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We propose three new characterizations and corresponding distance-based weighted test criteria for the two-sample problem, and for testing symmetry and independence with multivariate data. All quantities have the common feature of involving characteristic functions, and it is seen that these quantities are intimately related to some earlier methods, thereby generalizing them. The connection rests on a special choice of the weight function involved. Equivalent expressions of the distances in terms of densities are given as well as a Bayesian interpretation of the weight function is involved. The asymptotic behavior of the tests is investigated both under the null hypothesis and under alternatives, and affine invariant versions of the test criteria are suggested. Numerical studies are conducted to examine the performances of the criteria. It is shown that the normal weight function, which is the hitherto most often used, is seriously suboptimal. The procedures are biased in the sense that the corresponding test criteria degenerate in high dimension and hence a bias correction is required as the dimension tends to infinity.

We consider model-free monitoring procedures for strict stationarity of a given time series. The new criteria are formulated as L2-type statistics incorporating the multivariate empirical characteristic function. Asymptotic results are obtained for the closed-end scenario and Monte Carlo results are presented. The new methods are also employed in order to test for possible stationarity breaks in time-series data from the financial sector.

In this article we introduce a three-parameter extension of the bivariate exponential-geometric (BEG) law (Kozubowski and Panorska, 2005). We refer to this new distribution as bivariate gamma-geometric (BGG) law. A bivariate random vector $(X,N)$ follows BGG law if $N$ has geometric distribution and $X$ may be represented (in law) as a sum of $N$ independent and identically distributed gamma variables, where these variables are independent of $N$. Statistical properties such as moment generation and characteristic functions, moments and variance-covariance matrix are provided. The marginal and conditional laws are also studied. We show that BBG distribution is infinitely divisible, just as BEG model is. Further, we provide alternative representations for the BGG distribution and show that it enjoys a geometric stability property. Maximum likelihood estimation and inference are discussed and a reparametrization is proposed in order to obtain orthogonality of the parameters. We present an application to the real data set where our model provides a better fit than BEG model. Our bivariate distribution induces a bivariate L\'evy process with correlated gamma and negative binomial processes, which extends the bivariate L\'evy motion proposed by Kozubowski et al. (2008). The marginals of our L\'evy motion are mixture of gamma and negative binomial processes and we named it ${BMixGNB}$ motion. Basic properties such as stochastic self-similarity and covariance matrix of the process are presented. The bivariate distribution at fixed time of our ${BMixGNB}$ process is also studied and some results are derived, including a discussion about maximum likelihood estimation and inference., Comment: 22 pages, 9 figures

In this article we propose two consistent hypothesis tests of L2-type for weakly dependent observations based on the empirical characteristic function. We consider a symmetry test and a goodness-of-fit test for the marginal distribution of a time series. The asymptotic behaviour under the null as well as fixed and certain local alternatives is investigated. Since the limit distributions of the test statistics depend on unknown parameters in a complicated way, we suggest to apply certain parametric bootstrap methods in order to determine critical values of the tests.

We provide general results on the consistency of certain bootstrap methods applied to degree-2 degenerate statistics of U-type and V-type. While it follows from well known results that the original statistic converges in distribution to a weighted sum of centred chi-squared random variables, we use a coupling idea of Dehling and Mikosch to show that the bootstrap counterpart converges to the same distribution. The result is applied to a goodness-of-fit test based on the empirical characteristic function.

We devise a new method of estimating a distribution in a deconvolution model with panel data and an unknown distribution of the additive errors. We prove strong consistency under a minimal condition concerning the zero sets of the involved characteristic functions.

This paper studies the exact distributions of the MLEs of the regression coefficient matrices in a GMANOVA–MANOVA model with normal error. The unique conditions for linear functions of the MLEs of regression coefficient matrices are presented, and the exact density functions or characteristic functions for these linear functions are derived.

The joint density for a particular trivariate chi-squared distribution given by the diagonal elements of a complex Wishart matrix is derived. This distribution has applications in the processing of multilook synthetic aperture radar data. The expression for the density is in the form of an infinite series that converges rapidly and is simple and fast to compute. The expression is shown to reduce to known forms for a number of special cases and is validated by simulation. The characteristic function is also derived and used to relate joint moments of the trivariate distribution to the parameters of the density function.

In this paper we consider the problem of testing the hypothesis about the sub-mean vector. For this propose, the asymptotic expansion of the null distribution of Rao's U-statistic under a general condition is obtained up to order of n-1. The same problem in the k-sample case is also investigated. We find that the asymptotic distribution of generalized U-statistic in the k-sample case is identical to that of the generalized Hotelling's T2 distribution up to n-1. A simulation experiment is carried out and its results are presented. It shows that the asymptotic distributions have significant improvement when comparing with the limiting distributions both in the small sample case and the large sample case. It also demonstrates the equivalence of two testing statistics mentioned above.

T. Gneiting (1998, J. Multivariate Analysis64, 131–147) proved a relation between the primitives of the classes Φd(2) and Φd(1) of 2- and 1-symmetric characteristic functions on Rd, respectively. We will give a straightforward proof of his relation, answering a question of his. To do this we use the calculus of generalized hypergeometric functions.

Radial positive definite functions are of importance both as the characteristic functions of spherically symmetric probability distributions, and as the correlation functions of isotropic random fields. The Euclid's hat functionhn(‖x‖),x∈Rn, is the self-convolution of an indicator function supported on the unit ball in Rn. This function is evidently radial and positive definite, and so are its scale mixtures that form the classHn. Our main results characterize the classesHn,n⩾1, andH∞=∩n⩾1Hn. This leads to an analogue of Pólya's criterion for radial functions on Rn,n⩾2: Ifϕ:[0,∞)→R is such thatϕ(0)=1,ϕ(t) is continuous, limt→∞ϕ(t)=0, and(−1)kdkdtk[−ϕ′(t)]is convex fork=[(n−2)/2], the greatest integer less than or equal to (n−2)/2, thenϕ(‖x‖) is a characteristic function in Rn. Along the way, side results on multiply monotone and completely monotone functions occur. We discuss the relations ofHnto classes of radial positive definite functions studied by Askey (Technical Report No. 1262, Math. Res. Center, Univ. of Wisconsin–Madison), Mittal (Pacific J. Math.64(1976), 517–538), and Berman (Pacific J. Math.78(1978), 1–9), and close with hints at applications in geostatistics.

Ann-dimensional random vector is said to have anα-symmetric distribution,α>0, if its characteristic function is of the formϕ((|u1|α+…+|un|α)1/α). We study the classesΦn(α) of all admissible functionsϕ:[0, ∞)→R. It is known that members ofΦn(2) andΦn(1) are scale mixtures of certain primitivesΩnandωn, respectively, and we show thatωnis obtained fromΩ2n−1byn−1 successive integrations. Consequently, curious relations between 1- and 2- (or spherically) symmetric distributions arise. An analogue of Askey's criterion gives a partial solution to a question of D. St. P. Richards: Ifϕ(0)=1,ϕis continuous, limt→∞ϕ(t)=0, andϕ(2n−2)(t) is convex, thenϕ∈Φn(1). The paper closes with various criteria for the unimodality of anα-symmetric distribution.

The Kotz-type distributions form an important class of multivariate elliptical distributions. These distributions are studied in Fang et al. [Symmetric Multivariate and Related Distributions, Chap. 3.2. Chapman and Hall, London]. In the particular case when the shape parameters s equals 1, Iyengar and Tong [Sankhya Ser. A51 13-29 ] determined explicitly the characteristic function of the distributions. Streit [C.R. Math. Rep. Acad. Sci. Canada13 121-124] derived a general formula for the characteristic functions valid for all s >12. In the present paper, the structure of the characteristic functions for a Kotz-type multivariate distribution for all values of the parameters is obtained. The relationship to the characteristic function of a lognormal distribution is noted.

A general easily checkable Cochran theorem is obtained for a normal random operator Y . This result does not require that the covariance, Σ Y , of Y is nonsingular or is of the usual form A ⊗ Σ ; nor does it assume that the mean, μ, of Y is equal to zero. Indeed, { Y ′ W i Y } (with nonnegative definite W i ′s) is a family of independent Wishart random operators Y ′ W i Y of parameter ( m i , Σ , λ i ) if and only if for some nonnegative definite A and for all i ≠ j : (a)( W i ⊗ I )( Σ Y − A ⊗ Σ )( W i ⊗ I ) = 0; (b) AW i AW i = AW i , r ( AW i ) = m i , (c) λ i = μ′ W i μ = μ′ W i AW i μ; and (d) ( W i ⊗ I ) Σ Y ( W j ⊗ I ) = 0. The usual multivariate versions of Cochran′s theorem are contained in a special case of our result where Σ Y = A ⊗ Σ . The A in our version of Cochran′s theorem can actually be constructed from Σ , Σ Y , and the sum of the W i ′s.

For any distribution belonging to the Kagan class a general formula expressing its characteristic function in a neighbourhood of the origin in terms of the marginal characteristic functions is obtained. Also some applications are given.

The validity of the one-term Edgeworth expansion is proved for the multivariate mean of a random sample drawn without replacement under a limiting non-latticeness condition on the population. The theorem is applied to deduce the one-term expansion for the univariate statistics which can be expressed in a certain linear plus quadratic form. An application of the results to the theory of bootstrap is mentioned. A one-term expansion is also proved in the univariate lattice case.

The aim of this paper is to describe the equivalence classes (e.c.) of the following equivalence relation on the set Pn of probability measures on Rn: μ ∼ ν if μ ∗ μ̄ = ν ∗ ν̄, where μ̄(A) = μ(− A) for Borel sets A ⊂ Rn. μ ∼ ν iff |φμ(t)| = |φν(t)| for t ∈ Rn, where φμ is the characteristic function of μ. Sufficient conditions for the triviality of an e.c. (i.e., up to translations the e.c. consists of a unique pair (μ, μ̄)) are given. Gaussian measures are the only infinitely divisible probability measures whose e.c.'s are trivial. Distinct symmetrical probability measures with equal moments of all orders may have characteristic functions with the same absolute value. The corresponding problem is also considered on Rn/Zn.

Let F̂n be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n from a (smooth) distribution function F. Sufficient conditions are given for the central limit theorem to hold for the target statistic F̂n(Un) where {Un} is a sequence of U-statistics.

For fixed p (0 ≤ p ≤ 1), let { L 0 , R 0 } = {0, 1} and X 1 be a uniform random variable over { L 0 , R 0 }. With probability p let { L 1 , R 1 } = { L 0 , X 1 } or = { X 1 , R 0 } according as X 1 ≥ 1 2 (L 0 + R 0 ) or 1 2 (L 0 + R 0 ) ; with probability 1 − p let { L 1 , R 1 } = { X 1 , R 0 } or = { L 0 , X 1 } according as X 1 ≥ 1 2 (L 0 + R 0 ) or 1 2 (L 0 + R 0 ) , and let X 2 be a uniform random variable over { L 1 , R 1 }. For n ≥ 2, with probability p let { L n , R n } = { L n − 1 , X n } or = { X n , R n − 1 } according as X n ≥ 1 2 (L n − 1 + R n − 1 ) or 1 2 (L n − 1 + R n − 1 ) , with probability 1 − p let { L n , R n } = { X n , R n − 1 } or = { L n − 1 , X n } according as X n ≥ 1 2 (L n − 1 + R n − 1 ) or 1 2 (L n − 1 + R n − 1 ) , and let X n + 1 be a uniform random variable over { L n , R n }. By this iterated procedure, a random sequence { X n } n ≥ 1 is constructed, and it is easy to see that X n converges to a random variable Y p (say) almost surely as n → ∞. Then what is the distribution of Y p ? It is shown that the Beta, (2, 2) distribution is the distribution of Y 1 ; that is, the probability density function of Y 1 is g ( y ) = 6 y (1 − y ) I 0,1 ( y ). It is also shown that the distribution of Y 0 is not a known distribution but has some interesting properties (convexity and differentiability).

Series expansions and second-order asymptotic expansions are obtained for the characteristic functions of three statistics useful for testing the independence of two multivariate regression equations with different design matrices.

An n -dimensional random vector X is said (Cambanis, S., Keener, R., and Simons, G. (1983). J. Multivar. Anal. , 13 213–233) to have an α-symmetric distribution, α > 0, if its characteristic function is of the form φ (| ξ 1 | α + … + | ξ n | α ). Using the Radon transform, integral representations are obtained for the density functions of certain absolutely continuous α-symmetric distributions. Series expansions are obtained for a class of apparently new special functions which are encountered during this study. The Radon transform is also applied to obtain the densities of certain radially symmetric stable distributions on R n . A new class of “zonally” symmetric stable laws on R n is defined, and series expansions are derived for their characteristic functions and densities.

Asymptotic expansions of the distributions of two test criteria concerning a covariance matrix are derived under local alternatives in terms of noncentral χ 2 variates, and under the fixed alternative in terms of standard normal distribution function and its derivatives, respectively. Some numerical comparisons with the likelihood ratio criteria are made with these test criteria.

For any class Q of distributions on Rd, let L(Q) be the class of limit distributions of bn−1(X1 + … + Xn) − an, where {Xn} are independent Rd-valued random variables, each with distribution in Q, bn > 0, an ∈ Rd, and {bn−1Xj} is a null array. When Q is the class of all distributions on Rd. L(Q) = L0 is the usual class L. Define Lm = L(Lm−1) and L∞ = ∩Lm. It is shown that this definition of Lm is equivalent to Urbanik's definition. Description of Lévy measures and representation of characteristic functions of members in these classes are given. Other characterizations of the class L∞ are made. Conditions for convergence in terms of the representations are given. Continuity properties of distributions of class L are studied.

This paper deals with multivariate stable distributions. Press has given an explicit algebraic representation of characteristic functions of such distributions [I. Multiwariate Analysis 2 (1972), 444-4621. We present counter-examples and correct proofs of some of the statements of Press. The properties of multivariate stable distributions, connected with the spectral measure r, present in the expression of the characteristic function, are studied. In this paper stable distributions in k-dimensional Euclidean space R, , k > 2 are considered. Press [6] gave an explicit algebraic representation of characteristic functions (ch.f.) of multivariate stable distributions. Unfortunately, in the proof of the main result in [6] there are some mistakes, and even the result is wrong. Only part of the multivariate stable distributions has ch.f. which can be expressed by means of such algebraic representation. We present counter-examples and correct proofs of some of the statements of Press. The properties of stable distributions, connected with spectral measure r, present in the expression of ch.f. are studied.

Let {X(t), 0 ≤ t ≤ T} and {Y(t), 0 ≤ t ≤ T} be two additive processes over the interval [0, T] which, as measures over D[0, T], are absolutely continuous with respect to each other. Let μX and μY be the measures over D[0, T] determined by the two processes. The characteristic function of ln(dμXdμY) with respect to μY is obtained in terms of the determining parameters of the two processes.

Publisher Summary This chapter discusses an asymptotically precise estimate of the accuracy of Gaussian approximation in Hubert space. It presents an asymptotically precise estimate of type, n(a,r)=|P(|Sn−a

Let f be an infinitely divisible characteristic function without normal factor. We establish some sufficient conditions for f to belong to the class of characteristic functions without indecomposable factor. These conditions concern the support of the measure appearing in the Levy-Khintchine representation of f .

Three theorems are obtained that relate the asymptotic behavior of a distribution function with the behavior of its characteristic function at the origin. These theorems generalize one dimensional results that have been obtained by the author and by others.

Let A(t) be a complex Wishart process defined in terms of the M×N complex Gaussian matrix X(t) by A(t)=X(t)X(t)H. The covariance matrix of the columns of X(t) is Σ. If X(t), the underlying Gaussian process, is a correlated process over time, then we have dependence between samples of the Wishart process. In this paper, we study the joint statistics of the Wishart process at two points in time, t1, t2, where t1

The (univariate) t-distribution and symmetric V.G. distribution are competing models [D.S. Madan, E. Seneta, The variance gamma (V.G.) model for share market returns, J. Business 63 (1990) 511–524; T.W. Epps, Pricing Derivative Securities, World Scientific, Singapore, 2000 (Section 9.4)] for the distribution of log-increments of the price of a financial asset. Both result from scale-mixing of the normal distribution. The analogous matrix variate distributions and their characteristic functions are derived in the sequel and are dual to each other in the sense of a simple Duality Theorem. This theorem can thus be used to yield the derivation of the characteristic function of the t-distribution and is the essence of the idea used by Dreier and Kotz [A note on the characteristic function of the t-distribution, Statist. Probab. Lett. 57 (2002) 221–224]. The present paper generalizes the univariate ideas in Section 6 of Seneta [Fitting the variance-gamma (VG) model to financial data, stochastic methods and their applications, Papers in Honour of Chris Heyde, Applied Probability Trust, Sheffield, J. Appl. Probab. (Special Volume) 41A (2004) 177–187] to the general matrix generalized inverse gaussian (MGIG) distribution.

This paper deals with uniform rates of convergence for the empirical distribution function and the empirical characteristic function for a broad class of stationary linear processes. In particular, the class X(n) = Σi=0∞ δ(i) z(n−1) is considered under the conditions that (a) the disturbances z(n) are independent and identically distributed with a finite first absolute moment, (b) the distribution function F of X(n) has bounded density, and (c) the parameters δ(i) are bounded in absolute value by some function g which satisfies Σi=1∞ ig(i) < ∞. It is proved that the empirical distribution function F̂(x) based on X(n), n = 1, …, N, converges to the theoretical distribution, uniformly in x, at a rate O(N−12logN) a.s. For the empirical characteristic function a uniform (over a certain range of the argument) rate of convergence result is proved. A pointwise central limit theorem is also obtained. The above theorems constitute versions (weaker in part) of theorems that already exist in the i.i.d. case and for certain sequences of dependent random variables.

In this paper we obtain an asymptotic expansion for the distribution of Hotelling'sT2-statisticT2under nonnormality when the sample size is large. In the derivation we find an explicit Edgeworth expansion of the multivariatet-statistic. Our method is to use the Edgeworth expansion and to expand the characteristic function ofT2.

The theory of compact group actions on locally compact abelian groups provides a unifying theory under which different invariance conditions studied in several contexts by a number of statisticians are subsumed as special cases. For example, Schoenberg's characterization of radially symmetric characteristic functions on R n is extended to this general context and the integral representations are expressed in terms of the generalized spherical Bessel functions of Gross and Kunze. These same Bessel functions are also used to obtain a variant of the Levy-Khinchine formula of Parthasarathy, Ranga Rao, and Varadhan appropriate to invariant distributions.