arXiv subjects

Trisha Maitra

Publications and source records attributed to Trisha Maitra.

9 recordsLinked to original sources

arXiv2020

Increasing Domain Infill Asymptotics for Stochastic Differential Equations Driven by Fractional Brownian Motion

Although statistical inference in stochastic differential equations (SDEs) driven by Wiener process has received significant attention in the literature, inference in those driven by fractional Brownian motion seem to have seen much less development in comparison, despite their importance in modeling long range dependence. In this article, we consider both classical and Bayesian inference in such fractional Brownian motion based SDEs. In particular, we consider asymptotic inference for two parameters in this regard; a multiplicative parameter associated with the drift function, and the so-called "Hurst parameter" of the fractional Brownian motion, when the time domain tends to infinity. For unknown Hurst parameter, the likelihood does not lend itself amenable to the popular Girsanov form, rendering usual asymptotic development difficult. As such, we develop increasing domain infill asymptotic theory, by discretizing the SDE. In this setup, we establish consistency and asymptotic normality of the maximum likelihood estimators, as well as consistency and asymptotic normality of the Bayesian posterior distributions. However, classical or Bayesian asymptotic normality with respect to the Hurst parameter could not be established. We supplement our theoretical investigations with simulation studies in a non-asymptotic setup, prescribing suitable methodologies for classical and Bayesian analyses of SDEs driven by fractional Brownian motion. Applications to a real, close price data, along with comparison with standard SDE driven by Wiener process, is also considered. As expected, it turned out that our Bayesian fractional SDE triumphed over the other model and methods, in both simulated and real data applications.

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arXiv2020

On Classical and Bayesian Asymptotics in Stochastic Differential Equations with Random Effects having Mixture Normal Distributions

Delattre et al. (2013) considered a system of stochastic differential equations (SDEs) in a random effects setup. Under the independent and identical (iid) situation, and assuming normal distribution of the random effects, they established weak consistency of the maximum likelihood estimators (M LEs) of the population parameters of the random effects. In this article, respecting the increasing importance and versatility of normal mixtures and their ability to approximate any standard distribution, we consider the random effects having mixture of normal distributions and prove asymptotic results associated with the MLEs in both independent and identical (iid) and independent but not identical (non-iid) situations. Besides, we consider iid and non-iid setups under the Bayesian paradigm and establish posterior consistency and asymptotic normality of the posterior distribution of the population parameters, even when the number of mixture components is unknown and treated as a random variable. Although ours is an independent work, we later noted that Delattre et al. (2016) also assumed the SDE setup with normal mixture distribution of the random effect parameters but considered only the iid case and proved only weak consistency of the M LE under an extra, strong assumption as opposed to strong consistency that we are able to prove without the extra assumption. Furthermore, they did not deal with asymptotic normality of M LE or the Bayesian asymptotics counterpart which we investigate in details. Ample simulation experiments and application to a real, stock market data set reveal the importance and usefulness of our methods even for small samples.

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arXiv2018

On Classical and Bayesian Asymptotics in State Space Stochastic Differential Equations

In this article we investigate consistency and asymptotic normality of the maximum likelihood and the posterior distribution of the parameters in the context of state space stochastic differential equations (SDEs). We then extend our asymptotic theory to random effects models based on systems of state space SDEs, covering both independent and identical and independent but non-identical collections of state space SDEs. We also address asymptotic inference in the case of multidimensional linear random effects, and in situations where the data are available in discretized forms. It is important to note that asymptotic inference, either in the classical or in the Bayesian paradigm, has not been hitherto investigated in state space SDEs.

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arXiv2018

Asymptotic Theory of Bayes Factor in Stochastic Differential Equations: Part I

Research on asymptotic model selection in the context of stochastic differential equations (SDEs) is almost non-existent in the literature. In particular, when a collection of SDEs is considered, the problem of asymptotic model selection has not been hitherto investigated. Indeed, even though the diffusion coefficients may be considered known, questions on appropriate choice of the drift functions constitute a non-trivial model selection problem. In this article, we develop the asymptotic theory for comparisons between collections of SDEs with respect to the choice of drift functions using Bayes factors when the number of equations (individuals) in the collection of SDEs tend to infinity while the time domains remain bounded for each equation. Our asymptotic theory covers situations when the observed processes associated with the SDEs are independently and identically distributed (iid), as well as when they are independently but not identically distributed (non-iid). In particular, we allow incorporation of available time-dependent covariate information into each SDE through a multiplicative factor of the drift function; we also permit different initial values and domains of observations for the SDEs. Our model selection problem thus encompasses selection of a set of appropriate time-dependent covariates from a set of available time-dependent covariates, besides selection of the part of the drift function free of covariates. For both iid and non-iid set-ups we establish almost sure exponential convergence of the Bayes factor. Furthermore, we demonstrate with simulation studies that even in non-asymptotic scenarios Bayes factor successfully captures the right set of covariates.

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arXiv2018

Asymptotic Theory of Bayes Factor in Stochastic Differential Equations: Part II

The problem of model selection in the context of a system of stochastic differential equations (SDEs) has not been touched upon in the literature. Indeed, properties of Bayes factors have not been studied even in single SDE based model comparison problems. In this article, we first develop an asymptotic theory of Bayes factors when two SDEs are compared, assuming the time domain expands. Using this we then develop an asymptotic theory of Bayes factors when systems of SDEs are compared, assuming that the number of equations in each system, as well as the time domain, increase indefinitely. Our asymptotic theory covers situations when the observed processes associated with the SDEs are independently and identically distributed (iid), as well as when they are independently but not identically distributed (non-iid). Quite importantly, we allow inclusion of available time-dependent covariate information into each SDE through a multiplicative factor of the drift function in a random effects set-up; different initial values for the SDEs are also permitted. Thus, our general model-selection framework includes simultaneously the variable selection problem associated with time-varying covariates, as well as choice of the part of the drift function free of covariates. As we show, the Bayes factor is inconsistent for comparing individual SDEs, in the sense that the log-Bayes factor converges only in expectation, while the relevant variance does not converge to zero. Nevertheless, it has been possible to exploit this result to establish almost sure exponential convergence of the Bayes factor when, in addition, the number of individuals are also allowed to increase indefinitely. We carry out simulated and real data analyses to demonstrate that Bayes factor is a suitable candidate for covariate selection in our SDE models even in non-asymptotic situations.

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arXiv2018

A Short Note on Almost Sure Convergence of Bayes Factors in the General Set-Up

Although there is a significant literature on the asymptotic theory of Bayes factor, the set-ups considered are usually specialized and often involves independent and identically distributed data. Even in such specialized cases, mostly weak consistency results are available. In this article, for the first time ever, we derive the almost sure convergence theory of Bayes factor in the general set-up that includes even dependent data and misspecified models. Somewhat surprisingly, the key to the proof of such a general theory is a simple application of a result of Shalizi (2009) to a well-known identity satisfied by the Bayes factor.

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arXiv2017

On Asymptotic Inference in Stochastic Differential Equations with Time-Varying Covariates

In this article, we introduce a system of stochastic differential equations (SDEs) consisting of time-dependent covariates and consider both fixed and random effects set-ups. We also allow the functional part associated with the drift function to depend upon unknown parameters. In this general set-up of SDE system we establish consistency and asymptotic normality of the M LE through verification of the regularity conditions required by existing relevant theorems. Besides, we consider the Bayesian approach to learning about the population parameters, and prove consistency and asymptotic normality of the corresponding posterior distribution. We supplement our theoretical investigation with simulated and real data analyses, obtaining encouraging results in each case.

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arXiv2016

On Asymptotics Related to Classical Inference in Stochastic Differential Equations with Random Effects

Delattre et al. (2013) considered n independent stochastic differential equations (SDEs), where in each case the drift term is associated with a random effect, the distribution of which depends upon unknown parameters. Assuming the independent and identical (iid) situation the authors provide independent proofs of weak consistency and asymptotic normality of the maximum likelihood estimators (MLEs) of the hyper-parameters of their random effects parameters. In this article, as an alternative route to proving consistency and asymptotic normality in the SDE set-up involving random effects, we verify the regularity conditions required by existing relevant theorems. In particular, this approach allowed us to prove strong consistency under weaker assumption. But much more importantly, we further consider the independent, but non-identical set-up associated with the random effects based SDE framework, and prove asymptotic results associated with the MLEs.

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arXiv2016

On Bayesian Asymptotics in Stochastic Differential Equations with Random Effects

Delattre et al. (2013) investigated asymptotic properties of the maximum likelihood estimator of the population parameters of the random effects associated with n independent stochastic differential equations (SDEs) assuming that the SDEs are independent and identical (iid). In this article, we consider the Bayesian approach to learning about the population parameters, and prove consistency and asymptotic normality of the corresponding posterior distribution in the iid set-up as well as when the SDEs are independent but non-identical.

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