1,436,895 research outputs found
Beyond first-order asymptotics for Cox regression
To go beyond standard first-order asymptotics for Cox regression, we develop
parametric bootstrap and second-order methods. In general, computation of
-values beyond first order requires more model specification than is
required for the likelihood function. It is problematic to specify a censoring
mechanism to be taken very seriously in detail, and it appears that
conditioning on censoring is not a viable alternative to that. We circumvent
this matter by employing a reference censoring model, matching the extent and
timing of observed censoring. Our primary proposal is a parametric bootstrap
method utilizing this reference censoring model to simulate inferential
repetitions of the experiment. It is shown that the most important part of
improvement on first-order methods - that pertaining to fitting nuisance
parameters - is insensitive to the assumed censoring model. This is supported
by numerical comparisons of our proposal to parametric bootstrap methods based
on usual random censoring models, which are far more unattractive to implement.
As an alternative to our primary proposal, we provide a second-order method
requiring less computing effort while providing more insight into the nature of
improvement on first-order methods. However, the parametric bootstrap method is
more transparent, and hence is our primary proposal. Indications are that
first-order partial likelihood methods are usually adequate in practice, so we
are not advocating routine use of the proposed methods. It is however useful to
see how best to check on first-order approximations, or improve on them, when
this is expressly desired.Comment: Published at http://dx.doi.org/10.3150/13-BEJ572 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On the First Order Regression Procedure of Estimation for Incomplete Regression Models
This article discusses some properties of the first order regression method for imputation of missing values on an explanatory variable in linear regression model and presents an estimation strategy based on hypothesis testing
Modified First Order Regression, eine Simulationsstudie
In diesem Bericht werden verschiedene Imputationsmechanismen fuer fehlende Kovariablen in einem linearen Regressionsmodell mit zwei Kovariablen untersucht. Hierbei ist eine der Kovariablen vollstaendig beobachtet, die andere nur teilweise. Die betrachteten Imputationsmechanismen sind Zero Order Regression (ZOR), First Order Regression (FOR), First Order Regression plus random noise (FOR+) und Modified First Order Regression (MFOR)
Max-affine regression via first-order methods
We consider regression of a max-affine model that produces a piecewise linear
model by combining affine models via the max function. The max-affine model
ubiquitously arises in applications in signal processing and statistics
including multiclass classification, auction problems, and convex regression.
It also generalizes phase retrieval and learning rectifier linear unit
activation functions. We present a non-asymptotic convergence analysis of
gradient descent (GD) and mini-batch stochastic gradient descent (SGD) for
max-affine regression when the model is observed at random locations following
the sub-Gaussianity and an anti-concentration with additive sub-Gaussian noise.
Under these assumptions, a suitably initialized GD and SGD converge linearly to
a neighborhood of the ground truth specified by the corresponding error bound.
We provide numerical results that corroborate the theoretical finding.
Importantly, SGD not only converges faster in run time with fewer observations
than alternating minimization and GD in the noiseless scenario but also
outperforms them in low-sample scenarios with noise
Improved Coefficient and Variance Estimation in Stable First-Order Dynamic Regression Models
In dynamic regression models the least-squares coefficient estimators are biased in finite samples, and so are the usual estimators for the disturbance variance and for the variance of the coefficient estimators. By deriving the expectation of the initial terms in an expansion of the usual expression for the asymptotic coefficient variance estimator and by comparing these with an approximation to the true variance we find an approximation to the bias in variance estimation from which a bias corrected estimator for the variance readily follows. This is also achieved for a bias corrected coefficient estimator and allows to compare analytically the second-order approximation to the mean squared error of the least-squares estimator and its counterpart for the first-order bias corrected coefficient estimator. Two rather strong results on efficiency gains through bias correction for AR(1) models follow. Illustrative simulation results on the magnitude of bias in coefficient and variance estimation and on the scope for effective bias correction and efficiency improvement are presented for some relevant particular cases of the ARX(1) class of models.
Estimation of Parameters in Multiple Regression With Missing X-Observations using Modified First Order Regression Procedure
This paper considers the estimation of coefficients in a linear regression model with missing observations in the independent variables and introduces a modification of the standard first order regression method for imputation of missing values. The modification provides stochastic values for imputation. Asymptotic properties of the estimators for the regression coefficients arising from the proposed modification are derived when either both the number of complete observations and the number of missing values grow large or only the number of complete observations grows large and the number of missing observations stays fixed. Using these results, the proposed procedure is compared with two popular procedures - one which utilizes only the complete observations and the other which employs the standard first order regression imputation method for missing values. It is suggested that an elaborate simulation experiment will be helpful to evaluate the gain in efficiency especially in case of discrete regressor variables and to examine some other interesting issues like the impact of varying degree of multicollinearity in explanatory variables. Applications to some concrete data sets may also shed some light on these aspects. Some work on these lines is in progress and will be reported in a future article to follow
- …