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Global convergence of the log-concave MLE when the true distribution is geometric

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  • Fadoua Balabdaoui

Abstract

Let X 1 , ..., X n be i.i.d. from a discrete probability mass function (pmf) p . In Balabdaoui et al. [(2013), 'Asymptotic Distribution of the Discrete Log-Concave mle and Some Applications', JRSS-B , in press], the pointwise limit distribution of the log-concave maximum-likelihood estimator (MLE) was derived in both the well- and misspecified settings. In the well-specified setting, the geometric distribution was excluded, classified as being degenerate. In this article, we establish the global asymptotic theory of the log-concave MLE of a geometric pmf in all ℓ q distances for q ∈{1, 2, ...}∪{∞}. We also show how these asymptotic results could be used in testing whether a pmf is geometric.

Suggested Citation

  • Fadoua Balabdaoui, 2014. "Global convergence of the log-concave MLE when the true distribution is geometric," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(1), pages 21-59, March.
    • Handle: RePEc:taf:gnstxx:v:26:y:2014:i:1:p:21-59
    DOI: 10.1080/10485252.2013.826801
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    References listed on IDEAS

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    1. Cécile Durot & Laurence Reboul, 2010. "Goodness‐of‐Fit Test for Monotone Functions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(3), pages 422-441, September.
    2. repec:dau:papers:123456789/4650 is not listed on IDEAS
    3. Madeleine Cule & Richard Samworth & Michael Stewart, 2010. "Maximum likelihood estimation of a multi‐dimensional log‐concave density," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(5), pages 545-607, November.
    4. Fadoua Balabdaoui & Jon A. Wellner, 2010. "Estimation of a k‐monotone density: characterizations, consistency and minimax lower bounds," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(1), pages 45-70, February.
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