1.
Empirical spectral processes for locally stationary time series - Dahlhaus, Rainer; Polonik, Wolfgang
A time-varying empirical spectral process indexed by classes of functions is defined for locally stationary time series. We derive weak convergence in a function space, and prove a maximal exponential inequality and a Glivenko–Cantelli-type convergence result. The results use conditions based on the metric entropy of the index class. In contrast to related earlier work, no Gaussian assumption is made. As applications, quasi-likelihood estimation, goodness-of-fit testing and inference under model misspecification are discussed. In an extended application, uniform rates of convergence are derived for local Whittle estimates of the parameter curves of locally stationary time series models.
2.
Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency - Dümbgen, Lutz; Rufibach, Kaspar
We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least (log(n)/n)1/3 and typically (log(n)/n)2/5, whereas the difference between the empirical and estimated distribution function vanishes with rate op(n−1/2) under certain regularity assumptions.
3.
Adaptive estimation of linear functionals in the convolution model and applications - Butucea, C.; Comte, F.
We consider the model Zi=Xi+ɛi, for i.i.d. Xi’s and ɛi’s and independent sequences (Xi)i∈ℕ and (ɛi)i∈ℕ. The density fɛ of ɛ1 is assumed to be known, whereas the one of X1, denoted by g, is unknown. Our aim is to estimate linear functionals of g, 〈ψ, g〉 for a known function ψ. We propose a general estimator of 〈ψ, g〉 and study the rate of convergence of its quadratic risk as a function of the smoothness of g, fɛ and ψ. Different contexts with dependent data, such as stochastic volatility and AutoRegressive Conditionally Heteroskedastic models, are also considered. An estimator...
4.
Nonparametric two-sample tests for increasing convex order - Baringhaus, Ludwig; Grübel, Rudolf
Given two independent samples of non-negative random variables with unknown distribution functions F and G, respectively, we introduce and discuss two tests for the hypothesis that F is less than or equal to G in increasing convex order. The test statistics are based on the empirical stop-loss transform, critical values are obtained by a bootstrap procedure. It turns out that for the resampling a size switching is necessary. We show that the resulting tests are consistent against all alternatives and that they are asymptotically of the given size α. A specific feature of the problem is the behavior of the...
5.
Optimal designs for dose-finding experiments in toxicity studies - Dette, Holger; Pepelyshev, Andrey; Wong, Weng Kee
We construct optimal designs for estimating fetal malformation rate, prenatal death rate and an overall toxicity index in a toxicology study under a broad range of model assumptions. We use Weibull distributions to model these rates and assume that the number of implants depend on the dose level. We study properties of the optimal designs when the intra-litter correlation coefficient depends on the dose levels in different ways. Locally optimal designs are found, along with robustified versions of the designs that are less sensitive to misspecification in the initial values of the model parameters. We also report efficiencies of commonly...
6.
Approximation of the distribution of a stationary Markov process with application to option pricing - Pagès, Gilles; Panloup, Fabien
We build a sequence of empirical measures on the space $\mathbb{D}(\mathbb{R}_{+},\mathbb{R}^{d})$ of ℝd-valued cadlag functions on ℝ+ in order to approximate the law of a stationary ℝd-valued Markov and Feller process (Xt). We obtain some general results on the convergence of this sequence. We then apply them to Brownian diffusions and solutions to Lévy-driven SDE’s under some Lyapunov-type stability assumptions. As a numerical application of this work, we show that this procedure provides an efficient means of option pricing in stochastic volatility models.
7.
On continuous-time autoregressive fractionally integrated moving average processes - Tsai, Henghsiu
In this paper, we consider a continuous-time autoregressive fractionally integrated moving average (CARFIMA) model, which is defined as the stationary solution of a stochastic differential equation driven by a standard fractional Brownian motion. Like the discrete-time ARFIMA model, the CARFIMA model is useful for studying time series with short memory, long memory and antipersistence. We investigate the stationarity of the model and derive its covariance structure. In addition, we derive the spectral density function of a stationary CARFIMA process.
8.
Discrete approximation of a stable self-similar stationary increments process - Dombry, C.; Guillotin-Plantard, N.
The aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known about the context in which such processes can arise. To our knowledge, discretization and convergence theorems are available only in the case of stable Lévy motions and fractional Brownian motions. This paper yields new results in this direction. Our main result is the convergence of the random rewards schema first introduced by Cohen and Samorodnitsky, which we consider in a more general setting. Strong relationships with...
9.
Nonparametric estimation for Lévy processes from low-frequency observations - Neumann, Michael H.; Reiß, Markus
We suppose that a Lévy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the Lévy–Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific C2-criterion this estimator is rate-optimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line.
10.
Random systems of polynomial equations. The expected number of roots under smooth analysis - Armentano, Diego; Wschebor, Mario
We consider random systems of equations over the reals, with m equations and m unknowns Pi(t)+Xi(t)=0, t∈ℝm, i=1, …, m, where the Pi’s are non-random polynomials having degrees di’s (the “signal”) and the Xi’s (the “noise”) are independent real-valued Gaussian centered random polynomial fields defined on ℝm, with a probability law satisfying some invariance properties.
¶
For each i, Pi and Xi have degree di.
¶
The problem is the behavior of the number of roots for large m. We prove that under specified conditions on the relation signal over noise, which imply that in a certain sense this relation is neither too large...
11.
Asymptotics for diffusion first-passage laws - McGill, Paul
By using Berg’s Abelian theorem, Csáki extracted a sharp asymptotic estimate for a diffusion first-passage law from its Laplace transform. We extend the method and give a simple formulation for Itô diffusions.
12.
Multicolor urn models with reducible replacement matrices - Bose, Arup; Dasgupta, Amites; Maulik, Krishanu
Consider the multicolored urn model where, after every draw, balls of the different colors are added to the urn in a proportion determined by a given stochastic replacement matrix. We consider some special replacement matrices which are not irreducible. For three- and four-color urns, we derive the asymptotic behavior of linear combinations of the number of balls. In particular, we show that certain linear combinations of the balls of different colors have limiting distributions which are variance mixtures of normal distributions. We also obtain almost sure limits in certain cases in contrast to the corresponding irreducible cases, where only weak...
13.
The asymptotic structure of nearly unstable non-negative integer-valued AR(1) models - Drost, Feike C.; van den Akker, Ramon; Werker, Bas J.M.
This paper considers non-negative integer-valued autoregressive processes where the autoregression parameter is close to unity. We consider the asymptotics of this ‘near unit root’ situation. The local asymptotic structure of the likelihood ratios of the model is obtained, showing that the limit experiment is Poissonian. To illustrate the statistical consequences we discuss efficient estimation of the autoregression parameter and efficient testing for a unit root.
14.
Test for tail index change in stationary time series with Pareto-type marginal distribution - Kim, Moosup; Lee, Sangyeol
The tail index, indicating the degree of fatness of the tail distribution, is an important component of extreme value theory since it dominates the asymptotic distribution of extreme values such as the sample maximum. In this paper, we consider the problem of testing for a change in the tail index of time series data. As a test, we employ the cusum test and investigate its null limiting distribution. Further, we derive the null limiting distribution of the cusum test based on the residuals from autoregressive models. Simulation results are provided for illustration.
15.
A cluster identification framework illustrated by a filtering model for earthquake occurrences - Wu, Zhengxiao
A general dynamical cluster identification framework including both modeling and computation is developed. The earthquake declustering problem is studied to demonstrate how this framework applies.
¶
A stochastic model is proposed for earthquake occurrences that considers the sequence of occurrences as composed of two parts: earthquake clusters and single earthquakes. We suggest that earthquake clusters contain a “mother quake” and her “offspring.” Applying the filtering techniques, we use the solution of filtering equations as criteria for declustering. A procedure for calculating maximum likelihood estimations (MLE’s) and the most likely cluster sequence is also presented.
16.
Tie-respecting bootstrap methods for estimating distributions of sets and functions of eigenvalues - Hall, Peter; Lee, Young K.; Park, Byeong U.; Paul, Debashis
Bootstrap methods are widely used for distribution estimation, although in some problems they are applicable only with difficulty. A case in point is that of estimating the distributions of eigenvalue estimators, or of functions of those estimators, when one or more of the true eigenvalues are tied. The m-out-of-n bootstrap can be used to deal with problems of this general type, but it is very sensitive to the choice of m. In this paper we propose a new approach, where a tie diagnostic is used to determine the locations of ties, and parameter estimates are adjusted accordingly. Our tie diagnostic...
17.
Toward optimal multistep forecasts in non-stationary autoregressions - Ing, Ching-Kang; Lin, Jin-Lung; Yu, Shu-Hui
This paper investigates multistep prediction errors for non-stationary autoregressive processes with both model order and true parameters unknown. We give asymptotic expressions for the multistep mean squared prediction errors and accumulated prediction errors of two important methods, plug-in and direct prediction. These expressions not only characterize how the prediction errors are influenced by the model orders, prediction methods, values of parameters and unit roots, but also inspire us to construct some new predictor selection criteria that can ultimately choose the best combination of the model order and prediction method with probability 1. Finally, simulation analysis confirms the satisfactory finite sample...
18.
Subsampling needlet coefficients on the sphere - Baldi, P.; Kerkyacharian, G.; Marinucci, D.; Picard, D.
In a recent paper, we analyzed the properties of a new kind of spherical wavelets (called needlets) for statistical inference procedures on spherical random fields; the investigation was mainly motivated by applications to cosmological data. In the present work, we exploit the asymptotic uncorrelation of random needlet coefficients at fixed angular distances to construct subsampling statistics evaluated on Voronoi cells on the sphere. We illustrate how such statistics can be used for isotropy tests and for bootstrap estimation of nuisance parameters, even when a single realization of the spherical random field is observed. The asymptotic theory is developed in detail...
19.
Portfolio optimization when expected stock returns are determined by exposure to risk - Lindberg, Carl
It is widely recognized that when classical optimal strategies are applied with parameters estimated from data, the resulting portfolio weights are remarkably volatile and unstable over time. The predominant explanation for this is the difficulty of estimating expected returns accurately. In this paper, we modify the n stock Black–Scholes model by introducing a new parametrization of the drift rates. We solve Markowitz’ continuous time portfolio problem in this framework. The optimal portfolio weights correspond to keeping 1/n of the wealth invested in stocks in each of the n Brownian motions. The strategy is applied out-of-sample to a large data set....
20.
Estimating the joint distribution of independent categorical variables via model selection - Durot, C.; Lebarbier, E.; Tocquet, A.-S.
Assume one observes independent categorical variables or, equivalently, one observes the corresponding multinomial variables. Estimating the distribution of the observed sequence amounts to estimating the expectation of the multinomial sequence. A new estimator for this mean is proposed that is nonparametric, non-asymptotic and implementable even for large sequences. It is a penalized least-squares estimator based on wavelets, with a penalization term inspired by papers of Birgé and Massart. The estimator is proved to satisfy an oracle inequality and to be adaptive in the minimax sense over a class of Besov bodies. The method is embedded in a general framework which...
1.
