Local Polynomial Variance-Function Estimation

Ruppert, David; Wand, M. P.; Holst, Ulla; HöSJER, Ola

doi:10.1080/00401706.1997.10485117

Cited by 113 publications

(60 citation statements)

References 27 publications

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“…The LIDAR data was obtained from atmospheric monitoring of pollutants and consists of the range, or distance traveled before light is reflected back to a source and log ratio, the logarithm of the ratio of received light from two laser sources. For more details see Ruppert, Wand, Holst, and Hössier (1997). Figure 1 shows the effect of four mean-one Gamma priors for the precision of the truncated polynomial parameters.…”

Section: Priors On the Precision Parametersmentioning

confidence: 97%

Spatially Adaptive Bayesian Penalized Splines With Heteroscedastic Errors

Crainiceanu

Ruppert

Carroll

et al. 2007

Journal of Computational and Graphical Statistics

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Penalized splines have become an increasingly popular tool for nonparametric smoothing because of their use of low-rank spline bases, which makes computations tractable while maintaining accuracy as good as smoothing splines. This article extends penalized spline methodology by both modeling the variance function nonparametrically and using a spatially adaptive smoothing parameter. This combination is needed for satisfactory inference and can be implemented effectively by Bayesian MCMC. The variance process controlling the spatially adaptive shrinkage of the mean and the variance of the heteroscedastic error process are modeled as log-penalized splines. We discuss the choice of priors and extensions of the methodology, in particular, to multivariate smoothing. A fully Bayesian approach provides the joint posterior distribution of all parameters, in particular, of the error standard deviation and penalty functions. MATLAB, C, and FORTRAN programs implementing our methodology are publicly available.

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Section: Priors On the Precision Parametersmentioning

confidence: 97%

Spatially Adaptive Bayesian Penalized Splines With Heteroscedastic Errors

Crainiceanu

Ruppert

Carroll

et al. 2007

Journal of Computational and Graphical Statistics

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“…In their generalization, however, the nonparametric kernel regression was applied directly to estimate the multivariate regression function, which may not be effective due to the “curse of dimensionality”. In regressions with univariate predictor and response, Ruppert et al (1997) and Fan and Yao (1998) applied local linear regression to the squared residuals and demonstrated that such nonparametric estimate performs asymptotically as well as if the conditional mean was given a priori. Song and Yang (2009) derived asymptotically exact and conservative confidence bands for the heteroscedastic variance functions.…”

Section: Introductionmentioning

confidence: 99%

Semiparametric estimation of conditional heteroscedasticitity via single-index modeling

Zhu

Dong

Li³

2013

STAT SINICA

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We consider a single-index structure to study heteroscedasticity in regression with high-dimensional predictors. A general class of estimating equations is introduced, the resulting estimators remain consistent even when the structure of the variance function is misspecified. The proposed estimators also possess an adaptive property in an asymptotic sense. That is, they estimate the conditional variance function asymptotically as well as if the conditional mean function was given a priori. Numerical studies confirm our theoretical observations and demonstrate that our proposed estimator is superior to existing estimators with less bias and smaller standard deviation.

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“…The function α (·) can be estimated by a local linear regression estimator, resulting in an estimated function α̂ (·). The squared residuals

{false{r_{g i}^{2} false}}_{g = 1}^{N}

is then further smoothed on

{false{X_{g i} false}}_{g = 1}^{N}

to obtain an estimate ξ̂ 2 ( x ) of the variance function σ 2 (·), where r gi = Ŷ gi − α̂ ( X gi ) (Ruppert et al, 1997). …”

Section: Simulations and Comparisonsmentioning

confidence: 99%

“…But in the absence of other available techniques, they had to impose that the treatment effect { α g } is also a smooth function of the intensity level so that they can apply nonparametric methods to estimate genewise variance (Ruppert et al, 1997). However, this assumption is not valid in most microarray applications, and the estimator of genewise variance incurs big biases unless { α g } is sparse, a situation that Fan et al (2004) hoped.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric estimation of genewise variance for microarray data

Fan¹,

Feng²,

Niu³

2010

Ann. Statist.

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Estimation of genewise variance arises from two important applications in microarray data analysis: selecting significantly differentially expressed genes and validation tests for normalization of microarray data. We approach the problem by introducing a two-way nonparametric model, which is an extension of the famous Neyman-Scott model and is applicable beyond microarray data. The problem itself poses interesting challenges because the number of nuisance parameters is proportional to the sample size and it is not obvious how the variance function can be estimated when measurements are correlated. In such a high-dimensional nonparametric problem, we proposed two novel nonparametric estimators for genewise variance function and semiparametric estimators for measurement correlation, via solving a system of nonlinear equations. Their asymptotic normality is established. The finite sample property is demonstrated by simulation studies. The estimators also improve the power of the tests for detecting statistically differentially expressed genes. The methodology is illustrated by the data from MicroArray Quality Control (MAQC) project.

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Local Polynomial Variance-Function Estimation

Cited by 113 publications

References 27 publications

Spatially Adaptive Bayesian Penalized Splines With Heteroscedastic Errors

Spatially Adaptive Bayesian Penalized Splines With Heteroscedastic Errors

Semiparametric estimation of conditional heteroscedasticitity via single-index modeling

Nonparametric estimation of genewise variance for microarray data

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