Simulation algorithms for integrals of a class of sampling distributions arising in population genetics

Efficient stochastic algorithms are presented in order to simulate allele configurations distributed according to a family π _A , 0<A<∞, of exchangeable sampling distributions arising in population genetics. Each distribution π _A has two parameters n and k, the sample size and the number of alleles, respectively. For A→0, the distribution π _A is induced from neutral sampling, whereas for A→∞, it is induced from Maxwell–Boltzmann sampling. Three different Monte Carlo methods (independent sampling procedures) are provided, based on conditioning, sequential methods and a generalization of Pitmans ‘Chinese restaurant process’. Moreover, an efficient Markov chain Monte Carlo method is provided. The algorithms are applied to the homozygosity test and to the Ewens–Watterson–Slatkin test in order to test the hypothesis of selective neutrality.     

Efficient and flexible model-based clustering of jumps in diffusion processes

Jump–diffusion processes involving diffusion processes with discontinuous movements, called jumps, are widely used to model time-series data that commonly exhibit discontinuity in their sample paths. The existing jump–diffusion models have been recently extended to multivariate time-series data. The models are, however, still limited by a single parametric jump-size distribution that is common across different subjects. Such strong parametric assumptions for the shape and structure of a jump-size distribution may be too restrictive and unrealistic for multiple subjects with different characteristics. This paper thus proposes an efficient Bayesian nonparametric method to flexibly model a jump-size distribution while borrowing information across subjects in a clustering procedure using a nested Dirichlet process. For efficient posterior computation, a partially collapsed Gibbs sampler is devised to fit the proposed model. The proposed methodology is illustrated through a simulation study and an application to daily stock price data for companies in the S&P 100 index from June 2007 to June 2017.     

On asymptotics of multivariate integrals with applications to records

Let { X _n ,n≥1} be a sequence of iid. Gaussian random vectors in R ^d , d≥2, with nonsingular distribution function F. In this paper the asymptotics for the sequence of integrals I _F,n (G _n )?n∫ _{R ^d} G _n ^n?1( X ) dF( X ) is considered with G _n some distribution function on R ^d . In the case G _n =F the integral I _F,n (F)/n is the probability that a record occurs in X ₁,…, X _n at index n. [1] Gnedin, A.V. 1998. Records from a Multivariate Normal Sample. Statist. Probab. Lett., 39: 11–15. [Crossref], [Web of Science ®] , [Google Scholar] obtained lower and upper asymptotic bounds for this case, whereas [2] Ledford, W.A. and Twan, A.J. 1998. On the Tail Concomitant Behaviour for Extremes. Adv. Appl. Probab., 30: 197–215. [Crossref], [Web of Science ®] , [Google Scholar] showed the rate of convergence if d=2. In this paper we derive the exact rate of convergence of I _F,n (G _n ) for d≥2 under some restrictions on the distribution function G _n . Some related results for multivariate Gaussian tails are discussed also.     

On an application of the diffusion approximation of the generalized birth and death process for stochastic population projection

Assuming that both birth and death rates are density and time dependent, a diffusion approximation of the generalized birth and death process has been considered in this paper to obtain a suitable stochastic population model describing the population size and its moments. A simple method of estimating the parameters of the model Is discussed. The predictions of the expected size of the population, and the variance are made and compared with the corresponding census figures as well as with another deterministic projection series made for the corresponding period.     

Model selection for Lévy measures in diffusion processes with jumps from discrete observations

We deal with parametric inference and selection problems for jump components in discretely observed diffusion processes with jumps. We prepare several competing parametric models for the Lévy measure that might be misspecified, and select the best model from the aspect of information criteria. We construct quasi-information criteria (QIC), which are approximations of the information criteria based on continuous observations.     

Building adaptive estimating equations when inverse of covariance estimation is difficult

Summary. To construct an optimal estimating function by weighting a set of score functions, we must either know or estimate consistently the covariance matrix for the individual scores. In problems with high dimensional correlated data the estimated covariance matrix could be unreliable. The smallest eigenvalues of the covariance matrix will be the most important for weighting the estimating equations, but in high dimensions these will be poorly determined. Generalized estimating equations introduced the idea of a working correlation to minimize such problems. However, it can be difficult to specify the working correlation model correctly. We develop an adaptive estimating equation method which requires no working correlation assumptions. This methodology relies on finding a reliable approximation to the inverse of the variance matrix in the quasi-likelihood equations. We apply a multivariate generalization of the conjugate gradient method to find estimating equations that preserve the information well at fixed low dimensions. This approach is particularly useful when the estimator of the covariance matrix is singular or close to singular, or impossible to invert owing to its large size.     

A table of normal integrals

Integrals of functions of the univariate, bivariate, trivariate and multivariate normal densities are given. Both indefinite and definite integrals are included.     

The role of embedded integration rules in Bayesian statistics

Numerical approximations are often used to implement the Bayesian paradigm in analytically intractable parametric models. We focus on embedded integration rules which are an attractive numerical integration tool and present theoretical results which justify their use in a Bayesian integration strategy.     

THE SIGN TEST FOR STOCHASTIC PROCESSES

It is shown that the sign test may be applied to the residuals from the use of model fitting procedures, such as conditional least squares, to make inferences on the predictable part of a stochastic process. Minimal assumptions on the distribution of the process, apart from those already required for the model fitting procedure, are needed. The results are illustrated with an application to first order autoregressive processes.     

Fisher information and maximum-likelihood estimation of covariance parameters in Gaussian stochastic processes

The inverse of the Fisher information matrix is commonly used as an approximation for the covariance matrix of maximum-likelihood estimators. We show via three examples that for the covariance parameters of Gaussian stochastic processes under infill asymptotics, the covariance matrix of the limiting distribution of their maximum-likelihood estimators equals the limit of the inverse information matrix. This is either proven analytically or justified by simulation. Furthermore, the limiting behaviour of the trace of the inverse information matrix indicates equivalence or orthogonality of the underlying Gaussian measures. Even in the case of singularity, the estimator of the process variance is seen to be unbiased, and also its variability is approximated accurately from the information matrix.     

Logistic processes involving markovian minimization

The class of logistic processes involving geometric minimization introduced by Arnold (1989) is extended by replacing the Bernoulli sequence used in the definition of the process by a Markovian (0,1) sequence. A more flexible range of dependence structures is thus introduced. Parameter estimation and related processes are discussed.     

Spatiotemporal prediction for log-Gaussian Cox processes

Space–time point pattern data have become more widely available as a result of technological developments in areas such as geographic information systems. We describe a flexible class of space–time point processes. Our models are Cox processes whose stochastic intensity is a space–time Ornstein–Uhlenbeck process. We develop moment-based methods of parameter estimation, show how to predict the underlying intensity by using a Markov chain Monte Carlo approach and illustrate the performance of our methods on a synthetic data set.     

Is a pure jump process fitting the high frequency data better than a jump-diffusion process?

Two families of processes: pure jump processes and jump-diffusion processes are widely used in literatures. Recently, empirical findings demonstrate that the underlying processes of high frequency data sets are pure-jump processes of infinite variation in many situations. Statistical tests are also proposed to make the empirical findings theoretically grounded. In this paper, we extend the work of Jing et al. (2012) in two aspects: (1) the jump process in the null hypothesis and the alternative hypothesis could be different; (2) the null hypothesis covers more flexible processes which are more relevant in finance when considering models for asset prices or nominal interest rates. Theoretically, the test is proven to be very powerful and can control the type I error probabilities well under the nominal level.