Approximations to the distribution of the sample correlation matrix

In this article, multivariate density expansions for the sample correlation matrix R are derived. The density of R is expressed through multivariate normal and through Wishart distributions. Also, an asymptotic expansion of the characteristic function of R is derived and the main terms of the first three cumulants of R are obtained in matrix form. These results make it possible to obtain asymptotic density expansions of multivariate functions of R in a direct way.     

A note on the asymptotic distribution of the maximum likelihood estimator in a non-regular case

We consider the asymptotic distribution of the maximum likelihood estimator (MLE), when the log-likelihood ratio statistic weakly converges to the non-degenerated Gaussian process. We provide a simple expression for the density function of the asymptotic distribution by fundamental stochastic results. This note is helpful to investigate asymptotic properties of the MLE in a certain non-regular case.     

EDGEWORTH EXPANSION AND BOOTSTRAP APPROXIMATION FOR THE STUDENTIZED MLE FROM RANDOMLY CENSORED EXPONENTIAL SAMPLES

In this paper,the author studies the asymptotic accuracies of the one-term Edgeworth expansions and the bootstrap approximation for the studentized MLE from randomly censored exponential population.It is shown that the Edgeworth expansions and the bootstrap approximation are asymptotically close to the exact distribution of the studentized MLE with a rate.     

Derivation of transitional asymptotic expansions from integral representations

A method for deriving transitional asymptotic expansions from integral representations is described and applied to Anger function and modified Hankel function. The method consists in deriving asymptotic expansions of the function considered as well as its first derivativeat the transition point using conventional methods such as Laplace’s method or the method of steepest descents. Since both the functions considered satisfy a second order linear differential equation, it is possible to obtain asymptotic expansions of higher order derivatives of the functions from the first two expansions. Thus asymptotic expressions for all the derivatives at the transition point are known and a Taylor expansion of the function in the neighbourhood of the transition point can be written. The method is also applicable to the generalized exponential integral, Weber’s parabolic cylinder function and Poiseuille function.     

A systematization of the saddle point method. Application to the Airy and Hankel functions

The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples.     

Asymptotic Expansions of Large Deviations for Sums of Nonidentically Distributed Random Variables

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in Cramer zones and Linnik power zones for the distribution function of sums of independent nonidentically distributed random variables (r.v.). In this scheme of summation of r.v., the results are obtained first by mainly using the general lemma on large deviations considering asymptotic expansions for an arbitrary r.v. with regular behaviour of its cumulants [11]. Asymptotic expansions in the Cramer zone for the distribution function of sums of identically distributed r.v. were investigated in the works [1,2]. Note that asymptotic expansions for large deviations were first obtained in the probability theory by J. Kubilius [3].     

Asymptotic Expansions for Approximate Eigenvalues of Integral Operators with Nonsmooth Kernels

We consider approximation of eigenvalues of integral operators with Green's function kernels using the Nyström method and the iterated collocation method and obtain asymptotic expansions for approximate eigenvalues. We show that the Richardson extrapolation is applicable to find eigenvalue approximations of higher order and illustrate our results by numerical examples.     

Unified treatment of several asymptotic formulas for the gamma function

We consider a class of the asymptotic expansions for the gamma function, and derive a formula for determining the coefficients of the asymptotic expansions. Thus, we give a unified treatment of several asymptotic expansions for the gamma function due to Laplace, Ramanujan–Karatsuba, Gosper, Mortici and Batir.     

Methods de laplace et de la phase stationnaire sur l'espace de wiener

We obtain, using large deviations principles, full asymptotic expansions of functionals of Laplace type on Wiener space for general (i.e. degenerate) diffusions extending the results of Schilder [12], Azencott [2] and Doss [7, 20]. The variational hypothesis (non degeneracy of the minima) used here is shown to be optimal. The first term of the expansion is explicitly computed. Using the integration by parts of Malliavin calculus the stationary phase method is also developed. The results of this paper are the basic fact used (in [5]) to obtain the asymptotic expansion for small time of the density of a degenerate diffusion, they are also relevant for semi-classical expansions     

On asymptotics of the q-exponential and q-gamma functions

In this work we present a derivation for the complete asymptotic expansions of Euler?s q-exponential function and Jackson?s q-gamma function via Mellin transform. These formulas are valid everywhere, uniformly on any compact subset of the complex plane.     

Error Bounds for Uniform Asymptotic Expansions—Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals. By using a class of rational functions, they express these quantities in terms of Cauchy-type integrals; these expressions are natural generalizations of integral representations of the coefficients and the remainders in the Taylor expansions of analytic functions. By using the new representation, a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.     

Error Bounds for the Large‐Argument Asymptotic Expansions of the Lommel and Allied Functions

In this paper, we reconsider the large‐z asymptotic expansion of the Lommel function and its derivative. New representations for the remainder terms of the asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re‐expansions for these remainder terms and provide their error estimates. Applications to the asymptotic expansions of the Anger–Weber‐type functions, the Scorer functions, the Struve functions, and their derivatives are provided. The sharpness of our error bounds is discussed in detail, and numerical examples are given.     

A Method for Incorporating Transcendentally Small Terms into the Method of Matched Asymptotic Expansions

The essential ideas behind a method for incorporating exponentially small terms into the method of matched asymptotic expansions are demonstrated using an Ackerberg–O'Malley resonance problem and a spurious solutions problem of Carrier and Pearson. One begins with the application of the standard method of matched asymptotic expansions to obtain at least the leading terms in outer and inner (Poincaré-type) expansions; some, although not all, matching can be carried out at this stage. This is followed by the introduction of supplementary expansions whose gauge functions are transcendentally small compared to those in the standard expansions. Analysis of terms in these expansions allows the matching to be completed. Furthermore, the method allows for the inclusion of globally valid transcendentally small contributions to the asymptotic solution; it is well known that such terms may be numerically significant.     

Higher order asymptotic behaviour of partial maxima of random sample from generalized Maxwell distribution under power normalization

In this article, the higher order asymptotic expansions of cumulative distribution function and probability density function of extremes for generalized Maxwell distribution are established under nonlinear normalization. As corollaries, the convergence rates of the distribution and density of maximum are obtained under nonlinear normalization.     

Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise from an application of the method of steepest descents.     

A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function ₂F₁(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given.     

Merging asymptotic expansions for semistable random variables

Merging asymptotic expansions are established for distribution functions from the domain of geometric partial attraction of a semistable law. The length of the expansion depends on the exponent of the semistable law and on the characteristic function of the underlying distribution. We obtain sufficient conditions for the quantile function in order to get real infinite asymptotic expansion. The results are generalizations of the existing theory in the stable case.     

Remarks on asymptotic expansions for the gamma function

We unify several asymptotic expansions for the gamma function due to Laplace, Ramanujan–Karatsuba, Gosper, Mortici, Nemes and Batir. Furthermore we present new asymptotic expansions for the gamma function.     

A Systematic “Saddle Point Near a Pole” Asymptotic Method with Application to the Gauss Hypergeometric Function

In recent works [ 1 ] and [ 2 ], we have proposed more systematic versions of the Laplace’s and saddle point methods for asymptotic expansions of integrals. Those variants of the standard methods avoid the classical change of variables and give closed algebraic formulas for the coefficients of the expansions. In this work we apply the ideas introduced in [ 1 ] and [ 2 ] to the uniform method “saddle point near a pole.” We obtain a computationally more systematic version of that uniform asymptotic method for integrals having a saddle point near a pole that, in many interesting examples, gives a closed algebraic formula for the coefficients. The asymptotic sequence is given, in general, in terms of exponential integrals of fractional order (or incomplete gamma functions). In particular, when the order of the saddle point is two, the basic approximant is given in terms of the error function (as in the standard method). As an application, we obtain new asymptotic expansions of the Gauss Hypergeometric function ₂F₁(a, b, c; z) for large b and c with c > b + 1 .     

Laplace's method on a computer algebra system with an application to the real valued modified Bessel functions

We examine a Maple implementation of two distinct approaches to Laplace's method used to obtain asymptotic expansions of Laplace-type integrals. One algorithm uses power series reversion, whereas the other expands all quantities in Taylor or Puiseux series. These algorithms are used to derive asymptotic expansions for the real valued modified Bessel functions of pure imaginary order and real argument that mimic the well-known corresponding expansions for the unmodified Bessel functions.