A. W. van der Vaart, Asymptotic Statistics
From Marginalia
The following marginalia are for van der Vaart's Asymptotic statistics, released in 2000 (see the google book page). (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, emphasizes that are i.i.d. with mean .
Chapter 2 : Stochastic convergence
 (pg 5) If , then is understood componentwise.
 (pg 6) Lemma 2.2 (Portmanteau) (i): Why only continuity points? Simplest situation is when is a point mass at . Consider a sequence a.s., where and . Then and .
 (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) (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, counterexample: 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: and .
 (pg 12) Beware: bounded in probability has been defined very quickly on page 8, it is another name for uniformly tight.
 (pg 12) is less tricky to use than .
 (pg 12) Useful result: If then .
 Errata (pg 13, line 10) In the last line of the proof of Lemma 2.12, "the sequence is tight" should read "the sequence is uniformly tight".
 (pg ??) Euler's Identity:
 (pg ??) Counterexample; A distribution which is not uniquely determined by its moments
 (Source: Billingsley's book "Probability and Measure")
 (Standard) Lognormal distribution:
 Perturbed version:
 Moments:
 Moment generating function: Furthermore, despite that all moments of the lognormal distribution are welldefined, the moment generating function does not exist! To see that, note that if is standard lognormal then is standard Normal. Then
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 nonregular exponential family is the .
 An example of a regular nonfull rank exponential family is the .
 Peter noted that it may be neccesary to have a more general definition of the exponential family so it includes the 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 is marginalized out.
 (pg ??) Huber's psi example: was not defined initially, but it is density under . Unclear how appears, but Maria noted it could be due to integration by parts.
 It was noted that the concept of Zestimator 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 .
 (pg 83) Problem 3: Should start with instead of .
Chapter 6 : Contiguity
 (pg 85, line 11) Both P and Q are absolutely continuous w.r.t. the measure
 Errata (pg 86, Figure 6.1) The labels and are swapped.
 (pg 86, line 2) Many authors will write 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 to denote the RNderivative 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 RNderivative is used to reweigh the sample. Finite variance requires squareintegrability of the RadonNykodim derivative.
 (pg 88, line 11) These three equalities are reflected in the Lemma below, but note that the middle equality is reexpressed 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 should be , 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 nonconvergent sequence that has a convergent subsequence: .
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:
So,
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
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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 = {0521496039}, }