On the mode-change problem for random measures

The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for sufficiently general set-indexed random processes, namely for random measures. A method is shown for solving this problem both in the general form and for the intensity of compound Poisson random measures. The results obtained are novel for the change-point problem, too.     

The effect of memory on functional large deviations of infinite moving average processes

The large deviations of an infinite moving average process with exponentially light tails are very similar to those of an i.i.d. sequence as long as the coefficients decay fast enough. If they do not, the large deviations change dramatically. We study this phenomenon in the context of functional large, moderate and huge deviation principles.     

A normal inverse Gaussian model for a risky asset with dependence

We present a new construction of the normal inverse Gaussian (NIG) fractal activity time model for a risky asset. The construction uses superpositions of diffusion processes and allows for specified exact NIG marginal distributions of the returns and flexible and tractable dependence structure including short or long range dependence. In the case of finite superposition, the fractal activity time is asymptotically self-similar, which is a desired feature seen in practice. The support for the distributional and dependence features of the risky asset model is provided by the data of currency exchange rates.     

On the chernoff-savage theorem for dependent sequences

Summary Given a sequence of ϕ-mixing random variables not necessarily stationary, a Chernoff-Savage theorem for two-sample linear rank statistics is proved using the Pyke-Shorack [5] approach based on weak convergence properties of empirical processes in an extended metric. This result is a generalization of Fears and Mehra [4] in that the stationarity is not required and that the condition imposed on the mixing numbers is substantially relaxed. A similar result is shown to hold for strong mixing sequences under slightly stronger conditions on the mixing numbers. Research partially supported by the National Research Council of Canada under Grant No. A-3954.     

Existence and uniqueness of stationary Lévy-driven CARMA processes

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining a general Lévy-driven continuous-parameter ARMA process with index set R$\mathbb{R}$ are determined. Under these conditions the solution is shown to be unique and an explicit expression is given for the process as an integral with respect to the background driving Lévy process. The results generalize results obtained earlier for second-order processes and for processes defined by the Ornstein–Uhlenbeck equation.     

Asymptotic behavior of central order statistics from stationary processes

In this paper, we show that central order statistics from strictly stationary and ergodic sequences are strongly consistent estimators of population quantiles provided that the quantiles are unique. We generalize this result to strictly stationary but not necessarily ergodic sequences. We also describe three types of possible asymptotic behavior of central order statistics in the case when the corresponding population quantile is not unique. We give applications of the presented results to linear processes with both absolutely continuous and discrete innovations.     

Nonparametric comparison of regression functions

In this work, we provide a new methodology for comparing regression functions m₁ and m₂ from two samples. Since apart from smoothness no other (parametric) assumptions are required, our approach is based on a comparison of nonparametric estimators and of m₁ and m₂, respectively. The test statistics incorporate weighted differences of and computed at selected points. Since the design variables may come from different distributions, a crucial question is where to compare the two estimators. As our main results we obtain the limit distribution of (properly standardized) under the null hypothesis H₀:m₁=m₂ and under local and global alternatives. We are also able to choose the weight function so as to maximize the power. Furthermore, the tests are asymptotically distribution free under H₀ and both shift and scale invariant. Several such ’s may then be combined to get Maximin tests when the dimension of the local alternative is finite. In a simulation study we found out that our tests achieve the nominal level and already have excellent power for small to moderate sample sizes.     

On estimating the memory for finitarily Markovian processes

Finitarily Markovian processes are those processes for which there is a finite K () such that the conditional distribution of X₁ given the entire past is equal to the conditional distribution of X₁ given only . The least such value of K is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of K, both in the backward sense that we have just described and in the forward sense, where one observes successive values of {Xn} for n?0 and asks for the least value K such that the conditional distribution of Xn+1 given is the same as the conditional distribution of Xn+1 given . We allow for finite or countably infinite alphabet size.     

Fixed-gain estimation in continuous time

We prove that the parameter estimation error of continuous-time linear stochastic systems that is obtained in connection with a fixed-gain estimation method can be written as a stochastic integral plus a residual term, the moments of which are of order+o(1) where is the forgetting factor.     

State space models on special manifolds

This paper concerns modeling time series observations in state space forms considered on the Stiefel and Grassmann manifolds. We develop a state space model relating the time series observations to a sequence of unobserved state or parameter matrices assuming the matrix Langevin noise processes on the Stiefel manifolds. We show a Bayes method for estimating the state matrices by the posterior modes. We consider a further extended state space model where two sequences of unobserved state matrices are involved. A simple state space model on the Grassmann manifolds with matrix Langevin noise processes is also investigated.     

Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series

We develop a doubly spectral representation of a stationary functional time series, and study the properties of its empirical version. The representation decomposes the time series into an integral of uncorrelated frequency components (Cramér representation), each of which is in turn expanded in a Karhunen–Loève series. The construction is based on the spectral density operator, the functional analogue of the spectral density matrix, whose eigenvalues and eigenfunctions at different frequencies provide the building blocks of the representation. By truncating the representation at a finite level, we obtain a harmonic principal component analysis of the time series, an optimal finite dimensional reduction of the time series that captures both the temporal dynamics of the process, as well as the within-curve dynamics. Empirical versions of the decompositions are introduced, and a rigorous analysis of their large-sample behaviour is provided, that does not require any prior structural assumptions such as linearity or Gaussianity of the functional time series, but rather hinges on Brillinger-type mixing conditions involving cumulants.     

Nonparametric time series prediction: A semi-functional partial linear modeling

There is a recent interest in developing new statistical methods to predict time series by taking into account a continuous set of past values as predictors. In this functional time series prediction approach, we propose a functional version of the partial linear model that allows both to consider additional covariates and to use a continuous path in the past to predict future values of the process. The aim of this paper is to present this model, to construct some estimates and to look at their properties both from a theoretical point of view by means of asymptotic results and from a practical perspective by treating some real data sets. Although the literature on the use of parametric or nonparametric functional modeling is growing, as far as we know, this is the first paper on semiparametric functional modeling for the prediction of time series.     

Strong consistency of density estimation by orthogonal series methods for dependent variables with applications

Among several widely use methods of nonparametric density estimation is the technique of orthogonal series advocated by several authors. For such estimate when the observations are assumed to have been taken from strong mixing sequence in the sense of Rosenblatt [7] we study strong consistency by developing probability inequality for bounded strongly mixing random variables. The results obtained are then applied to two estimates of the functional Δ(f)=∫f ² (x)dx were strong consistency is established. One of the suggested two estimates of Δ(f) was recently studied by Schuler and Wolff [8] in the case of independent and identically distributed observations where they established consistency in the second mean of the estimate. Research supported in part by the National Research Council of Canada and in part by McMaster University Research Board. Now at Memphis State University, Memphis, Tennessee 38152, U.S.A.     

Asymptotic results on a class of adaptive multi-treatment designs

The play-the-winner (PW) rule is an important method in clinical trials where patients can be assigned to one of the two treatments. In the PW rule, the probability of the next patient to be assigned to a particular treatment only depends on the response of the current patient. In this paper, we consider a general kind of PW rule for multi-treatment adaptive designs, in which the probability that a treatment is assigned to the next patient depends upon both the response of the previous patient and an estimated parameter, e.g., the observed success rate. Using this kind of adaptive designs, more information of previous stages are used to update the model at each stage, and more patients may be assigned to better treatments. The strong consistency and the asymptotic normality are established for the allocation proportions.     

Partial sum process to check regression models with multiple correlated response: With an application for testing a change-point in profile data

We consider regression models with multiple correlated responses for each design point. Under the null hypothesis, a linear regression is assumed. For the least-squares residuals of this linear regression, we establish the limit of the partial sums. This limit is a projection on a certain subspace of the reproducing Kernel Hilbert space of a multivariate Brownian motion. Based on this limit, we propose a significance test of Kolmogorov-Smirnov type to test the null hypothesis and show that this result can be used to study a change-point problem in the case of linear profile data (panel data). We compare our proposed method, which does not rely on any distributional assumptions, with the likelihood ratio test in a simulation study.     

Characterization of the partial autocorrelation function of nonstationary time series

The second order properties of a process are usually characterized by the autocovariance function. In the stationary case, the parameterization by the partial autocorrelation function is relatively recent. We extend this parameterization to the nonstationary case. The advantage of this function is that it is subject to very simple constraints in comparison with the auto- covariance function which must be nonnegative definite. As in the stationary case, this parameterization is well adapted to autoregressive models or to the identification of deterministic processes.     

Multivariate semi-logistic distributions

Three new multivariate semi-logistic distributions (denoted by MSL⁽¹⁾, MSL⁽²⁾, and GMSL respectively) are studied in this paper. They are more general than Gumbel’s (1961) [1] and Arnold’s (1992) [2] multivariate logistic distributions. They may serve as competitors to these commonly used multivariate logistic distributions. Various characterization theorems via geometric maximization and geometric minimization procedures of the three MSL⁽¹⁾, MSL⁽²⁾ and GMSL are proved. The particular multivariate logistic distribution used in the multiple logistic regression model is introduced. Its characterization theorem is also studied. Finally, some further research work on these MSL is also presented. Some probability density plots and contours of the bivariate MSL⁽¹⁾, MSL⁽²⁾ as well as Gumbel’s and Arnold’s bivariate logistic distributions are presented in the Appendix.     

Testing multivariate normality in incomplete data of small sample size

In longitudinal studies with small samples and incomplete data, multivariate normal-based models continue to be a powerful tool for analysis. This has included a broad scope of biomedical studies. Testing the assumption of multivariate normality (MVN) is critical. Although many methods are available for testing normality in complete data with large samples, a few deal with the testing in small samples. For example, Liang et al. (J. Statist. Planning and Inference 86 (2000) 129) propose a projection procedure for testing MVN for complete-data with small samples where the sample sizes may be close to the dimension. To our knowledge, no statistical methods for testing MVN in incomplete data with small samples are yet available. This article develops a test procedure in such a setting using multiple imputations and the projection test. To utilize the incomplete data structure in multiple imputation, we adopt a noniterative inverse Bayes formulae (IBF) sampling procedure instead of the iterative Gibbs sampling to generate iid samples. Simulations are performed for both complete and incomplete data when the sample size is less than the dimension. The method is illustrated with a real study on an anticancer drug.     

Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing

Let be a fractional ARIMA(p,d,q) process with partial autocorrelation function α(·). In this paper, we prove that if d∈(−1/2,0) then |α(n)|∼|d|/n as n→∞. This extends the previous result for the case 0<d<1/2.