is proposed that the PLP intensity be reparametrized in terms of (β,η), where β is the elasticity of the mean number of events with respect to time and η is the mean number of events for the period in which the system was actually observed. It is shown that β and η are orthogonal and that the likelihood becomes proportional to a product of gamma densities. Therefore, the family of natural conjugate priors is also a product of gammas. The idea is extended to the case that several realizations of the same PLP are observed along overlapping periods of time. Some Monte Carlo simulations are provided to study the frequentist behavior of the Bayesian estimates and to compare them with the maximum likelihood estimates. The results are applied to a real problem concerning the determination of the optimal periodicity of preventive maintenance for a set of power transformers. Prior distributions are elicited for β and η based on their operational interpretation and engineering expertise.     

Annual arrival cycle of Atlantic tropical cyclones

Tropical cyclones occur in the North Atlantic Ocean and the Gulf of Mexico throughout the year. This paper statistically quantifies their annual arrival cycle. A Poisson process with a periodic time-varying intensity function is used to model the annual cycle of storm arrivals. The kernel method is used to estimate the time-varying intensity function. A data set that contains storm occurrence dates for the 120-year period 1871-1990 inclusive is analyzed. The estimated intensity function indicates that the peak of the hurricane season is around 12 September and that the true intensity function may be multi-modal. Adjustments are made in the analysis to account for storms that occurred but went unrecorded over the earlier years of the data record.     

A note on the scale parameter of the dirichlet process

This paper gives an interpretation for the scale parameter of a Dirichlet process when the aim is to estimate a linear functional of an unknown probability distribution. We provide exact first and second posterior moments for such functionals under both informative and noninformative prior specifications. The noninformative case provides a normal approximation to the Bayesian bootstrap.     

On hypothesis tests for disk-type intensities of spatial poisson processes

We consider the problem of testing whether realizations of a spatial Poisson process come from a circular intensity model or from an elliptical intensity model. We describe the behavior of the Generalized Likelihood Ratio and Wald tests, in the asymptotics of large samples. The power functions are studied under local alternatives and results of numerical simulations comparing them to the Neyman–Pearson envelope are presented.     

A Bootstrap Algorithm for Data from a Periodic Multiplicative Intensity Function

The periodic multiplicative intensity model is considered. A new bootstrap method for non stationary counting processes which intensity function has some periodicity properties is presented. Its main advantage is that it does not destroy the temporal order and the original periodicity of the underlying counting process. The proposed algorithm is used to construct a bootstrap version of the maximum likelihood hazard function estimator. The consistency of the bootstrap method is shown. A possible modification of the proposed bootstrap method is discussed. The bootstrap simultaneous confidence intervals for the hazard function are presented. The telecommunication network traffic real data example is discussed.     

Hierarchical Bayesian analysis of a discrete time series of Poisson counts

A problem involving non-stationary, discrete-time series of counts from a Poisson process with a varying but smooth intensity function is studied. A smoothness prior for the underlying intensity process is modelled using the hierarchical Bayesian approach, which is shown to provide an AR(1) representation for the intensity process. Since conjugate priors are not assumed, analytic derivation of estimates and predictions of the Poisson series are not available. Some reasonably good approximations are given and illustrated using data on British road casualties before and after the introduction of the seatbelt law.     

Hypotheses testing for a multidimensional parameter of inhomogeneous Poisson processes

We observe a realization of an inhomogeneous Poisson process whose intensity function depends on an unknown multidimensional parameter. We consider the asymptotic behaviour of the Rao score test for a simple null hypothesis against the multilateral alternative. By using the Edgeworth type expansion (under the null hypothesis) for a vector of stochastic integrals with respect to the Poisson process, we refine the (classic) threshold of the test (obtained by the central limit theorem), which improves the first type probability of error. The expansion allows us to describe the power of the test under the local alternative, i.e. a sequence of alternatives, which converge to the null hypothesis with a certain rate. The rates can be different for components of the parameter.     

Kernel estimations for multivariate density functional with bootstrap

In this article the bootstrap method is discussed for the kernel estimation of the multivariate density function. We have considered sample mean functional and constructed its consistency and asymptotic normality by bootstrap estimator. It has been shown that the bootstrap works for kernel estimates of multivariate density functional. The convergence rate with bootstrap for density has been proved. Finally, two simulations of application are given.     

Variance estimation for integrated population models

State-space models are widely used in ecology. However, it is well known that in practice it can be difficult to estimate both the process and observation variances that occur in such models. We consider this issue for integrated population models, which incorporate state-space models for population dynamics. To some extent, the mechanism of integrated population models protects against this problem, but it can still arise, and two illustrations are provided, in each of which the observation variance is estimated as zero. In the context of an extended case study involving data on British Grey herons, we consider alternative approaches for dealing with the problem when it occurs. In particular, we consider penalised likelihood, a method based on fitting splines and a method of pseudo-replication, which is undertaken via a simple bootstrap procedure. For the case study of the paper, it is shown that when it occurs, an estimate of zero observation variance is unimportant for inference relating to the model parameters of primary interest. This unexpected finding is supported by a simulation study.     

Local Smoothing Using the Bootstrap

We consider the problem of data-based choice of the bandwidth of a kernel density estimator, with an aim to estimate the density optimally at a given design point. The existing local bandwidth selectors seem to be quite sensitive to the underlying density and location of the design point. For instance, some bandwidth selectors perform poorly while estimating a density, with bounded support, at the median. Others struggle to estimate a density in the tail region or at the trough between the two modes of a multimodal density. We propose a scale invariant bandwidth selection method such that the resulting density estimator performs reliably irrespective of the density or the design point. We choose bandwidth by minimizing a bootstrap estimate of the mean squared error (MSE) of a density estimator. Our bootstrap MSE estimator is different in the sense that we estimate the variance and squared bias components separately. We provide insight into the asymptotic accuracy of the proposed density estimator.