Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Real and Complex Analysis

In an effort to improve the small sample properties of generalized method of moments (GMM) estimators, a number of alternative estimators have been suggested. These include empirical likelihood (EL), continuous updating, and exponential tilting estimators. We show that these estimators share a common structure, being members of a class of generalized empirical likelihood (GEL) estimators. We use this structure to compare their higher order asymptotic properties. We find that GEL has no asymptotic bias due to correlation of the moment functions with their Jacobian, eliminating an important source of bias for GMM in models with endogeneity. We also find that EL has no asymptotic bias from estimating the optimal weight matrix, eliminating a further important source of bias for GMM in panel data models. We give bias corrected GMM and GEL estimators. We also show that bias corrected EL inherits the higher order property of maximum likelihood, that it is higher order asymptotically efficient relative to the other bias corrected estimators.

/pdf/higher-order-properties-of-gmm-and-generalized-empirical-2il0xtokgo.pdf

Higher Order Properties of Gmm and Generalized Empirical Likelihood Estimators

Different continuous-time models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuous-time model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then mean-reverts strongly when far away from the mean. The volatility is higher when away from the mean.

Testing Continuous-Time Models of the Spot Interest Rate

We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under which our estimator is consistent and asympotically normal. Its rate and asymptotic bias and variance are the same as those without microstructure noise. To use our method in data analysis, we propose a data-based cross-validation method to determine the bandwidth in the Nadaraya–Watson estimator. Via simulation, we study several methods of bandwidth choices, and compare our estimator to several existing estimators. In terms of mean squared error, our new estimator outperforms existing estimators.

Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise

Suppose we observe a stationary Markov chain with unknown transition distribution The empirical estimator for the expectation of a function of two successive observations is known to be efficient For reversible Markov chains, an appropriate symmetrization is efficient For functions of more than two arguments, these estimators cease to be efficient We determine the influence function of efficient estimators of expectations of functions of several observations, both for completely unknown and for reversible Markov chains We construct simple efficient estimators in both cases

Estimating joint distributions of Markov chains

We consider a nonparametric regression model $Y=r(X)+\varepsilon$ with a random covariate $X$ that is independent of the error $\varepsilon$. Then the density of the response $Y$ is a convolution of the densities of $\varepsilon$ and $r(X)$. It can therefore be estimated by a convolution of kernel estimators for these two densities, or more generally by a local von Mises statistic. If the regression function has a nowhere vanishing derivative, then the convolution estimator converges at a parametric rate. We show that the convergence holds uniformly, and that the corresponding process obeys a functional central limit theorem in the space $C_0(\mathbb {R})$ of continuous functions vanishing at infinity, endowed with the sup-norm. The estimator is not efficient. We construct an additive correction that makes it efficient.

Uniform convergence of convolution estimators for the response density in nonparametric regression

A family of probability measures can be identified with a subset of the Hilbert space generated by the Hellinger distance. The family is smooth at a fixed probability measure if the corresponding subset admits a tangent space with respect to the Hausdorff distance at the point corresponding to the fixed probability measure. This smoothness concept was introduced by Chernoff and LeCam. An asymptotic version can be defined for hypotheses on independent, not necessarily identically distributed observations. Assuming that the hypothesis is asymptotically smooth in this sense, we obtain an asymptotic bound for the power of tests under contiguous alternatives. The bound is sharp. If a test-sequence attains the bound for a single alternative, its asymptotic power is uniquely determined.

Testing hypotheses on independent, not identically distributed models

A quasi-likelihood model for a stochastic process is defined by parametric models for the conditional mean and variance processes given the past. The customary estimator for the parameter is the maximum quasi-likelihood estimator. We discuss some ways of improving this estimator. For simplicity we restrict attention to a Markov chain with conditional mean ϑx given the previous state x, and conditional variance a function of ϑ but not of x. Then the maximum quasi-likelihood estimator is the least squares estimator. It remains consistent if the conditional variance is misspecified. In this case, a better estimator is a weighted least squares estimator, with weights involving nonparametric predictors for the conditional variance. If the conditional variance is correctly specified, a better estimator is given by a convex combination of the least squares estimator and a function of an ‘empirical’ estimator for the variance.

https://link.springer.com/content/pdf/10.1007%2F978-3-642-57984-4_42.pdf

Wolfgang Wefelmeyer

Papers

Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise

Estimating joint distributions of Markov chains

Uniform convergence of convolution estimators for the response density in nonparametric regression

Testing hypotheses on independent, not identically distributed models

Improving Maximum Quasi-Likelihood Estimators