- |
    Ashwini Maurya mauryaas@msu.edu\
    Department of Statistics and Probability\
    Michigan State University\
    East Lansing, MI 48824, USA
bibliography:
- 'sample.bib'
title: 'A Well-Conditioned and Sparse Estimation of Covariance and Inverse Covariance Matrices Using a Joint Penalty'
---

Sparsity, Eigenvalue Penalty, Penalized Estimation

Introduction
============

With the recent surge in data technology and storage capacity, today’s statisticians often encounter data sets where sample size $n$ is small and number of variables $p$ is very large: often hundreds, thousands and even million or more. Examples include gene expression data and web search problems \[@Clark1:7, @pass:21\]. For many of the high dimensional data problems, the choice of classical statistical methods becomes inappropriate for making valid inference. The recent developments in asymptotic theory deal with increasing $p$ as long as both $p$ and $n$ tend to infinity at some rate depending upon the parameters of interest.

The estimation of covariance and inverse covariance matrix is a problem of primary interest in multivariate statistical analysis. Some of the applications include: **(i)** Principal component analysis (PCA) \[@Johnstone:14, @Zou:37\]:, where the goal is to project the data on “best" $k$-dimensional subspace, and where best means the projected data explains as much of the variation in original 