Estimation of location parameters for spherically symmetric distributions

Estimation of the location parameters of a p×1 random vector with a spherically symmetric distribution is considered under quadratic loss. The conditions of Brandwein and Strawderman [Ann. Statist. 19(1991) 1639-1650] under which estimators of the form dominate are (i) where -h is superharmonic, (ii) is nonincreasing in R, where has a uniform distribution in the sphere centered at with a radius R, and (iii) . In this paper, we not only drop their condition (ii) to show the dominance of over but also obtain a new bound for a which is sometimes better than that obtained by Brandwein and Strawderman. Specifically, the new bound of a is 0<a<[μ₁/(p²μ_-1)][1-(p-1)μ₁/(pμ_-1μ₂)]^-1 with for i=-1,1,2. The generalization to concave loss functions is also considered. Additionally, we investigate estimators of the location parameters when the scale is unknown and the observation contains a residual vector.     

A unified approach to testing for and against a set of linear inequality constraints in the product multinomial setting

A problem that is frequently encountered in statistics concerns testing for equality of multiple probability vectors corresponding to independent multinomials against an alternative they are not equal. In applications where an assumption of some type of stochastic ordering is reasonable, it is desirable to test for equality against this more restrictive alternative. Similar problems have been considered heretofore using the likelihood ratio approach. This paper aims to generalize the existing results and provide a unified technique for testing for and against a set of linear inequality constraints placed upon on any probability vectors corresponding to r independent multinomials. The paper shows how to compute the maximum likelihood estimates under all hypotheses of interest and obtains the limiting distributions of the likelihood ratio test statistics. These limiting distributions are of chi bar square type and the expression of the weighting values is given. To illustrate our theoretical results, we use a real life data set to test against second-order stochastic ordering.     

The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility

In this paper, firstly we show that the determining equations of the (1+1) dimension nonlinear differential equation with arbitrary order for the nonclassical method can be derived by the compatibility between the original equation and the invariant surface condition. Then we generalize this result to the system of the (m+1) dimension differential equations. The nonlinear Klein–Gordon equation, the (2+1)-dimensional Boussinesq equation and the generalized Nizhnik–Novikov–Veselov equation serve as examples illustrating this method.     

On functions with the cauchy difference bounded by a functional. III

We are going to discuss special cases of a conditional functional inequality

Parameter estimation and asymptotic stability in stochastic filtering

In this paper, we study the problem of estimating a Markov chain X$X$ (signal) from its noisy partial information Y$Y$, when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process P{_Xn∣_Yn,…,_Y1}$\mathbb{P}\{X_n\mid Y_n,\dots ,Y_1\}$, referred to hereafter as the optimal filter. Following a standard Bayesian technique, we treat the parameters as a non-dynamic component of the Markov chain. As a result, the new Markov chain is not going to be mixing, even if the original one is. We show that, under certain conditions, the optimal filters are still going to be asymptotically stable with respect to the initial conditions. Thus, by computing the optimal filter of the new system, we can estimate the signal adaptively.     

Search for CP violation in the lepton sector

One of the major open issues in neutrino physics is the possible existence of CP violation in the neutrino sector. Such an observation would have an important impact in various domains of physics, from high energy physics to cosmology. Its search requires future accelerator neutrino facilities producing intense and pure neutrino beams such as “beta-beams”. Here we review the different beta-beam scenarios proposed so far and discuss the present status, with a particular emphasis on the original baseline scenario and its feasibility. Alternative strategies for the CP violation search are to be pursued as well. A possibility is to search for CP violation effects in astrophysical environments. Here we present recent analytical and numerical results obtained in the context of core-collapse supernovae. In particular, we point out the conditions under which there can be CP violating effects in dense media and show numerical results on the supernova (anti-)neutrino fluxes and on the electron fraction, relevant for the r-process nucleosynthesis.     

Change point estimators by local polynomial fits under a dependence assumption

We study a random design regression model generated by dependent observations, when the regression function itself (or its ν-th derivative) may have a change or discontinuity point. A method based on the local polynomial fits with one-sided kernels to estimate the location and the jump size of the change point is applied in this paper. When the jump location is known, a central limit theorem for the estimator of the jump size is established; when the jump location is unknown, we first obtain a functional limit theorem for a local dilated-rescaled version estimator of the jump size and then give the asymptotic distributions for the estimators of the location and the jump size of the change point. The asymptotic results obtained in this paper can be viewed as extensions of corresponding results for independent observations. Furthermore, a simulated example is given to show that our theory and method perform well in practice.     

Tail probability approximations for Student's t-statistics

In this paper, we derive saddlepoint approximations for Student's t-statistics for strongly nonlattice random variables without moment conditions. Under very mild conditions, we show that saddlepoint equations always have solutions. Supported in part by the grants R-155-050-055-133/101 and R-155-000-035-112 at the National University of Singapore     

Tails of multivariate Archimedean copulas

A complete and user-friendly directory of tails of Archimedean copulas is presented which can be used in the selection and construction of appropriate models with desired properties. The results are synthesized in the form of a decision tree: Given the values of some readily computable characteristics of the Archimedean generator, the upper and lower tails of the copula are classified into one of three classes each, one corresponding to asymptotic dependence and the other two to asymptotic independence. For a long list of single-parameter families, the relevant tail quantities are computed so that the corresponding classes in the decision tree can easily be determined. In addition, new models with tailor-made upper and lower tails can be constructed via a number of transformation methods. The frequently occurring category of asymptotic independence turns out to conceal a surprisingly rich variety of tail dependence structures.     

Positive and real-positive solutions to the equation axa*=c in C*-algebras

We obtain new representations for the general positive and real-positive solutions of the equation axa*=c in a C*-algebra using the characterization of positivity based on a matrix representation of an element and the generalized Schur complement. Applications to the equation AXA*=C for operators between Hilbert spaces and for finite matrices are given.