/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

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

Hogh, Linton and Nielsen (2006) showed that the famous result in the award winning paper of Froot and Stein (1998) is not correct in the sense that their result does not follow from their assumptions. In this paper we show that the economic intuition behind the paper of Froot and Stein (1998) is correct and that their result can be obtained when the market is reformulated in a continuous time setting and modern market theory is employed.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S1748499500000488

Froot and Stein Revisited Once Again

One of the most studied questions in economics and finance is whether equity returns or premiums can be predicted by empirical models. While many authors favor the historical mean or other simple parametric methods, this article focuses on nonlinear relationships. A straightforward bootstrap-test confirms that non- and semiparametric techniques help to obtain better forecasts. It is demonstrated how economic theory directly guides a model in an innovative way. The inclusion of prior knowledge enables for American data a further notable improvement in the prediction of excess stock returns of 35% compared to the fully nonparametric model, as measured by the more complex validated R2 as well as using classical out-of-sample validation. Statistically, a bias and dimension reduction method is proposed to import more structure in the estimation process as an adequate way to circumvent the curse of dimensionality.

Nonparametric prediction of stock returns guided by prior knowledge

It is of considerable interest to forecast future mesothelioma mortality. No measures for exposure are available so it is not straight forward to apply a dose-response model. It is proposed to model the counts of deaths directly using a Poisson regression with an age-period-cohort structure, but without offset. Traditionally the age-period-cohort is viewed to suffer from an identification problem. It is shown how to re-parameterize the model in terms of freely varying parameters, so as to avoid this problem. It is shown how to conduct inference and how to construct distribution forecasts.

Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality

H?gh, Linton and Nielsen (2006) showed that the famous result in the reward winning paper of Froot and Stein (1998) is not correct in the sense that their result does not follow from their assumptions. In this paper we show that the economic intuition behind the paper of Froot and Stein (1998) is correct and that their result can be obtained when the market is reformulated in a discrete time setting, where modern market theory is employed.

Nielsen (1999) showed the surprising fact that a nonparametric one-dimensional hazard as a function of time can be estimated x/~-consistently if a high quality marker is observed. In this paper we show that the hazard relevant for predicting remaining duration time, given the current status of a high quality marker, can be estimated v~-consistently if a Markov type property holds for the high quality marker.

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Jens Perch Nielsen

Papers

Froot and Stein Revisited Once Again

Nonparametric prediction of stock returns guided by prior knowledge

Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality

Froot and Stein Revisited Once Again

Super-Efficient Prediction Based on High-Quality Marker Information