Find out more Skip Navigation Oxford Journals Contact Us My Basket My Account Biometrika About This Journal Contact This Journal Subscriptions View Current Issue (Volume 103 Issue 3 September 2016) Archive Note that Sn(θ) are the working estimating equations defined in (10), and let S(θ) be its expectation. Carroll Journal of the American Statistical Association Vol. 104, No. 487 (September 2009), pp. 1129-1143 Published by: Taylor & Francis, Ltd. Let T0 = (t1, t2, t3, t4) be the set of target ages; and Ti = (ti,1, ti,2, ti,3, ti,4) be the actual measurement ages of the ith subject.

The generated datasets follow the distribution characterized by Models (16), (17), and (18).We generated 20 synthetic data sets, and repeatedly estimated the coefficients using the naive approach, the interpolation approach and Think you should have access to this item via your institution? We further denote Vn = var{Sn(θ0)} and Dn=∂∂θ0Sn(θ0). SEMIPARAMETRIC JOINT ESTIMATING EQUATIONS2.1 Preliminaries and the Case That x Is ObservedSuppose {yi, xi} is a random sample from Model (1) with sample size n, where x = (x1 … xp)⊤

Then the new sample estimating equations are ∑i=1n∑j=1mΨτk(yi−x∼i,j⊤βτk)x∼i,jf(ν)(x∼i,j∣yi,wi)=0,k=1,…,kn,which is a weighted quantile regression with response yi over the covariates x̃i,j with weights f (ν)(xi,j|yi, wi).Note that the original quantile regression estimating In this way, we only need to solve the estimating equations (7) for the grid of internal knots, τk’s. Please try the request again. For each subject i, generate m = 500 x̃i,j’s from the log-normal distribution in (18), the conditional distribution of xi given wi.Step 3′(b).

The system returned: (22) Invalid argument The remote host or network may be down. See also Carriquiry (2003) for other methods. The iteration converges to the solution of the approximated working estimation equations (11), which we denote as θ̂n. Louis RePEc also has a blog.

We call (10) the working estimating equations, which can be viewed as a spline approximation of the unbiased semiparametric joint estimating equations defined in (7). The proposed method successfully corrected the bias, and brings the estimates fairly close to the true values with all the quantile levels. The difference between x⊤ (τ) and x⊤(τ) is then bounded by∣x⊤Lθ(τ)−x⊤Lξl(τ)∣≤{∣ξl[1]−θ[1]∣,τ<1/(kn+1)∣ξl[kn]−θ[kn]∣,τ>kn/(kn+1)2∣ξl[⌊τkn⌋]−θ[⌊τkn⌋]∣+∣ξl[⌊τkn⌋+1]−θ[⌊τkn⌋+1]∣,else.Since |θ − ξl|O(kn/n) implies maxk |θ[k] − ξl[k]| = O(kn/n), then the boundness above implies supτ|x⊤ (τ) − x⊤(τ)| = Our results show that use of these methods will still lead to consistent estimation of the quantile function, although asymptotic distributions would have to be addressed separately if such deconvolution approaches

This paper aims at developing statistical methods and theory that yield consistent quantile estimation in the presence of covariate measurement error.There is some work on measurement error in quantile regression. With a sufficient numbers of knots, that is, kn → ∞ and ε → 0, the difference between β(τ), and its spline approximation (τ) is negligible (de Boor 2001). The convergence of the proposed algorithm follows from classical results on the EM algorithm (McLachlan and Krishnan 2008, page 19).The estimation algorithm involves a turning parameter, the number of quantile levels. To mimic the NCPP data, we chose the same sample size, n = 232, and use the original birth weight and actual measurement ages ti,2, ti,3, and ti,4 for all the

When x is measured with error and instead only a surrogate wi is observed, naively replacing xi by the observed wi will result in substantial bias. The equivalent version of this assumption is that 0 < f (εi) < ∞, and this is commonly assumed in the quantile regression literature; see, for example, Portnoy (2003) and Koenker Finally, we apply our methodology to part of the National Collaborative Perinatal Project growth data, a longitudinal study with an unusual measurement error structure.Keywords: Correction for attenuation, Growth curves, Longitudinal data, Thus, (3) is a slight abuse of notation, but since everything else involving observed data is an estimating equation that will have a zero, we will use the estimating equation nomenclature.

We hence reduce the infinite dimensional estimating equations (7) to a finite dimensional case Sn(θ)=n−1∑i=1n∫xΨ(yi−x⊤θ)⊗x·f(x∣yi,wi;θ)dx=0,(10) where Ψ(yi − x⊤θ) = {Ψτ1(yi −x⊤βτ1),…, Ψτkn(yi − x⊤ × βτkn)}⊤ is a kn-dimensional vector, Technical details are given in an Appendix.2. Its reciprocal is known as the sparsity function (Welsh 1988 and Koenker and Xiao 2004). In what follows, we show that kn−1I=o(1).

These choices correspond to moderate and larger attenuation which equal R = 4/5 and R = 2/3, respectively.Regression model scale It is important to note that the model (13) for the Come back any time and download it again. The estimation algorithm remains unchanged except that we need to replace the f (x|wi) in Step 2 by f̃(x|wi). Let S(θ) be the expectation of Sn(θ).

They are unknown parameters in the estimating equations, while the previously defined β0(τ) and β0,τ are the corresponding true values.Equation (6) is derived from the fact that the conditional quantile function SIMULATION STUDY5.1 Model SetupTo understand the effects of measurement errors and to demonstrate the performance of our method, we used a location-scale quantile regression model yi=β1+β2xi+(γ1+γ2xi)εi,(13) where εi = Normal(0, 1). Set initial values of θ based on uncorrected quantile regression.Step 2. Download Info If you experience problems downloading a file, check if you have the proper application to view it first.

The solid dark-gray lines are estimated coefficients from the misspecified f (x|w), and the long-dashed dark-gray lines are those from the correctly estimated ones. The method is based on constructing estimating equations jointly for all quantile levels τ ∈ (0, 1), and avoids specifying a distribution for the response given the true covariates. Printed from https://ideas.repec.org/ Share: MyIDEAS: Log in (now much improved!) to save this article Quantile Regression with Classical Additive Measurement Errors Contents:Author info Abstract Bibliographic info Download info Related research References ESTIMATION ALGORITHM3.1 PreliminariesThe crux of all measurement error problems which have a likelihood flavor is the estimating of the distribution of x given w.

The standard error of ...6.3 Further Investigation via Simulated NCPP-Like DataTo understand the observed differences seen in the NCPP example, we generated synthetic data sets based on the estimated models (16), It is well known that such errors can sometimes lead to substantial attenuation of estimated effects in mean regression (Carroll et al. 2006). Using similar arguments, we can also show that kn−1III=o(1). Zvi Griliches & Jerry A.

Let Sn(βτ) be the p× 1 subset of Sn(θ) that corresponds to the τ th quantile, and let S(βτ) = E{Sn(βτ)} be its expectation.We make the following assumptions.Assumption 3The true coefficient Therefore, f (y|x), and consequently, f (x|yi, wi) does not have a parametric form. Pay attention to names, capitalization, and dates. × Close Overlay Journal Info Journal of the American Statistical Association Description: The Journal of the American Statistical Association (JASA) has long been considered The integration in (4) makes the function continuous in its argument.

We apply our method to obtain consistent estimation of the βτ’s.Model for f (x|w) Suppose is the jth measurement of the ith subject taken at age ti,j. We repeated exactly the same estimation procedure (d) assuming U is Gaussian. Let β̂n(τ) = (τ) be the natural linear spline extended from θ̂n. We use f {x|y, w; β(τ)} and f {y|x; β(τ)} to indicate their dependence on the entire unknown quantile process β(τ).

Our simulations does not obey this restriction, although due to the nature of the data, our empirical example does obey the restriction.Recall that β0(τ) is the true quantile coefficient function, and The top panel plots the mean of the estimated slope functions with the red solid line representing the true slope function. To get a better understanding of this, we can rewrite the conditional density f (x|yi, wi) under the surrogacy condition byf(x∣yi,wi)=f(yi∣x)f(x∣wi)∫xf(yi∣x)f(x∣wi)dx.(5)In the spirit of quantile regression, we leave the error distribution Hu and Schennach (2008) and Schennach (2008) proved nonparametric identification of a nonparametric quantile function under various settings where there is an instrumental variable measured on all sampling units.

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