DOCSLIB.ORG

Lecture 27: Optimal Estimators and Functional Delta Method 1 Achieving Optimal Estimators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 27: Optimal Estimators and Functional Delta Method 1 Achieving Optimal Estimators Stat210B: Theoretical Statistics Lecture Date: April 19, 2007 Lecture 27: Optimal Estimators and Functional Delta Method Lecturer: Michael I. Jordan Scribe: Guilherme V. Rocha 1 Achieving Optimal Estimators From the last class, we know that the best limiting distribution we can hope for a parameter ψ(θ) is T (0, ψ˙θIθψ˙ ). The next question to ask is whether such bound can be achieved. N θ This is the theme of our next result: Lemma 1. (Lemma 8.14 in van der Vaart, 1998) Assume that the experiment (Pθ : θ Θ) is differentiable in quadratic mean at θ0 with non-singular Fisher ∈ information matrix Iθ. Let ψ be differentiable at θ0. Let Tn be an estimator sequence in the experiments (P n : θ Rk) such that: θ ∈ n 1 ˙ 1 ˙ √n (Tn ψ(θ)) = ψθI− ℓθ(Xi) + oPθ (1), − √n θ i=1 X then Tn is the best regular estimator for ψ(θ) at θ. Conversely, every best regular estimator sequence satisfies this expansion. Proof. Let n 1 ∆ := ℓ˙ (X ). n,θ √n θ i i=1 X We know that ∆n,θ converges in distribution to a ∆θ with a (0,Iθ) distribution. N From Theorem 7.2 in van der Vaart (1998) we know that: n dPθ+ h √n T 1 T log n = h ∆θ h Iθh + oPθ (1). dPθ ! − 2 Using Slutsky’s theorem, we get: √n (Tn ψ(θ)) n ˙ 1 dP− h θ ψ I− ∆ θ+ θ θ θ √n T 1 T log dP n h ∆θ 2 h Iθh θ ! − ˙ 1 ˙ T ˙ 0 ψθIθ− ψθ ψθh 1 T , T T ∼ N h Iθh ψ˙ h h I h − 2 θ θ h Using Le Cam’s third lemma we can conclude that the sequence √n(Tn ψ(θ)) under θ + converges in − √n 1 T h distribution to (ψ˙θh, ψ˙θI− ψ˙ ). Since ψ is differentiable, we have that √n ψ(θ + ) ψ(θ) ψ˙θh as N θ θ √n − → h h n . We conclude that, under θ + , √n(Tn ψ(θ + )) does not involve h, that is, Tn is regular. → ∞ √n − √n 1 2 Lecture 27: Optimal Estimators and Functional Delta Method To prove the converse, let Tn and Sn be a two best regular estimator sequences. Along subsequences, it can be shown that: h h √n Sn ψ θ + θ+ ˙ − √n √n S ψθh h − ˙ √n Tn ψ θ + T ψθh − √n − for a randomized estimator (S,T ) in the limiting experiment. Because Sn and Tn are best regular, S and T are best equivariant-in-law. Thus S = T = ψ˙θX almost surely and, as a result, √n(Sn Tn) converges in distribution to S T = 0. As a result, every two best regular estimator sequences are− asymptotically − equivalent. To get the result, apply this conclusion to Tn and: 1 1 S = ψ(θ) + ψ˙ I− ∆ n √n θ θ n,θ Remarks on Theorem 8.14: From theorem 5.39, an Maximum Likelihod Estimator θˆn satisfies: • n ˆ 1 1 ˙ √n θn θ = I− ℓθ(Xi) + oPθ (1) − √n θ i=1 X under regularity conditions. It follows that MLEs are asymptotically efficient. This result can be extended to a transforms of the MLE ψ(θ) for a differentiable ψ by using the delta method and observing that ψ(θˆn) satisfies the expansion in lemma 8.14. Lemma 8.14 suggests that Rao score functions leading to tests constructed from the scores are asymp- • totically efficient. 2 Functional Delta Method The functional delta method aims at extending the delta method to a nonparametric context. The high level idea is to interpret a statistic as a functional φ mapping from the space of probability distributions D to the real line R and use a notion of derivative of this functional to obtain the asymptotic distribution of φ(Fˆn). We have proven before that: Fˆ p supx n(x) F (x) 0, (Glivenko-Cantelli Theorem) k − F k → √n Fˆn F GF , (Donsker Theorem) − where Fˆn is the empirical distribution function based in n samples and GF is the Brownian Bridge. Our goal now is to find conditions on the functionals φ so we can extend the above modes of convergence to φ(Fˆn). As we will see in detail below, consistency of the sequence φ(Fˆn) follows easily from assuming φ to be continuous with respect to the supremum norm: this a natural extension of the continuous mapping theorem. The generalization of the delta method to functionals is more involved as different notions of differentiability of functionals exist. Before we jump into the continuous mapping theorem and the functional delta method, a few examples of statistical functionals are in order. Lecture 27: Optimal Estimators and Functional Delta Method 3 2.1 Examples of statistical functionals The mean: µ(F ) = XdF (X); • The variance: Var(FR) = (X µ(F ))2dF (X); • − R k Higher order moments: µk(F ) = (X µ(F )) dF (X); • − R The Kolmogorov-Smirnov statistics: K(F ) = supx F (x) F0(x) , where F0 is a fixed hypothesized • distribution; k − k 2 The Cr´amer-von Mises statistics: C(F ) = (F (x) F0(x)) dF0(x), where F0 is a fixed hypothesized • distribution; − R V-statistics: φ(F ) = EF (T (X1, X2,...,Xp)) where X1, X2,...,Xp are independent copies of F - • distributed random variables; 1 ∆ Quantile functional: φ(F ) = F − (p) = infx x : F (x) p ; • { ≥ } 2.2 Consistency of statistical functionals One possible assumption to ensure that φ(Fˆn) converges to φ(F ) is continuous with respect to the supremum norm. Formally, this is defined as: Definition 2. Continuity of a functional Let D be the space of distributions and φ : D R. We say φ is continuous (with respect to the supremum norm) at F if: → sup Fn(x) F (x) 0 φ(Fn) φ(F ). x k − k → ⇒ → The next result is an extension of the continuous mapping theorem to functionals and can be used to establish the consistency of statistical functionals. Theorem 3. Continuous Mapping Theorem for Statistical Functionals Let φ : D R be a contin- uous functional at F . It follows that: → p p If Fn F 0, then φ(Fn) φ(F ) 0. k − k∞ → − → Proof. From continuity of φ (with respect to the sup norm), we have that for every ε > 0, there exists δ > 0 such that: Fn F δ φ(Fn) φ(F ) ε. k − k∞ ≤ ⇒ k − k ≤ Hence, 0 P ( φ(Fn) φ(F ) > ε) P ( Fn F > δ(ε)) 0, ≤ k − k ≤ k − k∞ → p where the last convergence is due to Fn F 0 by hypothesis. k − k∞ → 4 Lecture 27: Optimal Estimators and Functional Delta Method 2.2.1 Examples of continuous statistical functionals The following two functionals are continuous with respect to the supremum norm: φ(F ) = F (a): The distribution function evaluated at a point. • To establish the continuity of this function, notice that for a sequence of distributions such that Fn F 0: k − k∞ → 0 Fn(a) F (a) sup Fn(x) F (x) 0 ≤ k − k ≤ x k − k → 2 φ(F ) = (F (x) F0(x)) dF0(x): The Cr´amer-von Mises functional. • − Again, take a sequence of distributions such that Fn F 0. We have: R k − k∞ → 2 2 0 φ(Fn) φ(F ) = F F + 2F0(F Fn) dF0 Fn F 2F0 F Fn dF0 ≤ k − k k n − − k ≤ | − | | − − | Z Z 2 ≤ 2 sup Fn(x) F (x) dF0 0 | {z } ≤ x | − | → Z 2.3 Limiting distribution of statistical functionals Recall our goal to determine the limiting distribution of φ(Fˆn). Heuristically, we might hope to derive it if we can find a linear φF′ resulting in an approximation of the sort: φ(Fˆn) φ(F ) = φ′ (Fˆn F ) + some residual − F − Gˆ = √nφF′ ( n) + some residual As before, we must keep track of the behavior of the residual term as n grows. The fact that φ operates on the infinite dimensional space of distribution functions will require us to be more careful and resort to some concepts of functional analysis. Namely, we will be looking at the notions of Gateaux and Hadamard derivatives. We start by considering Gateaux derivatives. To see how they can be used, notice that from linearity of φF′ , we have that: n Gˆ 1 φ′ ( n) = φ′ (δXi F ), F √n F − i=1 X where δXi is the distribution function concentrating all mass at Xi. For each of the terms in the sum, we can consider a “directional” derivative of φ in the direction δXi F . That is the Gateaux derivative which in this case is defined as: − d φ′ (δXi F ) = [φ ((1 t)F + tδXi )] . F − dt − t=0 The expression for φ′ (δXi F ) above is known in the robust statistics literature as the influence function: F − d IFφ,F (x) = [φ ((1 t)F + tδx)] . dt − t=0 Lecture 27: Optimal Estimators and Functional Delta Method 5 To some extent, it measures how much the statistic φ(F ) is affected by adding a new observation at x to the sample. A related concept is the gross-error sensitivity defined as: γ∗ = sup IFφ,F (x). x For robustness, γ∗ must be bounded. Going back to the approximation of φ(Fˆn) φ(F ), we now write: − n 1 φ(Fˆn) φ(F ) = IFφ,F (Xi) + Rn. − n i=1 X We now have: E F (φ′ (δXi F )) = φ′ (δx P )dP (x) = 0, F − F − Z If we assume: 2 VarF (φ′ (δXi F )) = (IFφ,F (x)) dF (x) < . F − ∞ Z and the residual term Rn can be controlled somehow (more on this in later classes), a central limit theorem will hold for the (random variable) φ′ (δXi F ) and we can expect that: F − F 2 √n φ(Fˆn) φ(F ) (0,λ ).
Load more

Recommended publications
  • Chapter 8 Large Sample Theory
    Chapter 8 Large Sample Theory 8.1 The CLT, Delta Method and an Expo- nential Family Limit Theorem Large sample theory, also called asymptotic theory, is used to approximate the distribution of an estimator when the sample size n is large. This theory is extremely useful if the exact sampling distribution of the estimator is complicated or unknown. To use this theory, one must determine what the estimator is estimating, the rate of convergence, the asymptotic distribution, and how large n must be for the approximation to be useful. Moreover, the (asymptotic) standard error (SE), an estimator of the asymptotic standard deviation, must be computable if the estimator is to be useful for inference. Theorem 8.1: the Central Limit Theorem (CLT). Let Y1,...,Yn be 2 1 n iid with E(Y )= µ and VAR(Y )= σ . Let the sample mean Y n = n i=1 Yi. Then √n(Y µ) D N(0,σ2). P n − → Hence n Y µ Yi nµ D √n n − = √n i=1 − N(0, 1). σ nσ → P Note that the sample mean is estimating the population mean µ with a √n convergence rate, the asymptotic distribution is normal, and the SE = S/√n where S is the sample standard deviation. For many distributions 203 the central limit theorem provides a good approximation if the sample size n> 30. A special case of the CLT is proven at the end of Section 4. Notation. The notation X Y and X =D Y both mean that the random variables X and Y have the same∼ distribution.
    [Show full text]
  • Applying the Delta Method in Metric Analytics
    Applying the Delta Method in Metric Analytics: A Practical Guide with Novel Ideas Alex Deng Ulf Knoblich Jiannan Lu∗ Microsoft Corporation Microsoft Corporation Microsoft Corporation Redmond, WA Redmond, WA Redmond, WA [email protected] [email protected] [email protected] ABSTRACT variance of a normal distribution from independent and identically During the last decade, the information technology industry has distributed (i.i.d.) observations, we only need to obtain their sum adopted a data-driven culture, relying on online metrics to measure and sum of squares, which are the corresponding summary statis- 1 and monitor business performance. Under the setting of big data, tics and can be trivially computed in a distributed fashion. In data- the majority of such metrics approximately follow normal distribu- driven businesses such as information technology, these summary tions, opening up potential opportunities to model them directly statistics are often referred to as metrics, and used for measuring without extra model assumptions and solve big data problems via and monitoring key performance indicators [15, 18]. In practice, it closed-form formulas using distributed algorithms at a fraction of is often the changes or differences between metrics, rather than the cost of simulation-based procedures like bootstrap. However, measurements at the most granular level, that are of greater inter- certain attributes of the metrics, such as their corresponding data est. In the context of randomized controlled experimentation (or generating processes and aggregation levels, pose numerous chal- A/B testing), inferring the changes of metrics can establish causal- lenges for constructing trustworthy estimation and inference pro- ity [36, 44, 50], e.g., whether a new user interface design causes cedures.
    [Show full text]
  • Chapter 6 Asymptotic Distribution Theory
    RS – Chapter 6 Chapter 6 Asymptotic Distribution Theory Asymptotic Distribution Theory • Asymptotic distribution theory studies the hypothetical distribution -the limiting distribution- of a sequence of distributions. • Do not confuse with asymptotic theory (or large sample theory), which studies the properties of asymptotic expansions. • Definition Asymptotic expansion An asymptotic expansion (asymptotic series or Poincaré expansion) is a formal series of functions, which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. (In asymptotic distribution theory, we do use asymptotic expansions.) 1 RS – Chapter 6 Asymptotic Distribution Theory • In Chapter 5, we derive exact distributions of several sample statistics based on a random sample of observations. • In many situations an exact statistical result is difficult to get. In these situations, we rely on approximate results that are based on what we know about the behavior of certain statistics in large samples. • Example from basic statistics: What can we say about 1/ x ? We know a lot about x . What do we know about its reciprocal? Maybe we can get an approximate distribution of 1/ x when n is large. Convergence • Convergence of a non-random sequence. Suppose we have a sequence of constants, indexed by n f(n) = ((n(n+1)+3)/(2n + 3n2 + 5) n=1, 2, 3, ..... 2 Ordinary limit: limn→∞ ((n(n+1)+3)/(2n + 3n + 5) = 1/3 There is nothing stochastic about the limit above. The limit will always be 1/3. • In econometrics, we are interested in the behavior of sequences of real-valued random scalars or vectors.
    [Show full text]
  • The 'Delta Method'
    B Appendix The ‘Delta method’ ... Suppose you have done a study, over 4 years, which yields 3 estimates of survival (say, φ1, φ2, and φ3). But, suppose what you are really interested in is the estimate of the product of the three survival values (i.e., the probability of surviving from the beginning of the study to the end of theb study)?b Whileb it is easy enough to derive an estimate of this product (as [φ φ φ ]), how do you derive 1 × 2 × 3 an estimate of the variance of the product? In other words, how do you derive an estimate of the variance of a transformation of one or more random variables (inb thisb case,b we transform the three random variables - φi - by considering their product)? One commonly used approach which is easily implemented, not computer-intensive, and can be robustly applied inb many (but not all) situations is the so-called Delta method (also known as the method of propagation of errors). In this appendix, we briefly introduce the underlying background theory, and the implementation of the Delta method, to fairly typical scenarios. B.1. Mean and variance of random variables Our primary interest here is developing a method that will allow us to estimate the mean and variance for functions of random variables. Let’s start by considering the formal approach for deriving these values explicitly, based on the method of moments. For continuous random variables, consider a continuous function f (x) on the interval [ ∞, +∞]. The first three moments of f (x) can be written − as +∞ M0 = f (x)dx ∞ Z− +∞ M1 = x f (x)dx ∞ Z− +∞ 2 M2 = x f (x)dx ∞ Z− In the particular case that the function is a probability density (as for a continuous random variable), then M0 = 1 (i.e., the area under the PDF must equal 1).
    [Show full text]
  • Chapter 5 the Delta Method and Applications
    Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X1,X2,... of independent and identically distributed (univariate) random variables with finite variance σ2. In this case, the central limit theorem states that √ d n(Xn − µ) → σZ, (5.1) where µ = E X1 and Z is a standard normal random variable. In this chapter, we wish to consider the asymptotic distribution of, say, some function of Xn. In the simplest case, the answer depends on results already known: Consider a linear function g(t) = at+b for some known constants a and b. Since E Xn = µ, clearly E g(Xn) = aµ + b =√g(µ) by the linearity of the expectation operator. Therefore, it is reasonable to ask whether n[g(Xn) − g(µ)] tends to some distribution as n → ∞. But the linearity of g(t) allows one to write √ √ n g(Xn) − g(µ) = a n Xn − µ . We conclude by Theorem 2.24 that √ d n g(Xn) − g(µ) → aσZ. Of course, the distribution on the right hand side above is N(0, a2σ2). None of the preceding development is especially deep; one might even say that it is obvious that a linear transformation of the random variable Xn alters its asymptotic distribution 85 by a constant multiple. Yet what if the function g(t) is nonlinear? It is in this nonlinear case that a strong understanding of the argument above, as simple as it may be, pays real dividends.
    [Show full text]
  • Stat 8931 (Aster Models) Lecture Slides Deck 4 [1Ex] Large Sample
    Stat 8931 (Aster Models) Lecture Slides Deck 4 Large Sample Theory and Estimating Population Growth Rate Charles J. Geyer School of Statistics University of Minnesota October 8, 2018 R and License The version of R used to make these slides is 3.5.1. The version of R package aster used to make these slides is 1.0.2. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License (http://creativecommons.org/licenses/by-sa/4.0/). The Delta Method The delta method is a method (duh!) of deriving the approximate distribution of a nonlinear function of an estimator from the approximate distribution of the estimator itself. What it does is linearize the nonlinear function. If g is a nonlinear, differentiable vector-to-vector function, the best linear approximation, which is the Taylor series up through linear terms, is g(y) − g(x) ≈ rg(x)(y − x); where rg(x) is the matrix of partial derivatives, sometimes called the Jacobian matrix. If gi (x) denotes the i-th component of the vector g(x), then the (i; j)-th component of the Jacobian matrix is @gi (x)=@xj . The Delta Method (cont.) The delta method is particularly useful when θ^ is an estimator and θ is the unknown true (vector) parameter value it estimates, and the delta method says g(θ^) − g(θ) ≈ rg(θ)(θ^ − θ) It is not necessary that θ and g(θ) be vectors of the same dimension. Hence it is not necessary that rg(θ) be a square matrix. The Delta Method (cont.) The delta method gives good or bad approximations depending on whether the spread of the distribution of θ^ − θ is small or large compared to the nonlinearity of the function g in the neighborhood of θ.
    [Show full text]
  • Taylor Approximation and the Delta Method
    Taylor Approximation and the Delta Method Alex Papanicolaou∗ April 28, 2009 1 Taylor Approximation 1.1 Motivating Example: Estimating the odds Suppose we observe X1;:::;Xn independent Bernoulli(p) random variables. Typically, we are p interested in p but there is also interest in the parameter 1−p , which is known as the odds. For example, if the outcomes of a medical treatment occur with p = 2=3, then the odds of getting better is 2 : 1. Furthermore, if there is another treatment with success probability r, we might also be p r interested in the odds ratio 1−p = 1−r , which gives the relative odds of one treatment over another. If we wished to estimate p, we would typically estimate this quantity with the observed success P p^ probabilityp ^ = i Xi=n. To estimate the odds, it then seems perfectly natural to use 1−p^ as an p estimate for 1−p . But whereas we know the variance of our estimatorp ^ is p(1 − p) (check this p^ be computing var(^p)), what is the variance of 1−p^? Or, how can we approximate its sampling distribution? The Delta Method gives a technique for doing this and is based on using a Taylor series approxi- mation. 1.2 The Taylor Series (r) dr Definition: If a function g(x) has derivatives of order r, that is g (x) = dxr g(x) exists, then for any constant a, the Taylor polynomial of order r about a is r X g(k)(a) T (x) = (x − a)k: r k! k=0 While the Taylor polynomial was introduced as far back as beginning calculus, the major theorem from Taylor is that the remainder from the approximation, namely g(x) − Tr(x), tends to 0 faster than the highest-order term in Tr(x).
    [Show full text]
  • TESTING for the PARETO DISTRIBUTION Suppose That X1
    1 M3S3/M4S3 STATISTICAL THEORY II WORKED EXAMPLE: TESTING FOR THE PARETO DISTRIBUTION Suppose that X1;:::;Xn are i.i.d random variables having a Pareto distribution with pdf θcθ f (xjθ) = x > c Xjθ xθ+1 and zero otherwise, for known constant c > 0, and parameter θ > 0. (i) Find the ML estimator, θbn, of θ, and ¯nd the asymptotic distribution of p n(θbn ¡ θT ) where θT is the true value of θ. (ii) Consider testing the hypotheses of H0 : θ = θ0 H1 : θ 6= θ0 for some θ0 > 0. Determine the likelihood ratio, Wald and Rao tests of this hypothesis. SOLUTION (i) The ML estimate θbn is computed in the usual way: Yn Yn θcθ θncnθ Ln(θ) = fXjθ(xijθ) = θ+1 = θ+1 i=1 i=1 xi sn Yn where sn = xi. Then i=1 ln(θ) = n log θ + nθ log c ¡ (θ + 1) log sn n l_ (θ) = + n log c ¡ log s n θ n and solving l_n(θ) = 0 yields the ML estimate · ¸ " #¡1 log s ¡1 1 Xn θb = n ¡ log c = log x ¡ log c : n n n i i=1 The corresponding estimator is therefore " #¡1 1 Xn θb = log X ¡ log c : n n i i=1 Computing the asymptotic distribution directly is di±cult because of the reciprocal. However, consider Á = 1/θ; by invariance, the ML estimator of Á is 1 Xn 1 Xn Áb = log X ¡ log c = (log X ¡ log c) n n i n i i=1 i=1 2 which implies how we should compute the asymptotic distribution of θbn - we use the CLT on the random variables Yi = log Xi ¡ log c = log(Xi=c), and then use the Delta Method.
    [Show full text]
  • Uniformity and the Delta Method
    Uniformity and the delta method Maximilian Kasy∗ January 1, 2018 Abstract When are asymptotic approximations using the delta-method uniformly valid? We provide sufficient conditions as well as closely related necessary conditions for uniform neg- ligibility of the remainder of such approximations. These conditions are easily verified by empirical practitioners and permit to identify settings and parameter regions where point- wise asymptotic approximations perform poorly. Our framework allows for a unified and transparent discussion of uniformity issues in various sub-fields of statistics and economet- rics. Our conditions involve uniform bounds on the remainder of a first-order approximation for the function of interest. ∗Associate professor, Department of Economics, Harvard University. Address: Littauer Center 200, 1805 Cambridge Street, Cambridge, MA 02138. email: [email protected]. 1 1 Introduction Many statistical and econometric procedures are motivated and justified using asymptotic ap- proximations.1 Standard asymptotic theory provides approximations for fixed parameter values, letting the sample size go to infinity. Procedures for estimation, testing, or the construction of confidence sets are considered justified if they perform well for large sample sizes, for any given parameter value. Procedures that are justified in this sense might unfortunately still perform poorly for arbi- trarily large samples. This happens if the asymptotic approximations invoked are not uniformly valid. In that case there are parameter values for every sample size such that the approximation is poor, even though for every given parameter value the approximation performs well for large enough sample sizes. Which parameter values cause poor behavior might depend on sample size, so that poor behavior does not show up in standard asymptotics.
    [Show full text]
  • Calculating Confidence Intervals for Continuous and Discontinuous Functions of Estimated Parameters
    IZA DP No. 5816 Calculating Confi dence Intervals for Continuous and Discontinuous Functions of Estimated Parameters John Ham Tiemen Woutersen June 2011 DISCUSSION PAPER SERIES Forschungsinstitut zur Zukunft der Arbeit Institute for the Study of Labor Calculating Confidence Intervals for Continuous and Discontinuous Functions of Estimated Parameters John Ham University of Maryland, IRP (Madison), IFAU and IZA Tiemen Woutersen Johns Hopkins University Discussion Paper No. 5816 June 2011 IZA P.O. Box 7240 53072 Bonn Germany Phone: +49-228-3894-0 Fax: +49-228-3894-180 E-mail: [email protected] Any opinions expressed here are those of the author(s) and not those of IZA. Research published in this series may include views on policy, but the institute itself takes no institutional policy positions. The Institute for the Study of Labor (IZA) in Bonn is a local and virtual international research center and a place of communication between science, politics and business. IZA is an independent nonprofit organization supported by Deutsche Post Foundation. The center is associated with the University of Bonn and offers a stimulating research environment through its international network, workshops and conferences, data service, project support, research visits and doctoral program. IZA engages in (i) original and internationally competitive research in all fields of labor economics, (ii) development of policy concepts, and (iii) dissemination of research results and concepts to the interested public. IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available directly from the author.
    [Show full text]
  • MLE Lecture 4.Pdf
    Lecture 4 Estimating the Asymptotic Covariance Matrices of Maximum Likelihood Estimators The asymptotic variance-covariance matrix of MLEs can be estimated by −1 2 −1 ∂ ln L [I (θ )] = − E . ~ |θ =θˆ ∂θ ∂θ′ ~ ~ ML ~ ~ |θ =θˆ ~ ~ ML However, this estimator may not be available because of the difficulty in calculating the necessary expectations to get the information matrix. There do, however, exist two alternative consistent estimators. Alternative 1 (Direct use of Hessian) −1 2 −1 −1 ∂ ln L(θ ) [Iˆ(θˆ)] = [− H (θˆ)] = − ~ ~ ~ ∂θ ∂θ′ ~ ~ |θ =θˆ ~ ~ where plimθˆ = θ . ~ ~ Of course, θˆ could be chosen as the maximum likelihood estimator θˆ . ~ ~ ML Alternative 2 (BHHH or outer product of gradients) − −1 N 1 ˆ ˆ ˆ ˆ −1 I (θ ) = gˆi gˆi′ = [GG′] ~ ∑ i=1 ~ ~ ∂ln L X i ;θ ~ ˆ = ˆ ˆ ˆ ˆ = ~ where G g1 g2 gN and gi ~ ~ ~ ~ ∂θ ~ |θ =θ ~ ~ ML The first alternative requires the derivation of the second derivations of the log likelihood function (i.e., the Hessian matrix) while the second alternative only requires 1 the derivation of the first derivatives of the log likelihood function (i.e., the gradient vector). Alternative 3 (QMLE/Sandwich Variance-Covariance Matrix) For the Type 2 ML estimator (see Davidson and McKinnon (2004, p. 406)) it can be shown that 1/ 2 ˆ −1 −1 Var( p lim n (θ −θ 0 )) = H (θ 0 ) I(θ 0 )H (θ 0 ) . n→∞ ~ ~ ~ ~ ~ Therefore, a consistent estimate of the variance-covariance matrix of the Type 2 ML estimator is ˆ ˆ −1 ˆ ˆ ˆ −1 VaˆrQMLE (θ ) = [H (θ )] GG'[H (θ )] .
    [Show full text]
  • Large Sample Tools∗ STA 312: Fall 2012
    Large Sample Tools∗ STA 312: Fall 2012 Background Reading: Davison's Statistical models • For completeness, look at Section 2.1, which presents some basic applied statistics in an advanced way. • Especially see Section 2.2 (Pages 28-37) on convergence. • Section 3.3 (Pages 77-90) goes more deeply into simulation than we will. At least skim it. Overview Contents 1 Foundations2 2 Law of Large Numbers3 3 Consistency7 4 Central Limit Theorem8 5 Convergence of random vectors 10 6 Delta Method 11 ∗See last slide for copyright information. 1 1 Foundations Sample Space Ω, ! 2 Ω • Observe whether a single individual is male or female: Ω = fF; Mg • Pair of individuals; observe their genders in order: Ω = f(F; F ); (F; M); (M; F ); (M; M)g • Select n people and count the number of females: Ω = f0; : : : ; ng For limits problems, the points in Ω are infinite sequences. Random variables are functions from Ω into the set of real numbers P rfX 2 Bg = P r(f! 2 Ω: X(!) 2 Bg) Random Sample X1(!);:::;Xn(!) • T = T (X1;:::;Xn) • T = Tn(!) • Let n ! 1 to see what happens for large samples Modes of Convergence • Almost Sure Convergence • Convergence in Probability • Convergence in Distribution Almost Sure Convergence a:s: We say that Tn converges almost surely to T , and write Tn ! if P rf! : lim Tn(!) = T (!)g = 1: n!1 • Acts like an ordinary limit, except possibly on a set of probability zero. • All the usual rules apply. • Called convergence with probability one or sometimes strong convergence.
    [Show full text]