Case Of Unknown Variance Research Articles

Using a Multivariate Virtual Experiment for Uncertainty Evaluation with Unknown Variance

Virtual experiments are a digital representation of a real measurement and play a crucial role in modern measurement sciences and metrology. Beyond their common usage as a modeling and validation tool, a virtual experiment may also be employed to perform a parameter sensitivity analysis or to carry out a measurement uncertainty evaluation. For the latter to be compliant with statistical principles and metrological guidelines, the procedure to obtain an estimate and a corresponding measurement uncertainty requires careful consideration. We employ a Monte Carlo sampling procedure using a virtual experiment that allows one to perform a measurement uncertainty evaluation according to the Monte Carlo approach of JCGM-101 and JCGM-102, two widely applied guidelines for uncertainty evaluation in metrology. We extend and formalize a previously published approach for simple additive models to account for a large class of non-linear virtual experiments and measurement models for multidimensionality of the data and output quantities, and for the case of unknown variance of repeated measurements. With the algorithm developed here, a simple procedure for the evaluation of measurement uncertainty is provided that may be applied in various applications that admit a certain structure for their virtual experiment. Moreover, the measurement model commonly employed for uncertainty evaluation according to JCGM-101 and JCGM-102 is not required for this algorithm, and only evaluations of the virtual experiment are performed to obtain an estimate and an associated uncertainty of the measurand. We demonstrate the efficacy of the developed approach and the effect of the underlying assumptions for a generic polynomial regression example and an example of a simplified coordinate measuring machine and its virtual representation. The results of this work highlight that considerable effort, diligence, and statistical considerations need to be invested to make use of a virtual experiment for uncertainty evaluation in a way that ensures equivalence with the accepted guidelines.

High-dimensional properties for empirical priors in linear regression with unknown error variance

We study full Bayesian procedures for high-dimensional linear regression. We adopt data-dependent empirical priors introduced in Martin et al. (Bernoulli 23(3):1822–1847, 2017). In their paper, these priors have nice posterior contraction properties and are easy to compute. Our paper extend their theoretical results to the case of unknown error variance . Under proper sparsity assumption, we achieve model selection consistency, posterior contraction rates as well as Bernstein von-Mises theorem by analyzing multivariate t-distribution.

CONFIDENCE INTERVALS CONSTRUCTED BY MODEL AVERAGING AND BOOTSTRAP SMOOTHING

We assess the performance of confidence intervals constructed by model averaging and bootstrap smoothing, using a simple testbed situation of two nested linear regression models with either known error variance or unknown error variance. The assessment is carried out by deriving computationally convenient exact expressions for the coverage probability and scaled expected length where the scaling is with respect to the usual confidence interval, with the same minimum coverage, based on the full model. We assessed the properties of the confidence interval centered on the model averaged estimator proposed by Buckland et al. (1997) with half-width proportional to the standard error of the estimate put forward by Buckland et al. (1997) of their formula (9). We considered both the finite sample case and the asymptotic case. Our results show that this model averaged confidence interval cannot be generally recommended, although it performs well when the error degrees of freedom is 1. We also assessed the properties of the confidence intervals centered on the bootstrap smoothed estimator with half-width proportional to the estimate of (a) the standard deviation of this estimator and (b) the delta method approximation, derived by Efron (2014), to the standard deviation of this estimator. We show that, except for the case of unknown error variance and small error degrees of freedom, the confidence interval centered on the bootstrap smoothed estimator does not have scaled expected length that is substantially less than 1, when the simpler model is correct. For this reason, we sought a formula for the data-based width that leads to improved performance of the confidence interval centered on the bootstrap smoothed estimator. Since this search failed, it raises the following question. Is there any formula for the data-based width of such a confidence interval so that it performs well? To answer this question, we compute the performance bound due to Kabaila & Kong (2016).

Joint Scattering Center Enumeration and Parameter Estimation in GTD Model

In this article, a novel multistage CLEAN-based method based on generalized likelihood ratio test (GLRT) is proposed for scattering centers’ enumeration and their parameter estimation. To analyze the performance of the proposed scattering center enumeration algorithm, the probability density function (pdf) of the GLRT is derived for both the known noise variance case and the unknown noise variance case. In addition, we derived a closed form for the probability of correct enumeration of scattering centers, obtained in the case of the known model parameter. A method to reduce the computational complexity of the proposed algorithm has also been introduced. Despite the conventional methods which are based on the damped exponential models, the proposed method uses the geometrical theory of diffraction (GTD) model whose performance does not degrade as the bandwidth increases. The simulation results show an appropriate performance of this method at low signal-to-noise ratio (SNR) situations.

Output-weighted optimal sampling for Bayesian regression and rare event statistics using few samples.

For many important problems the quantity of interest is an unknown function of the parameters, which is a random vector with known statistics. Since the dependence of the output on this random vector is unknown, the challenge is to identify its statistics, using the minimum number of function evaluations. This problem can be seen in the context of active learning or optimal experimental design. We employ Bayesian regression to represent the derived model uncertainty due to finite and small number of input-output pairs. In this context we evaluate existing methods for optimal sample selection, such as model error minimization and mutual information maximization. We show that for the case of known output variance, the commonly employed criteria in the literature do not take into account the output values of the existing input-output pairs, while for the case of unknown output variance this dependence can be very weak. We introduce a criterion that takes into account the values of the output for the existing samples and adaptively selects inputs from regions of the parameter space which have an important contribution to the output. The new method allows for application to high-dimensional inputs, paving the way for optimal experimental design in high dimensions.

Variance Prior Forms for High-Dimensional Bayesian Variable Selection

Consider the problem of high dimensional variable selection for the Gaussian linear model when the unknown error variance is also of interest. In this paper, we show that the use of conjugate shrinkage priors for Bayesian variable selection can have detrimental consequences for such variance estimation. Such priors are often motivated by the invariance argument of Jeffreys (1961). Revisiting this work, however, we highlight a caveat that Jeffreys himself noticed; namely that biased estimators can result from inducing dependence between parameters a priori. In a similar way, we show that conjugate priors for linear regression, which induce prior dependence, can lead to such underestimation in the Bayesian high-dimensional regression setting. Following Jeffreys, we recommend as a remedy to treat regression coefficients and the error variance as independent a priori. Using such an independence prior framework, we extend the Spike-and-Slab Lasso of Rockova and George (2018) to the unknown variance case. This extended procedure outperforms both the fixed variance approach and alternative penalized likelihood methods on simulated data. On the protein activity dataset of Clyde and Parmigiani (1998), the Spike-and-Slab Lasso with unknown variance achieves lower cross-validation error than alternative penalized likelihood methods, demonstrating the gains in predictive accuracy afforded by simultaneous error variance estimation. The unknown variance implementation of the Spike-and-Slab Lasso is provided in the publicly available R package SSLASSO (Rockova and Moran, 2017).

Adaptive estimation of the sparsity in the Gaussian vector model

Consider the Gaussian vector model with mean value $\theta$. We study the twin problems of estimating the number $\Vert \theta \Vert_{0}$ of nonzero components of $\theta$ and testing whether $\Vert \theta \Vert_{0}$ is smaller than some value. For testing, we establish the minimax separation distances for this model and introduce a minimax adaptive test. Extensions to the case of unknown variance are also discussed. Rewriting the estimation of $\Vert \theta \Vert_{0}$ as a multiple testing problem of all hypotheses $\{\Vert \theta \Vert_{0}\leq q\}$, we both derive a new way of assessing the optimality of a sparsity estimator and we exhibit such an optimal procedure. This general approach provides a roadmap for estimating the complexity of the signal in various statistical models.

On confidence intervals centred on bootstrap smoothed estimators

Bootstrap smoothed (bagged) estimators have been proposed as an improvement on estimators found after preliminary data‐based model selection. Efron derived a widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. He also considered a confidence interval centred on the bootstrap smoothed estimator, with width proportional to the estimate of this standard deviation. Recently, Kabaila and Wijethunga assessed the performance of this confidence interval in the scenario of two nested linear regression models, the full model and the simpler model, for the case of known error variance and preliminary model selection using a hypothesis test. They found that the performance of this confidence interval was not substantially better than the usual confidence interval based on the full model, with the same minimum coverage. We extend this assessment to the case of unknown error variance by deriving a computationally convenient exact formula for the ideal (i.e., in the limit as the number of bootstrap replications diverges to infinity) delta method approximation to the standard deviation of the bootstrap smoothed estimator. Our results show that, unlike the known error variance case, there are circumstances in which this confidence interval has attractive properties.

Designing one-sided group sequential clinical trials to detect a mixture alternative

We consider the construction of one-sided group sequential designs where the stopping rule includes boundaries for early stopping to accept for futility and to reject for efficacy. The traditional assumption that all patients have the same likelihood of benefiting from the treatment is sometimes unrealistic and can underestimate the required sample size. This motivates us to power the design for an alternative where the treatment group observations come from a mixture of normal distributions. For the proposed setting, we use standardized test statistics based on sample means, and the test turns out to be an L-optimal similar test. Stopping boundaries and arm size for the design are determined by Type I and Type II error spending equations. We demonstrate the need for larger arm sizes when trying to detect a mixture alternative compared to trying to detect a pure shift alternative. The unknown variance case is discussed. With the mixture model, we discuss a more general definition of treatment effect. The maximum likelihood estimator for the treatment effect is discussed.

Uniformly valid confidence sets based on the Lasso

In a linear regression model of fixed dimension $p \leq n$, we construct confidence regions for the unknown parameter vector based on the Lasso estimator that uniformly and exactly hold the prescribed in finite samples as well as in an asymptotic setup. We thereby quantify estimation uncertainty as well as the "post-model selection error" of this estimator. More concretely, in finite samples with Gaussian errors and asymptotically in the case where the Lasso estimator is tuned to perform conservative model selection, we derive exact formulas for computing the minimal coverage probability over the entire parameter space for a large class of shapes for the confidence sets, thus enabling the construction of valid confidence regions based on the Lasso estimator in these settings. The choice of shape for the confidence sets and comparison with the confidence ellipse based on the least-squares estimator is also discussed. Moreover, in the case where the Lasso estimator is tuned to enable consistent model selection, we give a simple confidence region with minimal coverage probability converging to one. Finally, we also treat the case of unknown error variance and present some ideas for extensions.

Group sequential designs for stepped-wedge cluster randomised trials.

Background/Aims:The stepped-wedge cluster randomised trial design has received substantial attention in recent years. Although various extensions to the original design have been proposed, no guidance is available on the design of stepped-wedge cluster randomised trials with interim analyses. In an individually randomised trial setting, group sequential methods can provide notable efficiency gains and ethical benefits. We address this by discussing how established group sequential methodology can be adapted for stepped-wedge designs.Methods:Utilising the error spending approach to group sequential trial design, we detail the assumptions required for the determination of stepped-wedge cluster randomised trials with interim analyses. We consider early stopping for efficacy, futility, or efficacy and futility. We describe first how this can be done for any specified linear mixed model for data analysis. We then focus on one particular commonly utilised model and, using a recently completed stepped-wedge cluster randomised trial, compare the performance of several designs with interim analyses to the classical stepped-wedge design. Finally, the performance of a quantile substitution procedure for dealing with the case of unknown variance is explored.Results:We demonstrate that the incorporation of early stopping in stepped-wedge cluster randomised trial designs could reduce the expected sample size under the null and alternative hypotheses by up to 31% and 22%, respectively, with no cost to the trial’s type-I and type-II error rates. The use of restricted error maximum likelihood estimation was found to be more important than quantile substitution for controlling the type-I error rate.Conclusion:The addition of interim analyses into stepped-wedge cluster randomised trials could help guard against time-consuming trials conducted on poor performing treatments and also help expedite the implementation of efficacious treatments. In future, trialists should consider incorporating early stopping of some kind into stepped-wedge cluster randomised trials according to the needs of the particular trial.

Recursive Least Squares for Censored Regression

Censored observations are encountered naturally in many engineering tasks. Conventional estimation algorithms may suffer from significant performance degradation when the observations are undesirably censored. This work focuses on adaptively estimating the regression parameter in a censored regression (CR) model. We first consider that the noise variance and censored thresholds are known a priori, and solve the online CR problem by computing the maximum-likelihood estimate in an expectation-maximization framework. This strategy yields a recursive least-squares algorithm for the CR (CR-RLS), and we prove its convergence and present analytical results for the steady-state error. Next, we extend the CR-RLS to the case of unknown noise variance and censored thresholds. Theoretical analysis and numerical simulation indicate that the CR-RLS performs significantly better than other competing algorithms in terms of both the estimation accuracy and convergence rate. Especially, for different censored thresholds, the CR-RLS can always achieve good performance, and its steady-state solution is almost as accurate as that of the RLS algorithm with the uncensored (complete) observations.

A new class of Bayes minimax estimators of the normal mean matrix for the case of common unknown variances

ABSTRACTThis note considers the problem of estimating the mean matrix of a matrix-variate normal distribution with the covariance matrix when the loss function is , where is unknown. We find a large class of (proper and generalized) Bayes minimax estimators for the mean matrix. This class includes classes of estimators obtained by Tsukuma [Proper Bayes minimax estimators of the normal mean matrix with common unknown variances. J Statist Plan Inference. 2010;140:2596–2606].

Admissibility of the Usual Confidence Set for the Mean of a Univariate or Bivariate Normal Population: The Unknown Variance Case

Summary In the Gaussian linear regression model (with unknown mean and variance), we show that the standard confidence set for one or two regression coefficients is admissible in the sense of Joshi. This solves a long-standing open problem in mathematical statistics, and this has important implications on the performance of modern inference procedures post model selection or post shrinkage, particularly in situations where the number of parameters is larger than the sample size. As a technical contribution of independent interest, we introduce a new class of conjugate priors for the Gaussian location–scale model.

Variance-stabilizing and Confidence-stabilizing Transformations for the Normal Correlation Coefficient with Known Variances

Fosdick and Raftery (2012) recently encountered the problem of inference for a bivariate normal correlation coefficient ρ with known variances. We derive a variance-stabilizing transformation y(ρ) analogous to Fisher’s classical z-transformation for the unknown-variance case. Adjusting y for the sample size n produces an improved “confidence-stabilizing” transformation yn(ρ) that provides more accurate interval estimates for ρ than the known-variance MLE. Interestingly, the z transformation applied to the unknown-but-equal-variance MLE performs well in the known-variance case for smaller values of |ρ|. Both methods are useful for comparing two or more correlation coefficients in the known-variance case.

Identification of Linear Dynamic Systems using Dynamic Iterative Principal Component Analysis

The paper is concerned with identifying models from data that have errors in both outputs and inputs, popularly known as the errors-in-variables (EIV) problem. The total least squares formulation of the problem is known to offer a few well-known solutions. In this work, we present a novel and systematic approach to the identification of linear dynamic models for the EIV case in the principal component analysis (PCA) framework. A methodology for the systematic recovery of the process model, including the determination of order and delay, using what we term as dynamic, iterative PCA is presented. The core step consists of determining the structure of the constraint matrix by a systematic exploitation of the stacking and PCA order, input-output partitioning of the constraint matrix and an appropriate rotation. Optimal estimates of the (input-output) noise covariance matrices are also obtained. The proposed method can be applied to a broad class of linear processes including the case of unequal and unknown error variances. Simulation results are presented to demonstrate the effectiveness and consistency of the proposed method.

Confidence Sets Based on Thresholding Estimators in High-Dimensional Gaussian Regression Models

We study confidence intervals based on hard-thresholding, soft-thresholding, and adaptive soft-thresholding in a linear regression model where the number of regressors k may depend on and diverge with sample size n. In addition to the case of known error variance, we define and study versions of the estimators when the error variance is unknown. In the known-variance case, we provide an exact analysis of the coverage properties of such intervals in finite samples. We show that these intervals are always larger than the standard interval based on the least-squares estimator. Asymptotically, the intervals based on the thresholding estimators are larger even by an order of magnitude when the estimators are tuned to perform consistent variable selection. For the unknown-variance case, we provide nontrivial lower bounds and a small numerical study for the coverage probabilities in finite samples. We also conduct an asymptotic analysis where the results from the known-variance case can be shown to carry over asymptotically if the number of degrees of freedom n − k tends to infinity fast enough in relation to the thresholding parameter.

A Generalization of the Partition Problem

The problem of partitioning with respect to a control or a standard is an important statistical problem in the area of selection and ranking. In the last 60 + years a number of formulations have been proposed for this problem utilizing the two-stage sampling strategy of Stein (1945) and other multistage sampling strategies. One such formulation, which had generalized many other formulations, was proposed in Tong (1969) for the normally distributed populations. Since then the formulation in Tong (1969) has been used by a multitude of researchers and practitioners. The indifference zone in Tong's (1969) formulation is a region in which the experimenter is indifferent to differentiate any treatment mean from the control treatment mean. However, this indifference zone also serves the additional role of defining the boundaries for “bad” and “good” treatments compared to the control treatment. Though the formulation in Tong (1969) is straightforward and easy to implement, this additional role that the indifference zone plays could make the formulation somewhat undesirable to the practitioner. To illustrate, assume that in a clinical trial a drug with less than 10% improvement over the placebo is considered as a drug with insignificant improvement and the drug is termed a “bad treatment” and a drug with more than 30% improvement over the placebo is considered a drug with significant improvement and the drug is termed a “good treatment.” In such a scenario, Tong's formulation would define the indifference zone to be between 10% and 30% improvement over placebo, and any treatment with an effect size of between 10% and 30% improvement can be partitioned either a good treatment or a bad treatment without any penalty. Such a wide indifference zone makes the partition obtained to be rather misleading as the treatments termed good treatment or bad treatment could include treatments with effect size between 10% and 30%. One could potentially reduce the size of the indifference zone in Tong's formulation, but this would then alter the definition of the bad and good treatments, which are generally provided by the experts in the area. In this article, a generalization of Tong's formulation is proposed that will partition the treatments in the indifference zone as a separate identifiable group without altering the definition of the good and bad treatments. It is shown that the formulation in Tong (1969) is a limiting case of the generalized partition formulation proposed in this article. A fixed sample procedure is constructed for the case when the common population variance is known. In addition, a purely sequential procedure is proposed for the unknown variance case. The first-order and second-order asymptotic properties are derived and verified using Monte Carlo simulation studies. An example is provided to illustrate an application of the proposed generalization.

Adjusted least squares fitting of algebraic hypersurfaces

We consider the problem of fitting a set of points in Euclidean space by an algebraic hypersurface. We assume that points on a true hypersurface, described by a polynomial equation, are corrupted by zero mean independent Gaussian noise, and we estimate the coefficients of the true polynomial equation. The adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator. The adjusted least squares estimator is based on constructing a quasi-Hankel matrix, which is a bias-corrected matrix of moments. For the case of unknown noise variance, the estimator is defined as a solution of a polynomial eigenvalue problem. In this paper, we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for an arbitrary set of monomials in the polynomial equation.

Inadmissibility of the best equivariant predictive density in the unknown variance case

This work treats the problem of estimating the predictive density of a random vector when both the mean vector and the variance are unknown. We prove that the density of reference in this context is inadmissible under the Kullback–Leibler loss in a nonasymptotic framework. Our result holds even when the dimension of the vector is strictly lower than three, which is surprising compared to the known variance setting. Finally, we discuss the relationship between the prediction and the estimation problems.