Estimation of the Parameters of a Chirp Type Model with Stationary Residuals

K Perera

doi:10.1155/2017/6219149

Abstract

Let Xn1,…,Xnn be the observations from a chirp type statistical model Xnt, Xnt=Acos⁡ (ωt+Δ/nt2)+Bsin⁡ ωt+Δ/nt2+ϵt, where ϵt is a stationary noise. We consider a method of estimation of parameters, A, B, ω, Δ, and ν, (where ν is the variance of ϵt’s) which is basically an approximate least-squares method. The main advantage of the proposed approach is that no assumptions are required. We make use of the three theorems which were established associated with the kernel ∑t=1neiut+vt2 and then use them to prove, under certain conditions, the consistency of the estimators.

Highlights

In [1] 1973 Walker considered the problem of estimating the parameters of a sine wave, Xt = A cos ωt + B sin ωt + εt, (1)
As n → ∞, the estimators Ân, Bn, ωn, and]n converge in probability to the actual values A0, B0, ω0, and ]0, respectively
Our objective is to show that these estimators are consistent

Summary

Introduction

The parameters A, B, ω, and ] are assumed unknown and are to be estimated He showed that, as n → ∞, the estimators Ân, Bn, ωn, and]n converge in probability to the actual values A0, B0, ω0, and ]0, respectively. Different approaches to the estimation of chirp parameters in similar kinds of models are found in [8,9,10,11,12]. We change the model by assuming that the change in frequency over the course of the n observations is a number Δ independent of n. As in (3), that the change in frequency is linear This leads to the model or, more precisely, sequence of models.

Estimation of the Parameters

The Consistency of the Estimators

Δ 0 t2 n