# Estimation of the Number of Components of Non-Parametric Multivariate Finite Mixture Models

## TL;DR

This paper introduces a new consistent estimator for determining the number of components in non-parametric multivariate finite mixture models, leveraging spectral properties of an integral operator derived from data.

## Contribution

The paper proposes a novel spectral thresholding estimator for the number of mixture components, with proven consistency and finite sample performance guarantees.

## Key findings

- Estimator accurately recovers the number of components in simulations.
- Finite sample bounds demonstrate reliable performance.
- Monte Carlo results confirm effectiveness for moderate sample sizes.

## Abstract

We propose a novel estimator for the number of components (denoted by $M$) in a K-variate non-parametric finite mixture model, where the analyst has repeated observations of $K\geq2$ variables that are independent given a finitely supported unobserved variable. Under a mild assumption on the joint distribution of the observed and latent variables, we show that an integral operator $T$, that is identified from the data, has rank equal to $M$. Using this observation, and the fact that singular values are stable under perturbations, the estimator of $M$ that we propose is based on a thresholding rule which essentially counts the number of singular values of a consistent estimator of $T$ that are greater than a data-driven threshold. We prove that our estimator of $M$ is consistent, and establish non-asymptotic results which provide finite sample performance guarantees for our estimator. We present a Monte Carlo study which shows that our estimator performs well for samples of moderate size.