# Penalized Exponentially Tilted Likelihood for Growing Dimensional Models with Missing Data

## TL;DR

This paper introduces a new statistical method to estimate and select variables in models with missing data, ensuring accurate results even when some information is missing.

## Contribution

The novel penalized exponentially tilted likelihood method enables parameter estimation and variable selection in high-dimensional models with missing responses.

## Key findings

- The penalized ET likelihood ensures consistent parameter estimation and variable selection.
- The ET likelihood ratio statistic for hypothesis testing has the Wilks’ property under certain conditions.
- Simulation and real data analysis confirm the effectiveness of the proposed method.

## Abstract

This paper develops a penalized exponentially tilted (ET) likelihood to simultaneously estimate unknown parameters and select variables for growing dimensional models with missing response at random. The inverse probability weighted approach is employed to compensate for missing information and to ensure the consistency of parameter estimators. Based on the penalized ET likelihood, we construct an ET likelihood ratio statistic to test the contrast hypothesis of parameters. Under some wild conditions, we obtain the consistency, asymptotic properties, and oracle properties of parameter estimators and show that the constrained penalized ET likelihood ratio statistic for testing the contrast hypothesis possesses the Wilks’ property. Simulation studies are conducted to validate the finite sample performance of the proposed methodologies. Thyroid data taken from the First People’s Hospital of Yunnan Province is employed to illustrate the proposed methodologies.