Profile likelihood inference for semiparametric models
Susan Murphy
Pennsylvania State University
Abstract
Many models used in survial analysis are of high dimension. A
commonly used model of high dimension is the proportional hazards
model for right censored data. Other examples are the proportional
hazards model for current status data, and the proportional odds model
for right censored data. In all of these models interest is primarily
in a vector of regression coefficients and a function, such as the
cumulative baseline hazard, is a nuisance parameter. This talk
concerns the verification of the intuitive practice of profiling the
nuisance parameter out of the likelihood and using the resulting
profile likelihood as if it is a likelihood for the vector parameter.
That is, will maximizing the profile likelihood yield an
asymptotically normal estimator of the parameter of interest? Can the
profile likelihood be used to make likelihood ratio tests and
confidence intervals? Can minus the second derivative matrix of the
profile likelihood be used to estimate the information matrix? In the
parametric setting these properties follow from a quadratic
approximation to the likelihood. This work gives sufficient conditions
for the above approximation to hold for a high dimensional model and
shows how the quadratic approximation leads to an affirmative answer
to the above questions.
The conditions are sufficiently simple so as to be satisfied in a
variety of semiparametric models, for example, the proportional
hazards model for current status data, the proportional odds model for
right censored survival data, errors in variables for a logistic
regression model, and a semiparametric logistic regression model.