Current status linear regression

Groeneboom, Piet; Hendrickx, Kim

doi:10.1214/17-aos1589

Cited by 44 publications

(59 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let

double-struckS

be a local parametrization mapping

R^{d - 1}

to the sphere

S_{d - 1}

, that is, for each

bold-italicα \in scriptB false(α_{0}, δ_{0} false)

on the sphere

S_{d - 1}

, there exists a unique vector

bold-italicβ \in R^{d - 1}

such that

α = S (bold-italicβ) .

The minimization problem given in is equivalent to minimizing

\frac{1}{n} true \sum_{i = 1}^{n} {({}, Y_{i} - {\overset{ψ}{true}}_{n bold-italicα} ((), double-struckS {false(bold-italicβ false)}^{T} X_{i}))}^{2},

over all β , where

{\overset{ψ}{true}}_{n bold-italicα}

is the LSE of the link function with

bold-italicα = double-struckS false(bold-italicβ false)

. Analogously to the treatment of the score approach in the current status regression model proposed by Groeneboom and Hendrickx (), we consider the derivative of w.r.t. β , where we ignore the nondifferentiability of the LSE

{\overset{ψ}{true}}_{n bold-italicα}

.…”

Section: Score Estimators For the Regression Parameter αmentioning

confidence: 99%

“…This leads to the set of d − 1 equations,

\frac{1}{n} true \sum_{i = 1}^{n} {((), J_{S} false(bold-italicβ false))}^{T} X_{i} ({}, Y_{i} - {\overset{ψ}{true}}_{n bold-italicα} ((), double-struckS {false(bold-italicβ false)}^{T} X_{i})) = bold0,

where

J_{S}

is the Jacobian of the map

double-struckS

and where

bold0 \in R^{d - 1}

is the vector of zeros. Just as in the analogous case of the “simple score equation” in the work of Groeneboom and Hendrickx (), we cannot hope to solve Equation exactly due to the discrete nature of the score function in . Instead, we define the solution in terms of a “zero‐crossing” of the above equation.…”

Section: Score Estimators For the Regression Parameter αmentioning

confidence: 99%

“…Recently, Groeneboom and Hendrickx () developed several score estimators for the current status linear regression model

Y = {bold-italicβ}_{0}^{T} bold-italicZ + ε

, where the distribution function F 0 of ε is left unspecified. Instead of observing the response Y , a censoring variable T and censoring indicator Δ = 1 Y ≤ T are observed.…”

Section: Introductionmentioning

confidence: 99%

“…This model is a special case of the monotone single‐index model and can be formulated as

double-struckE false(normalΔ false∣ T, bold-italicZ false) = F_{0} false(T - {bold-italicβ}_{0}^{T} bold-italicZ false) = F_{0} false({bold-italicα}_{0}^{T} bold-italicX false)

, where α 0 = (1, − β 0 ) T and X =( T , Z ) T . The estimators in the work of Groeneboom and Hendrickx () are obtained by the root of a score function involving the maximum likelihood estimator (MLE) of the distribution function for fixed β . The authors prove

\sqrt{n}

‐consistency and asymptotic normality of their estimators and show that, under certain smoothness assumptions, the limiting variance of a score estimator is arbitrarily close to the efficient variance.…”

Section: Introductionmentioning

confidence: 99%

“…We consider extending the estimators in the work of Groeneboom and Hendrickx () to the more general single‐index regression problem and propose two different score equations involving the LSE

{\overset{ψ}{true}}_{n bold-italicα}

of the unknown link function ψ 0 , which minimizes

\frac{1}{n} true \sum_{i = 1}^{n} {({}, Y_{i} - ψ false(α^{T} X_{i} false))}^{2},

over all monotone‐increasing functions ψ for fixed α . We establish an

n^{1 false/ 3} false/ \log n

‐rate for the estimator

{\overset{ψ}{true}}_{n bold-italicα}

and propose a single‐index score estimator of α 0 that converges at the parametric

\sqrt{n}

‐rate to the true regression parameter α 0 .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Score estimation in the monotone single‐index model

Balabdaoui

Groeneboom

Hendrickx

2018

Scandinavian J Statistics

Self Cite

View full text Add to dashboard Cite

We consider estimation in the single‐index model where the link function is monotone. For this model, a profile least‐squares estimator has been proposed to estimate the unknown link function and index. Although it is natural to propose this procedure, it is still unknown whether it produces index estimates that converge at the parametric rate. We show that this holds if we solve a score equation corresponding to this least‐squares problem. Using a Lagrangian formulation, we show how one can solve this score equation without any reparametrization. This makes it easy to solve the score equations in high dimensions. We also compare our method with the effective dimension reduction and the penalized least‐squares estimator methods, both available on CRAN as R packages, and compare with link‐free methods, where the covariates are elliptically symmetric.

show abstract

“…Let

double-struckS

be a local parametrization mapping

R^{d - 1}

to the sphere

S_{d - 1}

, that is, for each

bold-italicα \in scriptB false(α_{0}, δ_{0} false)

on the sphere

S_{d - 1}

, there exists a unique vector

bold-italicβ \in R^{d - 1}

such that

α = S (bold-italicβ) .

The minimization problem given in is equivalent to minimizing

\frac{1}{n} true \sum_{i = 1}^{n} {({}, Y_{i} - {\overset{ψ}{true}}_{n bold-italicα} ((), double-struckS {false(bold-italicβ false)}^{T} X_{i}))}^{2},

over all β , where

{\overset{ψ}{true}}_{n bold-italicα}

is the LSE of the link function with

bold-italicα = double-struckS false(bold-italicβ false)

{\overset{ψ}{true}}_{n bold-italicα}

.…”

Section: Score Estimators For the Regression Parameter αmentioning

confidence: 99%

“…This leads to the set of d − 1 equations,

\frac{1}{n} true \sum_{i = 1}^{n} {((), J_{S} false(bold-italicβ false))}^{T} X_{i} ({}, Y_{i} - {\overset{ψ}{true}}_{n bold-italicα} ((), double-struckS {false(bold-italicβ false)}^{T} X_{i})) = bold0,

where

J_{S}

is the Jacobian of the map

double-struckS

and where

bold0 \in R^{d - 1}

Section: Score Estimators For the Regression Parameter αmentioning

confidence: 99%

“…Recently, Groeneboom and Hendrickx () developed several score estimators for the current status linear regression model

Y = {bold-italicβ}_{0}^{T} bold-italicZ + ε

, where the distribution function F 0 of ε is left unspecified. Instead of observing the response Y , a censoring variable T and censoring indicator Δ = 1 Y ≤ T are observed.…”

Section: Introductionmentioning

confidence: 99%

“…This model is a special case of the monotone single‐index model and can be formulated as

double-struckE false(normalΔ false∣ T, bold-italicZ false) = F_{0} false(T - {bold-italicβ}_{0}^{T} bold-italicZ false) = F_{0} false({bold-italicα}_{0}^{T} bold-italicX false)

\sqrt{n}

Section: Introductionmentioning

confidence: 99%

“…We consider extending the estimators in the work of Groeneboom and Hendrickx () to the more general single‐index regression problem and propose two different score equations involving the LSE

{\overset{ψ}{true}}_{n bold-italicα}

of the unknown link function ψ 0 , which minimizes

\frac{1}{n} true \sum_{i = 1}^{n} {({}, Y_{i} - ψ false(α^{T} X_{i} false))}^{2},

over all monotone‐increasing functions ψ for fixed α . We establish an

n^{1 false/ 3} false/ \log n

‐rate for the estimator

{\overset{ψ}{true}}_{n bold-italicα}

and propose a single‐index score estimator of α 0 that converges at the parametric

\sqrt{n}

‐rate to the true regression parameter α 0 .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Score estimation in the monotone single‐index model

Balabdaoui

Groeneboom

Hendrickx

2018

Scandinavian J Statistics

Self Cite

View full text Add to dashboard Cite

show abstract

Inverse‐Weighted Quantile Regression With Partially Interval‐Censored Data

Kim,

Choi,

Park

et al. 2024

Biometrical J

View full text Add to dashboard Cite

This paper introduces a novel approach to estimating censored quantile regression using inverse probability of censoring weighted (IPCW) methodology, specifically tailored for data sets featuring partially interval‐censored data. Such data sets, often encountered in HIV/AIDS and cancer biomedical research, may include doubly censored (DC) and partly interval‐censored (PIC) endpoints. DC responses involve either left‐censoring or right‐censoring alongside some exact failure time observations, while PIC responses are subject to interval‐censoring. Despite the existence of complex estimating techniques for interval‐censored quantile regression, we propose a simple and intuitive IPCW‐based method, easily implementable by assigning suitable inverse‐probability weights to subjects with exact failure time observations. The resulting estimator exhibits asymptotic properties, such as uniform consistency and weak convergence, and we explore an augmented‐IPCW (AIPCW) approach to enhance efficiency. In addition, our method can be adapted for multivariate partially interval‐censored data. Simulation studies demonstrate the new procedure's strong finite‐sample performance. We illustrate the practical application of our approach through an analysis of progression‐free survival endpoints in a phase III clinical trial focusing on metastatic colorectal cancer.

show abstract