Natural estimation of variances in a general finite discrete spectrum linear regression model

Hančová, Martina

doi:10.1007/s00184-007-0132-9

Cited by 4 publications

(12 citation statements)

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“…3 The BLUP estimation method for ν 3.1 Definition and computational form of estimators In the paper [26], we proposed the method of always non-negative natural estimators (NE) of FDSLRM variances ν. The main idea behind NE came from the fact that σ 2 j = Cov{Y j } = E Y 2 j ; j = 1, .…”

Section: Theoretical and Computational Bases For Fdslrm Krigingmentioning

confidence: 99%

“…To get sufficiently simple, explicit analytic expressions available for further theoretical study, we predicted the random vector Y by the ordinary least squares method leading to the following linear predictor of Y based on X [26]) the orthogonal projector onto the orthogonal complement of the column space of F and matrix W 0 ( 0 means positive definiteness) is known in linear algebra and statistics [65, chap. 6] as the Schur complement of F F in the block matrix F F F V V F V V .…”

Section: Theoretical and Computational Bases For Fdslrm Krigingmentioning

confidence: 99%

“…Following the analogy with original NE in [26], for the first component ν 0 = σ 2 0 of ν we could take as a BLUP based natural estimator the sum of squares of white noise residuals w…”

Section: Remarkmentioning

confidence: 99%

“…The principal goal of our paper is to present an alternative estimation method for FDSLRM variances, one of the time series kriging steps [21]. Our approach represents an improved two-stage modification of the previously developed method of natural estimators (NE) [26] which is now based on the idea of empirical (plugin) BLUPs. We will refer to our new method as EBLUP-NE for short.…”

Section: Introductionmentioning

confidence: 99%

“…Section 4 is devoted to the second stage of EBLUP-NE method which consists in replacing true, in practice unknown variance parameters ν by another, appropriate FDSLRM estimates of ν obtained by maximum likelihood or least squares [21]. Since the theory and computational aspects of these estimation procedures in FDSLRM fitting based on the projection theory in Hilbert spaces are more than 10 years old ( [57], [59], [26]), we revisit and update them in the light of recent advances in closely related linear mixed modeling and convex optimization.…”

Section: Introductionmentioning

confidence: 99%

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Estimating variances in time series linear regression models using empirical BLUPs and convex optimization

Hančová,

Vozáriková,

Gajdoš

et al. 2019

Preprint

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We propose a two-stage estimation method of variance components in time series models known as FDSLRMs, whose observations can be described by a linear mixed model (LMM). We based estimating variances, fundamental quantities in a time series forecasting approach called kriging, on the empirical (plug-in) best linear unbiased predictions of unobservable random components in FDSLRM. The method, providing invariant non-negative quadratic estimators, can be used for any absolutely continuous probability distribution of time series data. As a result of applying the convex optimization and the LMM methodology, we resolved two problems -theoretical existence and equivalence between least squares estimators, non-negative (M)DOOLSE, and maximum likelihood estimators, (RE)MLE, as possible starting points of our method and a practical lack of computational implementation for FDSLRM. As for computing (RE)MLE in the case of n observed time series values, we also discovered a new algorithm of order O(n), which at the default precision is 10 7 times more accurate and n 2 times faster than the best current Python(or R)-based computational packages, namely CVXPY, CVXR, nlme, sommer and mixed. We illustrate our results on three real data setselectricity consumption, tourism and cyber security -which are easily available, reproducible, sharable and modifiable in the form of interactive Jupyter notebooks. Keywords finite discrete spectrum linear regression model • linear mixed model • double ordinary least squares estimators • maximum likelihood estimators • efficient computational algorithms Mathematics Subject Classification (2010) 62M10 • 62J12 • 91B84 • 90C25

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Section: Theoretical and Computational Bases For Fdslrm Krigingmentioning

confidence: 99%

Section: Theoretical and Computational Bases For Fdslrm Krigingmentioning

confidence: 99%

“…Following the analogy with original NE in [26], for the first component ν 0 = σ 2 0 of ν we could take as a BLUP based natural estimator the sum of squares of white noise residuals w…”

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%