arXiv subjects

Muneya Matsui

Publications and source records attributed to Muneya Matsui.

35 records · Page 2Linked to original sources

arXiv2017

Arrival times of Cox process with independent increment with application to prediction problems

Properties of arrival times are studied for a Cox process with independent (and stationary) increments. Under a reasonable setting the directing random measure is shown to take over independent (and stationary) increments of the process, from which the sets of arrival times and their numbers in disjoint intervals are proved to be independent (and stationary). Moreover, we derive the exact joint distribution of these quantities with Gamma random measure, whereas for a general random measure the method of calculation is presented. Based on the derived properties we consider prediction problems for the shot noise process with Cox process arrival times which trigger additive processes off. We obtain a numerically tractable expression for the predictor which works reasonably in view of numerical experiments.

math.PR↗

arXiv2017

Log-convexity and the cycle index polynomials with relation to compound Poisson distributions

We extend the exponential formula by Bender and Canfield (1996), which relates log-concavity and the cycle index polynomials. The extension clarifies the log-convexity relation. The proof is by noticing the property of a compound Poisson distribution together with its moment generating function. We also give a combinatorial proof of extended "log-convex part" referring Bender and Canfield's approach, where the formula by Bruijn and Erdös (1953) is additionally exploited. The combinatorial approach yields richer structural results more than log-convexity. Furthermore, we consider normal and binomial convolutions of sequences which satisfy the exponential formula. The operations generate interesting examples which are not included in well known laws about log-concavity/convexity.

math.CO↗

arXiv2017

Componentwise different tail solutions for bivariate stochastic recurrence equations -- with application to GARCH(1,1) processes --

We study bivariate stochastic recurrence equations (SREs) motivated by applications to GARCH(1,1) processes. If coefficient matrices of SREs have strictly positive entries, then the Kesten result applies and it gives solutions with regularly varying tails. Moreover, the tail indices are the same for all coordinates. However, for applications, this framework is too restrictive. We study SREs when coefficients are triangular matrices and prove that the coordinates of the solution may exhibit regularly varying tails with different indices. We also specify each tail index together with its constant. The results are used to characterize regular variations of bivariate stationary GARCH(1,1) processes.

math.PR↗

arXiv2017

Distance covariance for stochastic processes

The distance covariance of two random vectors is a measure of their dependence. The empirical distance covariance and correlation can be used as statistical tools for testing whether two random vectors are independent. We propose an analogs of the distance covariance for two stochastic processes defined on some interval. Their empirical analogs can be used to test the independence of two processes.

math.ST↗

arXiv2016

Applications of Distance Correlation to Time Series

The use of empirical characteristic functions for inference problems, including estimation in some special parametric settings and testing for goodness of fit, has a long history dating back to the 70s (see for example, Feuerverger and Mureika (1977), Csorgo (1981a,1981b,1981c), Feuerverger (1993)). More recently, there has been renewed interest in using empirical characteristic functions in other inference settings. The distance covariance and correlation, developed by Szekely and Rizzo (2009) for measuring dependence and testing independence between two random vectors, are perhaps the best known illustrations of this. We apply these ideas to stationary univariate and multivariate time series to measure lagged auto- and cross-dependence in a time series. Assuming strong mixing, we establish the relevant asymptotic theory for the sample auto- and cross-distance correlation functions. We also apply the auto-distance correlation function (ADCF) to the residuals of an autoregressive processes as a test of goodness of fit. Under the null that an autoregressive model is true, the limit distribution of the empirical ADCF can differ markedly from the corresponding one based on an iid sequence. We illustrate the use of the empirical auto- and cross-distance correlation functions for testing dependence and cross-dependence of time series in a variety of different contexts.

math.ST↗

arXiv2015

Prediction of components in random sums

We consider predictions of the random number and the magnitude of each iid component in a random sum based on its distributional structure, where only a total value of the sum is available and where iid random components are non-negative. The problem is motivated by prediction problems in a Poisson shot noise process. In the context, although conditional moments are best possible predictors under the mean square error, only a few special cases have been investigated because of numerical difficulties. We replace the prediction problem of the process with that of a random sum, which is more general, and establish effective numerical procedures. The methods are based on conditional technique together with the Panjer recursion and the Fourier transform. In view of numerical experiments, procedures work reasonably. An application in the compound mixed Poisson process is also suggested.

math.PR↗

arXiv2015

The extremogram and the cross-extremogram for a bivariate GARCH(1,1) process

In this paper, we derive some asymptotic theory for the extremogram and cross-extremogram of a bivariate GARCH(1,1) process. We show that the tails of the components of a bivariate GARCH(1,1) process may exhibit power law behavior but, depending on the choice of the parameters, the tail indices of the components may differ. We apply the theory to 5-minute return data of stock prices and foreign exchange rates. We judge the fit of a bivariate GARCH(1,1) model by considering the sample extremogram and cross-extremogram of the residuals. The results are in agreement with the iid hypothesis of the two-dimensional innovations sequence. The cross-extremograms at lag zero have a value significantly distinct from zero. This fact points at some strong extremal dependence of the components of the innovations.

math.ST↗

arXiv2014

Fractional absolute moments of heavy tailed distributions

Several convenient methods for calculation of fractional absolute moments are given with application to heavy tailed distributions. We use techniques of fractional differentiation to obtain formulae for $E[|X-μ|^γ]$ with $1<γ<2$ and $μ\in\mathbb{R}$. The main focus is on heavy tailed distributions, several examples are given with analytical expressions of fractional absolute moments. As applications, we calculate the fractional moment errors for both prediction and parameter estimation problems.

math.ST↗

arXiv2013

Prediction in a non-homogeneous Poisson cluster model

A non-homogeneous Poisson cluster model is studied, motivated by insurance applications. The Poisson center process which expresses arrival times of claims, triggers off cluster member processes which correspond to number or amount of payments. The cluster member process is an additive process. Given the past observations of the process we consider expected values of future increments and their mean squared errors, aiming the application in claims reserving problems. Our proposed process can cope with non-homogeneous observations such as the seasonality of claims arrival or the reducing property of payment processes, which are unavailable in the former models where both center and member processes are time homogeneous. Hence results presented in this paper are significant extensions toward applications. We also give numerical examples to show how non-homogeneity appears in predictions.

math.PR↗

arXiv2012

Distributions of exponential integrals of independent increment processes related to generalized gamma convolutions

It is known that in many cases distributions of exponential integrals of Levy processes are infinitely divisible and in some cases they are also selfdecomposable. In this paper, we give some sufficient conditions under which distributions of exponential integrals are not only selfdecomposable but furthermore are generalized gamma convolution. We also study exponential integrals of more general independent increment processes. Several examples are given for illustration.

math.ST↗

arXiv2009

Classes of infinitely divisible distributions on R^d related to the class of selfdecomposable distributions

This paper studies new classes of infinitely divisible distributions on R^d. Firstly, the connecting classes with a continuous parameter between the Jurek class and the class of selfdecomposable distributions are revisited. Secondly, the range of the parameter is extended to construct new classes and characterizations in terms of stochastic integrals with respect to Levy processes are given. Finally, the nested subclasses of those classes are discussed and characterized in two ways: One is by stochastic integral representations and another is in terms of Levy measures.

math.PR↗

arXiv2008

Generalized fractional Ornstein-Uhlenbeck processes

We introduce an extended version of the fractional Ornstein-Uhlenbeck (FOU) process where the integrand is replaced by the exponential of an independent Lévy process. We call the process the generalized fractional Ornstein-Uhlenbeck (GFOU) process. Alternatively, the process can be constructed from a generalized Ornstein-Uhlenbeck (GOU) process using an independent fractional Brownian motion (FBM) as integrator. We show that the GFOU process is well-defined by checking the existence of the integral included in the process, and investigate its properties. It is proved that the process has a stationary version and exhibits long memory. We also find that the process satisfies a certain stochastic differential equation. Our underlying intention is to introduce long memory into the GOU process which has short memory without losing the possibility of jumps. Note that both FOU and GOU processes have found application in a variety of fields as useful alternatives to the Ornstein-Uhlenbeck (OU) process.

math.PR↗

arXiv2006

Integral representations of one dimensional projections for multivariate stable densities

We consider the numerical evaluation of one dimensional projections of general multivariate stable densities introduced by Abdul-Hamid and Nolan (1998). In their approach higher order derivatives of one dimensional densities are used, which seem to be cumbersome in practice. Furthermore there are some difficulties for even dimensions. In order to overcome these difficulties we obtain the explicit finite-interval integral representation of one dimensional projections for all dimensions. For this purpose we utilize the imaginary part of complex integration, whose real part corresponds to the derivative of the one dimensional inversion formula. We also give summaries on relations between various parameterizations of stable multivariate density and its one dimensional projection.

math.ST↗

arXiv2006

Skewness and kurtosis as locally best invariant tests of normality

Consider testing normality against a one-parameter family of univariate distributions containing the normal distribution as the boundary, e.g., the family of $t$-distributions or an infinitely divisible family with finite variance. We prove that under mild regularity conditions, the sample skewness is the locally best invariant (LBI) test of normality against a wide class of asymmetric families and the kurtosis is the LBI test against symmetric families. We also discuss non-regular cases such as testing normality against the stable family and some related results in the multivariate cases.

math.ST↗

arXiv2006

Goodness-of-Fit Tests for Symmetric Stable Distributions -- Empirical Characteristic Function Approach

We consider goodness-of-fit tests of symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard symmetric stable distribution with the characteristic exponent $α$ estimated from the data. We treat $α$ as an unknown parameter, but for theoretical simplicity we also consider the case that $α$ is fixed. For estimation of parameters and the standardization of data we use maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE) which minimizes the weighted integral. We derive the asymptotic covariance function of the characteristic function process with parameters estimated by MLE and EISE. For the case of MLE, the eigenvalues of the covariance function are numerically evaluated and asymptotic distribution of the test statistic is obtained using complex integration. Simulation studies show that the asymptotic distribution of the test statistics is very accurate. We also present a formula of the asymptotic covariance function of the characteristic function process with parameters estimated by an efficient estimator for general distributions.

math.ST↗

arXiv2005

Fisher Information Matrix of General Stable Distributions Close to the Normal Distribution

We investigate behavior of the Fisher information matrix of general stable distributions. DuMouchel (1975, 1983) proved that the Fisher information of characteristic exponent αdiverges to infinity as αapproaches 2. Nagaev and Shkol'nik (1988) made more detailed analysis and derived asymptotic behavior of the Fisher information matrix of αdiverging to infinity as αapproaches 2 in the symmetric case. Extending their work in this paper we have obtained behavior of the Fisher information matrix of general stable distributions as αapproaches 2 by detailed study of behavior of the corresponding density and its score functions. We clarify the limiting values of the 4*4 Fisher Information matrix with respect to the location μ, the scale σ, the characteristic exponent αand the skewness parameter β.

math.ST↗

arXiv2004

Some improvements in numerical evaluation of symmetric stable density and its derivatives

We propose improvements in numerical evaluation of symmetric stable density and its partial derivatives with respect to the parameters. They are useful for more reliable evaluation of maximum likelihood estimator and its standard error. Numerical values of the Fisher information matrix of symmetric stable distributions are also given. Our improvements consist of modification of the method of Nolan (1997) for the boundary cases, i.e., in the tail and mode of the densities and in the neighborhood of the Cauchy and the normal distributions.

math.ST↗