Two-step estimation for inhomogeneous spatial point processes

Summary.  The paper is concerned with parameter estimation for inhomogeneous spatial point processes with a regression model for the intensity function and tractable second-order properties ( K -function). Regression parameters are estimated by using a Poisson likelihood score estimating function and in the second step minimum contrast estimation is applied for the residual clustering parameters. Asymptotic normality of parameter estimates is established under certain mixing conditions and we exemplify how the results may be applied in ecological studies of rainforests.     

Discretely Observed Diffusions: Approximation of the Continuous-time Score Function

We discuss parameter estimation for discretely observed, ergodic diffusion processes where the diffusion coefficient does not depend on the parameter. We propose using an approximation of the continuous-time score function as an estimating function. The estimating function can be expressed in simple terms through the drift and the diffusion coefficient and is thus easy to calculate. Simulation studies show that the method performs well.     

A new method of edge correction for estimating the nearest neighbor distribution

Analysis of data in the form of a set of points irregularly distributed within a region of space usually involves the study of some property of the distribution of inter-event distances. One such function is G, the distribution of the distance from an event to its nearest neighbor. In practice, point processes are commonly observed through a bounded window, thus making edge effects an important component in the estimation of G. Several estimators have been proposed, all dealing with the edge effect problem in different ways. This paper proposes a new alternative for estimating the nearest neighbor distribution and compares it to other estimators. The result is an estimator with relatively small mean squared error for a wide variety of stationary processes.     

Efficient sequential estimation in multiplicative counting processes

The problem of efficient sequential estimation is counting processes with multiplicative intensity processes is considered. A sequential version of Cramér-Rao type information inequality is obtained and all the 'efficient' triples (S, f, g) are characterized: the variance of an unbiased estimator f for g attains the lower bound under a sampling plan S. Applications to Poisson processes, Markov processes, birth and death processes and Markov branching processes with immigration are also considered.     

Non-ergodic martingale estimating functions and related asymptotics

Well-known estimation methods such as conditional least squares, quasilikelihood and maximum likelihood (ML) can be unified via a single framework of martingale estimating functions (MEFs). Asymptotic distributions of estimates for ergodic processes use constant norm (e.g. square root of the sample size) for asymptotic normality. For certain non-ergodic-type applications, however, such as explosive autoregression and super-critical branching processes, one needs a random norm in order to get normal limit distributions. In this paper, we are concerned with non-ergodic processes and investigate limit distributions for a broad class of MEFs. Asymptotic optimality (within a certain class of non-ergodic MEFs) of the ML estimate is deduced via establishing a convolution theorem using a random norm. Applications to non-ergodic autoregressive processes, generalized autoregressive conditional heteroscedastic-type processes, and super-critical branching processes are discussed. Asymptotic optimality in terms of the maximum random limiting power regarding large sample tests is briefly discussed.     

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.     

Parameter estimation for multivariate diffusion processes with the time inhomogeneously positive semidefinite diffusion matrix

Statistical inference for the diffusion coefficients of multivariate diffusion processes has been well established in recent years; however, it is not the case for the drift coefficients. Furthermore, most existing estimation methods for the drift coefficients are proposed under the assumption that the diffusion matrix is positive definite and time homogeneous. In this article, we put forward two estimation approaches for estimating the drift coefficients of the multivariate diffusion models with the time inhomogeneously positive semidefinite diffusion matrix. They are maximum likelihood estimation methods based on both the martingale representation theorem and conditional characteristic functions and the generalized method of moments based on conditional characteristic functions, respectively. Consistency and asymptotic normality of the generalized method of moments estimation are also proved in this article. Simulation results demonstrate that these methods work well.     

Smoothed Extreme Value Estimators of Non-Uniform Point Processes Boundaries with Application to Star-Shaped Supports Estimation

We address the problem of estimating the edge of a bounded set in ? ^d given a random set of points drawn from the interior. Our method is based on a transformation of estimators dedicated to uniform point processes and obtained by smoothing some of its bias corrected extreme points. An application to the estimation of star-shaped supports is presented.     

Power-transformed linear regression on quantile residual life for censored competing risks data

ABSTRACT

This paper proposes a power-transformed linear quantile regression model for the residual lifetime of competing risks data. The proposed model can describe the association between any quantile of a time-to-event distribution among survivors beyond a specific time point and the covariates. Under covariate-dependent censoring, we develop an estimation procedure with two steps, including an unbiased monotone estimating equation for regression parameters and cumulative sum processes for the Box–Cox transformation parameter. The asymptotic properties of the estimators are also derived. We employ an efficient bootstrap method for the estimation of the variance–covariance matrix. The finite-sample performance of the proposed approaches are evaluated through simulation studies and a real example.     

Robust surface estimation in multi-response multistage statistical optimization problems

As the ordinary least squares (OLS) method is very sensitive to outliers as well as to correlated responses, a robust coefficient estimation method is proposed in this paper for multi-response surfaces in multistage processes based on M-estimators. In this approach, experimental designs are used in which the intermediate response variables may act as covariates in the next stages. The performances of both the ordinary multivariate OLS and the proposed robust multi-response surface approach are analyzed and compared through extensive simulation experiments. Sum of the squared errors in estimating the regression coefficients reveals the efficiency of the proposed robust approach.     

Estimation of product moments of a stationary stochastic process with application to estimation of cumulants and cumulant spectral densities

Inference concerning the structure of stationary stochastic processes can be investigated by looking at properties of various cumulant spectral densities of order two and higher. However, except for cases when cumulants and product moments are identical, estimation of higher-order cumulant spectral densities has been restricted by the dependence of higher-order cumulants on lower-order product moments. By first estimating product moments and then using an identity between product moments and cumulants, asymptotically unbiased and consistent estimates of cumulants are obtained. This in turn leads to asymptotically unbiased and consistent estimators of higher-order cumulant spectral densities. In addition, asymptotic normality of product-moment estimators is exhibited under weak dependence.     

Optimal estimating function for estimation and prediction in semi-parametric models

In a seminal paper, Godambe [1985. The foundations of finite sample estimation in stochastic processes. Biometrika 72, 419–428.] introduced the ‘estimating function’ approach to estimation of parameters in semi-parametric models under a filtering associated with a martingale structure. Later, Godambe [1987. The foundations of finite sample estimation in stochastic processes II. Bernoulli, Vol. 2. V.N.V. Science Press, 49–54.] and Godambe and Thompson [1989. An extension of quasi-likelihood Estimation. J. Statist. Plann. Inference 22, 137–172.] replaced this filtering by a more flexible conditioning. Abraham et al. [1997. On the prediction for some nonlinear time-series models using estimating functions. In: Basawa, I.V., et al. (Eds.), IMS Selected Proceedings of the Symposium on Estimating Functions, Vol. 32. pp. 259–268.] and Thavaneswaran and Heyde [1999. Prediction via estimating functions. J. Statist. Plann. Inference 77, 89–101.] invoked the theory of estimating functions for one-step ahead prediction in time-series models. This paper addresses the problem of simultaneous estimation of parameters and multi-step ahead prediction of a vector of future random variables in semi-parametric models by extending the inimitable approach of 13 and 14. The proposed technique is in conformity with the paradigm of the modern theory of estimating functions leading to finite sample optimality within a chosen class of estimating functions, which in turn are used to get the predictors. Particular applications of the technique give predictors that enjoy optimality properties with respect to other well-known criteria.     

An additive–multiplicative mean model for panel count data with dependent observation and dropout processes

This paper discusses regression analysis of panel count data with dependent observation and dropout processes. For the problem, a general mean model is presented that can allow both additive and multiplicative effects of covariates on the underlying point process. In addition, the proportional rates model and the accelerated failure time model are employed to describe possible covariate effects on the observation process and the dropout or follow‐up process, respectively. For estimation of regression parameters, some estimating equation‐based procedures are developed and the asymptotic properties of the proposed estimators are established. In addition, a resampling approach is proposed for estimating a covariance matrix of the proposed estimator and a model checking procedure is also provided. Results from an extensive simulation study indicate that the proposed methodology works well for practical situations, and it is applied to a motivating set of real data.     

The Performance of Two Data-Generation Processes for Data with Specified Marginal Treatment Odds Ratios

Monte Carlo simulation methods are increasingly being used to evaluate the property of statistical estimators in a variety of settings. The utility of these methods depends upon the existence of an appropriate data-generating process. Observational studies are increasingly being used to estimate the effects of exposures and interventions on outcomes. Conventional regression models allow for the estimation of conditional or adjusted estimates of treatment effects. There is an increasing interest in statistical methods for estimating marginal or average treatment effects. However, in many settings, conditional treatment effects can differ from marginal treatment effects. Therefore, existing data-generating processes for conditional treatment effects are of little use in assessing the performance of methods for estimating marginal treatment effects. In the current study, we describe and evaluate the performance of two different data-generation processes for generating data with a specified marginal odds ratio. The first process is based upon computing Taylor Series expansions of the probabilities of success for treated and untreated subjects. The expansions are then integrated over the distribution of the random variables to determine the marginal probabilities of success for treated and untreated subjects. The second process is based upon an iterative process of evaluating marginal odds ratios using Monte Carlo integration. The second method was found to be computationally simpler and to have superior performance compared to the first method.     

Estimation of the intensity function of an inhomogeneous Poisson process with a change‐point

Recent work on point processes includes studying posterior convergence rates of estimating a continuous intensity function. In this article, convergence rates for estimating the intensity function and change‐point are derived for the more general case of a piecewise continuous intensity function. We study the problem of estimating the intensity function of an inhomogeneous Poisson process with a change‐point using non‐parametric Bayesian methods. An Markov Chain Monte Carlo (MCMC) algorithm is proposed to obtain estimates of the intensity function and the change‐point which is illustrated using simulation studies and applications. The Canadian Journal of Statistics 47: 604–618; 2019 © 2019 Statistical Society of Canada     

A regression approach for estimating the parameters of the covariance function of a stationary spatial random process

We consider the problem of estimating the parameters of the covariance function of a stationary spatial random process. In spatial statistics, there are widely used parametric forms for the covariance functions, and various methods for estimating the parameters have been proposed in the literature. We develop a method for estimating the parameters of the covariance function that is based on a regression approach. Our method utilizes pairs of observations whose distances are closest to a value h>0$h>0$ which is chosen in a way that the estimated correlation at distance h is a predetermined value. We demonstrate the effectiveness of our procedure by simulation studies and an application to a water pH data set. Simulation studies show that our method outperforms all well-known least squares-based approaches to the variogram estimation and is comparable to the maximum likelihood estimation of the parameters of the covariance function. We also show that under a mixing condition on the random field, the proposed estimator is consistent for standard one parameter models for stationary correlation functions.     

Nonparametric estimation of the mean function of a stochastic process with missing observations

In an attempt to identify similarities between methods for estimating a mean function with different types of response or observation processes, we explore a general theoretical framework for nonparametric estimation of the mean function of a response process subject to incomplete observations. Special cases of the response process include quantitative responses and discrete state processes such as survival processes, counting processes and alternating binary processes. The incomplete data are assumed to arise from a general response-independent observation process, which includes right- censoring, interval censoring, periodic observation, and mixtures of these as special cases. We explore two criteria for defining nonparametric estimators, one based on the sample mean of available data and the other inspired by the construction of Kaplan-Meier (or product-limit) estimator [J. Am. Statist. Assoc. 53 (1958) 457] for right-censored survival data. We show that under regularity conditions the estimated mean functions resulting from both criteria are consistent and converge weakly to Gaussian processes, and provide consistent estimators of their covariance functions. We then evaluate these general criteria for specific responses and observation processes, and show how they lead to familiar estimators for some response and observation processes and new estimators for others. We illustrate the latter with data from an recently completed AIDS clinical trial.     

Spectral analysis for processes with almost periodic covariances

The structure of random processes with almost periodic covariances is described from a spectral perspective. Under appropriate conditions methods for spectral estimation are described for such processes which are neither stationary nor locally stationary. Some spectral mass is then located off the main diagonal in this spectral plane. A method for estimating the support of the spectral mass is described in the Gaussian case. A number of open questions are mentioned.     

Estimation of association parameters in copula models for bivariate left-truncated and right-censored data

We investigate the problem of estimating the association between two related survival variables when they follow a copula model and bivariate left-truncated and right-censored data are available. By expressing truncation probability as the functional of marginal survival functions, we propose a two-stage estimation procedure for estimating the parameters of Archimedean copulas. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate the finite sample properties of the proposed estimators. The proposed method is applied to a bivariate RNA data.     

A series of single array 2m factorial search designs for even m

By means of a search design one is able to search for and estimate a small set of non‐zero elements from the set of higher order factorial interactions in addition to estimating the lower order factorial effects. One may be interested in estimating the general mean and main effects, in addition to searching for and estimating a non‐negligible effect in the set of 2‐ and 3‐factor interactions, assuming 4‐ and higher‐order interactions are all zero. Such a search design is called a ‘main effect plus one plan’ and is denoted by MEP.1. Construction of such a plan, for 2^m factorial experiments, has been considered and developed by several authors and leads to MEP.1 plans for an odd number m of factors. These designs are generally determined by two arrays, one specifying a main effect plan and the other specifying a follow‐up. In this paper we develop the construction of search designs for an even number of factors m, m≠6. The new series of MEP.1 plans is a set of single array designs with a well structured form. Such a structure allows for flexibility in arriving at an appropriate design with optimum properties for search and estimation.