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

Convergence of Probability Measures

Non-Uniform Random Variate Generation.

This is a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods. Authors’ address: School of Computer Science, McGill University, 3480 University Street, Montreal, Canada H3A 2K6. The authors’ research was sponsored by NSERC Grant A3456 and FCAR Grant 90-ER-0291. 1. The main paradigms The purpose of this chapter is to review the main methods for generating random variables, vectors and processes. Classical workhorses such as the inversion method, the rejection method and table methods are reviewed in section 1. In section 2, we discuss the expected time complexity of various algorithms, and give a few examples of the design of generators that are uniformly fast over entire families of distributions. In section 3, we develop a few universal generators, such as generators for all log concave distributions on the real line. Section 4 deals with random variate generation when distributions are indirectly specified, e.g, via Fourier coefficients, characteristic functions, the moments, the moment generating function, distributional identities, infinite series or Kolmogorov measures. Random processes are briefly touched upon in section 5. Finally, the latest developments in Markov chain methods are discussed in section 6. Some of this work grew from Devroye (1986a), and we are carefully documenting work that was done since 1986. More recent references can be found in the book by Hörmann, Leydold and Derflinger (2004). Non-uniform random variate generation is concerned with the generation of random variables with certain distributions. Such random variables are often discrete, taking values in a countable set, or absolutely continuous, and thus described by a density. The methods used for generating them depend upon the computational model one is working with, and upon the demands on the part of the output. For example, in a ram (random access memory) model, one accepts that real numbers can be stored and operated upon (compared, added, multiplied, and so forth) in one time unit. Furthermore, this model assumes that a source capable of producing an i.i.d. (independent identically distributed) sequence of uniform [0, 1] random variables is available. This model is of course unrealistic, but designing random variate generators based on it has several advantages: first of all, it allows one to disconnect the theory of non-uniform random variate generation from that of uniform random variate generation, and secondly, it permits one to plan for the future, as more powerful computers will be developed that permit ever better approximations of the model. Algorithms designed under finite approximation limitations will have to be redesigned when the next generation of computers arrives. For the generation of discrete or integer-valued random variables, which includes the vast area of the generation of random combinatorial structures, one can adhere to a clean model, the pure bit model, in which each bit operation takes one time unit, and storage can be reported in terms of bits. Typically, one now assumes that an i.i.d. sequence of independent perfect bits is available. In this model, an elegant information-theoretic theory can be derived. For example, Knuth and Yao (1976) showed that to generate a random integer X described by the probability distribution {X = n} = pn, n ≥ 1, any method must use an expected number of bits greater than the binary entropy of the distribution, ∑

Non-uniform random variate generation

Preface. Preface to First Edition. 1. Financial Time Series and Their Characteristics. 2. Linear Time Series Analysis and Its Applications. 3. Conditional Heteroscedastic Models. 4. Nonlinear Models and Their Applications. 5. High-Frequency Data Analysis and Market Microstructure. 6. Continuous-Time Models and Their Applications. 7. Extreme Values, Quantile Estimation, and Value at Risk. 8. Multivariate Time Series Analysis and Its Applications. 9. Principal Component Analysis and Factor Models. 10. Multivariate Volatility Models and Their Applications. 11. State-Space Models and Kalman Filter. 12. Markov Chain Monte Carlo Methods with Applications. Index.

http://inis.jinr.ru/sl/M_Mathematics/MV_Probability/MVspf_Stochastics%20in%20finance/Tsay%20Analysis.pdf

Analysis of Financial Time Series

This article considers the application of two families of nonlinear autoregressive models, the logistic (LSTAR) and exponential (ESTAR) autoregressive models. This includes the specification of the model based on simple statistical tests: linearity testing against smooth transition autoregression, determining the delay parameter and choosing between LSTAR and ESTAR models are discussed. Estimation by nonlinear least squares is considered as well as evaluating the properties of the estimated model. The proposed techniques are illustrated by examples using both simulated and real time series.

Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models

: A simple and relatively efficient method for simulating one- dimensional and two-dimensional nonhomogeneous Poisson processes is presented. The method is applicable for any rate function and is based on controlled deletion of points in a Poisson process whose rate function dominates the given rate function. In its simplest implementation, the method obviates the need for numerical integration of the rate function, for ordering of points, and for generation of Poisson variates.

Simulation of Nonhomogeneous Poisson Processes by Thinning

Multivariate Adaptive Regression Splines (MARS) is a new methodology, due to Friedman, for nonlinear regression modeling. MARS can be conceptualized as a generalization of recursive partitioning that uses spline fitting in lieu of other simple fitting functions. Given a set of predictor variables, MARS fits a model in the form of an expansion in product spline basis functions of predictors chosen during a forward and backward recursive partitioning strategy. MARS produces continuous models for high-dimensional data that can have multiple partitions and predictor variable interactions. Predictor variable contributions and interactions in a MARS model may be analyzed using an ANOVA style decomposition. By letting the predictor variables in MARS be lagged values of a time series, one obtains a new method for nonlinear autoregressive threshold modeling of time series. A significant feature of this extension of MARS is its ability to produce models with limit cycles when modeling time series data that...

Nonlinear Modeling of Time Series Using Multivariate Adaptive Regression Splines (MARS)

supported in part by National Science Foundation under
Grant NSF-ENG-79-01438 and NSF-ENG-79-10825; and by the Office of Naval
Research under Grant NR-42-284 and NR-42-469.

Stationary discrete autoregressive‐moving average time series generated by mixtures

Discrete Time Series Generated by Mixtures Ii: Asymptotic Properties

: An approach to modelling and residual analysis of nonlinear autoregressive time series in exponential variables is presented; the approach is illustrated by analysis of a long series of wind velocity data which has first been detrended and then transformed into a stationary series with an exponential marginal distribution The stationary series is modelled with a newly developed type of second order autoregressive process with random coefficients, called the NEAR(2) model; it has a second order autoregressive correlation structure but is nonlinear because its coefficients are random The exponential distributional assumptions involved in this model highlight a very broad four parameter structure which combines five exponential random variables into a sixth exponential random variable; other applications of this structure are briefly considered Dependency in the NEAR(2) process not accounted for by standard autocorrelations is explored by developing a residual analysis for time series having autoregressive correlation structure; this involves defining linear uncorrelated residuals which are dependent, and then assessing this higher order dependence by standard time series computations Application of this residual analysis to the wind velocity data illustrates both the utility and difficulty of nonlinear time series modelling

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1985.tb01344.x

Peter A. W. Lewis

Papers

Simulation of Nonhomogeneous Poisson Processes by Thinning

Nonlinear Modeling of Time Series Using Multivariate Adaptive Regression Splines (MARS)

Stationary discrete autoregressive‐moving average time series generated by mixtures

Discrete Time Series Generated by Mixtures Ii: Asymptotic Properties

Modelling and Residual Analysis of Nonlinear Autoregressive Time Series in Exponential Variables