Electronic Journal of Statistics

Recursive construction of confidence regions

Tomasz Bielecki, Tao Chen, and Igor Cialenco

Full-text: Open access

Abstract

Assuming that one-step transition kernel of a discrete time, time-homogenous Markov chain model is parameterized by a parameter $\theta\in\boldsymbol{\Theta}$, we derive a recursive (in time) construction of confidence regions for the unknown parameter of interest, say $\theta^{*}\in\boldsymbol{\Theta}$. It is supposed that the observed data used in the construction of the confidence regions is generated by a Markov chain whose transition kernel corresponds to $\theta^{*}$. The key step in our construction is the derivation of a recursive scheme for an appropriate point estimator of $\theta^{*}$. To achieve this, we start by what we call the base recursive point estimator, using which we design a quasi-asymptotically linear recursive point estimator (a concept introduced in this paper). For the latter estimator we prove its weak consistency and asymptotic normality. The recursive construction of confidence regions is needed not only for the purpose of speeding up the computation of the successive confidence regions, but, primarily, for the ability to apply the dynamic programming principle in the context of robust adaptive stochastic control methodology.

Article information

Source
Electron. J. Statist., Volume 11, Number 2 (2017), 4674-4700.

Dates
Received: May 2017
First available in Project Euclid: 18 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1510974130

Digital Object Identifier
doi:10.1214/17-EJS1362

Mathematical Reviews number (MathSciNet)
MR3724972

Zentralblatt MATH identifier
06816629

Subjects
Primary: 62M05: Markov processes: estimation 62F10: Point estimation 62F12: Asymptotic properties of estimators 62F25: Tolerance and confidence regions 60J05: Discrete-time Markov processes on general state spaces 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

Keywords
Recursive confidence regions stochastic approximation recursive point estimators statistical inference for Markov chains ergodic processes quasi-asymptotically linear estimator

Rights
Creative Commons Attribution 4.0 International License.

Citation

Bielecki, Tomasz; Chen, Tao; Cialenco, Igor. Recursive construction of confidence regions. Electron. J. Statist. 11 (2017), no. 2, 4674--4700. doi:10.1214/17-EJS1362. https://projecteuclid.org/euclid.ejs/1510974130


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