Nonlinear models with measurement errors subject to single-indexed distortion

Abstract

a b s t r a c tWe study nonlinear regression models whose both response and predictors are measured with errors and distorted as single-index models of some observable confounding variables, and propose a multicovariate-adjusted procedure. We first examine the relationship between the observed primary variables (observed response and observed predictors) and the confounding variables by appropriately estimating the single index. We then develop a semiparametric profile nonlinear least square estimation procedure for the parameters of interest after we calibrate the error-prone response and predictors. Asymptotic properties of the proposed estimators are established. To avoid estimating the asymptotic covariance matrix that contains the infinite-dimensional nuisance distorting functions and the single index, and to improve the accuracy of the proposed estimation, we also propose an empirical likelihood-based statistic, which is shown to be asymptotically chi-squared. A simulation study is conducted to evaluate the performance of the proposed methods and a real dataset is analyzed as an illustration.

Statistical inference on partial linear additive models with distortion measurement errors

“…A test statistic based on the difference between the sums of squared residuals under the null and alternative hypotheses is proposed. The estimation procedure and testing procedure in this paper can be applied to the partial linear singleindex model [52], the case of single-index distortion [27,47,53], the case of additive distortion [51], and others. The methodologies of model checking [43,44] can also be considered to check the adequacy of PLAMs with the distortion measurement errors.…”

“…Conditions (A1)-(A2) are used in the estimation of PLAMs, see for example, [13]. Conditions (A3)-(A6) are used in the study of covariate-adjusted models, see [3,33] and [52][53][54]. The bounded assumptions in (A2)-(A3) for the supports of Z j , U entail no loss of generality as the covariates can always be carried out monotone increasing transformations of Z j and U, even if their supports before transformation are unbounded.…”

“…For example, in a study of the relationship between fibrinogen level and serum transferrin level among hemodialysis patients, Kaysen et al [15] analyzed the fibrinogen level and serum transferrin level by firstly normalizing them by BMI, which indicates that there may exist a multiplicative fashion relationship between the unobserved primary variables and the confounding variables. For the statistical modeling of the distortion measurement errors data, readers can refer to the parametric models [3,[27][28][29][32][33][34][35][36]46,47,51,53], the partial linear models [21], the partial linear single index models [52], the dimension reduction models [54]. However, these existing literature do not consider estimation proposals for the PLAMs under the distortion measurement errors setting.…”

Statistical inference on restricted partial linear regression models with partial distortion measurement errors

“…Z HANG , Z HU and Z HU () also considered a general semi‐parametric model by using the dimension reduction techniques. Recently, Zhang, Gai and Wu () considered the estimation for linear and non‐linear models with multivariate confounding variables. Z HANG , L I and F ENG () proposed an empirical process test statistic based on residuals to solve the problem of model checking on parametric distortion measurement error regression models.…”

Nonparametric covariate-adjusted regression

“…32) such that in view of (7.31), P {C n (λ) c } ≤ 2N exp(−τ λ) for some absolute constant τ > 0. By taking N = n and λ = 2τ −1 log n, it follows from (7.30) and (7.32) that max x∈[a,b] |R n (x)| = O P {h 1/2 (n/ log n) −1/2 + n −1 log n + (nh) −1 }.…”