Wavelet estimators for the derivatives of the density function from data contaminated with heteroscedastic measurement errors

Wang, Jinru; Zhang, Qingqing; Kou, Junke

doi:10.1080/03610926.2016.1148735

Cited by 8 publications

(5 citation statements)

References 22 publications

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“…When observations can be made directly on

X

, the kernel density estimation procedure is often called on for this purpose. Starting with the classical kernel estimator, Wang et al (2009) followed the classical SIMEX algorithm, constructed an average of the kernel estimator

{\hat{f}}_{B,n}(x)={B}^{-1}{\sum}_{b=1}^B\left[{n}^{-1}{\sum}_{i=1}^n{K}_h\left(x-{Z}_i-\sqrt{\lambda }{V}_{i,b}\right)\right]

with

B

pseudo‐datasets

{\left\{{Z}_i+\sqrt{\lambda }{V}_{i,b}\right\}}_{i=1}^n

b=1,2,\dots, B

, where

{K}_h\left(\cdotp \right)={h}^{-1}K\left(\cdotp /h\right)

K

is a kernel density function, and

h

is a sequence of positive numbers often called bandwidths. By the law of large numbers,

{\overset{f}{true}}_{B, n} false(x false) \to

…”

Section: Motivating Examplesmentioning

confidence: 99%

“…By the law of large numbers,

{\hat{f}}_{B,n}(x)\to {n}^{-1}{\sum}_{i=1}^n\int {K}_h\left(x-{Z}_i-\sqrt{\lambda }{\sigma}_uu\right)\phi (u) du={\tilde{f}}_n(x)

in probability as

B\to \infty

. After some algebra, Wang et al (2009) proposed to estimate

{f}_X(x)

using

{\hat{f}}_n(x)={n}^{-1}{\sum}_{i=1}^n{\left(\sqrt{\lambda }{\sigma}_u\right)}^{-1}\phi \left(\left(x-{Z}_i\right)/\sqrt{\lambda }{\sigma}_u\right)

which approximates the limit

{\tilde{f}}_n(x)

for sufficiently large

n

. In fact, before initiating the simulation step, Cook and Stefanski (1994) suggested one should try to calculate the conditional expectation

E false[{\overset{f}{true}}_{B, n} false(x false...

…”

Section: Motivating Examplesmentioning

confidence: 99%

“…It is interesting to note that if we deliberately choose the kernel function

K

to be the standard norm density, we can show that

{\tilde{f}}_n(x)={\left(n\sqrt{\lambda {\sigma}_u^2+{h}^2}\right)}^{-1}{\sum}_{i=1}^n\phi \left(\left(x-{Z}_i\right)/\sqrt{\lambda {\sigma}_u^2+{h}^2}\right)

which can also be directly used for extrapolation. Because there is no approximation done here,

{\tilde{f}}_n(x)

should potentially perform better than the estimator

{\hat{f}}_n(x)

as proposed in Wang et al (2009).…”

Section: Motivating Examplesmentioning

confidence: 99%

See 2 more Smart Citations

Extrapolation estimation for nonparametric regression with measurement error

Song

Ayub

Shi

2023

Scandinavian J Statistics

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For the nonparametric regression models with covariates contaminated with the normal measurement errors, this paper proposes an extrapolation algorithm to estimate the regression functions. By applying the conditional expectation directly to the kernel‐weighted least squares of the deviations between the local linear approximation and the observed responses, the proposed algorithm successfully bypasses the simulation step in the classical simulation extrapolation, thus significantly reducing the computational time. It is noted that the proposed method also provides an exact form of the extrapolation function, although the extrapolation estimate generally cannot be obtained by simply setting the extrapolation variable to negative one in the fitted extrapolation function, if the bandwidth is less than the SD of the measurement error. Large sample properties of the proposed estimation procedure are discussed, as well as simulation studies and a real data example being conducted to illustrate its applications.

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“…When observations can be made directly on

X

{\hat{f}}_{B,n}(x)={B}^{-1}{\sum}_{b=1}^B\left[{n}^{-1}{\sum}_{i=1}^n{K}_h\left(x-{Z}_i-\sqrt{\lambda }{V}_{i,b}\right)\right]

with

B

pseudo‐datasets

{\left\{{Z}_i+\sqrt{\lambda }{V}_{i,b}\right\}}_{i=1}^n

b=1,2,\dots, B

, where

{K}_h\left(\cdotp \right)={h}^{-1}K\left(\cdotp /h\right)

K

is a kernel density function, and

h

is a sequence of positive numbers often called bandwidths. By the law of large numbers,

{\overset{f}{true}}_{B, n} false(x false) \to

…”

Section: Motivating Examplesmentioning

confidence: 99%

“…By the law of large numbers,

{\hat{f}}_{B,n}(x)\to {n}^{-1}{\sum}_{i=1}^n\int {K}_h\left(x-{Z}_i-\sqrt{\lambda }{\sigma}_uu\right)\phi (u) du={\tilde{f}}_n(x)

in probability as

B\to \infty

. After some algebra, Wang et al (2009) proposed to estimate

{f}_X(x)

using

{\hat{f}}_n(x)={n}^{-1}{\sum}_{i=1}^n{\left(\sqrt{\lambda }{\sigma}_u\right)}^{-1}\phi \left(\left(x-{Z}_i\right)/\sqrt{\lambda }{\sigma}_u\right)

which approximates the limit

{\tilde{f}}_n(x)

for sufficiently large

n

. In fact, before initiating the simulation step, Cook and Stefanski (1994) suggested one should try to calculate the conditional expectation

E false[{\overset{f}{true}}_{B, n} false(x false...

…”

Section: Motivating Examplesmentioning

confidence: 99%

“…It is interesting to note that if we deliberately choose the kernel function

K

to be the standard norm density, we can show that

{\tilde{f}}_n(x)={\left(n\sqrt{\lambda {\sigma}_u^2+{h}^2}\right)}^{-1}{\sum}_{i=1}^n\phi \left(\left(x-{Z}_i\right)/\sqrt{\lambda {\sigma}_u^2+{h}^2}\right)

which can also be directly used for extrapolation. Because there is no approximation done here,

{\tilde{f}}_n(x)

should potentially perform better than the estimator

{\hat{f}}_n(x)

as proposed in Wang et al (2009).…”

Section: Motivating Examplesmentioning

confidence: 99%

See 1 more Smart Citation

Extrapolation estimation for nonparametric regression with measurement error

Song

Ayub

Shi

2023

Scandinavian J Statistics

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show abstract

“…Derivative estimation has been studied in many statistical model, see [16][17][18][19][20][21]. For example, [17] develops an adaptive estimator of d-th derivatives of unknown function in the standard deconvolution model (M = 1) and proves that it achieves near optimal rates of convergence under the mean integrated squared error (MISE) over a wide range of smoothness classes.…”

Section: Introductionmentioning

confidence: 99%

Wavelet Estimation of Function Derivatives from a Multichannel Deconvolution Model

Guo

2022

Journal of Function Spaces

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This paper considers a multichannel deconvolution model with Gaussian white noises. The goal is to estimate the d -th derivatives of an unknown function in the model. For super-smooth case, we construct an adaptive linear wavelet estimator by wavelet projection method. For regular-smooth case, we provide an adaptive nonlinear wavelet estimator by hard-thresholded method. In order to measure the global performances of our estimators, we show upper bounds on convergence rates using the L p -risk ( 1 ≤ p < ∞ ).

show abstract

“…Chesneau & Fadili [9] constructed a wavelet estimator of the density and investigated its MISE ( 2 risk) performance over Besov balls. The risk (1 ≤ < ∞) of wavelet deconvolution estimator was extended by Wang, Zhang & Kou [10]. However, we do not know whether the density function is smooth or not in some practical applications.…”

mentioning

confidence: 99%