arXiv subjects

Peter F. Craigmile

Publications and source records attributed to Peter F. Craigmile.

5 recordsLinked to original sources

arXiv2024

Statistical Inference for Generalized Integer Autoregressive Processes

A popular and flexible time series model for counts is the generalized integer autoregressive process of order $p$, GINAR($p$). These Markov processes are defined using thinning operators evaluated on past values of the process along with a discretely-valued innovation process. This class includes the commonly used INAR($p$) process, defined with binomial thinning and Poisson innovations. GINAR processes can be used in a variety of settings, including modeling time series with low counts, and allow for more general mean-variance relationships, capturing both over- or under-dispersion. While there are many thinning operators and innovation processes given in the literature, less focus has been spent on comparing statistical inference and forecasting procedures over different choices of GINAR process. We provide an extensive study of exact and approximate inference and forecasting methods that can be applied to a wide class of GINAR($p$) processes with general thinning and innovation parameters. We discuss the challenges of exact estimation when $p$ is larger. We summarize and extend asymptotic results for estimators of process parameters, and present simulations to compare small sample performance, highlighting how different methods compare. We illustrate this methodology by fitting GINAR processes to a disease surveillance series.

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arXiv2022

Optimal Design Emulators: A Point Process Approach

Design of experiments is a fundamental topic in applied statistics with a long history. Yet its application is often limited by the complexity and costliness of constructing experimental designs, which involve searching a high-dimensional input space and evaluating computationally expensive criterion functions. In this work, we introduce a novel approach to the challenging design problem. We will take a probabilistic view of the problem by representing the optimal design as being one element (or a subset of elements) of a probability space. Given a suitable distribution on this space, a generative point process can be specified from which stochastic design realizations can be drawn. In particular, we describe a scenario where the classical entropy-optimal design for Gaussian Process regression coincides with the mode of a particular point process. We conclude with outlining an algorithm for drawing such design realizations, its extension to sequential designs, and applying the techniques developed to constructing designs for Stochastic Gradient Descent and Gaussian process regression.

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arXiv2021

Modeling Nonstationary Time Series using Locally Stationary Basis Processes

Methods of estimation and forecasting for stationary models are well known in classical time series analysis. However, stationarity is an idealization which, in practice, can at best hold as an approximation, but for many time series may be an unrealistic assumption. We define a class of locally stationary processes which can lead to more accurate uncertainty quantification over making an invalid assumption of stationarity. This class of processes assumes the model parameters to be time-varying and parameterizes them in terms of a transformation of basis functions that ensures that the processes are locally stationary. We develop methods and theory for parameter estimation in this class of models, and propose a test that allow us to examine certain departures from stationarity. We assess our methods using simulation studies and apply these techniques to the analysis of an electroencephalogram time series.

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arXiv2017

Marginally Interpretable Generalized Linear Mixed Models

Two popular approaches for relating correlated measurements of a non-Gaussian response variable to a set of predictors are to fit a marginal model using generalized estimating equations and to fit a generalized linear mixed model by introducing latent random variables. The first approach is effective for parameter estimation, but leaves one without a formal model for the data with which to assess quality of fit or make predictions for future observations. The second approach overcomes the deficiencies of the first, but leads to parameter estimates that must be interpreted conditional on the latent variables. Further complicating matters, obtaining marginal summaries from a generalized linear mixed model often requires evaluation of an analytically intractable integral or use of attenuation factors that are not exact. We define a class of marginally interpretable generalized linear mixed models that lead to parameter estimates with a marginal interpretation while maintaining the desirable statistical properties of a conditionally-specified model. We discuss the form of these models under various common link functions and also address computational issues associated with these models. For logistic mixed effects models, we introduce an accurate and efficient method for evaluating the logistic-normal integral.

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arXiv2014

Maximum Likelihood Estimation for Stochastic Differential Equations Using Sequential Kriging-Based Optimization

Stochastic Differential Equations (SDEs) are used as statistical models in many disciplines. However, intractable likelihood functions for SDEs make inference challenging, and we need to resort to simulation-based techniques to estimate and maximize the likelihood function. While sequential Monte Carlo methods have allowed for the accurate evaluation of likelihoods at fixed parameter values, there is still a question of how to find the maximum likelihood estimate. In this article we propose an efficient Gaussian-process-based method for exploring the parameter space using estimates of the likelihood from a sequential Monte Carlo sampler. Our method accounts for the inherent Monte Carlo variability of the estimated likelihood, and does not require knowledge of gradients. The procedure adds potential parameter values by maximizing the so-called expected improvement, leveraging the fact that the likelihood function is assumed to be smooth. Our simulations demonstrate that our method has significant computational and efficiency gains over existing grid- and gradient-based techniques. Our method is applied to modeling the closing stock price of three technology firms.

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