# Estimating the fundamental frequency using modified Newton-Raphson algorithm

## TL;DR

This paper introduces a modified Newton-Raphson algorithm for estimating the fundamental frequency, achieving super efficiency and comparable convergence rates to least squares estimators, validated through simulations and real data analysis.

## Contribution

It proposes a novel modified Newton-Raphson algorithm that improves frequency estimation accuracy and efficiency in fundamental frequency models with stationary errors.

## Key findings

- The estimator has lower asymptotic variance than least squares.
- The algorithm achieves an $O_p(n^{-3/2})$ convergence rate.
- Numerical and real data analyses confirm the method's effectiveness.

## Abstract

In this paper, we propose a modified Newton-Raphson algorithm to estimate the frequency parameter in the fundamental frequency model in presence of an additive stationary error. The proposed estimator is super efficient in nature in the sense that its asymptotic variance is less than the asymptotic variance of the least squares estimator. With a proper step factor modification, the proposed modified Newton-Raphson algorithm produces an estimator with the rate $O_p(n^{-\frac{3}{2}})$, the same rate as the least squares estimator. Numerical experiments are performed for different sample sizes, different error variances and for different models. For illustrative purposes, two real data sets are analyzed using the fundamental frequency model and the estimators are obtained using the proposed algorithm. It is observed the model and the proposed algorithm work quite well in both cases.