Related papers: Higher-order asymptotics for the parametric comple…
We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size…
The family of rank estimators, including Han's maximum rank correlation (Han, 1987) as a notable example, has been widely exploited in studying regression problems. For these estimators, although the linear index is introduced for…
In this monograph, we prove an asymptotic approximation for integrals of probability densities over sets in finite dimensional euclidean space, which are far away from the origin (asymptotic sets). We use this approximation to investigate…
In this paper we continue our earlier investigations into the asymptotic behaviour of infinite systems of coupled differential equations. Under the mild assumption that the so-called characteristic function of our system is completely…
In arXiv:0907.0079 by Cator and Lopuhaa, an asymptotic expansion for the MCD estimators is established in a very general framework. This expansion requires the existence and non-singularity of the derivative in a first-order Taylor…
A fast and numerically stable algorithm is described for computing the discrete Hankel transform of order $0$ as well as evaluating Schl\"{o}milch and Fourier--Bessel expansions in $\mathcal{O}(N(\log N)^2/\log\!\log N)$ operations. The…
Let $v_n$ be the maximum expected length of an increasing subsequence, which can be selected by an online nonanticipating policy from a random sample of size $n$. Refining known estimates, we obtain an asymptotic expansion of $v_n$ up to a…
The martingale posterior framework is a generalization of Bayesian inference where one elicits a sequence of one-step ahead predictive densities instead of the likelihood and prior. Posterior sampling then involves the imputation of unseen…
The Minimum Description Length (MDL) principle offers a formal framework for applying Occam's razor in machine learning. However, its application to neural networks such as Transformers is challenging due to the lack of a principled,…
We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly…
Let $\alpha_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence $(X_1,X_2,\ldots)$ of $p$-dimensional random vectors. Suppose $$\alpha_n= \mathcal{N} _p (M_n,Q_n)$$ where…
We present expressions for the coefficients which arise in asymptotic expansions of multiple integrals of Laplace type (the first term of which is known as Laplace's approximation) in terms of asymptotic series of the functions in the…
In this paper we study the asymptotics (as $n\to \infty$) of the sequences of Laguerre polynomials with varying complex parameters $\alpha$ depending on the degree $n$. More precisely, we assume that $\alpha_n = n A_n, $ and $ \lim_n A_n=A…
We consider the lossless compression bound of any individual data sequence. If we fit the data by a parametric model, the entropy quantity $nH({\hat \theta}_n)$ obtained by plugging in the maximum likelihood estimate is an underestimate of…
The problem of finding \emph{distance} between \emph{pattern} of length $m$ and \emph{text} of length $n$ is a typical way of generalizing pattern matching to incorporate dissimilarity score. For both Hamming and $L_1$ distances only a…
We provide an asymptotic expansion of the maximal mean squared error (MSE) of the sample median to be attained on shrinking gross error neighborhoods about an ideal central distribution. More specifically, this expansion comes in powers of…
Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated , in particular when X 1 is not…
We study the properties of the MDL (or maximum penalized complexity) estimator for Regression and Classification, where the underlying model class is countable. We show in particular a finite bound on the Hellinger losses under the only…
We derive the first exact, rigorous but practical, globally valid remainder terms for asymptotic expansions about saddles and contour endpoints of arbitrary order degeneracy derived from the method of steepest descents. The exact remainder…
Some existing models of the atherosclerosis development are discussed and a new improved mathematical model, which takes into account new experimental results about diverse roles of macrophages in atherosclerosis, is proposed. Using technic…