Regression models for functional data by reproducing kernel Hilbert spaces methods

Non-parametric regression models are developed when the predictor is a function-valued random variable X={_Xt}t∈T$X=\{X_t{\}}_{t\in T}$. Based on a representation of the regression function f(X)$f(X)$ in a reproducing kernel Hilbert space such models generalize the classical setting used in statistical learning theory. Two applications corresponding to scalar and categorical response random variable are performed on stock-exchange and medical data. The results of different regression models are compared.     

Euler(p,q) Processes and Their Application to Non Stationary Time Series with Time Varying Frequencies

We introduce Euler(p, q) processes as an extension of the Euler(p) processes for purposes of obtaining more parsimonious models for non stationary processes whose periodic behavior changes approximately linearly in time. The discrete Euler(p, q) models are a class of multiplicative stationary (M-stationary) processes and basic properties are derived. The relationship between continuous and discrete mixed Euler processes is shown. Fundamental to the theory and application of Euler(p, q) processes is a dual relationship between discrete Euler(p, q) processes and ARMA processes, which is established. The usefulness of Euler(p, q) processes is examined by comparing spectral estimation with that obtained by existing methods using both simulated and real data.     

Contrast Estimation for Parametric Stationary Determinantal Point Processes

We study minimum contrast estimation for parametric stationary determinantal point processes. These processes form a useful class of models for repulsive (or regular, or inhibitive) point patterns and are already applied in numerous statistical applications. Our main focus is on minimum contrast methods based on the Ripley's K‐function or on the pair correlation function. Strong consistency and asymptotic normality of theses procedures are proved under general conditions that only concern the existence of the process and its regularity with respect to the parameters. A key ingredient of the proofs is the recently established Brillinger mixing property of stationary determinantal point processes. This work may be viewed as a complement to the study of Y. Guan and M. Sherman who establish the same kind of asymptotic properties for a large class of Cox processes, which in turn are models for clustering (or aggregation).     

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.     

Classification rules for triply multivariate data with an AR(1) correlation structure on the repeated measures over time

In this article we study the problem of classification of three-level multivariate data, where multiple q$q$-variate observations are measured on u$u$-sites and over p$p$-time points, under the assumption of multivariate normality. The new classification rules with certain structured and unstructured mean vectors and covariance structures are very efficient in small sample scenario, when the number of observations is not adequate to estimate the unknown variance–covariance matrix. These classification rules successfully model the correlation structure on successive repeated measurements over time. Computation algorithms for maximum likelihood estimates of the unknown population parameters are presented. Simulation results show that the introduction of sites in the classification rules improves their performance over the existing classification rules without the sites.     

Autoregressive moving-average convolution processes on the unit circle

ARMA convolution models for processes in continuous space (in this case the unit circle) and discrete time are derived as a natural extension of the usual Box-Jenkins models. Both weakly time-stationary and nonstationary processes are considered. Sufficient conditions for the existence of weakly time-stationary AR_cMA_c processes are derived, and the covariance functions for some processes are computed. It is demonstrated that the usual scalar and multivariate ARMA processes can be embedded within the larger class of AR_CMA_c models. A possible application of these models to sea-surface temperature prediction is discussed.     

Finite mixtures of multivariate Poisson distributions with application

In the present paper we examine finite mixtures of multivariate Poisson distributions as an alternative class of models for multivariate count data. The proposed models allow for both overdispersion in the marginal distributions and negative correlation, while they are computationally tractable using standard ideas from finite mixture modelling. An EM type algorithm for maximum likelihood (ML) estimation of the parameters is developed. The identifiability of this class of mixtures is proved. Properties of ML estimators are derived. A real data application concerning model based clustering for multivariate count data related to different types of crime is presented to illustrate the practical potential of the proposed class of models.     

M‐estimation for general ARMA Processes with Infinite Variance

Abstract. General autoregressive moving average (ARMA) models extend the traditional ARMA models by removing the assumptions of causality and invertibility. The assumptions are not required under a non‐Gaussian setting for the identifiability of the model parameters in contrast to the Gaussian setting. We study M‐estimation for general ARMA processes with infinite variance, where the distribution of innovations is in the domain of attraction of a non‐Gaussian stable law. Following the approach taken by Davis et al. (1992) and Davis (1996) , we derive a functional limit theorem for random processes based on the objective function, and establish asymptotic properties of the M‐estimator. We also consider bootstrapping the M‐estimator and extend the results of Davis & Wu (1997) to the present setting so that statistical inferences are readily implemented. Simulation studies are conducted to evaluate the finite sample performance of the M‐estimation and bootstrap procedures. An empirical example of financial time series is also provided.     

On moment stability properties for a class of state-dependent stochastic networks

We consider a class of stochastic networks with state-dependent arrival and service rates. The state dependency is described via multi-dimensional birth/death processes, where the birth/death rates are dependent upon the current population size in the system. Under the uniform (in state) stability condition, we establish several moment stability properties of the system:

• (i) 

  the existence of a moment generating function in a neighborhood of zero, with respect to the unique invariant measure of the state process;

• (ii) 

  the convergence of the expected value of unbounded functionals of the state process to the expectation under the invariant measure, at an exponential rate;

• (iii) 

  uniform (in time and initial condition) estimates on exponential moments of the process;

• (iv) 

  growth estimates of polynomial moments of the process as a function of the initial conditions.

Our approach provides elementary proofs of these stability properties without resorting to the convergence of the scaled process to a stable fluid limit model.     

Stochastic comparison of generalized combined risk processes

Usually, two different types of shock models (extreme and cumulative shock models) are employed to model the dynamic risk processes. In extreme shock models, only the impact of the current fatal shock is usually taken into account, whereas, in cumulative shock models, the impact of the preceding shocks is accumulated as well. However, in practice, the effect of the corresponding shock can be realized in those two ways in one model (i.e., it can be fatal or, otherwise it is accumulated). This observation justifies the consideration of a ‘combined shock model’ in the risk modeling and analysis. In this paper, we generalize the study of the dynamic risk processes that were previously considered in the literature. The main theme of this paper is to find the optimal allocation policies for the generalized combined risk processes via the stochastic comparisons of survival functions. It will be seen that the obtained results hold for ‘general counting processes’ of shocks. In addition, we consider the problem of maximizing a gain function under certain risks and obtain reasonable decisions based on a variability measure. Furthermore, the meaningful explanations for the results on the policy ordering will be provided.     

A minimum asymptotic mean squared error controller for an IMA(1,1) noise process with a starting offset, and its resetting design

In a discrete-part manufacturing process, the noise is often described by an IMA(1,1) process and the pure unit delay transfer function is used as the feedback controller to adjust it. The optimal controller for this process is the well-known minimum mean square error (MMSE) controller. The starting level of the IMA(1,1) model is assumed to be on target when it starts. Considering such an impractical assumption, we adopt the starting offset. Since the starting offset is not observable, the MMSE controller does not exist. An alternative to the MMSE controller is the minimum asymptotic mean square error controller, which makes the long-run mean square error minimum.Another concern in this article is the un-stability of the controller, which may produce high adjustment costs and/or may exceed the physical bounds of the process adjustment. These practical barriers will prevent the controller to adjust the process properly. To avoid this dilemma, a resetting design is proposed. That is, the resetting procedure in use of the controller is to adjust the process according to the controller when it remains within the reset limit, and to reset the process, otherwise.The total cost for the manufacturing process is affected by the off-target cost, the adjustment cost, and the reset cost. Proper values for the reset limit are selected to minimize the average cost per reset interval (ACR) considering various process parameters and cost parameters. A time non-homogeneous Markov chain approach is used for calculating the ACR. The effect of adopting the starting offset is also studied here.     

A process capability index for discrete processes

Perakis and Xekalaki 2002, A process capability index that is based on the proportion of conformance. Journal of Statistical Computation and Simulation, 72(9), 707–718. introduced a process capability index that is based on the proportion of conformance of the process under study and has several appealing features. One of its advantages is that it can be used not only for continuous processes, as is the case with the majority of the indices considered in the literature, but also for discrete processes as well. In this article, the use of this index is investigated for discrete data under two alternative models, which are frequently considered in statistical process control. In particular, distributional properties and estimation of the index are considered for Poisson processes and for processes resulting in modeling attribute data. The performance of the suggested estimators and confidence limits is tested via simulation.     

On the specification and estimation of large scale simultaneous structural macroeconometric models

Summary This paper surveys the state of the art of the analysis and application of large scale structural simultaneous econometric models (SSEM). First, the importance of such models in empirical economics and especially for economic policy analysis is emphasized. We then focus on the methodological issues in the application of these models like questions about identification, nonstationarity of variables, adequate estimation of the parameters, and the inclusion of identities. In the light of the latest development in econometrics, we identify the main unsolved problems in this area, recommend a combined data-theory-driven procedure for the specification of such models, and give suggestions how one could overcome some of the indicated problems.     

Reliability of Supply Between Production Lines

ABSTRACT

We compare four strategies for ensuring a reliable just-in-time supply from a seat production line, which is prone to machine failure, to a car assembly line, which is assumed to operate at a constant speed over single shifts. The strategies are as follows: holding buffer stock; duplication of the least reliable machine; duplication of the production line as a stand-by; and running two production lines concurrently. Times between machine failures are assumed to have independent exponential distributions. A general distribution of repair times is allowed for by using phase-type representations. We show the stationary distribution for these models, and compare stationary distributions with average times within levels over shifts conditional on all machines working at the start of a shift. We compute moments of sojourn times within an arbitrary subset of states, which are relevant when cost is a non-linear function of downtime. We use first passage time results to obtain probabilities of line failure within a shift, and use these results to compare the four strategies.     

Assessing the bias of maximum likelihood estimates of contaminated garch models

It is well known that Gaussian maximum likelihood estimates of time series models are not robust. In this paper we prove this is also the case for the Generalized Autoregressive Conditional Heteroscedastic (GARCH) models. By expressing the Gaussian maximum likelihood estimates as Ψ estimates and by assuming the existence of a contaminated process, we prove they possess zero breakdown point and unbounded influence curves. By simulating GARCH processes under several proportions of contaminations we assess how much biased the maximum likelihood estimates may become and compare these results to a robust alternative. The t-student maximum likelihood estimates of GARCH models are also considered.     

Estimation methods for time‐dependent AUC models with survival data

The performance of clinical tests for disease screening is often evaluated using the area under the receiver‐operating characteristic (ROC) curve (AUC). Recent developments have extended the traditional setting to the AUC with binary time‐varying failure status. Without considering covariates, our first theme is to propose a simple and easily computed nonparametric estimator for the time‐dependent AUC. Moreover, we use generalized linear models with time‐varying coefficients to characterize the time‐dependent AUC as a function of covariate values. The corresponding estimation procedures are proposed to estimate the parameter functions of interest. The derived limiting Gaussian processes and the estimated asymptotic variances enable us to construct the approximated confidence regions for the AUCs. The finite sample properties of our proposed estimators and inference procedures are examined through extensive simulations. An analysis of the AIDS Clinical Trials Group (ACTG) 175 data is further presented to show the applicability of the proposed methods. The Canadian Journal of Statistics 38:8–26; 2010 © 2009 Statistical Society of Canada     

DYNAMIC FACTOR MODELS

This paper introduces nonlinear dynamic factor models for various applications related to risk analysis. Traditional factor models represent the dynamics of processes driven by movements of latent variables, called the factors. Our approach extends this setup by introducing factors defined as random dynamic parameters and stochastic autocorrelated simulators. This class of factor models can represent processes with time varying conditional mean, variance, skewness and excess kurtosis. Applications discussed in the paper include dynamic risk analysis, such as risk in price variations (models with stochastic mean and volatility), extreme risks (models with stochastic tails), risk on asset liquidity (stochastic volatility duration models), and moral hazard in insurance analysis.

We propose estimation procedures for models with the marginal density of the series and factor dynamics parameterized by distinct subsets of parameters. Such a partitioning of the parameter vector found in many applications allows to simplify considerably statistical inference. We develop a two- stage Maximum Likelihood method, called the Finite Memory Maximum Likelihood, which is easy to implement in the presence of multiple factors. We also discuss simulation based estimation, testing, prediction and filtering.     

Collapsibility of regression coefficients and its extensions

Collapsibility with respect to a measure of association implies that the measure of association can be obtained from the marginal model. We first discuss model collapsibility and collapsibility with respect to regression coefficients for linear regression models. For parallel regression models, we give simple and different proofs of some of the known results and obtain also certain new results. For random coefficient regression models, we define (average) A$A$-collapsibility and obtain conditions under which it holds. We consider Poisson regression and logistic regression models also, and derive conditions for collapsibility and A$A$-collapsibility, respectively. These results generalize some of the results available in the literature. Some suitable examples are also discussed.     

MCMC Bayesian Estimation in FIEGARCH Models

Bayesian inference for fractionally integrated exponential generalized autoregressive conditional heteroscedastic (FIEGARCH) models using Markov chain Monte Carlo (MCMC) methods is described. A simulation study is presented to assess the performance of the procedure, under the presence of long-memory in the volatility. Samples from FIEGARCH processes are obtained upon considering the generalized error distribution (GED) for the innovation process. Different values for the tail-thickness parameter ν are considered covering both scenarios, innovation processes with lighter (ν > 2) and heavier (ν < 2) tails than the Gaussian distribution (ν = 2). A comparison between the performance of quasi-maximum likelihood (QML) and MCMC procedures is also discussed. An application of the MCMC procedure to estimate the parameters of a FIEGARCH model for the daily log-returns of the S&P500 U.S. stock market index is provided.     

Bivariate degradation modelling with marginal heterogeneous stochastic processes

In this paper, we consider that the degradation of two performance characteristics of a product can be modelled by stochastic processes and jointly by copula functions, but different stochastic processes govern the behaviour of each performance characteristic (PC) degradation. Different heterogeneous and homogeneous models are presented considering copula functions and different combinations of the most used stochastic processes in degradation analysis as marginal distributions. This is an important aspect to consider because the behaviour of the degradation of each PC may be different in its nature. As the joint distributions of the proposed models result in complex distributions, the estimation of the parameters of interest is performed via MCMC. A simulation study is performed to compare heterogeneous and homogeneous models. In addition, the proposed models are implemented to crack propagation data of two terminals of an electronic device, and some insights are provided about the product reliability under heterogeneous models.