A uniform condition of local asymptotic normality

1986 ◽  
Vol 34 (1) ◽  
pp. 1427-1432 ◽  
Author(s):  
I. A. Ibragimov ◽  
R. Z. Khas'minskii
Keyword(s):  
Asymptotic Normality ◽  
Local Asymptotic Normality ◽  
Uniform Condition

scholarly journals Polynomials under Ornstein–Uhlenbeck noise and an application to inference in stochastic Hodgkin–Huxley systems

2020 ◽  
Author(s):  
Reinhard Höpfner
Keyword(s):  
Membrane Potential ◽  
Asymptotic Normality ◽  
Time Interval ◽  
Local Asymptotic Normality ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Estimation Problems ◽  
Long Time ◽  
Asymptotically Efficient

Abstract We discuss estimation problems where a polynomial $$s\rightarrow \sum _{i=0}^\ell \vartheta _i s^i$$ s → ∑ i = 0 ℓ ϑ i s i with strictly positive leading coefficient is observed under Ornstein–Uhlenbeck noise over a long time interval. We prove local asymptotic normality (LAN) and specify asymptotically efficient estimators. We apply this to the following problem: feeding noise $$dY_t$$ d Y t into the classical (deterministic) Hodgkin–Huxley model in neuroscience, with $$Y_t=\vartheta t + X_t$$ Y t = ϑ t + X t and X some Ornstein–Uhlenbeck process with backdriving force $$\tau $$ τ , we have asymptotically efficient estimators for the pair $$(\vartheta ,\tau )$$ ( ϑ , τ ) ; based on observation of the membrane potential up to time n, the estimate for $$\vartheta $$ ϑ converges at rate $$\sqrt{n^3\,}$$ n 3 .

scholarly journals Power in High‐Dimensional Testing Problems

Econometrica ◽  
2019 ◽  
Vol 87 (3) ◽  
pp. 1055-1069 ◽  
Author(s):  
Anders Bredahl Kock ◽  
David Preinerstorfer
Keyword(s):  
Asymptotic Normality ◽  
Sample Size ◽  
Parameter Space ◽  
Sufficient Conditions ◽  
High Dimensional ◽  
Local Asymptotic Normality ◽  
Asymptotic Power ◽  
Asymptotic Size ◽  
Power Of Tests ◽  
Power Enhancement

Fan, Liao, and Yao (2015) recently introduced a remarkable method for increasing the asymptotic power of tests in high‐dimensional testing problems. If applicable to a given test, their power enhancement principle leads to an improved test that has the same asymptotic size, has uniformly non‐inferior asymptotic power, and is consistent against a strictly broader range of alternatives than the initially given test. We study under which conditions this method can be applied and show the following: In asymptotic regimes where the dimensionality of the parameter space is fixed as sample size increases, there often exist tests that cannot be further improved with the power enhancement principle. However, when the dimensionality of the parameter space increases sufficiently slowly with sample size and a marginal local asymptotic normality (LAN) condition is satisfied, every test with asymptotic size smaller than 1 can be improved with the power enhancement principle. While the marginal LAN condition alone does not allow one to extend the latter statement to all rates at which the dimensionality increases with sample size, we give sufficient conditions under which this is the case.

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