Dimension reduction based on conditional multiple index density function

Zhang, Jun; He, Baohua; Lü, Tao; Wen, Songqiao

doi:10.1214/17-bjps370

Cited by 3 publications

(2 citation statements)

References 39 publications

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“…It is seen that

E false(false| Y - sans-serifg false(bold-italicX, {bold-italicα}_{0} false) false| false| bold-italicX false) = σ false({bold-italicβ}_{ϕ_{0}}^{normalT} bold-italicX false)

, that is, a SIM. One can use the dimension‐reduction method for

{false{false| Y_{i} - sans-serifg false({bold-italicX}_{i}, \overset{bold-italicα}{true} false) false|, {bold-italicX}_{i} false}}_{i = 1}^{n}

to obtain an initial estimator of ϕ 0 (see, e.g., Cui et al, ; Feng, Wang, & Zhu, ; Xia & Härdle, ; Zhang, He, Lu, & Wen, ). After we obtain the initial estimator, we then update ϕ 0 by the Newton–Raphson approximation:

{\hat{ϕ}}_{new} = {\hat{ϕ}}_{old} - V_{n}^{- 1} ({\hat{ϕ}}_{old}) W_{n} ({\hat{ϕ}}_{old}),

where

{scriptW}_{n} false({\overset{ϕ}{true}}_{old} false)

is defined in by substituting ϕ with

{\overset{ϕ}{true}}_{old}

and

{scriptV}_{n} false({\overset{ϕ}{true}}_{old} false)

is defined as

V_{n} ({\hat{ϕ}}_{old}) = - \sum_{i = 1}^{}

…”

Section: Estimation Methodologymentioning

confidence: 99%

Nonlinear regression models with single‐index heteroscedasticity

Zhang

Gai

Lin

et al. 2019

Statistica Neerlandica

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We consider nonlinear heteroscedastic single‐index models where the mean function is a parametric nonlinear model and the variance function depends on a single‐index structure. We develop an efficient estimation method for the parameters in the mean function by using the weighted least squares estimation, and we propose a “delete‐one‐component” estimator for the single‐index in the variance function based on absolute residuals. Asymptotic results of estimators are also investigated. The estimation methods for the error distribution based on the classical empirical distribution function and an empirical likelihood method are discussed. The empirical likelihood method allows for incorporation of the assumptions on the error distribution into the estimation. Simulations illustrate the results, and a real chemical data set is analyzed to demonstrate the performance of the proposed estimators.

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“…It is seen that

E false(false| Y - sans-serifg false(bold-italicX, {bold-italicα}_{0} false) false| false| bold-italicX false) = σ false({bold-italicβ}_{ϕ_{0}}^{normalT} bold-italicX false)

, that is, a SIM. One can use the dimension‐reduction method for

{false{false| Y_{i} - sans-serifg false({bold-italicX}_{i}, \overset{bold-italicα}{true} false) false|, {bold-italicX}_{i} false}}_{i = 1}^{n}

{\hat{ϕ}}_{new} = {\hat{ϕ}}_{old} - V_{n}^{- 1} ({\hat{ϕ}}_{old}) W_{n} ({\hat{ϕ}}_{old}),

where

{scriptW}_{n} false({\overset{ϕ}{true}}_{old} false)

is defined in by substituting ϕ with

{\overset{ϕ}{true}}_{old}

and

{scriptV}_{n} false({\overset{ϕ}{true}}_{old} false)

is defined as

V_{n} ({\hat{ϕ}}_{old}) = - \sum_{i = 1}^{}

…”

Section: Estimation Methodologymentioning

confidence: 99%

Nonlinear regression models with single‐index heteroscedasticity

Zhang

Gai

Lin

et al. 2019

Statistica Neerlandica

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“…Step 1: Compute the estimated projection direction̂[ ] by fitting a single-index model with "synthesis" data Cui, Härdle, and Zhu (2011);Li, Lai, and Lian (2015); Li, Peng, Dong, and Tong (2014); Lian, Liang, and Carroll (2015); Liang, Liu, Li, and Tsai (2010); Peng and Huang (2011);Yang, Tong, and Li (2019); Zhang, Feng, and Xu (2015); Zhang, He, Lu, and Wen (2018).…”

Section: (B) Under the Null Hypothesismentioning

confidence: 99%

Model checking for multiplicative linear regression models with mixed estimators

Zhang

2021

Statistica Neerlandica

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In this paper, we introduce the mixed estimators based on product least relative error estimation and least squares estimation in a multiplicative linear regression model. The asymptotic properties for the mixed estimators are established. We present some explicit expressions of the optimal estimator of the mixed estimators, and we also suggest some numerical solutions in the simulation studies and real data analysis. Studying model checking problems for multiplicative linear regression models, we propose four test statistics. One is the score-type test statistic, the second one is the residual-based empirical process test statistic marked by proper functions of the covariates. The third one is the integrated conditional moment test statistic by using linear projection weighting function, and the fourth one is the adaptive model test statistic. These test statistics are all related to the mixed estimators.The asymptotic properties of these test statistics are established, and some bootstrap procedures for calculating the critical values are also proposed. Simulation studies are conducted to demonstrate the performance of the proposed estimation procedures, and a real example is analyzed to illustrate its practical usage.

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