A. W. van der Vaart, Asymptotic Statistics

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The following marginalia are for van der Vaart's Asymptotic statistics, released in 2000 (see the google book page).
VdV00-cover.jpg
(Please send further errata or marginalia to Dan Roy.) These marginalia were produced as a byproduct of a reading group at UCL/Gatsby.

Contents


Marginalia

Chapter 1 : Introduction

  • (pg 1) Here, P_\mu emphasizes that X_n are i.i.d. with mean \mu.

Chapter 2 : Stochastic convergence

  • (pg 5) If X_n \in \mathbb{R}^k, then X_n \leq x is understood component-wise.
  • (pg 6) Lemma 2.2 (Portmanteau) (i): Why only continuity points? Simplest situation is when X is a point mass at x. Consider a sequence X_n = x_n a.s., where x_n > x and x_n\to x. Then \forall n\, P(X_n\le x)=0 and P(X\le x)=1.
  • (pg 6) almost surely convergence of random variables (measurable functions from the basic space) is pointwise convergence.
  • (pg 7) Continuous mapping theorem is fundamental.
  • Errata (pg 7, line -16) In part "(v)+(vi) \Rightarrow (vii)", ignore "By (iv)", or replace "By (iv)" with "By (v)" and replace "by (v)." with "by (vi)."
  • (pg 8) Prohorov's theorem: We can avoid using CDF in the proof (hence without Helly's lemma), using instead relative compactness arguments in the space of measures.
  • (pg 11) Slutsky's Lemma, counter-example: two variables converging each in law to two distributions, not to a constant, and the limit of their sum is not the sum of the limits: X_n \sim \mathcal{N}(0,1) and Y_n = -X_n.
  • (pg 12) Beware: bounded in probability has been defined very quickly on page 8, it is another name for uniformly tight.
  • (pg 12) O_p is less tricky to use than o_p.
  • (pg 12) Useful result: If  Y = O_p(n^{-\alpha}), \alpha > 0 then Y = o_p(n^{-\alpha+t}), t > 0.
  • Errata (pg 13, line -10) In the last line of the proof of Lemma 2.12, "the sequence g(X_n) is tight" should read "the sequence g(X_n) is uniformly tight".
  • (pg ??) Euler's Identity: e^{ix}=\cos{x}+i\sin{x}
  • (pg ??) Counterexample; A distribution which is not uniquely determined by its moments
    • (Source: Billingsley's book "Probability and Measure")
    • (Standard) Log-normal distribution: f_0(x)=(2 \pi)^{-1/2} x^{-1} e^{-\frac{(\log{x})^2}{2}}
    • Perturbed version: f_a(x)=f_0(x)(1+a \sin(2\pi \log{x}))\
    • Moments: E_0(x^n)=E_a(x^n)=e^{\frac{n^2}{2}}, n\in \mathbb{N}
    • Moment generating function: Furthermore, despite that all moments of the log-normal distribution are well-defined, the moment generating function does not exist! To see that, note that if X is standard log-normal then Y = \exp(X) is standard Normal. Then

      
\begin{align}
E\left\{\exp(tX) \right\} 
&= E\left\{t\exp(Y)\right\} \\
&=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \exp\left\{t \exp(y) - \frac{y^2}{2} \right\}dy \\
&\ge \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \exp\left\{t \left(1 + y + \frac{y^2}{2} + \frac{y^3}{6}\right) - \frac{y^2}{2} \right\}dy = \infty \, .
\end{align}

Notes

Here are some notes section 2.1: File:Notes.pdf

Chapter 3 : Delta Method

  • (pg 26) Note that differential of a function is a linear map, expressed in a given base by the Jacobian matrix.
  • (pg 26) No need for the second derivative in the proof of Theorem 3.1, as is the case in many other proofs.

Chapter 4 : Moment Estimators

Section 4.2

In this section of exponential families, Van Der Vaart defines the regular exponential family and the full rank one. So it is useful to have examples to clarify this further:

  • An example of a full rank non-regular exponential family is the U(0,\theta).
  • An example of a regular non-full rank exponential family is the N(\theta,\theta^2).
  • Peter noted that it may be neccesary to have a more general definition of the exponential family so it includes the U(0,\theta) i.e. a definition in terms of a finite dimension sufficient statistic.

Chapter 5 : M and Z Estimators

  • (pg 56) Problem 13: Exponential frailty model example: Unclear how equation at bottom of page is derived. Unclear how z is marginalized out.
  • (pg ??) Huber's psi example: p was not defined initially, but it is density under P. Unclear how p'/p appears, but Maria noted it could be due to integration by parts.
  • It was noted that the concept of Z-estimator may be Van Der Vaart's own terminology because this doesn't appear elsewhere in the literature.
  • We need to work in a functional space because there could be cases of criterion functions for which you can't solve the equation for the estimator explicitly. So we need to keep in mind that since we are in a functional space, metrics are not equivalent here and so we need to consider convergence with respect to different metrics.
  • Peter noted that even though uniform convergence is a strong requirement, rather than try to achieve a weaker type of convergence for criterion functions what will be done is to look for a smaller class of functions which are uniformly convergent (Glivenko Cantelli class). And also, we need uniform convergence so that the convergence rate doesn't depend on \theta.
  • (pg 83) Problem 3: Should start with p-1 instead of 1-p.

Chapter 6 : Contiguity

  • (pg 85, line -11) Both P and Q are absolutely continuous w.r.t. the measure \mu = P + Q
  • Errata (pg 86, Figure 6.1) The labels \Omega_P and \Omega_Q are swapped.
  • (pg 86, line 2) Many authors will write dQ/dP to imply the necessary absolute continuity and to denote the resulting (a.e.. unique) density. So care should be taken adopting this convention of writing dQ/dP to denote the RN-derivative of the absolutely continuous part of Q w.r.t. P and P.
  • (pg 87, line 8) The reason "dividing by zero is not permitted" seems odd. The inequality is strict because the absolutely continuous part of Q w.r.t. P may not be equal to Q itself.
  • (pg 87, line 13) In Section 6.2, the expectation in the formula here is the basic idea behind importance sampling. (The importance distribution is P, and RN-derivative is used to reweigh the sample. Finite variance requires square-integrability of the Radon-Nykodim derivative.
  • (pg 88, line 11) These three equalities are reflected in the Lemma below, but note that the middle equality is re-expressed as a measure 1 event, rather than a null event. It's not clear why this nice symmetry was avoided.
  • (pg 88) In Lemma 6.4, parts (i) to (iv) are very much used in robust estimation: show consistency for one (sequence of) model, and use contiguity to show the consistency for any other model.
  • (pg 88) While reading Lemma 6.4, note that Portmanteau lemmas are stated on pg 6.
  • Errata (pg 90, line 9) Here E1_B should be E_P 1_B, as on page 87.
  • (pg 90, line 10) Monotone convergence is for sigma additivity, presumably.

Chapter 7 : Local Asymptotic Normality

Section 7.4

Example of non-convergent sequence that has a convergent subsequence: {0,1,0,1, \ldots}.

Chapter 8 : Efficiency of Estimators

Section 8.3

  • For the proof of theorem 8.3 we use the usual definition of the derivative of a function:


\begin{align}
\dot{\psi}(\theta)& = &\stackrel{lim}{_{\frac{1}{\sqrt{n}}\rightarrow 0}}\frac{\psi(\theta + h/\sqrt{n})-\psi(\theta)}{h/\sqrt{n}}\\
\end{align}


So,


\begin{align}
S_n&=&\sqrt{n}(T_n-\psi(\theta+\frac{h}{\sqrt{n}}))+\sqrt{n}(\psi(\theta+\frac{h}{\sqrt{n}}-\psi(\theta))\\
\end{align}


\begin{align}
S_n&\stackrel{d}{\rightarrow}& T+\dot{\psi}h\\
\end{align}

Section 8.4

  • In the proof of proposition 8.6 it is assumed that T (the statistic in the normal experiment) is equivariant in law. This is not a hypothesis of the proposition and it is not clear why it could be assumed.

Remaining chapters

No marginalia recorded. Please submit your own.

Bibtex

@book{vdV00,
author = {A. W. van der Vaart},
title = {Asymptotic statistics},
series = {Cambridge Series in Statistical and Probabilistic Mathematics}, 
publisher = {Cambridge University Press}
city = {Cambridge}
year = {1998},
isbn = {0-521-49603-9},
}
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