Bounds for the normal approximation of the maximum likelihood estimator

Anastasiou, Andreas; Reinert, Gesine

doi:10.3150/15-bej741

Cited by 22 publications

(64 citation statements)

References 26 publications

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“…These regularity conditions in the multi-parameter case resemble those in [3] where the parameter is scalar. The following theorem gives the qualitative result for the asymptotic normality of a vector MLE (see [6] for a proof).…”

Section: ) the Expected Fisher Information Matrix For One Random Vecmentioning

confidence: 79%

Multivariate normal approximation of the maximum likelihood estimator via the delta method

Anastasiou¹,

Gaunt²

2020

Braz. J. Probab. Stat.

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We use the delta method and Stein's method to derive, under regularity conditions, explicit upper bounds for the distributional distance between the distribution of the maximum likelihood estimator (MLE) of a d-dimensional parameter and its asymptotic multivariate normal distribution. Our bounds apply in situations in which the MLE can be written as a function of a sum of i.i.d. t-dimensional random vectors. We apply our general bound to establish a bound for the multivariate normal approximation of the MLE of the normal distribution with unknown mean and variance.MSC 2010 subject classifications: Primary 60F05, 62E17, 62F12 Keywords and phrases. Multi-parameter maximum likelihood estimation, multivariate normal distribution, Stein's method 1 imsart-bjps ver.

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Section: ) the Expected Fisher Information Matrix For One Random Vecmentioning

confidence: 79%

Multivariate normal approximation of the maximum likelihood estimator via the delta method

Anastasiou¹,

Gaunt²

2020

Braz. J. Probab. Stat.

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“…The bound in (21) is not as simple and sharp as the bound given in Theorem 2.1 of Anastasiou and Reinert (2014). This is expected since Corollary 2.1 is a special application of a result which was originally obtained to satisfy the assumption of local dependence for our random variables, while Anastasiou and Reinert (2014) used directly results of Stein's method for independent random variables. In addition, the assumption (A.D.1) used for the result of Corollary 2.1 is stronger than the condition (R3) of Anastasiou and Reinert (2014).…”

Section: The General Boundmentioning

confidence: 99%

Bounds for the normal approximation of the maximum likelihood estimator fromm-dependent random variables

Anastasiou

2017

Statistics & Probability Letters

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The asymptotic normality of the Maximum Likelihood Estimator (MLE) is a long established result. Explicit bounds for the distributional distance between the distribution of the MLE and the normal distribution have recently been obtained for the case of independent random variables. In this paper, a local dependence structure is introduced between the random variables and we give upper bounds which are specified for the Wasserstein metric.

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“…i ∨ y (2) i ], with the edges parallel to the coordinate axes in R n , and y (1) and y (2) are two opposite vertices of the box. Now we can state the mentioned lemma:…”

Section: Appendix A: On Condition (11)mentioning

confidence: 99%

Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators

Pinelis¹

2017

Electron. J. Statist.

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It is well known that, under general regularity conditions, the distribution of the maximum likelihood estimator (MLE) is asymptotically normal. Very recently, bounds of the optimal order O(1/ √ n) on the closeness of the distribution of the MLE to normality in the so-called bounded Wasserstein distance were obtained [2, 1], where n is the sample size. However, the corresponding bounds on the Kolmogorov distance were only of the order O(1/n 1/4 ). In this paper, bounds of the optimal order O(1/ √ n) on the closeness of the distribution of the MLE to normality in the Kolmogorov distance are given, as well as their nonuniform counterparts, which work better in tail zones of the distribution of the MLE. These results are based in part on previously obtained general optimal-order bounds on the rate of convergence to normality in the multivariate delta method. The crucial observation is that, under natural conditions, the MLE can be tightly enough bracketed between two smooth enough functions of the sum of independent random vectors, which makes the delta method applicable. It appears that the nonuniform bounds for MLEs in general have no precedents in the existing literature; a special case was recently treated by Pinelis and Molzon [20]. The results can be extended to M -estimators.AMS 2010 subject classifications: 62F10, 62F12, 60F05, 60E15.

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Bounds for the normal approximation of the maximum likelihood estimator

Cited by 22 publications

References 26 publications

Multivariate normal approximation of the maximum likelihood estimator via the delta method

Multivariate normal approximation of the maximum likelihood estimator via the delta method

Bounds for the normal approximation of the maximum likelihood estimator fromm-dependent random variables

Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators

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