Risk aversion asymptotics for power utility maximization

We consider the economic problem of optimal consumption and investment with power utility. We study the optimal strategy as the relative risk aversion tends to infinity or to one. The convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models. The limits are related to exponential and logarithmic utility. To derive these results, we combine approaches from optimal control, convex analysis and backward stochastic differential equations (BSDEs).     

On the most powerful quantile test of the scale parameter

Summary The most powerful test of the null hypothesisH ₀:σ=σ ₀ versus the alternative hypothesisH ₁:σ=σ ₁ based on a few selected sample quantiles is proposed here where σ is the scale parameter of the distribution and the location parameter μ is known. The quantiles are chosen from a large sample that is either complete or censored (singly-censored or doubly-censored). The relationship between the proposed test and the asymptotically best linear unbiased estimate (ABLUE) of the scale parameter is discussed.     

On solutions with generalized power asymptotics to systems of differential equations

In the paper we study methods for constructing particular solutions with nonexponential asymptotic behavior to a system of ordinary differential equations with infinitely differentiable right-hand sides. We construct the corresponding formal solutions in the form of generalized power series whose first terms are particular solutions to the so-called truncated system. We prove that these series are asymptotic expansions of real solutions to the complete system. We discuss the complex nature of the functions that are represented by these series in the analytic case.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 851–861, December, 1995.     

A quantile based test for comparing cumulative incidence functions of competing risks models

In the present note, we develop a nonparametric testing procedure for testing equality of cumulative incidence functions of competing risks models using quantile functions. Asymptotic properties of the test statistic are discussed. Simulation studies and real data examples illustrate the practical utility of the procedure.     

The local power of the gradient test

The asymptotic expansion of the distribution of the gradient test statistic is derived for a composite hypothesis under a sequence of Pitman alternative hypotheses converging to the null hypothesis at rate n ^−1/2, n being the sample size. Comparisons of the local powers of the gradient, likelihood ratio, Wald and score tests reveal no uniform superiority property. The power performance of all four criteria in one-parameter exponential family is examined.     

Exact asymptotics for Boundary crossings of the brownian bridge with trend with application to the Kolmogorov test

We consider a boundary crossing probability of a Brownian bridgeB ₀ and a piecewise linear boundary functionu(t)−γh(t). The main result of this paper is an asymptotic expansion for γ→∞ of the boundary crossing probability thatB ₀ (t) is larger than the piecewise linear boundary functionu(t)−γh(t) for somet. Such probabilities occur for instance in the context of change point problems when the Kolmogorov test is used. Examples are discussed showing that the approximation is rather accurate even for small positive γ values. Supported by the Swiss National Science Foundation Grant 20-55586.98.     

Estimation and test procedures for composite quantile regression with covariates missing at random

In this paper, we study the weighted composite quantile regression (WCQR) for general linear model with missing covariates. We propose the WCQR estimation and bootstrap test procedures for unknown parameters. Simulation studies and a real data analysis are conducted to examine the finite performance of our proposed methods.     

Resurrecting the asymptotics of linear recurrences

Once on the forefront of mathematical research in America, the asymptotics of the solutions of linear recurrence equations is now almost forgotten, especially by the people who need it most, namely combinatorists and computer scientists. Here we present this theory in a concise form and give a number of examples that should enable the practicing combinatorist and computer scientist to include this important technique in her (or his) asymptotics tool kit.     

On the asymptotics of the Riesz-Bochner kernel

В работе рассматрива ется асимптотика в ме трике пространстваL _p (T ^N ),T ^N ={x∈R ^N , 0<x _i <2π} ядра Р исса-Бохнера $$\Theta ^s \left( {x, \lambda } \right) = \left( {2\pi } \right)^{ - N} \mathop \Sigma \limits_{\left| n \right|^2< \lambda } \left( {1 - \frac{{\left| n \right|^2 }}{\lambda }} \right)^s e^{inx} \left( {x \in T^N , s \geqq 0, \lambda \geqq 0} \right)$$ при λ→∞. Доказывается, что есл иN≧4,p≧2N/(N?1) иs>N((N?1)/2N?1/p), то для произвольной точкиx∈T ^N существует п остояннаяC=C _p (x, s) такая, что выполняется неравен ство $$\parallel \Theta ^s \left( {x - y, \lambda } \right) - \left( {2\pi } \right)^{ - {N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}} 2^s \Gamma \left( {s + 1} \right)\lambda ^{{N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}} J_{{N \mathord{\left/ {\vphantom {N {2 + s}}} \right. \kern-\nulldelimiterspace} {2 + s}}} {{\left( {\left| {x - y} \right|\sqrt \lambda } \right)} \mathord{\left/ {\vphantom {{\left( {\left| {x - y} \right|\sqrt \lambda } \right)} {\left( {\left| {x - y} \right|\sqrt \lambda } \right)^{{N \mathord{\left/ {\vphantom {N {2 + s}}} \right. \kern-\nulldelimiterspace} {2 + s}}} \parallel _{L_p \left( {T^N } \right)} \leqq }}} \right. \kern-\nulldelimiterspace} {\left( {\left| {x - y} \right|\sqrt \lambda } \right)^{{N \mathord{\left/ {\vphantom {N {2 + s}}} \right. \kern-\nulldelimiterspace} {2 + s}}} \parallel _{L_p \left( {T^N } \right)} \leqq }}$$ где нормаL _p (T ^N ) берется по пе ременнойy, а черезJ _v обозначена функция Б есселя первого рода порядкаv. СлучаиN=2 иN=3 рассматриваются отдельно.     

Degree asymptotics of the numerical semigroup tree

A numerical semigroup is a subset Λ of the nonnegative integers that is closed under addition, contains 0, and omits only finitely many nonnegative integers (called the gaps of Λ). The collection of all numerical semigroups may be visually represented by a tree of element removals, in which the children of a semigroup Λ are formed by removing one element of Λ that exceeds all existing gaps of Λ. In general, a semigroup may have many children or none at all, making it difficult to understand the number of semigroups at a given depth on the tree. We investigate the problem of estimating the number of semigroups at depth g with h children, showing that as g becomes large, it tends to a proportion φ ^?h?2 of all numerical semigroups, where φ is the golden ratio.     

The asymptotics of the generalised Hermite-Bell polynomials

The Hermite-Bell polynomials are defined by for n=0,1,2,… and integer r≥2 and generalise the classical Hermite polynomials corresponding to r=2. We obtain an asymptotic expansion for as n→∞ using the method of steepest descents. For a certain value of x, two saddle points coalesce and a uniform approximation in terms of Airy functions is given to cover this situation. An asymptotic approximation for the largest positive zeros of is derived as n→∞. Numerical results are presented to illustrate the accuracy of the various expansions.     

Spectral asymptotics of hypersurfaces

Pseudodifferential operator techniques are employed to obtain higher-order spectral asymptotic results for integral operators on hypersurfaces in Euclidean space.Supported by a grant from the National Science Foundation.     

A Berry-Esseen theorem for the kernel quantile estimator with application to studying the deficiency of quantile estimators

A Berry-Esseen bound is established for the kernel quantile estimator under various conditions. The results improve an earlier result of Falk (1985,Ann. Statist.,13, 428–433) and rely on the local smoothness of the quantile function. This new Berry-Esseen bound is applied to studying the deficiency of the sample quantile estimator with respect to the kernel quantile estimator. A new result is obtained which is an extension of that in Falk (1985).     

A note on the quicksort asymptotics

In a recent paper, Bindjeme and Fill obtained a surprisingly easy exact formula for the L₂‐distance of the (normalized) number of comparisons of Quicksort under the uniform model to its limit. Shortly afterwards, Neininger proved a central limit theorem for the error. As a consequence, he obtained the asymptotics of the L₃‐distance. In this short note, we use the moment transfer approach to re‐prove Neininger's result. As a consequence, we obtain the asymptotics of the L_p‐distance for all Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,677–687, 2015