Consistency of the least squares estimates of trigonometric regression model parameters in the presence of linear random noise.

Abstract

Regression analysis is a huge part of mathematical and applied statistics. Nonlinear regression analysis is a significant extension and complication of classical linear regression analysis, due to the use of nonlinear or partially nonlinear in parameters models that describe more adequately than linear model phenomena requiring statistical analysis. A large number of applied problems in the numerical scientific, technical, and humanitarian fields of knowledge give impetus to the development of nonlinear regression analysis. The task of estimation the vector signal parameter in the «signal + noise» observation models is a well-known problem of statistics of stochastic processes, and in the case of a nonlinear signal parameter is the problem of nonlinear regression analysis. Among the variety of nonlinear regression analysis problems the problem of estimating amplitudes and angular frequencies of the sum of harmonic oscillations that are observed against the background of a random noise, takes significant place due to its numerous applications. Statistical model of such a type is said to be trigonometric regression, and the problem of statistical estimation is called the problem of detecting hidden periodicities. The paper is devoted to the study of time continuous trigonometric regression model where the random noise is a linear L´evy driven stationary of the fourth order stochastic process with zero mean, integrable and square integrable impulse transmission function. This assumption leads to the integrability of the noise covariance function and cumulant function of the fourth order. To estimate unknown amplitudes and angular frequencies of such a trigonometric model we use the least squares estimators in the Walker sense, that is special parametric set are considered to distinguish properly different angular frequencies in the sum of harmonic oscillations. Theorem on strong consistency of the least squares estimators is proved in the paper under the assumption on the random noise described above. To obtain such a result a very important lemma was proved on the uniform tending to zero almost surely of the average value of L´evy-driven linear stochastic process Fourier transform. This Lemma is the main tool of the strong consistency Theorem proof. To prove the Theorem we, firstly, find some expressions for the least squares estimates of amplitudes via corresponding estimates of angular frequencies. Secondly, we substitute these formulas into the functional of the least squares method. The last step of the proof consists of the L2- norm transformation of the difference between empirical trigonometric regression function and true regression function such that this norm tends to zero almost surely if and on if the estimators are strongly consistent.

Keywords

• the detection of hidden periodicities
• least squares estimator
• consistency
• lévy-driven linear stochastic process.