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We extend Stein’s lemma for averages that explicitly contain the Gaussian random variable at a power. We present two proofs for this extension of Stein’s lemma, with the first being a rigorous proof by mathematical induction. The alternative, second proof is a constructive formal derivation in which we express the average not as an integral, but as the action of a pseudodifferential operator defined via the Gaussian moment-generating function. In extended Stein’s lemma, the absolute values of the coefficients of the probabilist’s Hermite polynomials appear, revealing yet another link between Hermite polynomials and normal distribution.

Image denoising methods generally remove not only noise but also fine-scale textures and thus degrade the subjective image quality. In this paper, we propose a method of recovering the texture component that is lost under a state-of-the-art denoising method called weighted nuclear norm minimization (WNNM). We recover the image texture with a linear minimum mean squared error estimator (LMMSE filter), which requires statistical information about the texture and noise. This requirement is the key problem preventing the application of the LMMSE filter for texture recovery because such information is not easily obtained. We propose a new method of estimating the necessary statistical information using Stein’s lemma and several assumptions and show that our estimated information is more accurate than the simple estimation in terms of the Fréchet distance. Experimental results show that our proposed method can improve the objective quality of denoised images. Moreover, we show that our proposed method can also improve the subjective quality when an additional parameter is chosen for the texture to be added.

We consider the problem of detecting (testing) Gaussian stochastic sequences (signals) with imprecisely known means and covariance matrices. An alternative is independent identically distributed zero-mean Gaussian random variables with unit variances. For a given false alarm (1st-kind error) probability, the quality of minimax detection is given by the best miss probability (2nd-kind error probability) exponent over a growing observation horizon. We study the maximal set of means and covariance matrices (composite hypothesis) such that its minimax testing can be replaced with testing a single particular pair consisting of a mean and a covariance matrix (simple hypothesis) without degrading the detection exponent. We completely describe this maximal set. [ABSTRACT FROM AUTHOR]

Inspired by Shushi [1] and Adcock et al. [2], we consider Stein’s lemma for truncated generalized skew-elliptical random vectors. We provide two Stein’s lemmas. One is Stein’s lemma for truncated generalized skew-elliptical random vectors, the other is a special form of Stein’s lemma for truncated generalized skew-elliptical random vectors. Finally, the conditional tail expectation allocation, the lower-orthant conditional tail expectation at probability level q, the upper-orthant conditional tail expectation at probability level q, the truncated version of Wang’s premium, the multivariate tail conditional expectation and the multivariate tail covariance matrix as applications are given.

We consider the detection problem for Gaussian stochastic sequences (signals) with unknown covariance matrices in white Gaussian noise. For a given false alarm probability (1st-kind error probability), the quality of minimax detection is given by the best miss probability (2nd-kind error probability) exponent over a growing observation interval. The goal is finding the largest set of covariance matrices (composite hypothesis) such that its minimax testing can be replaced with testing a single particular covariance matrix (simple hypothesis) with no degradation of the detection exponent. We completely describe this maximal set of covariance matrices. We also consider some consequences concerning minimax detection of Gaussian stochastic signals against Gaussian white noise and detection of Gaussian stationary signals. [ABSTRACT FROM AUTHOR]

This paper generalizes Stein's Lemma recently obtained for elliptical class distributions to the generalized skew-elliptical family of distributions. Stein's Lemma provides a useful tool for deriving covariances between functions of component random variables. This Lemma has applications in finance, notably for portfolio selection and hence for the capital asset pricing model (CAPM), as well as technical applications such as the computation of moments. It also leads to important propositions concerning the mean and variance of generalized skew-elliptical variables. [ABSTRACT FROM AUTHOR]

The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes. [ABSTRACT FROM AUTHOR]

• In this paper, we develop a risk estimation framework for speech enhancement, where we optimize an unbiased estimate of the risk instead of the actual risk. The estimated risk is expressed solely as a function of the noisy observations and noise statistics. Hence, the denoiser obtained by minimizing the risk estimate does not require the clean speech prior. The stateof- the-art image denoising techniques optimize Steins unbiased risk estimate (SURE), which is an unbiased estimate of MSE, to obtain the optimum denoising function. Even though the MSE is a widely and successfully used distortion measure for signal denoising, in speech processing applications, distortion measures such as Itakura-Saito (IS), hyperbolic-cosine (cosh), weighted cosh, are known to be more perceptually relevant than MSE. Considering this into account, in this paper, we solve the speech denoising problem within the framework of perceptual risk estimation (wherein we derive unbiased estimates of speech-specific perceptual distortion measures and minimize them to obtain the corresponding denoising functions). We employ a DCT-domain pointwise shrinkage estimator for denoising where the optimum shrinkage estimator is obtained by minimizing the perceptual risk estimate. We evaluate the performance of the risk estimation-based techniques objective assessment in terms of segmental signal-to-noise ratio (SSNR), perceptual evaluation of speech quality (PESQ), short-time objective intelligibility (STOI), signal-to-distortion ratio (SDR), and subjective assessment by means of listening tests. Validation on several speech signals in real-world nonstationary noise scenarios and comparisons with benchmark techniques showed that, for input SNR greater than 5 dB, the proposed method results in better denoising performance than several benchmarking techniques. Among the risk estimation-based techniques, the quality of the denoised speech is higher (measured in terms of PESQ and subjective listening scores) for perceptual risk-based techniques than the MSE-based technique. Further, we want to emphasize that the proposed methodology is relatively simpler, from an implementation perspective since the shrinkage estimators are easy to compute, and does not require training making it ideal for deployment in practical applications, particularly for those involving hearing aids, mobile devices, etc. The goal in speech enhancement is to obtain an estimate of clean speech starting from the noisy signal by minimizing a chosen distortion measure (risk). Often, this results in an estimate that depends on the unknown clean signal or its statistics. Since access to such priors is limited or impractical, one has to rely on an estimate of the clean signal statistics. In this paper, we develop a risk estimation framework for speech enhancement, in which one optimizes an unbiased estimate of the risk instead of the actual risk. The estimated risk is expressed solely as a function of the noisy observations and the noise statistics. Hence, the corresponding denoiser does not require the clean speech prior. We consider several speech-specific perceptually relevant distortion measures and develop corresponding unbiased estimates. Minimizing the risk estimates gives rise to denoisers, which are nonlinear functions of the a posteriori SNR. Listening tests show that, within the risk estimation framework, Itakura-Saito and weighted hyperbolic cosine distortions are superior than the other measures. Comparisons in terms of perceptual evaluation of speech quality (PESQ), segmental SNR (SSNR), source-to-distortion ratio (SDR), and short-time objective intelligibility (STOI) also indicate a superior performance for these two distortion measures. For SNRs greater than 5 dB, the proposed approach results in better denoising performance — both in terms of objective and subjective assessment — than techniques based on the Wiener filter, log-MSE minimization, and Bayesian nonnegative matrix factorization. [ABSTRACT FROM AUTHOR]

Abstract Principal Hessian directions (pHd) based on the Hessian matrix is a moment-based method and a promising methodology for sufficient dimension reduction because of its easy implementation. However, it requires strong conditions on the distribution of the predictors, which must be nearly Gaussian. We investigate here whether and how this method is applicable when the distribution is a mixture multivariate skew elliptical (MMSE) distribution, and if not, how to adapt the technique. Further, we propose two new estimation algorithms for an extended version of pHd. The theoretical results also serve as a reminder for researchers and users to pay attention to the theoretical conditions on which pHd critically relies. Numerical studies are conducted to examine its performance in finite-sample cases. [ABSTRACT FROM AUTHOR]

This note consists of two parts. In the first part, we provide a pedagogic review on the multivariate generalized hyperbolic (MGH) distribution. We show that this probability family is close under margining, conditioning, and linear transforms; however, such property does not hold for its subclasses. In the second part, we obtain the Stein-type inequality in the context of MGH distribution. Moreover, we apply the Stein-type inequality to prove a lower bound for Var[h(X)]. Particularly, we present examples when X belongs to some well-known subclasses in MGH family. [ABSTRACT FROM AUTHOR]

A representation of the degrees of freedom akin to Stein's lemma is given for a class of estimators of a mean value parameter in Rn. Contrary to previous results our representation holds for a range of discontinues estimators. It shows that even though the discontinuities form a Lebesgue null set, they cannot be ignored when computing degrees of freedom. Estimators with discontinuities arise naturally in regression if data driven variable selection is used. Two such examples, namely best subset selection and lasso-OLS, are considered in detail in this paper. For lasso-OLS the general representation leads to an estimate of the degrees of freedom based on the lasso solution path, which in turn can be used for estimating the risk of lasso-OLS. A similar estimate is proposed for best subset selection. The usefulness of the risk estimates for selecting the number of variables is demonstrated via simulations with a particular focus on lasso-OLS. [ABSTRACT FROM AUTHOR]

In inverse regression-based methodologies for sufficient dimension reduction, ellipticity (or slightly more generally, the linearity condition) of the predictor vector is a widely used condition, though there is concern over its restrictiveness. In this paper, Stein's Lemma is generalized to the class of mixture multivariate skew-elliptical distributions in different scenarios to identify and estimate the central subspace. Within this class, necessary and sufficient conditions are explored for the simple covariance between the response (or its function) and the predictor vector to identify the central subspace. Further, we provides a way to do adjustments such that the central subspace can still be identifiable when this simple covariance fails to work. Simulations are used to assess the performance of the results and compare with existing methods. A data example is analysed for illustration. [ABSTRACT FROM AUTHOR]

We address the problem of channel estimation for cyclic-prefix (CP) Orthogonal Frequency Division Multiplexing (OFDM) systems. We focus on situations wherein knowledge of channel statistics are not available a priori, and model the channel as a vector of unknown deterministic parameters. Since computing the mean-square error (MSE) is not practicable in such a scenario, we propose a novel technique using Stein's lemma to obtain an unbiased estimate of the mean-square error, namely Stein's unbiased risk estimate (SURE). The channel estimate is obtained from noisy observations using linear and nonlinear denoising functions, whose parameters are chosen to minimize SURE. Based on computer simulations, it is shown that using SURE-based channel estimates for equalization offers an improvement in signal-to-noise performance over existing channel estimation schemes. [ABSTRACT FROM AUTHOR]

Abstract: When two random variables are bivariate normally distributed Stein's original lemma allows to conveniently express the covariance of the first variable with a function of the second. Landsman and Neslehova (2008) extend this seminal result to the family of multivariate elliptical distributions. In this paper we use the technique of conditioning to provide a more elegant proof for their result. In doing so, we also present a new proof for the classical linear regression result that holds for the elliptical family. [Copyright &y& Elsevier]

This article extends the classic Samuelson (1970) and Merton (1973) model of optimal portfolio allocation with one risky asset and a riskless one to include the effect of the skewness. Using an extended version of Stein's Lemma, we provide the explicit solution for optimal demand that holds for all expected utility maximizing investors when the risky asset is skew-normally and normally distributed. A closed expression is achieved for investors with constant absolute risk aversion. [ABSTRACT FROM AUTHOR]

We derive a large deviation result for the log-likelihood ratio for testing simple hypotheses in locally stationary Gaussian processes. This result allows us to find explicitly the rates of exponential decay of the error probabilities of type I and type II for Neyman-Pearson tests. Furthermore, we obtain the analogue of classical results on asymptotic efficiency of tests such as Stein's lemma and the Chernoff bound, as well as the more general Hoeffding bound concerning best possible joint exponential rates for the two error probabilities. [ABSTRACT FROM AUTHOR]

In this article, we consider a general form of univariate skewed distributions. We denote this form by GUS( λ; h( x)) or GUS with density s( x| λ, h( x)) = 2 f( x) G( λ h( x)), where f is a symmetric density, G is a symmetric differentiable distribution, and h( x) is an odd function. A special case of this general form, normal case, is derived and denoted by GUSN( λ; h( x)). Some representations and some main properties of GUS( λ; h( x)) are studied. The moments of GUSN( λ; h( x)) and SN( λ), the known skew normal distribution of Azzalini (1985), are compared and the relationship between them is given. As an application, we use it to construct a new form for skew t-distribution and skew Cauchy distribution. In addition, we extend Stein's lemma and study infinite divisibility of GUSN( λ; h( x)). [ABSTRACT FROM AUTHOR]

This paper addresses the detection of a stochastic process in noise from a finite sample under various sampling schemes. We consider two hypotheses. The noise only hypothesis amounts to model the observations as a sample of a i.i.d. Gaussian random variables (noise only). The signal plus noise hypothesis models the observations as the samples of a continuous time stationary Gaussian process (the signal) taken at known but random time-instants and corrupted with an additive noise. Two binary tests are considered, depending on which assumptions is retained as the null hypothesis. Assuming that the signal is a linear combination of the solution of a multidimensional stochastic differential equation (SDE), it is shown that the minimum Type II error probability decreases exponentially in the number of samples when the False Alarm probability is fixed. This behavior is described by error exponents that are completely characterized. It turns out that they are related to the asymptotic behavior of the Kalman Filter in random stationary environment, which is studied in this paper. Finally, numerical illustrations of our claims are provided in the context of sensor networks. [ABSTRACT FROM AUTHOR]

Abstract: Extensions of the Nourdin–Peccati analysis to -valued random variables are obtained by taking conditional expectation on the Wiener space. Several proof techniques are explored, from infinitesimal geometry, to quasi-sure analysis (including a connection to Stein''s lemma), to classical analysis on Wiener space. Partial differential equations for the density of an -valued centered random variable are obtained. Of particular importance is the function defined by the conditional expectation given Z of the auxiliary random matrix , , where D and are respectively the derivative operator and the generator of the Ornstein–Uhlenbeck semigroup on Wiener space. [Copyright &y& Elsevier]

Abstract: We consider a random variable satisfying almost-sure conditions involving where is ’s Malliavin derivative and is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. A lower- (resp. upper-) bound condition on is proved to imply a Gaussian-type lower (resp. upper) bound on the tail . Bounds of other natures are also given. A key ingredient is the use of Stein’s lemma, including the explicit form of the solution of Stein’s equation relative to the function , and its relation to . Another set of comparable results is established, without the use of Stein’s lemma, using instead a formula for the density of a random variable based on , recently devised by the author and Ivan Nourdin. As an application, via a Mehler-type formula for , we show that the Brownian polymer in a Gaussian environment, which is white-noise in time and positively correlated in space, has deviations of Gaussian type and a fluctuation exponent . We also show this exponent remains after a non-linear transformation of the polymer’s Hamiltonian. [Copyright &y& Elsevier]

Abstract: We study the degrees of freedom in shrinkage estimation of regression coefficients. Generalizing the idea of the Lasso, we consider the problem of estimating the coefficients by minimizing the sum of squares with the constraint that the coefficients belong to a closed convex set. Based on a differential geometric approach, we derive an unbiased estimator of the degrees of freedom for this estimation method, under a smoothness assumption on the boundary of the closed convex set. The result presented in this paper is applicable to estimation with a wide class of constraints. As an application, we obtain a type criterion and AIC for selecting tuning parameters. [Copyright &y& Elsevier]

The optimal exponential error rate for adaptive discrimination of two channels is discussed. In this problem, adaptive choice of input signal is allowed. This problem is discussed in various settings. It is proved that adaptive choice does not improve the exponential error rate in these settings. These results are applied to quantum state discrimination. [ABSTRACT FROM AUTHOR]

In the present work, a novel solution of the Reynolds Equation for hydrodynamically lubricated bearings with stochastic roughness is presented. On the basis of stochastic analysis and theory, an advanced form of Reynolds-type Equation is derived. The proposed equation formulation, utilizing Stein's Lemma to calculate variational derivatives of the pressure with respect to the roughness variables, presents some differences in comparison with the deterministic Reynolds Equation, as well as to other previously published approaches which consider the stochastic surface roughness. The proposed methodology is validated against relevant results in other literature works and applied to characterize the tribological performance of rough slider and journal bearings with different roughness profiles, over a wide range of bearing design and operational parameters. • Stochastic implementation of the surface roughness profiles as Gaussian variables. • Hydrodynamically lubricated bearings with surface roughness on the stator and rotor. • Utilization of Stein's Lemma, novel stochastic calculus theorem of probability theory. • Mathematical derivation of a novel Stochastic Reynolds-type Equation for rough bearings. • Comparison and validation of acquired equation with existing approaches in literature. [ABSTRACT FROM AUTHOR]

Abstract: Tracking the correct directions of monotonicity in multi-dimensional modeling plays an important role in interpreting functional associations. In the presence of multiple predictors, we provide empirical evidence that the observed monotone directions via parametric, nonparametric or semiparametric fit of commonly used multi-dimensional models may entirely violate the actual directions of monotonicity. This breakdown is caused primarily by the dependence structure of covariates, with negligible influence from the bias of function estimation. To examine the linkage between the dependent covariates and monotone directions, we first generalize Stein’s Lemma for random variables which are mutually independent Gaussian to two important cases: dependent Gaussian, and independent non-Gaussian. We show that in both two cases, there is an explicit one-to-one correspondence between the monotone directions of a multi-dimensional function and the signs of a deterministic surrogate vector. Moreover, we demonstrate that the second case can be extended to accommodate a class of dependent covariates. This generalization further enables us to develop a de-correlation transform for arbitrarily dependent covariates. The transformed covariates preserve modeling interpretability with little loss in modeling efficiency. The simplicity and effectiveness of the proposed method are illustrated via simulation studies and real data application. [Copyright &y& Elsevier]

Abstract: For the family of multivariate normal distribution functions, Stein''s Lemma presents a useful tool for calculating covariances between functions of the component random variables. Motivated by applications to corporate finance, we prove a generalization of Stein''s Lemma to the family of elliptical distributions. [Copyright &y& Elsevier]

When two random variables have a bivariate normal distribution, Stein's lemma (Stein, 1973, 1981), provides, under certain regularity conditions, an expression for the covariance of the first variable with a function of the second. An extension of the lemma due to Liu (1994) as well as to Stein himself establishes an analogous result for a vector of variables which has a multivariate normal distribution. The extension leads in turn to a generalization of Siegel's (1993) formula for the covariance of an arbitrary element of a multivariate normal vector with its minimum element. This article describes extensions to Stein's lemma for the case when the vector of random variables has a multivariate skew-normal distribution. The corollaries to the main result include an extension to Siegel's formula. This article was motivated originally by the issue of portfolio selection in finance. Under multivariate normality, the implication of Stein's lemma is that all rational investors will select a portfolio which lies on Markowitz's mean-variance efficient frontier. A consequence of the extension to Stein's lemma is that under multivariate skew-normality, rational investors will select a portfolio which lies on a single mean-variance-skewness efficient hyper-surface. [ABSTRACT FROM AUTHOR]

This paper reports the results of an investigation into the properties of a theoretical modification of beta proposed by Leland (1999) and based on earlier work of Rubinstein (1976). It is shown that when returns are elliptically symmetric, beta is the appropriate measure of risk and that there are other situations in which the modified beta will be similar to the traditional measure based on the capital asset pricing model. For the case where returns have a normal distribution, it is shown that the criterion either does not exist or reduces exactly to the conventional beta. It is therefore conjectured that the modified measure will only be useful for portfolios that have nonstandard return distributions which incorporate skewness. For such situations, it is shown how to estimate the measure using regression and how to compare the resulting statistic with a traditional estimated beta using Hotelling's test. An empirical study based on stocks from the FTSE350 does not find evidence to support the use of the new measure even in the presence of skewness.Journal of Asset Management (2007) 7, 388–403. doi:10.1057/palgrave.jam.2250054 [ABSTRACT FROM AUTHOR]

Inspired by Stein's lemma, we derive two expressions for the joint moments of elliptical distributions. We use two different methods to derive E [ X 1 2 f (X) ] for any measurable function f satisfying some regularity conditions. Then, by making use of this result, we obtain new formulae for expectations of product of normally distributed random variables, and also present simplified expressions of E [ X 1 2 f (X) ] for multivariate Student- t , logistic and Laplace distributions. [ABSTRACT FROM AUTHOR]