The Gibbs sampler, the algorithm of Metropolis and similar iterative simulation methods are potentially very helpful for summarizing multivariate distributions. Used naively, however, iterative simulation can give misleading answers. Our methods are simple and generally applicable to the output of any iterative simulation; they are designed for researchers primarily interested in the science underlying the data and models they are analyzing, rather than for researchers interested in the probability theory underlying the iterative simulations themselves. Our recommended strategy is to use several independent sequences, with starting points sampled from an overdispersed distribution. At each step of the iterative simulation, we obtain, for each univariate estimand of interest, a distributional estimate and an estimate of how much sharper the distributional estimate might become if the simulations were continued indefinitely. Because our focus is on applied inference for Bayesian posterior distributions in real problems, which often tend toward normality after transformations and marginalization, we derive our results as normal-theory approximations to exact Bayesian inference, conditional on the observed simulations. The methods are illustrated on a random-effects mixture model applied to experimental measurements of reaction times of normal and schizophrenic patients.

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Inference from Iterative Simulation Using Multiple Sequences

Anomaly detection is an important problem that has been researched within diverse research areas and application domains. Many anomaly detection techniques have been specifically developed for certain application domains, while others are more generic. This survey tries to provide a structured and comprehensive overview of the research on anomaly detection. We have grouped existing techniques into different categories based on the underlying approach adopted by each technique. For each category we have identified key assumptions, which are used by the techniques to differentiate between normal and anomalous behavior. When applying a given technique to a particular domain, these assumptions can be used as guidelines to assess the effectiveness of the technique in that domain. For each category, we provide a basic anomaly detection technique, and then show how the different existing techniques in that category are variants of the basic technique. This template provides an easier and more succinct understanding of the techniques belonging to each category. Further, for each category, we identify the advantages and disadvantages of the techniques in that category. We also provide a discussion on the computational complexity of the techniques since it is an important issue in real application domains. We hope that this survey will provide a better understanding of the different directions in which research has been done on this topic, and how techniques developed in one area can be applied in domains for which they were not intended to begin with.

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Anomaly detection: A survey

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

Convex Analysisの二,三の進展について

(2005). Applied Multivariate Statistical Analysis. Technometrics: Vol. 47, No. 4, pp. 517-517.

Applied Multivariate Statistical Analysis

Fundamental properties of conditional value-at-risk (CVaR), as a measure of risk with significant advantages over value-at-risk (VaR), are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. CVaR is able to quantify dangers beyond VaR and moreover it is coherent. It provides optimization short-cuts which, through linear programming techniques, make practical many large-scale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calculations, shown in several case studies, are illustrated further with an example of index tracking.

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Conditional value-at-risk for general loss distributions

We study a class of backtests for forecast distributions in which the test statistic depends on a spectral transformation that weights exceedance events by a function of the modeled probability level. The weighting scheme is specified by a kernel measure which makes explicit the user’s priorities for model performance. The class of spectral backtests includes tests of unconditional coverage and tests of conditional coverage. We show how the class embeds a wide variety of backtests in the existing literature, and further propose novel variants which are easily implemented, well-sized and have good power. In an empirical application, we backtest forecast distributions for the overnight P&L of ten bank trading portfolios. For some portfolios, test results depend materially on the choice of kernel.

/pdf/spectral-backtests-of-forecast-distributions-with-4f2cazu9hn.pdf

Spectral backtests of forecast distributions with application to risk management

/pdf/credit-risk-models-an-overview-2ksiqhnm9w.pdf

Credit Risk Models: An Overview

An approach to the modelling of volatile time series using a class of uniformity-preserving transforms for uniform random variables is proposed. V-transforms describe the relationship between quantiles of the stationary distribution of the time series and quantiles of the distribution of a predictable volatility proxy variable. They can be represented as copulas and permit the formulation and estimation of models that combine arbitrary marginal distributions with copula processes for the dynamics of the volatility proxy. The idea is illustrated using a Gaussian ARMA copula process and the resulting model is shown to replicate many of the stylized facts of financial return series and to facilitate the calculation of marginal and conditional characteristics of the model including quantile measures of risk. Estimation is carried out by adapting the exact maximum likelihood approach to the estimation of ARMA processes, and the model is shown to be competitive with standard GARCH in an empirical application to Bitcoin return data.

https://www.mdpi.com/2227-9091/9/1/14/pdf

Modelling Volatile Time Series with V-Transforms and Copulas

(2002) The Laplace Distribution and Generalizations: A Revisit With Applications to Communications, Economics, Engineering, and Finance Journal of the American Statistical Association: Vol 97, No 460, pp 1210-1211

The Laplace Distribution and Generalizations: A Revisit With Applications to Communications, Economics, Engineering, and Finance

An approach to modelling volatile financial return series using d-vine copulas combined with uniformity preserving transformations known as v-transforms is proposed By generalizing the concept of stochastic inversion of v-transforms, models are obtained that can describe both stochastic volatility in the magnitude of price movements and serial correlation in their directions In combination with parametric marginal distributions it is shown that these models can rival and sometimes outperform well-known models in the extended GARCH family

Alexander J. McNeil

Papers

Spectral backtests of forecast distributions with application to risk management

Credit Risk Models: An Overview

Modelling Volatile Time Series with V-Transforms and Copulas

The Laplace Distribution and Generalizations: A Revisit With Applications to Communications, Economics, Engineering, and Finance

Time series copula models using d-vines and v-transforms: an alternative to GARCH modelling