# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Estimation in a semiparametric partially linear errors-in-variables model. (English) Zbl 0977.62036
Summary: We consider the partially linear model relating a response $Y$ to predictors $(X,T)$ with mean function $X^T\beta +g(T)$ when the $X$’s are measured with additive error. The semiparametric likelihood estimate of {\it T.A. Severini} and {\it J.G. Staniswalis} [J. Am. Stat. Assoc. 89, No. 426, 501-511 (1994; Zbl 0798.62046)] leads to biased estimates of both the parameter $\beta$ and the function $g(\cdot)$ when measurement error is ignored. We derive a simple modification of their estimator whieh is a semiparametric version of the usual parametric correction for attenuation. The resulting estimator of $\beta$ is shown to be consistent and its asymptotic distribution theory is derived. Consistent standard error estimates using sandwich-type ideas are also developed.

##### MSC:
 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics 62G08 Nonparametric regression 62G20 Nonparametric asymptotic efficiency
XploRe
Full Text:
##### References:
 [1] CARROLL, R. J., RUPPERT, D. and STEFANSKI, L. A. 1995. Nonlinear Measurement Error Models. Chapman and Hall, New York. Z. · Zbl 0853.62048 [2] CHEN, H. 1988. Convergence rates for parametric components in a partly linear model. Ann. Statist. 16 136 146. Z. · Zbl 0637.62067 · doi:10.1214/aos/1176350695 [3] CUZICK, J. 1992a. Semiparametric additive regression. J. Roy. Statist. Soc. Ser. B 54 831 843. Z. JSTOR: · Zbl 0776.62036 · http://links.jstor.org/sici?sici=0035-9246%281992%2954%3A3%3C831%3ASAR%3E2.0.CO%3B2-Z&origin=euclid [4] CUZICK, J. 1992b. Efficient estimates in semiparametric additive regression models with unknown error distribution. Ann. Statist. 20 1129 1136. Z. · Zbl 0746.62037 · doi:10.1214/aos/1176348675 [5] ENGLE, R. F., GRANGER, C. W. J., RICE, J. and WEISS, A. 1986. Semiparametric estimates of the relation between weather and electricity sales. J. Amer. Statist. Assoc. 81 310 320. Z. [6] FAN, J. and TRUONG, Y. K. 1993. Nonparametric regression with errors in variables. Ann. Statist. 21 1900 1925. Z. · Zbl 0791.62042 · doi:10.1214/aos/1176349402 [7] FULLER, W. A. 1987. Measurement Error Models. Wiley, New York. Z. · Zbl 0800.62413 [8] HARDLE, W., KLINKE, S. and TURLACH, B. A. 1995. XploRe: An Interactive Statistical Computing Ënvironment. Springer, New York. Z. · Zbl 0826.62001 [9] HECKMAN, N. E. 1986. Spline smoothing in partly linear models. J. Roy. Statist. Soc. Ser. B 48 244 248. Z. JSTOR: · Zbl 0623.62030 · http://links.jstor.org/sici?sici=0035-9246%281986%2948%3A2%3C244%3ASSIAPL%3E2.0.CO%3B2-T&origin=euclid [10] KENDALL, M. and STUART, A. 1992. The Advanced Theory of Statistics 2, 4th ed. Griffin, London. Z. [11] LIANG, H. and HARDLE, W. 1997. Asymptotic normality of parametric part in partially linear ḧeteroscedastic regression models. DP 33, SFB 373, Humboldt Univ. Berlin. Z. [12] SEVERINI, T. A. and STANISWALIS, J. G. 1994. Quasilikelihood estimation in semiparametric models. J. Amer. Statist. Assoc. 89 501 511. Z. JSTOR: · Zbl 0798.62046 · doi:10.2307/2290852 · http://links.jstor.org/sici?sici=0162-1459%28199406%2989%3A426%3C501%3AQEISM%3E2.0.CO%3B2-T&origin=euclid [13] SPECKMAN, P. 1988. Kernel smoothing in partial linear models. J. Roy. Statist. Soc. Ser. B 50 413 436. JSTOR: · Zbl 0671.62045 · http://links.jstor.org/sici?sici=0035-9246%281988%2950%3A3%3C413%3AKSIPLM%3E2.0.CO%3B2-%23&origin=euclid [14] COLLEGE STATION, TEXAS 77843-3143 E-MAIL: carroll@stat.tamu.edu