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Convex Analysisの二,三の進展について

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

Convergence of Probability Measures

This paper develops the method of matching as an econometric evaluation estimator. A rigorous distribution theory for kernel-based matching is presented. The method of matching is extended to more general conditions than the ones assumed in the statistical literature on the topic. We focus on the method of propensity score matching and show that it is not necessarily better, in the sense of reducing the variance of the resulting estimator, to use the propensity score method even if propensity score is known. We extend the statistical literature on the propensity score by considering the case when it is estimated both parametrically and nonparametrically. We examine the benefits of separability and exclusion restrictions in improving the efficiency of the estimator. Our methods also apply to the econometric selection bias estimator. Matching is a widely-used method of evaluation. It is based on the intuitively attractive idea of contrasting the outcomes of programme participants (denoted Y1) with the outcomes of "comparable" nonparticipants (denoted Y0). Differences in the outcomes between the two groups are attributed to the programme. Let 1 and 11 denote the set of indices for nonparticipants and participants, respectively. The following framework describes conventional matching methods as well as the smoothed versions of these methods analysed in this paper. To estimate a treatment effect for each treated person iecI, outcome Yli is compared to an average of the outcomes Yoj for matched persons je10 in the untreated sample. Matches are constructed on the basis of observed characteristics X in Rd. Typically, when the observed characteristics of an untreated person are closer to those of the treated person ieI1, using a specific distance measure, the untreated person gets a higher weight in constructing the match. The estimated gain for each person i in the treated sample is

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Matching As An Econometric Evaluation Estimator

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

This book provides the most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management. Whether you are a financial risk analyst, actuary, regulator or student of quantitative finance, Quantitative Risk Management gives you the practical tools you need to solve real-world problems. Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book's methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives. Fully revised and expanded to reflect developments in the field since the financial crisis Features shorter chapters to facilitate teaching and learning Provides enhanced coverage of Solvency II and insurance risk management and extended treatment of credit risk, including counterparty credit risk and CDO pricing Includes a new chapter on market risk and new material on risk measures and risk aggregation

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Quantitative Risk Management: Concepts, Techniques, and Tools

Summary. Compared with the classical backfitting of Buja, Hastie and Tibshirani, the smooth backfitting estimator (SBE) of Mammen, Linton and Nielsen not only provides complete asymptotic theory under weaker conditions but is also more efficient, robust and easier to calculate. However, the original paper describing the SBE method is complex and the practical as well as the theoretical advantages of the method have still neither been recognized nor accepted by the statistical community. We focus on a clear presentation of the idea, the main theoretical results and practical aspects like implementation and simplification of the algorithm. We introduce a feasible cross-validation procedure and apply it to the problem of data-driven bandwidth choice for the SBE. By simulations it is shown that the SBE and our cross-validation work very well indeed. In particular, the SBE is less affected by sparseness of data in high dimensional regression problems or strongly correlated designs. The SBE has reasonable performance even in 100-dimensional additive regression problems.

Smooth backfitting in practice

We introduce a new kernel hazard estimator in a nonparametric model where the stochastic hazard depends on the current value of time and on the current value of a time dependent covariate or marker. We establish the pointwise and global convergence of our estimator.

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Kernel estimation in a nonparametric marker dependent hazard model

In this paper we estimate actuarial loss functions based on a symmetrized version of the semiparametric transformation approach to kernel smoothing. We apply this method to an actuarial study of automobile claims. The method gives a good overall impression while estimating actuarial loss functions, since it is capable of estimating both the initial mode and the heavy tail that is so typical for actuarial and other economic loss distributions. We study the properties of the transformation kernel density estimation and show the differences with the multiplicative bias corrected estimator. We add insight into the kernel smoothing transformation method through an extensive simulation study with a particular view to the performance of the estimation at the tail.

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Kernel density estimation of actuarial loss functions

For over a decade, nonparametric modelling has been successfully applied to studying nonlinear structures in financial time series. It is well known that the usual nonparametric models often have less than satisfactory performance when dealing with more than one lag. When the mean has an additive structure, however, better estimation methods are available which fully exploit such a structure. Although in the past such nonparametric applications had been focused more on the estimation of the conditional mean, it is equally if not more important to measure the future risk of the series along with the mean. For the volatility function, i.e. the conditional variance given the past, a multiplicative structure is more appropriate than an additive structure, as the volatility is a positive scale function and a multiplicative model provides a better interpretation of each lagged value's influence on such a function. In this paper we consider the joint estimation of both the additive mean and the multiplicative volatility. The technique used is marginally integrated local polynomial estimation. The procedure is applied to the deutschmark/US dollar daily exchange returns.

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Nonparametric Autoregression with Multiplicative Volatility and Additive mean

By adding the information of reported count data to a classical triangle of reserving data, we derive a suprisingly simple method for forecasting IBNR and RBNS claims. A simple relationship between development factors allows to involve and then estimate the reporting and payment delay. Bootstrap methods provide prediction errors and make possible the inference about IBNR and RBNS claims, separately.

Jens Perch Nielsen

Papers

Smooth backfitting in practice

Kernel estimation in a nonparametric marker dependent hazard model

Kernel density estimation of actuarial loss functions

Nonparametric Autoregression with Multiplicative Volatility and Additive mean

Double Chain Ladder