An alternative approximation to the variance of transformation score

An alternative approximation to the variance of transformation score is given, based on an asymptotic expansion of the transformation estimator. It is then compared with the variance approximation given by Lawrance (1987) in terms of standardized scores. Simulations show that the two standardized scores behave very similarly when model error standard deviation is small. However,when error standard deviation is not small, the new standardized score outperforms that of Lawrance,especially in the structured models.     

15 minutes of statistical fame

Tony Lawrance surveyed his email one morning to find an urgent request from the West Mercia Police Authority to give a statistical opinion on a government report recommending the merger of the current 43 police forces into as few as 12. The Authority asked for an independent objective assessment, "even if we don't like the answer." Persuaded by this freedom, he submitted his report within the 2-day deadline and returned, as he thought, to the day job. Well, not quite—a media storm followed.     

Graphical Models for Composable Finite Markov Processes

Abstract.  Composable Markov processes were introduced by Schweder (1970) in order to capture the idea that a process can be composed of different components where some of these only depend on a subset of the other components. Here we propose a graphical representation of this kind of dependence which has been called 'local dependence'. It is shown that the graph allows to read off further independencies characterizing the underlying Markov process. Also, some standard methods for inference are adapted to exploit the graphical representation, e.g. for testing local independence.     

On a threshold representation for complex load-sharing systems

Complex load-sharing systems are studied to incorporate dependencies among components through a load-sharing rule. As the load on the system increases, a series of cycles of Phase I/II failures occur where Phase I failure is a single component failure, which then causes a cascade of component failures (Phase II) due to the load transfer as these components fail. A threshold representation for the process of system failure is given. This representation is a gamma-type mixture representation when the component strengths are independent exponentials. In this case, for a given breaking pattern the mixture is over the gamma scale parameter and is based on a convolution of uniforms defined by the load-sharing parameters. Such convolutions can be approximated by normal densities which reduces the dimension of the parameter space. This representation can be generalized to independent component strengths with arbitrary distributions by transforming the strength and load-sharing to pseudo-strength and pseudo-load-sharing rules.     

Prior Distributions on Measure Space

A measure is the formal representation of the non-negative additive functions that abound in science. We review and develop the art of assigning Bayesian priors to measures. Where necessary, spatial correlation is delegated to correlating kernels imposed on otherwise uncorrelated priors. The latter must be infinitely divisible (ID) and hence described by the Lévy–Khinchin representation. Thus the fundamental object is the Lévy measure, the choice of which corresponds to different ID process priors. The general case of a Lévy measure comprising a mixture of assigned base measures leads to a prior process comprising a convolution of corresponding processes. Examples involving a single base measure are the gamma process, the Dirichlet process (for the normalized case) and the Poisson process. We also discuss processes that we call the supergamma and super-Dirichlet processes, which are double base measure generalizations of the gamma and Dirichlet processes. Examples of multiple and continuum base measures are also discussed. We conclude with numerical examples of density estimation.     

On Cramér–von Mises type test based on local time of switching diffusion process

We consider a Cramér–von Mises type test for hypothesis that the observed diffusion process has sign-type trend coefficient based on empirical density function. It is shown that the limit distribution of the proposed test statistic is defined by the integral type functional of continuous Gaussian process. We provide the Karhunen–Loève expansion of the corresponding limiting process. Approximation of the threshold is given through the representation for the limit statistic.     

Estimation for the autoregressive moving average process observed with noise

SUMMARY The autoregressive moving average process ARMA (p,q) observed with noise has another ARMA (p,k) representation, where k = max (p,q). Parameters for the ARMA (p,k) representation satisfy some non-linear restrictions. We develop restricted Newton-Raphson estimators of the ARMA (p,k) process which takes advantage of the information given in the non-linear restrictions. The asymptotic relative efficiency of the estimators indicates that the proposed restricted Newton-Raphson estimator is more efficient than the unrestricted Newton-Raphson estimator. In a Monte Carlo experiment, the proposed estimator is shown to perform better than the unrestricted estimator of the ARMA (p,k) process.     

Asymptotic properties of QMLE for seasonal threshold GARCH model with periodic coefficients

Statistical Methods & Applications - Periodic models for volatility process constitute an alternative representation for the seasonal patterns observed in data exhibits a strong seasonal...     

A NOTE ON THE INVERSE COVARIANCE MATRIX OF THE AUTOREGRESSIVE PROCESS1

The inverse covariance matrix of an autoregressive process of arbitrary order is found explicitly using the notion of the information matrix for the vector random variable, not the parameters. Any process for which a simple conditional representation exists, can be treated in the same way. The approach has merit in the teaching of statistics.     

ELICITING A DIRECTED ACYCLIC GRAPH FOR A MULTIVARIATE TIME SERIES OF VEHICLE COUNTS IN A TRAFFIC NETWORK

The problem of modelling multivariate time series of vehicle counts in traffic networks is considered. It is proposed to use a model called the linear multiregression dynamic model (LMDM). The LMDM is a multivariate Bayesian dynamic model which uses any conditional independence and causal structure across the time series to break down the complex multivariate model into simpler univariate dynamic linear models. The conditional independence and causal structure in the time series can be represented by a directed acyclic graph (DAG). The DAG not only gives a useful pictorial representation of the multivariate structure, but it is also used to build the LMDM. Therefore, eliciting a DAG which gives a realistic representation of the series is a crucial part of the modelling process. A DAG is elicited for the multivariate time series of hourly vehicle counts at the junction of three major roads in the UK. A flow diagram is introduced to give a pictorial representation of the possible vehicle routes through the network. It is shown how this flow diagram, together with a map of the network, can suggest a DAG for the time series suitable for use with an LMDM.     

Asymptotic estimate of variance with applications to stochastic differential equations arises in mathematical neuroscience

Matrix representation of a limit of variance for circular process is given. It is shown that the variance is asymptotically measured by the decrease in spectral energy in one step of a Markov chain. Then we apply this result to a stochastic differential equation with parametric noise (which arises in mathematical neuroscience) and demonstrate how the results can be used to analyze propagation of a signal in sound mechanism.     

Random spectral measure for non Gaussian moving averages

We study the distribution of phases and amplitudes for the spectral representation of weighted moving averages of a general noise measure. The simple independent structure, known for the Gaussian case, and involving Rayleigh amplitude and uniform phase distributions, is lost for the non Gaussian noise case. We show that the amplitude/phase distributions exhibit a rich and more complex structure depending not just on the covariance of the process but specifically on the form of the kernel and the noise distribution. We present a theoretical tool for studying these distributions that follows from a proof of the spectral theorem that yields an explicit expression for the spectral measure. The main interest is in noise measures based on second-order Lévy motions since such measures are easily available through independent sampling. We approximate the spectral stochastic measure by independent noise increments which allows us to obtain amplitude/phase distributions that is of fundamental interest for analyzing processes in the frequency domain. For the purpose of approximating the moving average process through sums of trigonometric functions, we assess the mean square error of discretization of the spectral representation. For a specified accuracy, the approximation is explicitly given. We illustrate the method for the moving averages driven by the Laplace motion.     

Exact and approximate sum representations for the Dirichlet process

The Dirichlet process can be regarded as a random probability measure for which the authors examine various sum representations. They consider in particular the gamma process construction of Ferguson (1973) and the “stick‐breaking” construction of Sethuraman (1994). They propose a Dirichlet finite sum representation that strongly approximates the Dirichlet process. They assess the accuracy of this approximation and characterize the posterior that this new prior leads to in the context of Bayesian nonpara‐metric hierarchical models.     

SPECTRAL REPRESENTATION AND ERGODICITY OF ASYMPTOTICALLY STATIONARY PROCESSES

A second order process with mean zero and covariance is asymptotically stationary if lim ds exists for every; this limit then defines the covariance function of the process. The paper establishes the spectral representation for the covariance function and a mean ergodic theorem for the process. When stationarity is assumed, the results reduce to the well-known corresponding theorems for stationary processes.     

Karhunen–Loève expansions of Lévy processes

We derive the basis functions and joint distribution of the stochastic coefficients of the Karhunen–Loève expansion of a square-integrable Lévy process. Further, we demonstrate a method for simulating the coefficients via a shot-noise representation.     

Strong approximations for time-varying infinite-server queues with non-renewal arrival and service processes

In real stochastic systems, the arrival and service processes may not be renewal processes. For example, in many telecommunication systems such as internet traffic where data traffic is bursty, the sequence of inter-arrival times and service times are often correlated and dependent. One way to model this non-renewal behavior is to use Markovian Arrival Processes (MAPs) and Markovian Service Processes (MSPs). MAPs and MSPs allow for inter-arrival and service times to be dependent, while providing the analytical tractability of simple Markov processes. To this end, we prove fluid and diffusion limits for MAP_t/MSP_t/∞ queues by constructing a new Poisson process representation for the queueing dynamics and leveraging strong approximations for Poisson processes. As a result, the fluid and diffusion limit theorems illuminate how the dependence structure of the arrival or service processes can affect the sample path behavior of the queueing process. Finally, our Poisson representation for MAPs and MSPs is useful for simulation purposes and may be of independent interest.     

Perfect Simulation for Length-interacting Polygonal Markov Fields in the Plane

Abstract.  The purpose of this paper was to construct perfect samplers for length-interacting Arak–Clifford–Surgailis polygonal Markov fields in the plane with nodes of order 2 ( V -shaped nodes). This is achieved by providing for the polygonal fields a hard core marked point process representation with individual points carrying polygonal loops as their marks, so that the coupling from the past and clan of ancestors routines can be adopted.     

Probabilistic Properties of Parametric Dual and Inverse Time Series Models Generated by ARMA Models

For the class of autoregressive-moving average (ARMA) processes, we examine the relationship between the dual and the inverse processes. It is demonstrated that the inverse process generated by a causal and invertible ARMA (p, q) process is a causal and invertible ARMA (q, p) model. Moreover, it is established that this representation is strong if and only if the generating process is Gaussian. More precisely, it is derived that the linear innovation process of the inverse process is an all-pass model. Some examples and applications to time reversibility are given to illustrate the obtained results.     

Optimal Detection of a Change Point in a Poisson Process for Different Observation Schemes

Abstract.  Change point problems are considered where at some unobservable time the intensity of a point process ( T_n ), n ∈  N , has a jump. For a given reward functional we detect the change point optimally for different information schemes. These schemes differ in the available information. We consider three information levels, namely sequential observation of ( T_n ), ex post decision after observing the point process up to a fixed time t * and a combination of both observation schemes. In all of these cases the detection problem is viewed as an optimal stopping problem which can be solved by deriving a semimartingale representation of the gain process and applying tools from filtering theory.