Estimation of the Number of Components of Non-Parametric Multivariate Finite Mixture Models
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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≥2 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.
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