Asymptotic distribution

In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical estimators. Definition A sequence of distributions corresponds to a sequence of random variables Zi for i = 1, 2, ..., I . In the simplest case, an asymptotic distribution exists if the probability distribution of Zi converges to a probability distribution (the asymptotic distribution) as i increases: see convergence in distribution. A special case of an asymptotic distribution is when the sequence of random variables is always zero or Zi = 0 as i approaches infinity. Here the asymptotic distribution is a degenerate distribution, corresponding to the value zero. However, the most usual sense in which the term asymptotic distribution is used arises where the rand

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Estimating an extreme Bayesian network via scalings

Mario Krali

A recursive max-linear vector models causal dependence between its components by expressing each node variable as a max-linear function of its parental nodes in a directed acyclic graph and some exogenous innovation. Motivated by extreme value theory, innovations are assumed to have regularly varying distribution tails. We propose a scaling technique in order to determine a causal order of the node variables. All dependence parameters are then estimated from the estimated scalings. Furthermore, we prove asymptotic normality of the estimated scalings and dependence parameters based on asymptotic normality of the empirical spectral measure. Finally, we apply our structure learning and estimation algorithm to financial data and food dietary interview data. (C) 2020 Elsevier Inc. All rights reserved.

ELSEVIER INC2021

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Frequency domain theory for functional time series: Variance decomposition and an invariance principle

Neda Mohammadi Jouzdani

This paper is concerned with frequency domain theory for functional time series, which are temporally dependent sequences of functions in a Hilbert space. We consider a variance decomposition, which is more suitable for such a data structure than the variance decomposition based on the Karhunen-Loeve expansion. The decomposition we study uses eigenvalues of spectral density operators, which are functional analogs of the spectral density of a stationary scalar time series. We propose estimators of the variance components and derive convergence rates for their mean square error as well as their asymptotic normality. The latter is derived from a frequency domain invariance principle for the estimators of the spectral density operators. This principle is established for a broad class of linear time series models. It is a main contribution of the paper.

INT STATISTICAL INST2020

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The Debiased Spatial Whittle likelihood

Sofia Charlotta Olhede

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.

WILEY2022

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