A note on exponential dispersion models which are invariant under length-biased sampling

Length-biased sampling (LBS) situations may occur in clinical trials, reliability, queueing models, survival analysis and population studies where a proper sampling frame is absent. In such situations items are sampled at rate proportional to their “length” so that larger values of the quantity being measured are sampled with higher probabilities. More specifically, if f(x) is a p.d.f. presenting a parent population composed of non-negative valued items then the sample is practically drawn from a distribution with p.d.f. g(x)=xf(x)/E(X) describing the length-biased population. In this case the distribution associated with g is termed a length-biased distribution. In this note, we present a unified approach for characterizing exponential dispersion models which are invariant, up to translations, under various types of LBS. The approach is rather simple as it reduces such invariance problems into differential equations in terms of the derivatives of the associated variance functions.     

TESTING FOR VARYING DISPERSION IN DISCRETE EXPONENTIAL FAMILY NONLINEAR MODELS

§ 1　 Introduction and modelsThe general form of exponential family nonlinear models isg(μi) =f(xi,﹀) , (1 )where,g(· ) is a monotonic link function,f is a known differentiable nonlinear functionand﹀ is a p-vectoroffixed population parameters;μi=E(yi) and the density of response yiisp(yi) =exp{[yiθi -b(θi) -c(yi) ] -12 a(yi,) } ,(2 )whereθi is the natural parameter, is the dispersion parameter.From [1 1 ] ,μi=b(θi) ,Vi=Var(yi) =- 1 b(θi) .If f(xi,β) =x Ti ﹀,then mod…     

Minimal invariant tori in the resonant regions for nearly integrable Hamiltonian systems

Consider a real analytical Hamiltonian system of KAM type that has degrees of freedom (2$">) and is positive definite in . Let . In this paper we show that for most rotation vectors in , in the sense of ()-dimensional Lebesgue measure, there is at least one ()-dimensional invariant torus. These tori are the support of corresponding minimal measures. The Lebesgue measure estimate on this set is uniformly valid for any perturbation.

     

The behavior of the Laplace transform of the invariant measure on the hypersphere of high dimension

We consider the sequence of the hyperspheres M _n , i.e., the homogeneous transitive spaces of the Cartan subgroup of the group and study the normalized limit of the corresponding sequence of invariant measures m _n on those spaces. In the case of compact groups and homogeneous spaces, for example, for the classical pairs (SO(n), S ^n-1), n = 1, 2, … , the limit of the corresponding measures is the classical infinite-dimensional Gaussian measure; this is the well-known Maxwell-Poincaré lemma. Simultaneously the Gaussian measure is a unique (up to a scalar) invariant measure with respect to the action of the infinite orthogonal group O(∞). This coincidence implies the asymptotic equivalence between grand and small canonical ensembles for the series of the pairs (SO(n), S ^n-1). Our main result shows that the situation for noncompact groups, for example for the case , is completely different: the limit of the measures m _n does not exist in the literal sense, and we show that only a normalized logarithmic limit of the Laplace transforms of those measures does exist. At the same time, there exists a measure which is invariant with respect to a continuous analogue of the Cartan subgroup of the group GL(∞), the so-called infinite-dimensional Lebesgue measure (see [7]). This difference is an evidence for non-equivalence between the grand and small canonical ensembles in the noncompact case. To my friend Dima Arnold     

On the existence of an invariant measure for Markov-Feller operators

Let X be a Polish space and P a Markov operator acting on the space of Borel measures on X. We will prove the existence of an invariant measure with respect to P, provided that P satisfies some condition of a Prokhorov type and that the family of functions is equi-continuous with respect to the Prokhorov distance at some point of the space X. Moreover, we will construct a counterexample which show that the above equi-continuity condition cannot be dropped.     

Stability of invariant linearly sufficient statistics in the general Gauss-Markov model

Necessary and sufficient conditions are derived for the inclusions and to be fulfilled where are some classes of invariant linearly sufficient statistics (Oktaba, Kornacki, Wawrzosek (1988)) corresponding to the Gauss-Markov models , respectively.     

An invariant measure for the loop space of a simply connected compact symmetric space

Let X denote a simply connected compact Riemannian symmetric space, U the universal covering of the identity component of the group of automorphisms of X, and LU the loop group of U. In this paper we prove the existence (and conjecture the uniqueness) of an LU-invariant probability measure on a distributional completion of the loop space of X.     

A Karhunen-Loeve decomposition of a Gaussian process generated by independent pairs of exponential random variables

We obtain the explicit Karhunen-Loeve decomposition of a Gaussian process generated as the limit of an empirical process based upon independent pairs of exponential random variables. The orthogonal eigenfunctions of the covariance kernel have simple expressions in terms of Jacobi polynomials. Statistical applications, in extreme value and reliability theory, include a Cramér-von Mises test of bivariate independence, whose null distribution and critical values are tabulated.     

Asymptotic estimates of Gerber-Shiu functions in the renewal risk model with exponential claims

This paper continues to study the asymptotic behavior of Gerber-Shiu expected discounted penalty functions in the renewal risk model as the initial capital becomes large. Under the assumption that the claim-size distribution is exponential, we establish an explicit asymptotic formula. Some straightforward consequences of this formula match existing results in the field.     

A decomposition for invariant tests of uniformity on the sphere

We introduce a -statistic on which can be based a test for uniformity on the sphere. It is a simple function of the geometric mean of distances between points of the sample and consistent against all alternatives. We show that this type of -statistic, whose kernel is invariant by isometries, can be separated into a set of statistics whose limiting random variables are independent. This decomposition is obtained via the so-called canonical decomposition of a group representation. The distribution of the limiting random variables of the components under the null hypothesis is given. We propose an interpretation of Watson type identities between quadratic functionals of Gaussian processes in the light of this decomposition.

     

Notes on a finite differencing scheme for the dispersion equation

Finite difference techniques applied to atmospheric dispersion problems often encounter time step limitations due to the variance in the characteristic length scales (horizontal to vertical) of both the field variables and the computational region. Methods to maximize the integration time step are explored and techniques are described which ensure numerical accuracy and stability of these optimized time step techniques.

To circumvent time step limitations arising from consideration of the vertical diffusion term in the dispersion equation, a column implicitization technique is suggested which, through correction terms added to the differencing equation to compensate for truncation errors, provides an efficient and economical atmospheric dispersion solver which is insensitive to the common time step limitations of explicit schemes when large aspect ratio computational volumes are required. Further, it is shown that a relaxed stability criteria proposed by Leonard and Clancy for explicit differencing of the horizontal terms in the dispersion equation, presents a further saving in computational time provided correction terms to the differencing equation are included to eliminate phase and amplitude errors resulting from the larger time steps employed.     

A note on generalized invariant subspaces for infinite-dimensional systems

In this paper, some generalized invariant subspaces for uncertainlinear infinite-dimensional systems in the sense that each uncertainparameters are in given real intervals are studied.     

A projection method of estimation for a subfamily of exponential families

Summary This paper is concerned with estimation for a subfamily of exponential-type, which is a parametric model with sufficient statistics. The family is associated with a surface in the domain of a sufficient statistic. A new estimator, termed a projection estimator, is introduced. The key idea of its derivation is to look for a one-to-one transformation of the sufficient statistic so that the subfamily can be associated with a flat subset in the transformed domain. The estimator is defined by the orthogonal projection of the transformed statistic onto the flat surface. Here the orthogonality is introduced by the inverse of the estimated variance matrix of the statistic on the analogy of Mahalanobis's notion (1936,Proc. Nat. Inst. Sci. Ind.,2, 49–55). Thus the projection estimator has an explicit representation with no iterations. On the other hand, the MLE and classical estimators have to be sought as numerical solutions by some algorithm with a choice of an initial value and a stopping rule. It is shown that the projection estimator is first-order efficient. The second-order property is also discussed. Some examples are presented to show the utility of the estimator.     

A note on exponential decay properties of ground states for quasilinear elliptic equations

We give an explicit formula for exponential decay properties of ground states for a class of quasilinear elliptic equations in the whole space .

     

Quasi-invariance of Lebesgue measure under the homeomorphic flow generated by SDE with non-Lipschitz coefficient

We consider the stochastic flow generated by Stratonovich stochastic differential equations with non-Lipschitz drift coefficients. Based on the author's previous works, we show that if the generalized divergence of the drift is bounded, then the Lebesgue measure on Rd is quasi-invariant under the action of the stochastic flow, and the explicit expression of the Radon-Nikodym derivative is also presented. Finally we show in a special case that the unique solution of the corresponding Fokker-Planck equation is given by the density of the stochastic flow.     

Existence and exponential stability of almost periodic solutions for a neutral multi-species Logarithmic population model

In this paper, we study the almost periodic solution for a neutral multi-species Logarithmic population model. By employing Banach’s fixed point theorem and using differential inequality technique, we present some sufficient conditions ensuring the existence, uniqueness and globally exponential stability of almost periodic solution for the model. The results obtained extend and improve the earlier publications. Finally, two examples are provided to show the correctness of our analysis.     

A new method for computing the matrix exponential operation based on vector valued rational approximations

In this paper a new method for computing the action of the matrix exponential on a vector e^Atb, where A is a complex matrix and t is a positive real number, is proposed. Our approach is based on vector valued rational approximation where the approximants are determined by the denominator polynomials whose coefficients are obtained by solving an inexpensive linear least-squares problem. No matrix multiplications or divisions but matrix-vector products are required in the whole process. A technique of scaling and recurrence enables our method to be more effective when the problem is for fixed A,b and many values of t. We also give a backward error analysis in exact arithmetic for the truncation errors to derive our new algorithm. Preliminary numerical results illustrate that the new algorithm performs well.     

A local proof for the characterization of Young measures generated by sequences in BV

This work presents a new proof of the recent characterization theorem for generalized Young measures generated by sequences in BV by Kristensen and Rindler (2010) [14]. The present argument is based on a localization technique together with a local Hahn–Banach argument in novel function spaces combined with an application of Alberti's Rank-One Theorem. This strategy avoids employing a relaxation theorem as in the previously known proof, and the new tools introduced in its course should prove useful in other contexts as well. In particular, we introduce “homogeneous” Young measures, separately at regular and singular points, which exhibit rather different behavior than the classical homogeneous Young measures. As an application, we show how for BV-Young measures with an “atomic” part one can find a generating sequence respecting this structure.     

A sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases

We describe a sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases. It includes interchanges that, whenever this is possible, avoid the use of any eliminations (with consequent fill-ins) when revising the factorization at an iteration. Test results on some medium scale problems are presented and comparisons made with the algorithm of Forrest and Tomlin.     

Beurling's theorem and invariant subspaces for the shift on Hardy spaces

Let G be a bounded open subset in the complex plane and let H~2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1-1 with respect to the Lebesgue measure on D and if the Riemann map belongs to the weak-star closure of the polynomials in H~∞(W). Our main theorem states: in order that for each M∈Lat (M_z), there exist u∈H~∞(G) such that M=∨{uH~2(G)}, it is necessary and sufficient that the following hold: (1) each component of G is a perfectly connected domain; (2) the harmonic measures of the components of G are mutually singular; (3) the set of polynomials is weak-star dense in H~∞(G). Moreover, if G satisfies these conditions, then every M∈Lat (M_z) is of the form uH~2(G), where u∈H~∞(G) and the restriction of u to each of the components of G is either an inner function or zero.