# Optimal Sampling for Generalized Linear Models under Measurement Constraints

## TL;DR

This paper introduces OSUMC, a novel response-free sampling method for generalized linear models under measurement constraints, optimizing sampling probabilities to improve statistical efficiency without requiring responses for most data points.

## Contribution

The paper develops a response-free sampling procedure for GLMs, establishes an unconditional asymptotic distribution, and proposes an algorithm to approximate the optimal sampling distribution under measurement constraints.

## Key findings

- OSUMC outperforms existing methods in empirical studies.
- Theoretical guarantees are established for the asymptotic distribution of estimators.
- The method effectively balances statistical efficiency and measurement cost.

## Abstract

Under "measurement constraints," responses are expensive to measure and initially unavailable on most of records in the dataset, but the covariates are available for the entire dataset. Our goal is to sample a relatively small portion of the dataset where the expensive responses will be measured and the resultant sampling estimator is statistically efficient. Measurement constraints require the sampling probabilities can only depend on a very small set of the responses. A sampling procedure that uses responses at most only on a small pilot sample will be called "response-free." We propose a response-free sampling procedure \mbox{(OSUMC)} for generalized linear models (GLMs). Using the A-optimality criterion, i.e., the trace of the asymptotic variance, the resultant estimator is statistically efficient within a class of sampling estimators. We establish the unconditional asymptotic distribution of a general class of response-free sampling estimators. This result is novel compared with the existing conditional results obtained by conditioning on both covariates and responses. Under our unconditional framework, the subsamples are no longer independent and new martingale techniques are developed for our asymptotic theory. We further derive the A-optimal response-free sampling distribution. Since this distribution depends on population level quantities, we propose the Optimal Sampling Under Measurement Constraints (OSUMC) algorithm to approximate the theoretical optimal sampling. Finally, we conduct an intensive empirical study to demonstrate the advantages of OSUMC algorithm over existing methods in both statistical and computational perspectives.