Annual Progress Seminar
Day, Date and Time : Friday, 27th Sept, 4 pm.
Venue : Ramanujan hall
Speaker : Mayukh Choudhury
Title: Asymptotic Inferences in Generalized Linear Models
Abstract: This talk will be bifurcated into two segments. In the first
part, we will mainly indulge in "Asymptotic Properties of Cross-Validated
Lasso Estimator in GLM". This is precisely the continuation of our last
talk. The penalty parameter in LASSO, is generally chosen in a data
dependent way in practice. Among them, the K-fold CV is the most
celebrated one. So far we have defined the K-fold CV Lasso estimator
$\hat{\lambda}_{n,K}$ and explored its asymptotic properties in terms of
establishing the consistency of the sequence $n^{-1}\hat{\lambda}_{n,K}$
and boundedness of the sequence $n^{-1/2}\hat{\lambda}_{n,K}$ . However,
to justify the distributional convergence of LASSO estimator, we usually
need the convergence of the sequence $n^{-1/2}\hat{\lambda}_{n}$. Towards
that, we will prove the sequence $n^{-1/2}\hat{\lambda}_{n,K}$ is Cauchy
under some additional conditions. Now boundedness together with Cauchy
will serve our purpose. With this we will summarise and conclude this
segment.
In the later half, we will talk about "Large Dimensional CLT in GLM over
Convex Sets and Balls". We will aim to approximate the distribution of
properly centered and scaled GLM estimator with Gaussian random vector
under finite fourth moment condition uniformly over convex sets and
Euclidean Balls precisely when the dimension of the parameter vector, d
can grow with n. For class of measurable convex sets we obtain that d can
grow as o(n^{2/5}) and that for Euclidean Balls, we get d=o(n^{1/2}).
These are the best possible rates that we can have, which are similar to
the findings of Fang and Koike (2024). Lastly, we will prove the Bootstrap
approximation results for the distribution of properly centered and scaled
GLM estimator when the covariance matrix of the Gaussian random vector is
usually unknown.