This paper is mainly concerned with the asymptotic theory of maximum likelihood estimation for continuous-time stochastic processes. The role of martingale limit theory in this theory is developed. Some analogues of classical statistical concepts and quantities are also suggested. Various examples that illustrate parts of the theory are worked through, producing new results in some cases. The role of diffusion approximations in estimation is also explored. MAXIMUM LIKELIHOOD ESTIMATION; CONTINUOUS-TIME STOCHASTIC PROCESSES; ASYMPTOTIC THEORY; MARTINGALE LIMIT THEORY; DIFFUSION APPROXIMATIONS

Maximum likelihood estimation for continuous-time stochastic processes

Five parameters of one of the most common neuronal models, the diffusion leaky integrate-and-fire model, also known as the Ornstein-Uhlenbeck neuronal model, were estimated on the basis of intracellular recording. These parameters can be classified into two categories. Three of them (the membrane time constant, the resting potential and the firing threshold) characterize the neuron itself. The remaining two characterize the neuronal input. The intracellular data were collected during spontaneous firing, which in this case is characterized by a Poisson process of interspike intervals. Two methods for the estimation were applied, the regression method and the maximum-likelihood method. Both methods permit to estimate the input parameters and the membrane time constant in a short time window (a single interspike interval). We found that, at least in our example, the regression method gave more consistent results than the maximum-likelihood method. The estimates of the input parameters show the asymptotical normality, which can be further used for statistical testing, under the condition that the data are collected in different experimental situations. The model neuron, as deduced from the determined parameters, works in a subthreshold regimen. This result was confirmed by both applied methods. The subthreshold regimen for this model is characterized by the Poissonian firing. This is in a complete agreement with the observed interspike interval data.

The parameters of the stochastic leaky integrate-and-fire neuronal model

The thesis is divided into two parts. In Part I after an introductory chapter, a general approach is offered for the construction of recursions from a Prediction Error or Gaussian Likelihood viewpoint. This approach also allows other methods (for example, Model Reference) to be seen in context. Next, an heuristic argument is used to intuit the asymptotic properties of a general class of recursions (including the Prediction Error Recursions) and Time Series Stochastic Approximation schemes. The third chapter of Part I contains simulations illustrating some of the above ideas.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/5859A3EF5EFE45F5D54951BAF1641D1F/S0004972700009229a.pdf/div-class-title-time-series-recursions-and-stochastic-approximation-div.pdf

Time series recursions and stochastic approximation

Asymptotic tests of composite hypotheses for non-ergodic type stochastic processes

Asymptotic inference for stochastic processes

This thesis is primarily concerned with the investigation of asymptotic properties of the maximum likelihood estimate (MLE) of parameters of a stochastic process. These asymptotic properties are related to martingale limit theory by recognizing the (known) fact that, under certain regularity conditions, the derivative of the logarithm of the likelihood function is a martingale. To this end, part of the thesis is devoted to using or developing martingale limit theory to provide conditions for the consistency and/or asymptotic normality of the MLE. Thus, Chapter 1 is concerned with the martingale limit theory, while the remaining chapters look at its application to three broad types of stochastic processes. Chapter 2 extends the classical development of asymptotic theory of MLE's (a la Cramer [/]) to stochastic processes which, basically, behave in a non-explosive way and for which non-random norming sequences can be used. In this chapter we also introduce a generalization of Fisher's measure of information to the stochastic process situation. Chapter 3 deals with the theory for general processes develops the notion of \"conditional\" exponential families of processes, as well as establishing the importance of using random norming sequences. In Chapter it we consider the asymptotic theory of maximum likelihood estimation for continuous time processes and establish results which are analogous to those for discrete time processes. In each of these chapters many applications are considered in an attempt to show how known and new results fit into the general

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Paul David Feigin

Papers

Maximum likelihood estimation for stochastic processes - a martingale approach