The Annals of Statistics

On Conditional Least Squares Estimation for Stochastic Processes

Lawrence A. Klimko and Paul I. Nelson

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An estimation procedure for stochastic processes based on the minimization of a sum of squared deviations about conditional expectations is developed. Strong consistency, asymptotic joint normality and an iterated logarithm rate of convergence are shown to hold for the estimators under a variety of conditions. Special attention is given to the widely studied cases of stationary ergodic processes and Markov processes with are asymptotically stationary and ergodic. The estimators and their limiting covariance matrix are worked out in detail for a subcritical branching process with immigration. A brief Monte Carlo study of the performance of the estimators is presented.

Article information

Ann. Statist., Volume 6, Number 3 (1978), 629-642.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M05: Markov processes: estimation
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62F10: Point estimation

Estimation ergodic Markov processes stationary processes consistency asymptotic normality iterated logarithm branching process with immigration time series


Klimko, Lawrence A.; Nelson, Paul I. On Conditional Least Squares Estimation for Stochastic Processes. Ann. Statist. 6 (1978), no. 3, 629--642. doi:10.1214/aos/1176344207.

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