Math 2.0

Statistics Definition

Local asymptotic normality

Builds on Radon–Nikodym derivativeConvergence in probabilityConvergence in distributionMultivariate normal distributionPositive definite matrixFisher informationScoreContiguity of measures

Definition

Let Θ⊆ℝk\Theta \subseteq \R^k be open, and for each nn let {Pn,θ:θ∈Θ}\set{P_{n,\theta} : \theta \in \Theta} be a family of probability measures on a measurable space (Ωn,ℱn)(\Omega_n, \mathcal{F}_n). Let θ∈Θ\theta \in \Theta. The sequence of models is locally asymptotically normal (LAN) at θ\theta if the following holds. There are random vectors Δn,θ\Delta_{n,\theta} on Ωn\Omega_n and a k×kk \times k positive definite matrix IθI_\theta such that:

  1. Δn,θ→d𝒩k(0,Iθ)\Delta_{n,\theta} \xrightarrow{d} \mathcal{N}_k(0, I_\theta) under Pn,θP_{n,\theta}, in the sense of convergence in distribution to a multivariate normal law;
  2. for every sequence hn→hh_n \to h in ℝk\R^k, log⁡dPn,θ+hn/ndPn,θ=hTΔn,θ−12hTIθh+oPn,θ(1).\log \frac{dP_{n, \theta + h_n/\sqrt{n}}}{dP_{n,\theta}} = h^T \Delta_{n,\theta} - \frac{1}{2} h^T I_\theta h + o_{P_{n,\theta}}(1). Here the log-likelihood ratio uses the Radon–Nikodym derivative of the absolutely continuous part. The term oPn,θ(1)o_{P_{n,\theta}}(1) is a sequence of random variables that tends to 00 in probability under Pn,θP_{n,\theta}.

Remarks

Example: for i.i.d. 𝒩(θ,1)\mathcal{N}(\theta, 1) observations, Pn,θ=𝒩(θ,1)⊗nP_{n,\theta} = \mathcal{N}(\theta, 1)^{\otimes n}, the log-likelihood ratio is exactly h Δn−h2/2h\, \Delta_n - h^2/2 with Δn=n−1/2∑i(Xi−θ)∼𝒩(0,1)\Delta_n = n^{-1/2}\sum_i (X_i - \theta) \sim \mathcal{N}(0, 1), so the model is LAN with Iθ=1I_\theta = 1 and no remainder.

More generally, if X1,…,XnX_1, \dots, X_n are i.i.d. from a model that is differentiable in quadratic mean at θ\theta with nonsingular Fisher information IθI_\theta, the model is LAN at θ\theta with Δn,θ=n−1/2∑is(θ;Xi)\Delta_{n,\theta} = n^{-1/2}\sum_i s(\theta; X_i), where ss is the score.

Non-example: for i.i.d. Uniform(0,θ)(0, \theta) observations and h≠0h \ne 0, Pn,θ+h/nP_{n, \theta + h/\sqrt{n}} and Pn,θP_{n,\theta} become asymptotically singular. The reason is that the scale at which they differ is 1/n1/n; clause 2 fails, and the model is not LAN.

Under LAN, Pn,θ+h/nP_{n,\theta + h/\sqrt{n}} and Pn,θP_{n,\theta} are mutually contiguous, since the limit law 𝒩(−12hTIθh, hTIθh)\mathcal{N}(-\frac{1}{2} h^T I_\theta h, \, h^T I_\theta h) of the log-likelihood ratio has EeL=1E e^{L} = 1.

Reading path

To understand this page from the ground up, read these 61 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Contiguity of measures concept
  3. Element definition
  4. Fisher information concept
  5. Integers object
  6. Matrix definition
  7. Ordered field definition
  8. Proposition concept
  9. Radon–Nikodym derivative concept
  10. Random experiment concept
  11. Rational numbers object
  12. Real numbers object
  13. Score concept
  14. Set definition
  15. Ring of integers statement
  16. Square matrix definition
  17. Transpose definition
  18. Disjunction definition
  19. Implication definition
  20. Predicate definition
  21. Ordered pair definition
  22. Union definition
  23. Symmetric matrix definition
  24. Conjugate transpose definition
  25. Sample space definition
  26. The rationals form a field statement
  27. Existential quantifier definition
  28. Universal quantifier definition
  29. Cartesian product definition
  30. Hermitian matrix definition
  31. Subset definition
  32. The rationals form an ordered field statement
  33. Uniqueness quantifier definition
  34. Binary relation definition
  35. Universal set definition
  36. Order on the reals definition
  37. Positive definite matrix definition
  38. Power set definition
  39. Complement definition
  40. Function definition
  41. Interval definition
  42. Sigma-algebra definition
  43. Indicator function definition
  44. Preimage of a set definition
  45. Event definition
  46. Borel sigma-algebra definition
  47. Probability axioms concept
  48. Probability measure definition
  49. Dirac measure definition
  50. Probability distribution definition
  51. Probability space definition
  52. Random variable definition
  53. Convergence in probability definition
  54. Cumulative distribution function definition
  55. Random variable with a density definition
  56. Convergence in distribution definition
  57. Probability density function definition
  58. Density of the normal distribution definition
  59. Normal distribution definition
  60. Multivariate normal distribution definition
  61. Local asymptotic normality definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Local_asymptotic_normality
About this page
TagYMDADSX0E1
URIm2://YMDADSX0E1
Kinddefinition
WritingW0 stub
MSC62F12, 62E20
DefinesLocal asymptotic normality
UsesRadon–Nikodym derivative, Convergence in probability, Convergence in distribution, Multivariate normal distribution, Positive definite matrix, Fisher information, Score, Contiguity of measures

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