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A Theorem on Uniform Convergence of Stochastic Functions with Applications

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  • Yuan, Ke-Hai

Abstract

In a variety of statistical problems one needs to manipulate a sequence of stochastic functions involving some unknown parameters. The asymptotic behavior of the estimated parameters often depends on the asymptotic properties of such functions. Especially, the consistency of the estimated parameters follows from the uniform convergence of the sequence of stochastic functions. A theorem on uniform convergence of a sequence of vector valued random functions is presented. The forms of these functions are very general and the assumptions are rather natural. If the sequence of random functions is generated by a sequence of random vectors, these random vectors are only required to be independently distributed and can be of different dimensions. As applications, we consider the consistency of the estimated regression parameters in logistic regression and in M-estimation in a linear model.

Suggested Citation

  • Yuan, Ke-Hai, 1997. "A Theorem on Uniform Convergence of Stochastic Functions with Applications," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 100-109, July.
    • Handle: RePEc:eee:jmvana:v:62:y:1997:i:1:p:100-109
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    References listed on IDEAS

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    1. Cheng, Ching-Shui & Li, Ker-Chau, 1984. "The strong consistency of M-estimators in linear models," Journal of Multivariate Analysis, Elsevier, vol. 15(1), pages 91-98, August.
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    Cited by:

    1. Juan Carlos Pardo-Fernández & M. Dolores Jiménez-Gamero, 2019. "A model specification test for the variance function in nonparametric regression," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 103(3), pages 387-410, September.
    2. Yuan, Ke-Hai, 2009. "Normal distribution based pseudo ML for missing data: With applications to mean and covariance structure analysis," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1900-1918, October.
    3. Yuan, Ke-Hai & Jennrich, Robert I., 1998. "Asymptotics of Estimating Equations under Natural Conditions," Journal of Multivariate Analysis, Elsevier, vol. 65(2), pages 245-260, May.
    4. Bai, Yang & Fung, Wing K. & Zhu, Zhong Yi, 2009. "Penalized quadratic inference functions for single-index models with longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 152-161, January.
    5. Ke-Hai Yuan & Robert Jennrich, 2000. "Estimating Equations with Nuisance Parameters: Theory and Applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(2), pages 343-350, June.

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