Class Of Linear Estimators Research Articles

Reading the Candlesticks: An OK Estimator for Volatility

Abstract We propose an Optimal candlesticK (OK) estimator for the spot volatility using high-frequency candlestick observations. Under a standard infill asymptotic setting, we show that the OK estimator is asymptotically unbiased and has minimal asymptotic variance within a class of linear estimators. Its estimation error can be coupled by a Brownian functional, which permits valid inference. Our theoretical and numerical results suggest that the proposed candlestick-based estimator is much more accurate than the conventional spot volatility estimator based on high-frequency returns. An empirical illustration documents the intraday volatility dynamics of various assets during the Fed chairman’s recent congressional testimony.

Adaptive and efficient estimation in the Gaussian sequence model

This paper deals with the problem of estimating the mean of an infinite-dimensional Gaussian vector by the principle of minimizing unbiased risk estimation (URE). The aim is to obtain an adaptive estimator to mimic the oracle with the smallest squared L2 loss/risk in a given class of linear estimators Λ. It is noticed that there is an essential balance between enlarging Λ for more efficient oracle risk and contracting Λ for the adaptivity to the oracle. This paper proves that this balance can be well achieved by minimizing URE in a list of intermediate classes between the projection and monotone classes. The resulting estimators are adaptive on the nonparametric space and have more efficient risks than the projection estimators under certain conditions.

Minimax efficient finite-difference stochastic gradient estimators using black-box function evaluations

Standard approaches to stochastic gradient estimation, with only noisy black-box function evaluations, use the finite-difference method or its variants. Though natural, it is open to our knowledge whether their statistical accuracy is the best possible. This paper argues so by showing that central finite-difference is a nearly minimax optimal zeroth-order gradient estimator for a suitable class of objective functions and mean squared risk, among both the class of linear estimators and the much larger class of all (nonlinear) estimators.

Empirical Bayes sequential estimation of a finite population mean

ABSTRACTIn this article, we consider the Bayes and empirical Bayes problem of the current population mean of a finite population when the sample data is available from other similar (m-1) finite populations. We investigate a general class of linear estimators and obtain the optimal linear Bayes estimator of the finite population mean under a squared error loss function that considered the cost of sampling. The optimal linear Bayes estimator and the sample size are obtained as a function of the parameters of the prior distribution. The corresponding empirical Bayes estimates are obtained by replacing the unknown hyperparameters with their respective consistent estimates. A Monte Carlo study is conducted to evaluate the performance of the proposed empirical Bayes procedure.

Properties of Estimators in Exponential Family Settings with Observation-based Stopping Rules.

Often, sample size is not fixed by design. A key example is a sequential trial with a stopping rule, where stopping is based on what has been observed at an interim look. While such designs are used for time and cost efficiency, and hypothesis testing theory has been well developed, estimation following a sequential trial is a challenging, still controversial problem. Progress has been made in the literature, predominantly for normal outcomes and/or for a deterministic stopping rule. Here, we place these settings in a broader context of outcomes following an exponential family distribution and, with a stochastic stopping rule that includes a deterministic rule and completely random sample size as special cases. It is shown that the estimation problem is usually simpler than often thought. In particular, it is established that the ordinary sample average is a very sensible choice, contrary to commonly encountered statements. We study (1) The so-called incompleteness property of the sufficient statistics, (2) A general class of linear estimators, and (3) Joint and conditional likelihood estimation. Apart from the general exponential family setting, normal and binary outcomes are considered as key examples. While our results hold for a general number of looks, for ease of exposition, we focus on the simple yet generic setting of two possible sample sizes, N=n or N=2n.

Optimal rules and robust Bayes estimation of a Gamma scale parameter

For estimating an unknown scale parameter of Gamma distribution, we introduce the use of an asymmetric scale invariant loss function reflecting precision of estimation. This loss belongs to the class of precautionary loss functions. The problem of estimation of scale parameter of a Gamma distribution arises in several theoretical and applied problems. Explicit form of risk-unbiased, minimum risk scale-invariant, Bayes, generalized Bayes and minimax estimators are derived. We characterized the admissibility and inadmissibility of a class of linear estimators of the form $$cX\,{+}\,d$$ , when $$X\sim \varGamma (\alpha ,\eta )$$ . In the context of Bayesian statistical inference any statistical problem should be treated under a given loss function by specifying a prior distribution over the parameter space. Hence, arbitrariness of a unique prior distribution is a critical and permanent question. To overcome with this issue, we consider robust Bayesian analysis and deal with Gamma minimax, conditional Gamma minimax, the stable and characterize posterior regret Gamma minimax estimation of the unknown scale parameter under the asymmetric scale invariant loss function in detail.

An improved mean estimator for judgment post-stratification

We prove that the standard nonparametric mean estimator for judgment post-stratification is inadmissible under squared error loss within a certain class of linear estimators. We derive alternate estimators that are admissible in this class, and we show that one of them is always better than the standard estimator. The reduction in mean squared error from using this alternate estimator can be as large as 10% for small set sizes and small sample sizes.

New Technique to Estimate the Asymmetric Trimming Mean

A trimming mean eliminates the extreme observations by removing observations from each end of the ordered sample. In this paper, we adopted the Hogg′s and Brys′s tail weight measures. In addition, a new algorithm was proposed as a linear estimator based on the quartile; we used a quartile to divide the data into three and four groups. Then two new estimators were proposed. These classes of linear estimators were examined via simulation method over a variety of asymmetric distributions. Sample sizes 50, 100, 150, and 200 were generated using R program. The results of 50 were tabulated, since we have similar results for the other sizes. These results were tabulated for 7 asymmetric distributions with total trimmed proportions 0.10 and 0.20 on both sides, respectively. The results for these estimators were ordered based on their relative efficiency.

Optimal channel estimation and training design for two-way relay networks

In this work, we consider the two-way relay network (TWRN) where two terminals exchange their information through a relay node in a bi-directional manner and study the training-based channel estimation under the amplify-and-forward (AF) relay scheme. We propose a two-phase training protocol for channel estimation: in the first phase, the two terminals send their training signals concurrently to the relay; and in the second phase, the relay amplifies the received signal and broadcasts it to both terminals. Each terminal then estimates the channel parameters required for data detection. First, we assume the channel parameters to be deterministic and derive the maximum-likelihood (ML) -based estimator. It is seen that the newly derived ML estimator is nonlinear and differs from the conventional least-square (LS) estimator. Due to the difficulty in obtaining a closed-form expression of the mean square error (MSE) for the ML estimator, we resort to the Crameacuter-Rao lower bound (CRLB) on the estimation MSE for design of optimal training sequence. Secondly, we consider stochastic channels and focus on the class of linear estimators. In contrast to the conventional linear minimum-mean-square-error (LMMSE) -based estimator, we introduce a new type of estimator that aims at maximizing the effective receive signal-to-noise ratio (SNR) after taking into consideration the channel estimation errors, thus referred to as the linear maximum SNR (LMSNR) estimator. Furthermore, we prove that orthogonal training design is optimal for both the CRLB- and the LMSNR-based design criteria. Finally, simulations are conducted to corroborate the proposed studies.

Asymptotics for the nonparametric estimation of the mean function of a random process

We study the nonparametric estimation of the mean function of a random process indexed by a compact metric space. We elaborate on the asymptotic variance and prove asymptotic normality for a general class of linear estimators. An application to simultaneous confidence intervals is proposed and investigated by simulation.

Estimation of the exponential mean time to failure under a weighted balanced loss function

This paper considers estimation of a exponential mean time to failure using a loss function that reflects both goodness of fit and precision of estimation. The admissibility and inadmissibility of a class of linear estimators of the form\(c\bar X + d\) are studied.

ADAPTIVE ESTIMATORS OF A MEAN MATRIX: TOTAL LEAST SQUARES VERSUS TOTAL SHRINKAGE

An unknown constant matrix M is observed with additive random error. The basic problem considered is to devise an estimator of M that trades off bias against variance so as to achieve relatively low quadratic risk. This paper develops an adaptive total least squares estimator and an adaptive total shrinkage estimator of M that minimize estimated risk over certain large classes of linear estimators. It is shown that the asymptotic risk of the adaptive total least squares estimator is the smallest attainable among reduced rank total least squares fits to the data matrix. The asymptotic risk of the adaptive total shrinkage estimator is shown to be smaller still. A close link is established between total shrinkage and the Efron-Morris estimator of M. In the asymptotics, the row dimension of M tends to infinity, and the column dimension stays fixed. The risks converge uniformly when the signal to-noise ratio and the measurement error variance are both bounded. A second problem treated is estimation of M under the assumption that a linear relation holds among its columns. In this formulation of the errors-in-variables linear regres sion model, rank constrained adaptive total least squares asymptotically domi nates the usual total least squares estimator of M, and rank constrained adaptive total shrinkage is better still.

Investigation of Linear Optimal Estimator

Features of the linear optimal estimator (LOE) that minimizes the root mean-square (RMS) criterion in the class of linear estimates in nonlinear problems are investigated. The equivalent linear models for measurements have been introduced for the LOE and the Cramer-Rao Bound. The features of the LOE in comparison with the optimal Bayesian estimator, which corresponds to the conditional mean, are studied. Some examples are considered to illustrate the results obtained.

Estimation of a Gamma Scale Parameter Under Asymmetric Squared Log Error Loss

Estimation of scale parameter under the squared log error loss function is considered with restriction to the principle of invariance and risk unbiasedness. An explicit form of minimum risk scale-equivariant estimator under this loss is obtained. The admissibility and inadmissibility of a class of linear estimators of the form (cT + d) are considered, where T follows a gamma distribution with an unknown scale parameter η and a known shape parameter ν. This includes the admissibility of the minimum risk equivariant estimator on η (MRE).

On an admissibility problem involving the moment generating function of the lognormal distribution

A new location invariant loss function is considered and the best invariant estimator of normal mean is obtained. This estimator is a function of the moment generating function of the lognormal distribution. The admissibility is studied of a class of linear estimators of the form cX + d, where X ∼ N(θ, σ2), with θ unknown and σ2 known. This yields the admissibility of the best invariant estimator of θ.

Sharp adaptation for inverse problems with random noise

We consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems, in particular multivariate ones. We propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellipsoid of functions within a wide scale (that includes ellipoids with polynomially and exponentially decreasing axes) and, at the same time, satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone non-increasing weights. The construction of the estimator is based on a properly penalized blockwise Stein's rule, with weakly geometically increasing blocks. As an application, we construct sharp adaptive estimators in the problems of deconvolution and tomography.

Admissible Estimation for Finite Population When the Parameter Space is Restricted

This paper considers the admissibility of the estimators for finite population when the parameter space is restricted. We obtain all admissible linear estimators of an arbitrary linear function of characteristic values of a finite population in the class of linear estimators under the criterion of the expectation of mean squared error.

Simultaneous Estimation of Several Stratum Means under Error-in-Variables Superpopulation Models

This paper considers simultaneous estimation of means from several strata under error-in-variables superpopulation models. Necessary and sufficient conditions for an estimator to be admissible in the class of linear estimators are obtained.

Admissible estimation for finite population under the Linex loss function

This paper considers the estimation problems of an arbitrary linear function of the characteristic values of a finite population under the Linex loss function. We obtain all admissible linear estimators when the variance σ 2 is known and all admissible linear estimators in the class of linear estimators when σ 2 is unknown.

Asymptotically efficient estimation of analytic functions in Gaussian noise

The problem of recovery of an unknown regression function f(x), x e 1 from noisy data is considered. The function f(-) is assumed to belong to a class of functions analytic in a strip of the complex plane around the real axis. The performance of an estimator is measured either by its deviation at a fixed point, or by its maximal error in the La-norm over a bounded interval. It is shown that in the case of equidistant observations, with an increasing design density, asymptotically minimax estimators of the unknown regression function can be found within the class of linear estimators. Such best linear estimators are explicitly obtained. Nonparametric regression models are widely studied in the statistical literature, since they are mathematically attractive and have many useful applications. Recent results by Ibragimov and Hasminskii (1981; 1982a; 1984a) and Stone (1980; 1982) marked a new approach to these models, with the emphasis on optimal (minimax) rates of convergence in estimating an unknown regression function in various functional classes. In some remarkable cases not only minimax rates of convergence, but also exact asymptotic constants have been found, and the corresponding asymptotically minimax estimators have been derived. Pinsker (1980) was the first to do this in the problem of regression estimation in continuous-time Gaussian white noise. He obtained asymptotically minimax estimators of the regression function in the L2-norm, with the underlying functional classes defined as ellipsoids in L2. These classes include as special cases Sobolev's classes as well as the classes of periodic analytic functions. In the ensuing papers of Nussbaum (1985) and Golubev and Nussbaum (1992), this study was extended to regression models in discrete time and Sobolev's classes of functions. Another example of asymptotically efficient nonparametric regression estimators was given by Ibragimov and Hasminskii (1984b) for the problem of estimating the unknown