Inference with the Whittle Likelihood: A Tractable Approach Using Estimating Functions

We provide a computationally and statistically efficient method for estimating the parameters of a stochastic covariance model observed on a regular spatial grid in any number of dimensions. Our proposed method, which we call the Debiased Spatial Whittle likelihood, makes important corrections to the well‐known Whittle likelihood to account for large sources of bias caused by boundary effects and aliasing. We generalize the approach to flexibly allow for significant volumes of missing data including those with lower‐dimensional substructure, and for irregular sampling boundaries. We build a theoretical framework under relatively weak assumptions which ensures consistency and asymptotic normality in numerous practical settings including missing data and non‐Gaussian processes. We also extend our consistency results to multivariate processes. We provide detailed implementation guidelines which ensure the estimation procedure can be conducted in O(nlogn) operations, where n is the number of points of the encapsulating rectangular grid, thus keeping the computational scalability of Fourier and Whittle‐based methods for large data sets. We validate our procedure over a range of simulated and realworld settings, and compare with state‐of‐the‐art alternatives, demonstrating the enduring practical appeal of Fourier‐based methods, provided they are corrected by the procedures developed in this paper.

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“…Hence the quantity given in Equation (5) can be seen as a composite likelihood (Bevilacqua & Gaetan, 2015; Varin et al, 2011). We also observe that

{normal∇}_{bold-italicθ} ℓ false(bold-italicθ false) = 0

such that our method fits within the general theory of estimating equations (Heyde, 1997; Jesus & Chandler, 2017).…”

Section: Methodssupporting

confidence: 67%

The Debiased Spatial Whittle Likelihood

Guillaumin

Sykulski

Olhede

et al. 2022

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…The Whittle likelihood of Whittle (1953) is a frequency-domain approximation to the exact likelihood. This method is considered a standard method in parametric spectral analysis on account of its order n log n computational efficiency (Choudhuri et al, 2004;Fuentes, 2007;Matsuda & Yajima, 2009;Krafty & Collinge, 2013;Jesus & Chandler, 2017). However, it has been observed that the Whittle likelihood, despite its desirable asymptotic properties, may exhibit poor properties when applied to real-world, finite-length time series, particularly in terms of estimation bias (Dahlhaus, 1988;Velasco & Robinson, 2000;Contreras-Cristan et al, 2006).…”

Section: Introductionmentioning

confidence: 99%

The debiased Whittle likelihood

et al. 2019

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The Whittle likelihood is a widely used and computationally efficient pseudo-likelihood. However, it is known to produce biased parameter estimates for large classes of models. We propose a method for de-biasing Whittle estimates for second-order stationary stochastic processes. The de-biased Whittle likelihood can be computed in the same O(n log n) operations as the standard approach. We demonstrate the superior performance of the method in simulation studies and in application to a large-scale oceanographic dataset, where in both cases the de-biased approach reduces bias by up to two orders of magnitude, achieving estimates that are close to exact maximum likelihood, at a fraction of the computational cost. We prove that the method yields estimates that are consistent at an optimal convergence rate of n −1/2 , under weaker assumptions than standard theory, where we do not require that the power spectral density is continuous in frequency. We describe how the method can be easily combined with standard methods of bias reduction, such as tapering and differencing, to further reduce bias in parameter estimates.

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“…The BND procedure and all three BFR( ) procedures estimate d well, regardless of the value of chosen or true value of the memory parameter. Because it is more challenging to estimate percentiles of Whittle likelihood estimators d d well (Robinson, 1978;Jesus and Chandler, 2017), we only consider the BND procedure to obtain adaptive estimates of d obtained by maximizing the Whittle likelihood. Figure 7 shows the average estimates of d obtained using the BND procedure, and indicates that BND procedure produces excellent estimates of d regardless of the true value of the memory parameter.…”

Section: Adaptive Point Estimation Of the Memory Parametermentioning

confidence: 99%

Likelihood Inference for Possibly Non-Stationary Processes via Adaptive Overdifferencing

Griffin¹,

Samorodnitsky²,

Matteson³

2020

Preprint

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We make a simple observation that facilitates valid likelihood-based inference for the parameters of the popular ARFIMA or FARIMA model without requiring stationarity by allowing the upper bound d for the memory parameter d to exceed 0.5. We observe that estimating the parameters of a single non-stationary ARFIMA model is equivalent to estimating the parameters of a sequence of stationary ARFIMA models. This enables improved inference because many standard methods perform poorly when estimates are close to the boundary of the parameter space. It also allows us to leverage the wealth of likelihood approximations that have been introduced for estimating the parameters of a stationary process. We explore how estimation of the memory parameter d depends on the upper bound d and introduce adaptive procedures for choosing d. Via simulations, we examine the performance of our adaptive procedures for estimating the memory parameter when the true value is as large as 2.5. Our adaptive procedures estimate the memory parameter well, can be used to obtain confidence intervals for the memory parameter that achieve nominal coverage rates, and perform favorably relative to existing alternatives.

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Inference with the Whittle Likelihood: A Tractable Approach Using Estimating Functions

Cited by 4 publications

References 35 publications

The Debiased Spatial Whittle Likelihood

The Debiased Spatial Whittle Likelihood

The debiased Whittle likelihood

Likelihood Inference for Possibly Non-Stationary Processes via Adaptive Overdifferencing

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