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The studies of analysis of variance components is one of the important topics in mathematical statistics for this subject of wide application. In this paper given best quadratic unbiased estimator of variance components for balanced data for linear one-way repeated measurement model (RMM). We computed the quadratic unbiased estimator, which has minimum variance (best quadratic unbiased estimate (BQUE)) by using analysis of variance (ANOVA) method of estimating the variance components.

Extragalactic foregrounds are known to constitute a limiting systematic in temperature-based CMB lensing with AdvACT, SPT-3G, Simons Observatory and CMB S4. Furthermore, since these foregrounds are emitted at cosmological distances, they are also themselves lensed. The correlation between this foreground lensing and CMB lensing causes an additional bias in CMB lensing estimators. In this paper, we quantify for the first time this "lensed foreground bias" for the standard CMB lensing quadratic estimator, the CMB shear and the CMB magnification estimators, in the case of Simons Observatory and in the absence of multi-frequency component separation. This percent-level bias is highly significant in cross-correlation of CMB lensing with LSST galaxies, and comparable to the statistical uncertainty in CMB lensing auto-spectrum. We discuss various mitigation strategies, and show that "lensed foreground bias-hardening" methods can reduce this bias at some cost in signal-to-noise. The code used to generate our theory curves is publicly available at https://github.com/EmmanuelSchaan/LensedForegroundBias ., Comment: Version accepted in PRD. Code available at https://github.com/EmmanuelSchaan/LensedForegroundBias

We demonstrate how to obtain optimal constraints on a primordial gravitational wave component in lensed Cosmic Microwave Background (CMB) data under ideal conditions. We first derive an estimator of the tensor-to-scalar ratio, $r$, by using an error-controlled close approximation to the exact posterior, under the assumption of Gaussian primordial CMB and lensing deflection potential. This combines fast internal iterative lensing reconstruction with optimal recovery of the unlensed CMB. We evaluate its performance on simulated low-noise polarization data targeted at the recombination peak. We carefully demonstrate our $r$-posterior estimate is optimal and shows no significant bias, making it the most powerful estimator of primordial gravitational waves from the CMB. We compare these constraints to those obtained from $B$-mode band-power likelihood analyses on the same simulated data, before and after map-level quadratic estimator delensing, and iterative delensing. Internally, iteratively delensed band powers are only slightly less powerful on average (by less than 10\%), promising close-to-optimal constraints from a stage IV CMB experiment., Matches version accepted by PRD (No significant changes to v1)

This chapter gives a fresh perspective on the best speech enhancement estimator in the frequency domain. We consider the general nonlinear case. In the first part, we deal with the single-channel scenario, where an example is studied with gamma distributions and a best quadratic estimator is derived. Then, in the second part, we focus on the multichannel scenario.

A new approach for optimal observer design of nonlinear systems, the so-called Nonlinear Quadratic Estimator is proposed. This approach employs the minimisation of a quadratic cost functional, thereby comprising two design parameters: Selective weighting of specific error state components and estimated upper bound minimisation. The new approach works without dual system transformation, achieving significant error state minimisation with optimised error dynamics and enabling a selective error state minimisation. Within this proceeding the observer and estimator design method, respectively, is derived from a Lyapunov stability condition of nonlinear, time-continuous systems in Takagi-Sugeno model structure, solved with linear matrix inequalities. Its capability is illustrated for an academical example of a nonlinear system with observer based stabilisation.

We present a measurement of the cosmic microwave background (CMB) gravitational lensing potential using data from the first two seasons of observations with SPTpol, the polarization-sensitive receiver currently installed on the South Pole Telescope (SPT). The observations used in this work cover 100 deg$^2$ of sky with arcminute resolution at 150 GHz. Using a quadratic estimator, we make maps of the CMB lensing potential from combinations of CMB temperature and polarization maps. We combine these lensing potential maps to form a minimum-variance (MV) map. The lensing potential is measured with a signal-to-noise ratio of greater than one for angular multipoles between $100< L, Comment: 16 pages, 8 figures

In this paper, we first point out that a result in Mukhopadhyay (1994) on the optimality of the usual estimator sy2 of finite population variance is not true. We then give a necessary and sufficient condition for ((1 − f)/n) sy2 (where f means the sampling fraction) as the estimator of the precision of the sample mean y s to be admissible in the class of quadratic estimators. Our result shows that there is virtual difference between the admissibility of estimators under error-in-variables superpopulation model and the usual superpopulation model. We also show that the improved estimator ((1 − f)/n) ((n − 1)/(n + 1)) sy2 over ((1 − f)/n) sy2 under the usual superpopulation model without measurement errors is admissible in the class of quadratic estimators.

An additive growth curve model with orthogonal design matrices is proposed in which observations may have different profile forms. The proposed model allows us to fit data and then estimate parameters in a more parsimonious way than the traditional growth curve model. Two-stage generalized least-squares estimators for the regression coefficients are derived where a quadratic estimator for the covariance of observations is taken as the first-stage estimator. Consistency, asymptotic normality and asymptotic independence of these estimators are investigated. Simulation studies and a numerical example are given to illustrate the efficiency and parsimony of the proposed model for model specifications in the sense of minimizing Akaike's information criterion (AIC)., Published in at http://dx.doi.org/10.3150/10-BEJ315 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Minimum norm quadratic estimators were introduced by C. R. Rao in the 70's to attempt to give an unified approach to variance component estimation. One such estimator, which is called invariant nonnegative minimum norm quadratic, was proposed but not explicitly determined. In this paper, a fnormal solution to the invariant nonnegative minimum norm quadratic estimator is presented and then applied to some special cases.

Samenvatting De beste kwadratische schattingsfunctie van de storingsvariantie in regressie-analyse. Dit artikel handelt over de schatting van de variantie σ2 van de storingen in de regressieanalyse onder klassieke veronderstellingen: niet-stochastische waarden aangenomen door de verklarende variabelen en normaliteit, onafhankelijkheid en homoskedasticiteit van de storingen. Bekend is dat de schatting volgens maximale aannemelijkheid neerkomt op net bepalen van de kwadratensom van de volgens kleinste-kwadraten geschatte storingen en deling door T (het aantal waarne-mingen); voorts, dat de schatting die minimale variantie heeft binnen de klasse van schattingsfuncties die zuiver zijn en kwadratisch in de afhankelijke variabele (de beste zuivere kwadratische schattingsfunctie) gevonden wordt door genoemde kwadratensom te delen door T–A, waarbij λ het aantal te schatten coefficienten is [d.w.z. het aantal verklarende variabelen (+ 1 indien een constante term aanwezig is)]. Hier wordt aangetoond, dat de schattingsfunctie van σ2 die een minimaal tweede moment heeft binnen de klasse van schattingsfuncties die kwadratisch zijn in de afhankelijke variabele (de beste kwadratische schattingsfunctie) gevonden wordt door de kwadratensom van de volgens kleinste kwadraten geschatte storingen te delen door T–Λ+ 2.