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Bias correction of quadratic spectral estimators

Lachlan Astfalck, Adam Sykulski, Edward Cripps

Abstract

The three cardinal, statistically consistent, families of non-parametric estimators to the power spectral density of a time series are lag-window, multitaper and Welch estimators. However, when estimating power spectral densities from a finite sample each can be subject to non-ignorable bias. Astfalck et al. (2024) developed a method that offers significant bias reduction for finite samples for Welch's estimator, which this article extends to the larger family of quadratic estimators, thus offering similar theory for bias correction of lag-window and multitaper estimators as well as combinations thereof. Importantly, this theory may be used in conjunction with any and all tapers and lag-sequences designed for bias reduction, and so should be seen as an extension to valuable work in these fields, rather than a supplanting methodology. The order of computation is larger than O(n log n) typical in spectral analyses, but not insurmountable in practice. Simulation studies support the theory with comparisons across variations of quadratic estimators.

Bias correction of quadratic spectral estimators

Abstract

The three cardinal, statistically consistent, families of non-parametric estimators to the power spectral density of a time series are lag-window, multitaper and Welch estimators. However, when estimating power spectral densities from a finite sample each can be subject to non-ignorable bias. Astfalck et al. (2024) developed a method that offers significant bias reduction for finite samples for Welch's estimator, which this article extends to the larger family of quadratic estimators, thus offering similar theory for bias correction of lag-window and multitaper estimators as well as combinations thereof. Importantly, this theory may be used in conjunction with any and all tapers and lag-sequences designed for bias reduction, and so should be seen as an extension to valuable work in these fields, rather than a supplanting methodology. The order of computation is larger than O(n log n) typical in spectral analyses, but not insurmountable in practice. Simulation studies support the theory with comparisons across variations of quadratic estimators.

Paper Structure

This paper contains 7 sections, 1 theorem, 20 equations, 2 figures.

Table of Contents

  1. Quadratic Spectral Estimators
  2. Statistical Properties
  3. Bias correction of quadratic spectral estimators
  4. Bias correction
  5. Computation
  6. Flexible basis functions
  7. Simulation study with canonical AR(4) model

Key Result

Proposition 1

Given $\{\mathrm{X}_t\}$ satisfies Assumption 1, and $\omega \neq 0, \pm \pi$, the bias of a quadratic spectral estimator, $I_\mathrm{Quad}(\omega)$ is of order

Figures (2)

  • Figure 1: Standard and debiased lag-window, multitaper and Welch estimates of a random ar(4) process. The standard and debiased estimates are shown by the solid grey and black lines, respectively; the true process spectrum is shown by the black dashed line and the expectation of the standard estimator by the grey dashed line. Plot insets correspond to the shaded dotted boxes. Parameterisations of the ar(4) process and the estimators are given in the main text.
  • Figure 2: Estimates of bias and root-mean-squared-error of the standard (solid) and debiased (dashed) lag-window, multitaper and Welch estimates as a function of the factor of variance reduction, $\mathrm{M}$. Each graphed point is obtained by calculating the metric from a 1000 member ensemble, for each frequency, and aggregating over frequency by the mean log value. Parameterisations of the ar(4) process and the estimators is given in the main text.

Theorems & Definitions (12)

  • Definition 1
  • Example 1: Periodogram
  • Example 2: Lag-window
  • Example 3: Welch's estimator
  • Example 4: Multitaper
  • Proposition 1
  • proof
  • Remark 1
  • Example 5: Parameterisation of Bartlett's estimator
  • Example 6: Parameterisation of an infinite-order flat-top lag-sequence
  • ...and 2 more