Curve registration by local regression

Functional data analysis involves the extension of familiar statistical procedures such as principal‐components analysis, linear modelling and canonical correlation analysis to data where the raw observation is a function x, (t). An essential preliminary to a functional data analysis is often the registration or alignment of salient curve features by suitable monotone transformations h_i(t). In effect, this conceptualizes variation among functions as being composed of two aspects: phase and amplitude. Registration aims to remove phase variation as a preliminary to statistical analyses of amplitude variation. A local nonlinear regression technique is described for identifying the smooth monotone transformations h_i, and is illustrated by analyses of simulated and actual data.     

First and Second Derivatives in Time Series Classification Using DTW

In our previous work, we developed a new distance function based on a derivative and showed that our algorithm is effective. In contrast to well-known measures from the literature, our approach considers the general shape of a time series rather than standard distance of function (value) comparison. The new distance was used in classification with the nearest neighbor rule. Now we improve on our previous technique by adding the second derivative. In order to provide a comprehensive comparison, we conducted a set of experiments, testing effectiveness on 47 time series datasets from a wide variety of application domains. Our experiments show that this new method provides a significantly more accurate classification on the examined datasets.     

A Generalization of Geometric Brownian Motion with Applications

Although geometric Brownian motion has a great variety of applications, it can not cover all the random phenomena. The purpose of this article is to propose a model that generalizes geometric Brownian motion. We present some interesting applications of this model in financial engineering and statistical inferences for the unknown parameters.     

Elastic statistical analysis of interval-valued time series

We investigate the problem of statistical analysis of interval-valued time series data – two nonintersecting real-valued functions, representing lower and upper limits, over a period of time. Specifically, we pay attention to the two concepts of phase (or horizontal) variability and amplitude (or vertical) variability, and propose a phase-amplitude separation method. We view interval-valued time series as elements of a function (Hilbert) space and impose a Riemannian structure on it. We separate phase and amplitude variability in observed interval functions using a metric-based alignment solution. The key idea is to map an interval to a point in R2, view interval-valued time series as parameterized curves in R2, and borrow ideas from elastic shape analysis of planar curves, including PCA, to perform registration, summarization, analysis, and modeling of multiple series. The proposed phase-amplitude separation provides a new way of PCA and modeling for interval-valued time series, and enables shape clustering of interval-valued time series. We apply this framework to three different applications, including finance, meteorology and physiology, proves the effectiveness of proposed methods, and discovers some underlying patterns in the data. Experimental results on simulated data show that our method applies to the point-valued time series.     

Alternative forms of fractional Brownian motion

It is pointed out that two contradictory definitions of fractional Brownian motion are well-established, one prevailing in the probabilistic literature, the other in the econometric literature. Each is associated with a different definition of nonstationary fractional time series, arising in functional limit theorems based on such series. These various definitions have occasionally led to some confusion. The paper discusses the definitions and attempts a clarification.     

Bounds for the expected value of the stochastic Divisia's price index

In this article, we estimate bounds for the expected value of the stochastic Divisia's price index, that is, we assume that prices and quantities of the given commodities are stochastic processes with continuous time. We consider some special case of the stochastic model in which prices and quantities are described by the geometric Brownian motion. It is shown that the precision of this estimation depends rather on the volatility of prices than quantities volatilities.     

Heuristic approach to some laws for Brownian motion

Csàki and Vincze have shown that for an elementary tied-down random walk, the pair (maximum, instant of maximum) has the same law as (time spent in (0, 1/2), time spent above 1/2). Formal passage to the limit indicates that the former pair has for a Brownian bridge the same law as (local time at 0, duration of positivity). A quadrivariate density of Karatzas and Shreve and an equivalence for Brownian motion with drift follow.     

Asymptotic behavior of the grenander estimator at density flat regions

Over forty years ago, Grenander derived the MLE of a monotone decreasing density f with known mode. Prakasa Rao obtained the asymptotic distribution of this estimator at a fixed point x where f' (x) < 0. Here, we obtain the asymptotic distribution of this estimator at a fixed point x when f is constant and nonzero in some open neighborhood of x. This limiting distribution is expressible as the convolution of a closed-form density and a rescaled standard normal density. Groeneboom (1983) derived the aforementioned closed-form density and we provide an alternative, more direct derivation.     

Estimation for the three-parameter inverse gaussian distribution

The three-parameter inverse Gaussian distribution is defined and moment estimators and maximum likelihood estimators are obtained. The moment estimators are found in closed form and their asymprotic normality is proven. A sufficient condition is provided for the existence of the maximum likelihood estimators.     

Ruin problem of a two-dimensional fractional Brownian motion risk process

This paper investigates ruin probability and ruin time of a two-dimensional fractional Brownian motion risk process. The net loss process of an insurance company is modeled by a fractional Brownian motion. The two-dimensional fractional Brownian motion risk process models the surplus processes of an insurance and a reinsurance company, where the net loss is divided between them in some specified proportions. The ruin problem considered is that of the two-dimensional risk process first entering the negative quadrant, that is, the simultaneous ruin problem. We derive both asymptotics of the ruin probability and approximations of the scaled conditional ruin time as the initial capital tends to infinity.     

Minimum Contrast Estimation for Fractional Diffusions

Abstract.  When the Hurst coefficient of a fractional Brownian motion     is greater than 1/2 it is possible to define a stochastic integral with respect to     , as the pathwise limit of Riemann sums, and thus to consider pathwise solutions to fractional diffusion equations. In this paper, we consider the vanishing drift case and assume that the solution X _t is parameterized by θ in a compact parameter space Θ . Our main interest is the estimation of θ based on discrete time, but with very frequent observations. It is shown that the estimation problem in this context is locally asymptotically mixed normal. The asymptotic behaviour of a certain class of minimum contrast estimators is then studied and asymptotic efficiency is discussed.     

Seasonal Adjustment of the Weekly Monetary Aggregates: A Model-Based Approach

A procedure is developed for seasonally adjusting weekly time series, based on a composite of regression and time series models. The procedure is applied to some weekly U.S. money supply series currently seasonally adjusted by the Federal Reserve.     

Some results on residual life and inactivity time at random time

In this article, we enhance the study of residual life at random time (RLRT) and inactivity time at random time (ITRT). To this aim, first we provide some stochastic orderings results among ITRT in two-sample problems when they failed at two different random times. Then, we develop some sufficient conditions which lead to the stochastic comparisons of RLRT and ITRT based on variance residual life order. The results are expected to be useful in reliability theory, forensic science, queue theory, and actuarial science.     

Priors and component structures in autoregressive time series models

New approaches to prior specification and structuring in autoregressive time series models are introduced and developed. We focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series. These new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends, quasi-periodic subprocesses and high frequency residual noise components of observed series. The class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters. Analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series, as it does for predictions of future values. Posterior analysis involves easily implemented iterative simulation methods, developed and described here. One motivating field of application is climatology, where the evaluation of latent structure, especially quasi-periodic structure, is of critical importance in connection with issues of global climatic variability. We explore the analysis of data from the southern oscillation index, one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators.     

Total shift during the first passages of Markov-modulated Brownian motion with bilateral ph-type jumps: Formulas driven by the minimal solution matrix of a Riccati equation

This article describes our study of the total shift during the first passages (one-sided and two-sided exit times) of Markov-modulated Brownian motion with bilateral ph-type jumps, which is referred to as MMBM. The total shift is defined as the value of a so-called shift process at the first passage epochs of the MMBM. The shift process, introduced by Bean and O’Reilly, behaves like a continuous Markovian fluid process; that is, it increases or decreases linearly with slopes regulated by the underlying Markov process that determines the path of the MMBM. Hence, the notion of total shift, which includes the first passage times of the MMBM as special cases, is useful for describing various performance measures of systems modeled by the MMBM. In this article, we present formulas for the Laplace–Stieltjes transform matrices of the total shift during various first passages of the MMBM. In particular, a Riccati equation is derived so that a matrix associated with the Laplace–Stieltjes transform of the total shift during the first return time of the MMBM is its minimal non-negative solution matrix. With this solution matrix, the Laplace–Stieltjes transform matrices can be obtained without much additional work. Furthermore, it is shown that the Riccati equation satisfies the conditions for the Newton scheme to have quadratic convergence, which enables us to use algorithms with quadratic convergence, such as Newton’s method and the Stochastic Doubling Algorithm, to compute the presented matrix-driven formulas. For the analyses, we take an approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows with bilateral ph-type jumps, referred to as MMFF, that weakly converge to the MMBM. Another contribution of this article is that duality results are derived in relation to the MMBM, which is an extension of the duality theorems developed by Ahn and Ramaswami for an MMFF without a jump.     

Maximum likelihood estimation of linear continuous time long memory processes with discrete time data

Summary.  We develop a new class of time continuous autoregressive fractionally integrated moving average (CARFIMA) models which are useful for modelling regularly spaced and irregu-larly spaced discrete time long memory data. We derive the autocovariance function of a stationary CARFIMA model and study maximum likelihood estimation of a regression model with CARFIMA errors, based on discrete time data and via the innovations algorithm. It is shown that the maximum likelihood estimator is asymptotically normal, and its finite sample properties are studied through simulation. The efficacy of the approach proposed is demonstrated with a data set from an environmental study.