Estimation in dynamic regression with an integrated process

This paper deals with a dynamic regression model y_t = αy_t−1 + βz_t + u_t, where z_t is an integrated process of order one abbreviated as z_t ∼ I(1). Generally speaking, nonstandard asymptotic theory is required to investigate asymptotic properties of statistics related to an integrated process and the asymptotic results are very different from standard ones. There are two distinctive properties in nonstandard asymptotics: the so-called ‘super-consistency’ or T-consistency (where T is a sample size) and the weak convergence to a functional of the Wiener process. In spite of z_t being involved in our model, however, it is shown that our asymptotic results are the same as in the standard asymptotics in classical dynamic regression models, or if the disturbance u_t is serially correlated the OLS estimators of α and β have √T-inconsistency. This is due to the cointegration between y_t−1 and z_t. Although this point was clarified by Park and Phillips (1989) in a general context, we examine this explicitly through our specific model and connect the standard asymptotic theory with the nonstandard one in our case. Furthermore we investigate the limiting properties of other statistics such as t-ratio, the Durbin-Watson test and h-test. We also propose a consistent estimator of α and β by making use of Durbin's 2-step method. Finally, we carry out simulation studies which support our theoretical results.     

Inverse prediction for heteroscedastic response using mixed models software

Distributions of a response y (height, for example) differ with values of a factor t (such as age). Given a response y_* for a subject of unknown t_*, the objective of inverse prediction is to infer the value of t_* and to provide a defensible confidence set for it. Training data provide values of y observed on subjects at known values of t. Models relating the mean and variance of y to t can be formulated as mixed (fixed and random) models in terms of sets of functions of t, such as polynomial spline functions. A confidence set on t_* can then be had as those hypothetical values of t for which y_* is not detected as an outlier when compared to the model fit to the training data. With nonconstant variance, the p-values for these tests are approximate. This article describes how versatile models for this problem can be formulated in such a way that the computations can be accomplished with widely available software for mixed models, such as SAS PROC MIXED. Coverage probabilities of confidence sets on t_* are illustrated in an example.     

Partly linear models on Riemannian manifolds

In partly linear models, the dependence of the response y on (x ^T, t) is modeled through the relationship y=x ^T β+g(t)+?, where ? is independent of (x ^T, t). We are interested in developing an estimation procedure that allows us to combine the flexibility of the partly linear models, studied by several authors, but including some variables that belong to a non-Euclidean space. The motivating application of this paper deals with the explanation of the atmospheric SO₂ pollution incidents using these models when some of the predictive variables belong in a cylinder. In this paper, the estimators of β and g are constructed when the explanatory variables t take values on a Riemannian manifold and the asymptotic properties of the proposed estimators are obtained under suitable conditions. We illustrate the use of this estimation approach using an environmental data set and we explore the performance of the estimators through a simulation study.     

On the orthogonal designs of order 40

Using a new method we construct all 17 remaining (unresolved for over 20 years) full orthogonal designs of order 40 in three variables. This implies that all full orthogonal designs OD(2^t5;x,y, 2^t5−x−y) exist for all t⩾3. The last two remaining orthogonal designs of order 40 in 2 variables are obtained as a special case of two of these designs.     

A characterization of invariant power-series abundance distributions

Consider a population the individuals in which can be classified into groups. Let y, the number of individuals in a group, be distributed according to a probability function f(y;ø_o) where the functional form f is known. The random variable y cannot be observed directly, and hence a random sample of groups cannot be obtained. Consider a random sample of N individuals from the population. Suppose the N individuals are distributed into S groups with x₁, x₂, …, x_S representatives respectively. The random variable x, the number of individuals in a group in the sample, will be a fraction of its population counterpart y, and the distributions of x and y need not have the same functional form. If the two random variables x and y have the same functional form for their distributions, then the particular common distribution is called an invariant abundance distribution. The paper provides a characterization of invariant abundance distributions in the class of power-series distributions.     

Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Comparing the BLUEs Under Two Linear Models

In this article, we consider two linear models, ?₁ = {y, X β, V ₁} and ?₂ = {y, X β, V ₂}, which differ only in their covariance matrices. Our main focus lies on the difference of the best linear unbiased estimators, BLUEs, of X β under these models. The corresponding problems between the models {y, X β, I _n } and {y, X β, V}, i.e., between the OLSE (ordinary least squares estimator) and BLUE, are pretty well studied. Our purpose is to review the corresponding considerations between the BLUEs of X β under ?₁ and ?₂. This article is an expository one presenting also new results.     

Mixed covering arrays of strength three with few factors

Covering arrays with mixed alphabet sizes, or mixed covering arrays, are useful generalizations of covering arrays that are motivated by software and network testing. Suppose that there are k factors, and that the ith factor takes values or levels from a set G_i of size g_i. A run is an assignment of an admissible level to each factor. A mixed covering array, MCA(N;t,k,g₁g₂…g_k), is a collection of N runs such that for any t distinct factors, i₁,i₂,…,i_t, every t-tuple from G_i1×G_i2×?×G_it occurs in factors i₁,i₂,…,i_t in at least one of the N runs. When g=g₁=g₂=?=g_k, an MCA(N;t,k,g₁g₂…g_k) is a CA(N;t,k,g). The mixed covering array number, denoted by MCAN(t,k,g₁g₂…g_k), is the minimum N for which an MCA(N;t,k,g₁g₂…g_k) exists. In this paper, we focus on the constructions of mixed covering arrays of strength three. The numbers MCAN(3,k,g₁g₂…g_k) are determined for all cases with k∈{3,4} and for most cases with k∈{5,6}.     

On the construction of orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set

Complete sets of orthogonal F-squares of order n = s^p, where g is a prime or prime power and p is a positive integer have been constructed by Hedayat, Raghavarao, and Seiden (1975). Federer (1977) has constructed complete sets of orthogonal F-squares of order n = 4t, where t is a positive integer. We give a general procedure for constructing orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set, where n is not necessarily a prime or prime power. In particular, we show how to construct sets of orthogonal F-squares of order n = 2s^p, where s is a prime or prime power and p is a positive integer. These sets are shown to be near complete and approach complete sets as s and/or p become large. We have also shown how to construct orthogonal arrays by these methods. In addition, the best upper bound on the number t of orthogonal F(n, λ₁), F(n, λ₂), …, F(n, λ₁) squares is given.     

On double hysteretic heteroskedastic model

ABSTRACT

This paper proposes a hysteretic autoregressive model with GARCH specification and a skew Student's t-error distribution for financial time series. With an integrated hysteresis zone, this model allows both the conditional mean and conditional volatility switching in a regime to be delayed when the hysteresis variable lies in a hysteresis zone. We perform Bayesian estimation via an adaptive Markov Chain Monte Carlo sampling scheme. The proposed Bayesian method allows simultaneous inferences for all unknown parameters, including threshold values and a delay parameter. To implement model selection, we propose a numerical approximation of the marginal likelihoods to posterior odds. The proposed methodology is illustrated using simulation studies and two major Asia stock basis series. We conduct a model comparison for variant hysteresis and threshold GARCH models based on the posterior odds ratios, finding strong evidence of the hysteretic effect and some asymmetric heavy-tailness. Versus multi-regime threshold GARCH models, this new collection of models is more suitable to describe real data sets. Finally, we employ Bayesian forecasting methods in a Value-at-Risk study of the return series.     

Estimation of an autoregressive semiparametric model with exogenous variables

In many autoregressive relationships, there are observed external influences. This paper deals with the estimation of the multivariate model X_t+1= φ(X_t,…,X_t−r+1) + ψ(Y_t) + ε_t, where φ(·) is an unknown nonlinear function, ∫ the exogenous variable concerning ψ(·). Two cases are considered: ψ(·) is linear ψ(Y_t) = AY_t, where A is an unknown parameter, and ψ(·) the nonlinear function corresponding to a series expansion. In the latter situation, the method of estimation is ‘seminonparametric’. We first isolate and estimate parametrically the exogenous part, and then estimate nonparametrically the endogenous part ψ(·).     

Asymptotic Inferences for an AR(1) Model with a Change Point and Possibly Infinite Variance

The basic model in this paper is an AR(1) model with a structural break in the AR parameter β at an unknown time k₀. That is, y_t = β₁y_{t ? 1}I{t ? k₀} + β₂y_{t ? 1}I{t > k₀} + ?_t, t = 1, 2, ???, T, where I{ · } denotes the indicator function. Suppose |β₁| < 1, |β₂| < 1, and {?_t, t ? 1} is a sequence of i.i.d. random variables which are in the domain of attraction of the normal law with zero mean and possibly infinite variance, then the limiting distributions for the least squares estimators of β₁ and β₂ are studied in the present paper, which extend some results in Chong (2001 Chong, T.L. (2001). Structural change in AR(1) models. Econometric Theory 17:87–155.[Crossref], [Web of Science ®] , [Google Scholar]).     

MODELING THE LIFETIME OF LONGITUDINAL ELEMENTS

In the study of the stochastic behaviour of the lifetime of an element as a function of its length, it is often observed that the failure time (or lifetime) decreases as the length increases. In probabilistic terms, such an idea can be expressed as follows. Let T be the lifetime of a specimen of length x, so the survival function, which denotes the probability that an element of length x survives till time t, will be given by S_T (t, x) = P(T > t/α(x), where α(x) is a monotonically decreasing function. In particular, it is often assumed that T has a Weibull distribution. In this paper, we propose a generalization of this Weibull model by assuming that the distribution of T is Generalized gamma (GG). Since the GG model contains the Weibull, Gamma and Lognormal models as special and limiting cases, a GG regression model is an appropriate tool for describing the size effect on the lifetime and for selecting among the embedded models. Maximum likelihood estimates are obtained for the GG regression model with α(x) = cx^b . As a special case this provide an alternative to the usual approach to estimation for the GG distribution which involves reparametrization. Related parametric inference issues are addressed and illustrated using two experimental data sets. Some discussion of censored data is also provided.     

Simulation of a stationary autoregression: A characterization of the normal distribution

Simulating a stationary AR(p), X_t = ∑^p_i=1α_iX_t−i + Z_t, when the innovations {Z_t} are assumed to be i.i.d. is straightforward. Starting the process in the stationary state, however, requires generation of (X₁,X₂,…,X_p) from the stationary p-dimensional distribution. When Z_t is normal this may be achieved by generating X_i as a linear function of X₁,X₂,…,X_i−1 and an independent normal variate for i = 2,3,…, p. It is shown that the ability to initialize a stationary AR(p) in this way characterizes the normal distribution.     

Computing first-order sensitivity indices with contribution to the sample mean plot

In this paper, we investigate the use of the contribution to the sample mean plot (CSM plot) as a graphical tool for sensitivity analysis (SA) of computational models. We first provide an exact formula that links, for each uncertain model input X_j, the CSM plot C_j(·) with the first-order variance-based sensitivity index S_j. We then build a new estimate for S_j using polynomial regression of the CSM plot. This estimation procedure allows the computation of S_j from given data, without any SA-specific design of experiment. Numerical results show that this new S_j estimate is efficient for large sample sizes, but that at small sample sizes it does not compare well with other S_j estimation techniques based on given data, such as the effective algorithm for computing global sensitivity indices method or metamodel-based approaches.     

Model-based clustering, classification, and discriminant analysis via mixtures of multivariate t-distributions

The last decade has seen an explosion of work on the use of mixture models for clustering. The use of the Gaussian mixture model has been common practice, with constraints sometimes imposed upon the component covariance matrices to give families of mixture models. Similar approaches have also been applied, albeit with less fecundity, to classification and discriminant analysis. In this paper, we begin with an introduction to model-based clustering and a succinct account of the state-of-the-art. We then put forth a novel family of mixture models wherein each component is modeled using a multivariate t-distribution with an eigen-decomposed covariance structure. This family, which is largely a t-analogue of the well-known MCLUST family, is known as the tEIGEN family. The efficacy of this family for clustering, classification, and discriminant analysis is illustrated with both real and simulated data. The performance of this family is compared to its Gaussian counterpart on three real data sets.     

Optimal designs for approximating the path of a stochastic process

We consider a centered stochastic process {X(t):t ∈ T} with known and continuous covariance function. On the basis of observations X(t₁), …, X(t_n) we approximate the whole path by orthogonal projection and measure the performance of the chosen design d = (t₁, …, t_n)′ by the corresponding mean squared L²-distance. For covariance functions on T² = [0, 1]², which satisfy a generalized Sacks-Ylvisaker regularity condition of order zero, we construct asymptotically optimal sequences of designs. Moreover, we characterize the achievement of a lower error bound, given by Micchelli and Wahba (1981), and study the question of whether this bound can be attained.     

Nonparametric regression estimation in a null recurrent time series

We derive an asymptotic theory of nonparametric estimation for a time series regression model Z_t=f(X_t)+W_t, where {X_t} and {Z_t} are observed nonstationary processes, and {W_t} is an unobserved stationary process. The class of nonstationary processes allowed for {X_t} is a subclass of the class of null recurrent Markov chains. This subclass contains the random walk, unit root processes and nonlinear processes. The process {W_t} is assumed to be linear and stationary.     

On the quantile process based on the autoregressive residuals

Let {X_t} be the stationary AR(p) process satisfying the difference equation X_t=β₁X_t−1 + … + β_pX_t−p+ε_t, where {ε_t} is a sequence of iid random variables with mean zero and finite variance. Motivated by a goodness of fit test on the true errors {ε_t}, we are led to study the asymptotic behavior of the quantile process based on residuals (the residual quantile process). Particularly, we concentrate on the deviations between the residual quantile process and the empirical process based on the true errors. In this asymptotic study, it is shown that the deviations converge to zero in probability uniformly over certain intervals with specific order as sample size increases. Here, these intervals are allowed to vary with the sample size n and converge to the unit interval as n goes to infinity. Then, based on our result and the strong approximation result of Csörgö and Révész (1978), we propose a goodness of fit test statistic of which limiting distribution is the same as of a functional form of a standard Brownian bridge.