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Convergence rate for predictive recursion estimation of finite mixtures

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  • Martin, Ryan

Abstract

Predictive recursion (PR) is a fast stochastic algorithm for nonparametric estimation of mixing distributions in mixture models. It is known that the PR estimates of both the mixing and mixture densities are consistent under fairly mild conditions, but currently very little is known about the rate of convergence. Here I first investigate asymptotic convergence properties of the PR estimate under model misspecification in the special case of finite mixtures with known support. Tools from stochastic approximation theory are used to prove that the PR estimates converge, to the best Kullback–Leibler approximation, at a nearly root-n rate. When the support is unknown, PR can be used to construct an objective function which, when optimized, yields an estimate of the support. I apply the known-support results to derive a rate of convergence for this modified PR estimate in the unknown support case, which compares favorably to known optimal rates.

Suggested Citation

  • Martin, Ryan, 2012. "Convergence rate for predictive recursion estimation of finite mixtures," Statistics & Probability Letters, Elsevier, vol. 82(2), pages 378-384.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:2:p:378-384
    DOI: 10.1016/j.spl.2011.10.023
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    References listed on IDEAS

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    1. Ishwaran H. & James L.F. & Sun J., 2001. "Bayesian Model Selection in Finite Mixtures by Marginal Density Decompositions," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1316-1332, December.
    2. Ryan Martin & Surya T. Tokdar, 2011. "Semiparametric inference in mixture models with predictive recursion marginal likelihood," Biometrika, Biometrika Trust, vol. 98(3), pages 567-582.
    3. Woo, Mi-Ja & Sriram, T.N., 2007. "Robust estimation of mixture complexity for count data," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4379-4392, May.
    4. Chen, Jiahua & Khalili, Abbas, 2008. "Order Selection in Finite Mixture Models With a Nonsmooth Penalty," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1674-1683.
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    1. Ryan Martin, 2021. "A Survey of Nonparametric Mixing Density Estimation via the Predictive Recursion Algorithm," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 97-121, May.

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