Search

The main contribution of this paper is the derivation of the asymptotic behaviour of the out-of-sample variance, the out-of-sample relative loss, and of their empirical counterparts in the high-dimensional setting, i.e., when both ratios $p/n$ and $p/m$ tend to some positive constants as $m\to\infty$ and $n\to\infty$, where $p$ is the portfolio dimension, while $n$ and $m$ are the sample sizes from the in-sample and out-of-sample periods, respectively. The results are obtained for the traditional estimator of the global minimum variance (GMV) portfolio, for the two shrinkage estimators introduced by \cite{frahm2010} and \cite{bodnar2018estimation}, and for the equally-weighted portfolio, which is used as a target portfolio in the specification of the two considered shrinkage estimators. We show that the behaviour of the empirical out-of-sample variance may be misleading is many practical situations. On the other hand, this will never happen with the empirical out-of-sample relative loss, which seems to provide a natural normalization of the out-of-sample variance in the high-dimensional setup. As a result, an important question arises if this risk measure can safely be used in practice for portfolios constructed from a large asset universe., 21 pages, 5 figures

Motivated by the problem of variable selection in spatially varying coefficients models for spatial econometrics data, a Bayesian spatially dynamic selection model based on spatial normal-gamma process (SNGP) is proposed, which pursues spatial varying sparsity in dynamic regression models. Theoretical properties of SNGP are discussed. Posterior samples are obtained by nimble, a powerful R package for Bayesian inference. A new tuning-free variable selection based on K -groups clustering is proposed for discriminating the signal and the noise. Simulation studies show that the proposed method has both good estimation performance and selection performance. Finally, the new method is applied to analyzing a county level income data of Louisiana.

Bayesian shrinkage priors have received a lot of attention recently because of their efficiency in computation and accuracy in estimation and variable selection. In this paper, we study the contraction properties of shrinkage priors in a logistic regression model where the number of covariates is high. For a shrinkage prior distribution that is heavy-tailed and concentrated around zero with high probability such as the horseshoe prior, the Dirichlet–Laplace prior, and the normal-gamma prior with appropriate choices of hyper-parameters, estimates of the logistic regression coefficient are shown to asymptotically concentrate around the true sparse vector in the L 2 -sense. It is shown that the proposed contraction rate is comparable with the point mass prior that is studied in Atchade (2017). The simulation study under the logistic regression model verifies the theoretical results by showing that the horseshoe prior and the Dirichlet–Laplace prior perform like the point mass prior for the estimation, variable selection and prediction, and yield much better results than Bayesian lasso and the non-informative normal prior.

Identifying elite inbred parent lines that produce high-performing hybrid Pacific oyster seed requires diallel or factorial test crosses among lines, each acting as both a male and a female parent. Previously, we used the generalized linear model with fixed effects (i.e. GLM) to partition variance in yield, among hybrid families produced by a diallel cross, into causal genetic components—principally, general combining ability (GCA), specific combining ability (SCA), and reciprocal effect (R). However, GLM is extremely sensitive to missing information, which arises from loss of hybrid families for random environmental causes or from variation in the reproductive success of parent lines. To resolve this issue, we apply a Bayesian hierarchical model, which partitions yield variance into the familiar causal genetic components, while providing Bayesian shrinkage estimates incorporating the uncertainty of missing data. Our study suggests that correlation between observed yields and those predicted by the Bayesian model is high (r2 ≥ 0.99), for observed offspring, regardless of diallel completeness. Additionally, in analyses of complete diallel crosses, line-specific GCA rankings from GLM and Bayesian models are consistent for parent lines. Finally, comparing simulated complete and incomplete diallel datasets, we show the accuracy of predicted yield for families that are present and of parent-line ranking by GCA and the reliability of parent-line selection for double-cross hybrids, especially when non-parental lines (i.e. the four hybrid parents used to predict the yield of double-cross hybrids) are present. Our study demonstrates that the Bayesian hierarchical model performs as well as GLM in analyzing complete diallel crosses and can properly deal with incomplete diallel crosses for which GLM does not work. Therefore, the Bayesian hierarchical model is powerful in diallel analysis to select superior parent lines for producing high-yielding, hybrid, Pacific oyster seed.

In this paper we derive the optimal linear shrinkage estimator for the high-dimensional mean vector using random matrix theory. The results are obtained under the assumption that both the dimension $p$ and the sample size $n$ tend to infinity in such a way that $p/n \to c\in(0,\infty)$. Under weak conditions imposed on the underlying data generating mechanism, we find the asymptotic equivalents to the optimal shrinkage intensities and estimate them consistently. The proposed nonparametric estimator for the high-dimensional mean vector has a simple structure and is proven to minimize asymptotically, with probability $1$, the quadratic loss when $c\in(0,1)$. When $c\in(1, \infty)$ we modify the estimator by using a feasible estimator for the precision covariance matrix. To this end, an exhaustive simulation study and an application to real data are provided where the proposed estimator is compared with known benchmarks from the literature. It turns out that the existing estimators of the mean vector, including the new proposal, converge to the sample mean vector when the true mean vector has an unbounded Euclidean norm., 20 pages, UPDATE2: revised version of the manuscript accepted for publication in Journal of Multivariate Analysis

We investigate improved loss estimation in the matrix mean estimation problem. Specifically, for estimators of a normal mean matrix, we consider estimation of the Frobenius loss. Based on the singular values of the observation, we develop loss estimators that dominate the unbiased loss estimator for a broad class of matrix mean estimators including the Efron–Morris estimator. This is an extension of the results of Johnstone (1988) for a normal mean vector. We also provide improved estimators of loss for reduced-rank estimators. Numerical results show the effectiveness of the proposed loss estimators.

The covariance operator plays an important role in modern statistical methods and is critical for inference. It is most often estimated by the empirical covariance operator. In spite of its simple and appealing properties, however, this estimator can be improved by a class of shrinkage operators. In this paper, we study shrinkage estimation of the covariance operator in reproducing kernel Hilbert spaces. A data-driven shrinkage estimator enjoying desirable theoretical and computational properties is proposed. The procedure is easily implemented and its numerical performance is investigated through simulations. In finite samples, the estimator outperforms the empirical covariance operator, especially when the data dimension is much larger than the sample size. We also show that the rate of convergence in Hilbert–Schmidt norm is of the order n − 1 ∕ 2 . Furthermore, we establish the minimax optimal rate of convergence over suitable classes of probability measures and demonstrate that these shrinkage operators are all minimax rate-optimal.

Parameter uncertainty and estimation errors often cause the presence of unstable asset weights and the poor performance of portfolio model. In addition, in the real world, most investors prefer to choose a small number of stocks to invest. In this paper, we propose some improved sparse and stable portfolio models by combining the shrinkage method and objective function L1 regularization method. An ‘optimal’ shrinkage constant is obtained by minimizes the expected distance between the shrinkage estimator and the true covariance matrix. Moreover, we investigate the combination of the constant correlation and objective function L1 regularization method. Empirical studies show that the proposed strategies have better out-of-sample performance than many other strategies for tested datasets.

Existing factor models struggle to model the covariance matrix for a large number of stocks and factors. Therefore, we introduce a new covariance matrix estimator that first shrinks the factor model coefficients and then applies nonlinear shrinkage to the residuals and factors. The estimator blends a regularized factor structure with conditional heteroskedasticity of residuals and factors and displays superior all-around performance against various competitors. We show that for the proposed double- shrinkage estimator, it is enough to use only the market factor or the most important latent factor(s). Thus there is no need for laboriously taking into account the factor zoo. Supplementary material for this article is available online.

The classification of high-dimensional data is a very important problem that has been studied for a long time. Many studies have proposed linear classifiers based on Fisher’s linear discriminant rule (LDA) which consists of estimating the unknown covariance matrix and the mean vector of each group. In particular, if the data dimension p is larger than the number of observations n ( p > n ) , the sample covariance matrix cannot be a good estimator of the covariance matrix due to the well-known rank deficiency. To solve this problem, many studies proposed methods by modifying the LDA classifier through diagonalization or regularization of covariance matrix. In this paper, we categorize existing methods into three cases and discuss the shortcomings of each method. To compensate for these shortcomings, our baseline idea is that we consider estimation of the high dimensional mean vector and covariance matrix altogether while existing methods focus on shrinkage estimator of either mean vector or covariance matrix. We provide theoretical result that the proposed method is successful in both sparse and dense situations of the mean vector structures. In contrast, some existing methods work well only under specific situations. We also present numerical studies that our methods outperform existing methods through various simulation studies and real data examples such as electroencephalogy (EEG), gene expression microarray, and Spectro datasets.

This paper proposes a Bayesian alternative to the synthetic control method for comparative case studies with a single or multiple treated units. We adopt a Bayesian posterior predictive approach to Rubin’s causal model, which allows researchers to make inferences about both individual and average treatment effects on treated observations based on the empirical posterior distributions of their counterfactuals. The prediction model we develop is a dynamic multilevel model with a latent factor term to correct biases induced by unit-specific time trends. It also considers heterogeneous and dynamic relationships between covariates and the outcome, thus improving precision of the causal estimates. To reduce model dependency, we adopt a Bayesian shrinkage method for model searching and factor selection. Monte Carlo exercises demonstrate that our method produces more precise causal estimates than existing approaches and achieves correct frequentist coverage rates even when sample sizes are small and rich heterogeneities are present in data. We illustrate the method with two empirical examples from political economy.

Many researchers seek factors that predict the cross-section of stock returns. In finance, the key is to replicate anomalies by long-short portfolios based on their factor scores, with microcaps alleviated via New York Stock Exchange (NYSE) breakpoints and value-weighted returns. In econometrics, the key is to include a covariance matrix estimator of stock returns for the (mimicking) portfolio construction. This paper marries these two strands of literature in order to test the zoo of cross-sectional anomalies by injecting size controls, basically NYSE breakpoints and value-weighted returns, into efficient sorting. Thus, we propose to use a covariance matrix estimator for ultra-high dimensions (up to 5,000) taking into account large, small and microcap stocks. We demonstrate that using a nonlinear shrinkage estimator of the covariance matrix substantially enhances the power of tests for cross-sectional anomalies: On average, ‘Student’ t-statistics more than double.

The study is empirically motivated to analyze the performance of new class of Bayesian shrinkage priors that are powerful in reducing time-varying parameters to static ones to avoid over-fitting problem in time-varying parameter models. We utilized newly improved shrinkage priors in a non-centered parameterization form following Bitto and Fruhwirth-Schnatter (2019) who introduced variance selection using normal-gamma prior which nests the Bayesian LASSO prior of Belmonte et al. (2014) and spike and slab prior as in Schnatter and Wagner (2010). Therefore, the priors are able to discriminate time-varying coefficients from the static ones and the coefficients that can be shrunk to zero. In the empirical exercise, we estimated the generalized Phillip curve for three inflation-targeting countries and produced evidence of significant time-variation in most of the predictors. We compare the performance of the priors in density forecast of inflation allowing for constant and stochastic volatility in the model estimation. Evidence from the estimates of the log predictive density score shows that the hierarchical Normal Gamma shrinkage prior produces the best results for Canada and South Africa whilst the Normal Bayesian LASSO produces the best results for New Zealand, and that adding stochastic volatility improves the performance of models in density forecast.

In this study, we develop a higher-order asymptotic theory of shrinkage estimation for general statistical models, which includes dependent processes, multivariate models, and regression models (i.e., non-independent and identically distributed models). We introduce a shrinkage estimator of the maximum likelihood estimator (MLE) and compare it with the standard MLE by using the third-order mean squared error. A sufficient condition for the shrinkage estimator to improve the MLE is given in a general setting. Our model is described as a curved statistical model p ( ⋅ ; θ ( u ) ) , where θ is a parameter of the larger model and u is a parameter of interest with dim u dim θ . This setting is especially suitable for estimating portfolio coefficients u based on the mean and variance parameters θ . We finally provide the results of our numerical study and discuss an interesting feature of the shrinkage estimator.

This paper compares three types of standard errors for optimal minimum distance (OMD) estimator where the structural parameter is recovered from the reduced form parameters estimated by a two-step GMM estimator. We demonstrate that the naive standard errors are severely biased and cause over-rejection, while the OMD estimator based on the bias-corrected variance matrix by Windmeijer (2005) and newly derived variance estimator yield much more accurate inference.

Shrinkage estimation of eigenvalues of covariance matrices for Gaussian spiked covariance models are known to be effective especially in high-dimensional problems. Various shrinkage methods have been proposed, and these typically employ nonlinear functions of eigenvalues of sample covariance matrices. Here we investigate Bayesian shrinkage methods for single-spiked covariance models. The choice of priors and the construction of the Bayes estimator are considered, and it is proved that the Bayes estimator of the present shrinkage prior dominates asymptotically that of the Jeffreys prior regarding the Kullback–Leibler risk. The Bayes estimators are obtained as the posterior mean of the covariance matrices. In numerical simulations, the Bayes methods based on the present shrinkage prior and the Jeffreys prior are compared with other non-Bayes methods.

A covariance matrix shrinkage method is proposed to make an improvement of the direction of arrival (DOA) estimation under a uniform linear array in a scenario where the number of sensors is large and the sample size is relatively small. The main contribution is that we provide a shrinkage target with Toeplitz structure and deduce a closed-form estimation of the shrinkage coefficient. The closed-form and the expectation of the shrinkage coefficient estimate are calculated based on the unbiased and consistent estimates of the trace and moments of a Wishart distributed covariance matrix. The statistical property of the shrinkage coefficient estimate is discussed through theoretical analysis and simulations, which demonstrate the shrinkage coefficient estimate can ensure that the proposed covariance matrix estimate is a good compromise between the sample covariance matrix (SCM) and the target. The root-mean-square-error (RMSE) simulations of DOA estimation show that the proposed method can improve the multiple signal classification (MUSIC) DOA estimation performance in the case of low signal-to-noise ratio (SNR) with small sample size, and also can provide a satisfactory performance at high SNR.

In this paper, we provide the proof of nonexistence of the James–Stein estimator in the whole parameter space for normal random matrices, equivalently, for multivariate linear regression models, which solves the open problem raised by S.F. Arnold [1] . By introducing the concepts of left and right James–Stein estimators, we obtain the left James–Stein estimator of mean matrix and show that the left James–Stein estimator has minimaxity and optimality in terms of the Efron–Morris type modification. We construct a new minimax combination estimator with lower risk by absorbing the advantages of the left James–Stein estimator and the existing modified Stein estimator. Risk comparisons through finite sample simulation studies illustrate that the proposed combination estimator has a better performance, under the mean-squared error or l 2 risk, compared with all existing estimators.

The multivariate Bayesian hierarchical model with independent means has been studied extensively and is widely used in practice. In contrast, the case of dependent means has received scant attention, even though multivariate observations are often correlated. In this paper, we investigate a multivariate heteroscedastic Bayesian hierarchical model in which an informative prior with equicorrelated means is assumed. We estimate the mean vector by the shrinkage estimator based on Stein’s unbiased risk estimation (SURE). It is shown that the squared error loss of the SURE estimator is close to that of an oracle estimator as the number of means grows. Our SURE estimator includes the SURE estimator under independence considered by Xie et al. (2012) as a special case. The finite-sample performance of our estimator is explored via simulations and two real data sets are used for illustration purposes.

This paper deals with the problem of estimating predictive densities of a matrix-variate normal distribution with known covariance matrix. Our main aim is to establish some Bayesian predictive densities related to matricial shrinkage estimators of the normal mean matrix. The Kullback–Leibler loss is used for evaluating decision-theoretic optimality of predictive densities. It is shown that a proper hierarchical prior yields an admissible and minimax predictive density. Also, some minimax predictive densities are derived from superharmonicity of prior densities.

We propose a regularized partially functional quantile regression model where the response variable is scalar while the explanatory variables involve both infinite-dimensional predictor processes viewed as functional data, and high-dimensional scalar covariates. Despite extensive work focusing on functional linear models, little effort has been devoted to the development of robust methodologies that tackle the scenarios of non-normal errors. This motivates our proposal of functional quantile regression that seeks an alternative and robust solution to least squares type procedures within the partially functional regression framework. We focus on estimating and selecting the important variables in the high-dimensional covariates, which is complicated by the infinite-dimensional functional predictor. We establish the asymptotic properties of the resulting shrinkage estimator, and empirical illustrations are given by simulation and an application to a brain imaging dataset.

We study estimation of several non-negative normal means under sum of squared errors loss when the common variance is unknown. In particular we consider a multivariate shrinkage version of the Katz estimator and show that it dominates the vector version of the Katz estimator which, in the known variance case, is generalized Bayes with respect to the uniform prior over the positive orthant.

Parameter shrinkage applied optimally can always reduce error and projection variances from those of maximum likelihood estimation. Many variables that actuaries use are on numerical scales, like age or year, which require parameters at each point. Rather than shrinking these towards zero, nearby parameters are better shrunk towards each other. Semiparametric regression is a statistical discipline for building curves across parameter classes using shrinkage methodology. It is similar to but more parsimonious than cubic splines. We introduce it in the context of Bayesian shrinkage and apply it to joint mortality modeling for related populations. Bayesian shrinkage of slope changes of linear splines is an approach to semiparametric modeling that evolved in the actuarial literature. It has some theoretical and practical advantages, like closed-form curves, direct and transparent determination of degree of shrinkage and of placing knots for the splines, and quantifying goodness of fit. It is also relatively easy to apply to the many nonlinear models that arise in actuarial work. We find that it compares well to a more complex state-of-the-art statistical spline shrinkage approach on a popular example from that literature., Comment: 39 pages, 8 graphs

Bayesian shrinkage estimation is a familiar concept that arises from minimizing the Bayes risk under the squared error loss. For observations over a period of time, Bayesian shrinkage estimators may be used to predict the value of a response variable for a subject, given previously observed values. In this article, we formulate such estimators under a change of probability measure within the copula framework. Such reformulation is demonstrated using the multivariate generalized beta of the second kind (GB2) distribution. Within this family of GB2 copulas, we are able to derive explicit form of Bayesian shrinkage estimators. Numerical illustrations show the application of these estimators in determining experience-rated insurance premium. We consider generalized Pareto as a special case.

We provide a measure of sparsity for expected returns within the context of classical factor models. Our measure is inversely related to the percentage of active predictors. Empirically, sparsity varies over time and displays an apparent countercyclical behavior. Proxies for financial conditions and for liquidity supply are key determinants of the variability in sparsity. Deteriorating financial conditions and illiquid times are associated with an increase in the number of characteristics that are useful to predict anomaly returns (i.e., the forecasting model becomes more dense). Looking at specific categories of characteristics, we find that variables classified as value, trading frictions and, in particular, profitability are robustly present throughout the sample. A strategy that exploits the dynamics of sparsity to time factors delivers substantial economic gain out-of-sample relative to both a random walk and a simple rolling window shrinkage estimator as well as standard models based on preselected, well-know characteristics like size, momentum, book-to-market, investment and accruals.

This paper examines the impact of the current oil prices fall on Europe. We estimate a Bayesian shrinkage VAR model and analyse the impulse response functions to investigate the reaction of European stock markets to the current oil prices collapse. Using data covering the March 2002 to May 2014 period, our main result is that European stock markets are negatively and significantly affected by the crude oil shock. We prove that this result is robust to reasonable changes in the Bayesian shrinkage VAR model of the variables order and inclusion of additional variables. The findings shed light that common features exist among the European stock markets. Furthermore, the results highlight that the most exposed stock market is the French one, and that the least affected market is the Austrian one.

Carbon accounting is at the heart of efforts to mitigate the effects of climate change. One approach for estimating population parameters for live tree stem carbon entails three primary steps: (1) construction of an individual tree, allometric carbon model, (2) application of the model to tree-level data for a probability sample of plots, and (3) use of a probability-based (design-based) estimator of mean carbon per unit area for a population of interest. Compliance with the IPCC good practice guidance requires satisfaction of two criteria, one related to minimizing bias and one related to minimizing uncertainty. For this carbon estimation procedure, the portion of uncertainty attributed to the variance of the probability-based estimator of the population mean using the plot-level predictions is usually correctly estimated, but the portion attributed to the variance of the allometric model estimator is usually ignored. The result is that the total variance of the population mean estimator cannot be asserted to comply with the IPCC good practice criteria because not only is it not minimized, it is not even correctly estimated. Within the framework of what is coming to be characterized as hybrid inference, model-based inferential methods were used to estimate the variance of the tree-level allometric model estimator which was then propagated through to the variance of the probability-based estimator of mean carbon per unit area. This combined estimator, consisting of a model-based estimator used to predict a variable for a probability sample of a population followed by a probability-estimator of the population total or mean using the sample predictions, is characterized as a hybrid estimator. For this study, two probability-based estimators of the mean were considered, simple random sampling estimators and model-assisted regression estimators that used airborne laser scanning (ALS) data as auxiliary information. The variance of the allometric model estimator incorporated variances of distributions of diameter and height measurement errors, covariances of model parameter estimators, model residual variance, and variances of distributions of wood densities and carbon content proportions. The novel features of the study included the hybrid inferential framework, consideration of six sources of uncertainty including the variances of distributions of wood densities and carbon content proportions, use of ALS data with model-assisted regression estimators of the population mean, and use of confidence intervals for the population mean as the basis for comparisons rather than intermediate products such as model prediction accuracy. The primary conclusions were that the variance of the allometric model estimator was negligible or marginally negligible relative to the variance of the probability estimator when using species-specific allometric models and simple random sampling estimators, but non-negligible when using species-specific models and model-assisted regression estimators and when using a non-specific model with either estimator.

We propose Bayesian shrinkage methods for coefficient estimation for high-dimensional vector autoregressive (VAR) models using scale mixtures of multivariate normal distributions for independently sampled additive noises. We also suggest an efficient selection procedure for the shrinkage parameter as a computationally feasible alternative to the traditional MCMC sampling methods for high-dimensional data. A shrinkage parameter is selected at the minimum point of a newly proposed score function which is asymptotically equivalent to the mean squared error of the model coefficients. The selected shrinkage parameter is presented in a closed form as a function of sample size, level of noise, and non-normality in data, and it can be efficiently estimated by using a suggested variation of cross validation. Consistency of both of the cross validation estimator and proposed shrinkage estimator is proved. The competitiveness of the proposed methods is demonstrated based on comprehensive experimental results using simulated data and high-dimensional plant gene expression data in the context of coefficient estimation and structural inference for VAR models. The proposed methods are applicable to high-dimensional stationary time series with or without near unit roots. The proposed methods are useful for high-dimensional VAR models with small data.A formula for an optimal shrinkage parameter is derived under a Bayes framework.The proposed methods are superior in computation time and performance.

Typically, central banks use a variety of individual models (or a combination of models) when forecasting inflation rates. Most of these require excessive amounts of data, time, and computational power; all of which are scarce when monetary authorities meet to decide over policy interventions. In this paper we use a rolling Bayesian combination technique that considers inflation estimates by the staff of the Central Bank of Colombia during 2002-2011 as prior information. Our results show that: 1) the accuracy of individual models is improved by using a Bayesian shrinkage methodology, and 2) priors consisting of staff's estimates outperform all other priors that comprise equal or zero-vector weights. Consequently, our model provides readily available forecasts that exceed all individual models in terms of forecasting accuracy at every evaluated horizon. Classification JEL: C22, C53, C11, E31.

Forecasting macroeconomic variables using many predictors is considered. Model selection and model reduction approaches are compared. Model selection includes heuristic optimisation of information criteria using: simulated annealing, genetic algorithms, MC3 and sequential testing. Model reduction employs the methods of principal components, partial least squares and Bayesian shrinkage regression. The problem of unbalanced datasets is discussed and potential solutions are suggested. An out-of-sample forecasting exercise provides evidence that these methods are useful in predicting the growth rates of quarterly GDP and monthly inflation.

In this work we construct an optimal shrinkage estimator for the precision matrix in high dimensions. We consider the general asymptotics when the number of variables p ? ∞ and the sample size n ? ∞ so that p / n ? c ? ( 0 , + ∞ ) . The precision matrix is estimated directly, without inverting the corresponding estimator for the covariance matrix. The recent results from random matrix theory allow us to find the asymptotic deterministic equivalents of the optimal shrinkage intensities and estimate them consistently. The resulting distribution-free estimator has almost surely the minimum Frobenius loss. Additionally, we prove that the Frobenius norms of the inverse and of the pseudo-inverse sample covariance matrices tend almost surely to deterministic quantities and estimate them consistently. Using this result, we construct a bona fide optimal linear shrinkage estimator for the precision matrix in case c < 1 . At the end, a simulation is provided where the suggested estimator is compared with the estimators proposed in the literature. The optimal shrinkage estimator shows significant improvement even for non-normally distributed data.

The paper considers spatial econometric model and presents a family of shrinkage estimators for the regression coefficients vector. The asymptotic distribution of the proposed family of estimators has been derived under the assumption that sample size is large. The risk properties of least squares estimator and proposed improved family of estimators have been investigated under quadratic loss function and dominance conditions have been obtained. For investigating the finite sample behavior of various estimators belonging to proposed family of shrinkage estimators, a simulation study has been carried out and results have been presented.

This paper introduces shrinkage for general parametric models. We show how to shrink maximum likelihood estimators towards parameter subspaces de…ned by general nonlinear restrictions. We derive the asymptotic distribution and risk of the generalized shrinkage estimator using a local asymptotic framework. We show that if the shrinkage dimension exceeds two, the asymptotic risk of the shrinkage estimator is strictly less than that of the MLE. This reduction holds globally in the parameter space. We show that the reduction in asymptotic risk is substantial, even for moderately large values of the parameters. The formula simplify in a very convenient way in the context of high dimensional models. We derive a simple bound for the asymptotic risk. We also provide a new large sample minimax e¢ ciency bound. We use the concept of local asymptotic minimax bounds, a generalization of the conventional asymptotic minimax bounds. The

The negative multinomial distribution is a multivariate generalization of the negative binomial distribution. In this paper, we consider the problem of estimating an unknown matrix of probabilities on the basis of observations of negative multinomial variables under the standardized squared error loss. First, a general sufficient condition for a shrinkage estimator to dominate the UMVU estimator is derived and an empirical Bayes estimator satisfying the condition is constructed. Next, a hierarchical shrinkage prior is introduced, an associated Bayes estimator is shown to dominate the UMVU estimator under some conditions, and some remarks about posterior computation are presented. Finally, shrinkage estimators and the UMVU estimator are compared by simulation., Comment: 31 pages; the code for numerical computation of the hierarchical Bayes estimator in Section 4 has been corrected; Tables 2, 3, and 4 and the second-to-the-last paragraph of Section 4 have been changed

Environmental and technical efficiency in the electricity sector is a major focus of regulatory policies aimed at hastening greenhouse gas reduction and lowering energy prices for final consumers. Understanding the impact of market and environmental regulations on efficiency is crucial for the design and choice of policy packages. At the national level, the productivity response to regulations depends on unobserved country-specific factors that current empirical analyses have not modelled. In this framework, we analyse the regulation effects on efficiency in the electricity sector for a panel of European Union countries. We explicitly consider the effects of the environmental regulation along with the market regulation. Our estimation strategy uses the Bayesian shrinkage estimator, which can deal with the bias aggregation problem and cross-country heterogeneity. It allows us to identify the different transmission channels through which regulatory policy is implemented. The results highlight divergent behaviours at the country level. The direct impacts of the market and environmental regulatory policies on productivity are negative—around −2.7% and −2.3%, respectively—but the countries vary in the degree to which they are able to compensate these negative effects.

One of the most important subject in multivariate analysis is parameters estimation. Among different methods, the shrinkage estimation is of interest. In this paper we consider the generalized Bayes shrinkage estimator of location parameter for spherical distribution under balance-type loss. We assume that the random vector having a spherical symmetric distribution with the known scalar variational component. Also, we find minimax and admissible estimator of location parameter based on generalized Bayes estimator. We investigate wavelet generalized Bayes estimator of location under balance-LINEX loss function. At the end, the performance evaluation of the proposed class of estimators is checked through a simulation study.

In the present study, the biased estimators for the scale parameter of a classical Rayleigh distribution are proposed by using two different shrinkage techniques giving a smaller mean square error than an unbiased estimator. Then the obtained biased estimators are compared with the unbiased estimator using their mean square error.

Bayesian regularization, a relatively new method for estimating model parameters, shrinks estimates towards the overall mean by shrinking the parameters. It has been proven to lower estimation and prediction variances from those of MLE for linear models, such as regression or GLM. It has a goodness-of-fit measure, and can readily be applied using available software. This can be used for any type of actuarial linear modeling, but it is slightly more complicated for mortality and loss reserving models that use row, column, and diagonal effects for array data. These are called age-period-cohort, or APC models by statisticians. The problem is that the row, column and diagonal effects are not what should be shrunk. These models can easily become over-parameterized, and actuaries often reduce parameters with smooth curves or cubic splines. We discuss an alternative smoothing method that uses regularization, with its reduction in estimation errors, and illustrate both its classical and Bayesian forms and their application to APC modeling. Typical actuarial models and some generalizations are used as examples.

This article shows how asset characteristics can be incorporated into the Bayesian portfolio selection framework. We use Gaussian process priors to model the belief that assets with similar characteristics are likely to have similar expected returns. The resulting Bayesian shrinkage estimator biases expected return estimates of assets with similar characteristics towards each other. A closed-form solution of the optimal portfolio weights in the Gaussian process prior framework is derived. Our simulation results and our empirical analysis suggest that portfolio selection with Gaussian process priors is a competitive alternative to benchmark portfolio selection approaches from the literature.

The current variance estimators for most evolutionary models were derived when a nucleotide substitution number estimator was approximated with a simple first order Taylor expansion. In this study, we derive three variance estimators for the F81, F84, HKY85 and TN93 nucleotide substitution models, respectively. They are obtained using the second order Taylor expansion of the substitution number estimator, the first order Taylor expansion of a squared deviation and the second order Taylor expansion of a squared deviation, respectively. These variance estimators are compared with the existing variance estimator in terms of a simulation study. It shows that the variance estimator, which is derived using the second order Taylor expansion of a squared deviation, is more accurate than the other three estimators. In addition, we also compare these estimators with an estimator derived by the bootstrap method. The simulation shows that the performance of this bootstrap estimator is similar to the estimator derived by the second order Taylor expansion of a squared deviation. Since the latter one has an explicit form, it is more efficient than the bootstrap estimator.

　　 This paper addresses the problem of estimating the normal mean matrix with an unknown covariance matrix. Motivated by an empirical Bayes method, we suggest a uni ed form of the Efron-Morris type estimators based on the Moore-Penrose inverse. This form not only can be de ned for any dimension and any sample size, but also can contain the Efron-Morris type or Baranchik type estimators suggested so far in the literature. Also, the uni ed form suggests a general class of shrinkage estimators. For shrinkage estimators within the general class, a uni ed expression of unbiased estimators of the risk functions is derived regardless of the dimension of covariance matrix and the size of the mean matrix. An analytical dominance result is provided for a positive-part rule of the shrinkage estimators.

We consider estimation in generalized linear models when there are many potential predictors and some of them may not have influence on the response of interest. In the context of two competing models where one model includes all predictors and the other restricts variable coefficients to a candidate linear subspace based on subject matter or prior knowledge, we investigate the relative performances of Stein type shrinkage, pretest, and penalty estimators ( L 1 GLM, adaptive L 1 GLM, and SCAD) with respect to the unrestricted maximum likelihood estimator (MLE). The asymptotic properties of the pretest and shrinkage estimators including the derivation of asymptotic distributional biases and risks are established. In particular, we give conditions under which the shrinkage estimators are asymptotically more efficient than the unrestricted MLE. A Monte Carlo simulation study shows that the mean squared error (MSE) of an adaptive shrinkage estimator is comparable to the MSE of the penalty estimators in many situations and in particular performs better than the penalty estimators when the dimension of the restricted parameter space is large. The Steinian shrinkage and penalty estimators all improve substantially on the unrestricted MLE. A real data set analysis is also presented to compare the suggested methods.

In this paper, ridge and non-ridge type shrinkage estimators and their positive parts are defined in the semiparametric regression model when the errors are dependent and some non-stochastic linear restrictions are imposed under a multicollinearity setting. The exact risk expressions in addition to biases are derived for the estimators under study and the region of optimality of each estimator is exactly determined. Also, necessary and sufficient conditions, for the superiority of the ridge type estimator over its counterpart, for selecting the ridge parameter k are obtained. Lastly, a simulation study and real data analysis are performed to illustrate the efficiency of proposed estimators based on the minimum risk criterion. In this regard, kernel smoothing and modified cross-validation methods for estimating the non-parametric function are used.

Optimal portfolios with a restriction on the number of assets, also referred to as cardinality-constrained portfolios, have been receiving attention in the literature due to its popularity among market practitioners and retail investors. In most cases, however, the interest is in proposing efficient optimization methods to solve the problem, with little or no attention to the characteristics of the resulting portfolio such as risk-adjusted performance and turnover. We address this question by implementing a tractable reformulation of the cardinality-constrained version of the minimum variance portfolio. We analyze the out-of-sample performance of cardinality-constrained portfolios according to alternative criteria and check the robustness of the results for portfolios with alternative number of assets and under alternative re-balancing frequencies. Our empirical application for the Brazilian equities market shows that cardinality-constrained minimum variance portfolios with very few assets, e.g. 3 stocks, can deliver statistically lower portfolio risk and higher Sharpe ratios in comparison to the market index. Similar results are obtained for constrained portfolios with 5 and 10 assets and under daily, weekly, and monthly re-balancing frequencies. Our evidence indicates that it is possible to obtain better risk-adjusted performance with fewer securities in the portfolio by using an improved allocation scheme. Carteiras ótimas com restrições no número de ativos, também conhecidas como carteiras com restrições de cardinalidade, tem recebido grande atenção na literatura em função da sua popularidade entre praticantes de mercado e pequenos investidores. Na maioria dos casos, entretanto, o interesse está em propor métodos eficientes de otimização para resolver o problema, com pouca ou nenhuma atenção às características das carteiras como desempenho ajustado ao risco ou turnover. Abordamos esta questão através da implementação de uma reformulação tratável do problema de minimização da variância da carteira sujeita à restrição de cardinalidade. Analisamos o desempenho fora-da-amostra das carteiras com restrição de cardinalidade segundo diversos critérios e checamos a robustez dos resultados para carteiras com diversos números de ativos e também diversas frequências de rebalanceamento. Nossa aplicação empírica para o mercado acionário brasileiro mostra de carteiras de variância mínima com restrições de cardinalidade envolvendo um pequeno número de ativos, por exemplo, três ações podem obter níveis menores de risco e maiores índices de Sharpe em relação ao índice de mercado. Resultados semelhantes são obtidos para carteiras restringidas com cinco e 10 ativos e com frequências de rebalanceamento diária, semanal e mensal. Nossa evidência indica que é possível obter um melhor desempenho ajustado ao risco com menos ativos na carteira através do uso de uma melhor estratégia de alocação.

We propose a shrinkage estimator for covariance matrices designed to minimize estimation error of the Global Minimum Variance (GMV) portfolio. Implementing the GMV portfolio requires estimating the asset covariance matrix and using this to obtain variance-minimizing portfolio weights. Standard estimation approaches for this application utilize shrinkage. These estimators use shrinkage weights that are not designed to directly minimize estimation error of the final object of interest: GMV portfolio weights. We develop a focused shrinkage approach to the problem. This method utilizes the form of the trading rule to derive a shrinkage estimator that directly controls estimation error of GMV portfolio weights. Extensive simulations are conducted to compare performance with nine standard competitors. Our estimator uniformly outperformed all competitors across portfolios of various sizes. The methods are applied to several standard portfolios of US and international assets. Similar improvements are found. Our estimator achieves the smallest out-of-sample portfolio variance in 25 of 28 data sets considered.

We propose a shrinkage estimator for parameters θ which improves the mean squared error of functions x (θ) over standard choices. When the restricted model estimator is in the class of minimum distance estimators, we project onto the restricted parameter space using a matrix based on the derivatives of x (θ). The proposed matrix is shown to minimize the bias among estimators in this class. This choice is then incorporated into a shrinkage procedure. We derive a risk bound for shrinkage estimators which allows for arbitrary projection matrices. Using this result, it is shown the proposed projection matrix can lead to substantially lower risk. The improvement is largest when the restricted model has nontrivial bias. This is shown with our motivating example: implementing a mean-variance portfolio choice rule. Our method is applied by shrinking toward a model with restricted covariance matrix. Extensive simulations demonstrate improved risk in this case. The estimator is also implemented in a portfolio choice application to futures data. Our approach outperforms standard procedures here as well.

In this paper, we investigate the shrinkage problem of image denoising for various methods under the additive white Gaussian noise (AWGN) model. Our main contribution is to derive the closed-form of the optimal shrinkage that minimizes the Stein's unbiased risk estimator (SURE) and thus allows direct blockwise shrinkage without additional optimizations. Simulation results show that the proposed method boosts the denoising performance for a variety of image denoising techniques including the moving average filter, the median filter, the wiener filter, the bilateral filter, the probabilistic non-local means, and the block matching 3D filter in terms of higher pixel signal noise ratio (PSNR) and structural similarity index (SSIM). We also case study the proposed shrinkage solution with respect to the classic NLM denoising, whose shrinkage solutions and equivalent forms have been widely researched, and further confirm its superiority. The proposed shrinkage solution can be used to improve arbitrary image denoising methods under the AWGN model, and it serves as a good remedy to save badly denoised images due to inappropriate parameters. HighlightsWe derive the closed-form optimal blockwise SURE shrinkage for image denoising.We propose an adaptive and fast blockwise shrinkage algorithm with a constant-time complexity.We demonstrate this shrinkage solution is effective to a wide collection of image denoising methods.

Estimation of normal mean vector has broad applications such as small area estimation, estimation of nonparametric functions and estimation of wavelet coefficients. In this paper, we propose a new shrinkage estimator based on conditional maximum likelihood estimator incorporating with Stein’s risk unbiased estimator (SURE) when data have the normality. We present some theoretical work and provide numerical studies to compare with some existing methods.

In this paper, we propose two novel source localization methods; one is the shrinkage estimator with the minimum mean squared error criterion, and the other is the shrinkage estimator with the minimum bias criterion. The mean squared error performance of the two-step weighted least squares deteriorates in the large noise variance regimes. In order to improve the two-step weighted least squares in the large noise variance regimes, the shrinkage factor is multiplied by the two-step weighted least squares estimator, and then the novel estimator is determined such that the mean squared error and squared bias are minimized. Simulation results show that the mean squared error performances of the proposed methods are better than those of the two-step weighted least squares method as well as the minimax estimator in a regime with large measurement noise variances.

Extracting a precise data-driven failure rate of electrical distribution components is a very prominent issue in asset management decision-making process, but lack of appropriate data would cause problems. In this paper, in order to overcome the deficiency and non-homogeneity of outage data, a shrinkage estimator is proposed for failure rate estimation of overhead lines, which compromises between individual and pooled failure rate estimations. This method is modeled through Hierarchical Bayesian Model (HBM). The functionality of HBM is compared with two other Bayesian models using Deviance Information Criterion (DIC) and real failure data of 34 electrical distribution feeders in Alborz Power Distribution Company.