Local asymptotic normality and mixed normality for Markov statistical models

1990 ◽  
Vol 86 (1) ◽  
pp. 105-129 ◽  
Author(s):  
Reinhard H�pfner ◽  
Jean Jacod ◽  
Lucia Ladelli
Keyword(s):  
Asymptotic Normality ◽  
Statistical Models ◽  
Local Asymptotic Normality

scholarly journals Local Asymptotic Normality Complexity Arising in a Parametric Statistical Lévy Model

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Wissem Jedidi
Keyword(s):  
Probability Density Function ◽  
Asymptotic Normality ◽  
Probability Density ◽  
Statistical Models ◽  
Lévy Process ◽  
Unknown Parameter ◽  
Stable Process ◽  
Levy Process ◽  
Local Asymptotic Normality ◽  
Statistical Experiments

We consider statistical experiments associated with a Lévy process X = X t t ≥ 0 observed along a deterministic scheme i u n ,   1 ≤ i ≤ n . We assume that under a probability ℙ θ , the r.v. X t ,   t > 0 , has a probability density function > o , which is regular enough relative to a parameter θ ∈ 0 , ∞ . We prove that the sequence of the associated statistical models has the LAN property at each θ , and we investigate the case when X is the product of an unknown parameter θ by another Lévy process Y with known characteristics. We illustrate the last results by the case where Y is attracted by a stable process.

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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