A bijective proof of the hook-length formula

A well-known theorem of Frame, Robinson, and, Thrall states that if λ is a partition of n, then the number of Standard Young Tableaux of shape λ is n! divided by the product of the hook-lengths. We give a new combinatorial proof of this formula by exhibiting a bijection between the set of unsorted Young Tableaux of shape λ, and the set of pairs (T, S), where T is a Standard Young Tableau of shape λ and S is a “Pointer” Tableau of shape λ.     

An everywhere convergent series representation of the distribution of Hotelling's generalized T0

A new series representation of the exact distribution of Hotelling's generalized T₀² statistic is obtained. Unlike earlier work, the series representation given here is everywhere convergent. Explicit formulae are given for both the null and the non-central distributions. Earlier results by [1], 215–225), which are convergent on the interval [0, 1), are also derived quite simply from our formulae. The paper therefore provides a solution to the long standing problem of the exact distribution of the T₀² statistic in the general case.     

The general form of the law of the iterated logarithm for generalized Kolmogorov-Smirnov-Renyi statistics

One result of Smirnov's important paper [Uspehi Mat. Nauk. 10, 179–206, (in Russian)] yields exponential bounds for the large deviations of his one-sided Smirnov statistic and the two-sided Kolmogorov statistic. In the present paper exponential bounds are given for the large deviations of a wide class of Kolmogorov-Smirnov-Renyi type statistics. As a by-product, exponential bounds for the large deviations of the corresponding limit distributions are obtained.     

Robust M-estimators of location vectors

Let X₁,…,X_n be i.i.d. random vectors in R^m, let θεR^m be an unknown location parameter, and assume that the restriction of the distribution of X₁−θ to a sphere of radius d belongs to a specified neighborhood of distributions spherically symmetric about 0. Under regularity conditions on and d, the parameter θ in this model is identifiable, and consistent M-estimators of θ (i.e., solutions of Σ_i=1ⁿψ(|X_i− |)(X_i− )=0) are obtained by using “re-descenders,” i.e., ψ's wh satisfy ψ(x)=0 for x≥c. An iterative method for solving for is shown to produce consistent and asymptotically normal estimates of θ under all distributions in . The following asymptotic robustness problem is considered: finding the ψ which is best among the re-descenders according to Huber's minimax variance criterion.     

On powerful distributional tests based on sample spacings

This paper is devoted to tests for uniformity based on sum-functions of m-spacings, where m diverges to infinity as the sample size, n, increases. It is shown that if m diverges at a slower rate than n^1/2 then the commonly used sum-function will detect alternatives distant (mn)^−1/4 from the uniform. This result fails if m diverges more quickly than n^1/2, and in that situation the statistic must be modified. The case where m/n → , 0 < < 1, is also considered, and it is shown that the test has adequate power against local and fixed alternatives if and only if is irrational.     

A Dynamic Mixture Model for Unsupervised Tail Estimation without Threshold Selection

Exceedances over high thresholds are often modeled by fitting a generalized Pareto distribution (GPD) on R⁺. It is difficult to select the threshold, above which the GPD assumption is enough solid and enough data is available for inference. We suggest a new dynamically weighted mixture model, where one term of the mixture is the GPD, and the other is a light-tailed density distribution. The weight function varies on R⁺ in such a way that for large values the GPD component is predominant and thus takes the role of threshold selection. The full data set is used for inference on the parameters present in the two component distributions and in the weight function. Maximum likelihood provides estimates with approximate standard deviations. Our approach has been successfully applied to simulated data and to the (previously studied) Danish fire loss data set. We compare the new dynamic mixture method to Dupuis' robust thresholding approach in peaks-over-threshold inference. We discuss robustness with respect to the choice of the light-tailed component and the form of the weight function. We present encouraging simulation results that indicate that the new approach can be useful in unsupervised tail estimation, especially in heavy tailed situations and for small percentiles.     

Extensions completes d'une theorie forcing complete

We give two examples of a universal theoryT such that the forcing companionT ^F (as defined by A. Robinson for infinite forcing) has some model which is not the elementary equivalent of a generic model ofT. Our examples answer in a negative way a question posed by E. Fisher and A. Robinson. In the first example, forT ^F, there is an extensionT′ which is complete and forcing comcomplete (that is, (T′) ^F=T′) not generated by a generic model ofT. In the second exampleT ^F has a complete extensionT′ which is not forcing complete.      

Semidifferentiable functions and necessary optimality conditions

We consider Lyapunov's equationPA+A ^T P+Q=0, whereQ is symmetric positive definite andA is in controllable companion form. We prove that a necessary and sufficient condition thatA be stable is that the first rowP ₁ of theP-matrix be a stablen–1 coefficient vector. This result is related to the minimum phase property of linear systems and is useful in designing robust controllers.     

On the rate of approximation in the central limit theorem for dependent random variables and random vectors

Large “O” and small “o” approximations of the expected value of a class of smooth functions (f C^r(R)) of the normalized partial sums of dependent random variable by the expectation of the corresponding functions of normal random variables have been established. The same types of approximations are also obtained for dependent random vectors. The technique used is the Lindberg-Levy method generalized by Dvoretzky to dependent random variables.     

Maximizing the descent statistic

For a subsetS, let the descent statistic (S) be the number of permutations that have descent setS. We study inequalities between the descent statistics of subsets. Each subset (and its complement) is encoded by a list containing the lengths of the runs. We define two preorders that compare different lists based on the descent statistic. Using these preorders, we obtain a complete order on lists of the form (k ⁱ ,P,k ^n–i , whereP is a palindrome, whose first entry is larger thank. We prove a conjecture due to Gessel, which determines the list that maximizes the descent statistic, among lists of a given size and given length. We also have a generalization of the boustrophedon transform of Millar, Sloane and Young.     

Rank-dependent moderate deviations of U-empirical measures in strong topologies

 We prove a rank-dependent moderate deviation principle for U-empirical measures, where the underlying i.i.d. random variables take values in a measurable (not necessarily Polish) space (S,𝒮). The result can be formulated on a suitable subset of all signed measures on (S ^m ,𝒮^{⊗ m} ). We endow this space with a topology, which is stronger than the usual τ-topology. A moderate deviation principle for Banach-space valued U-statistics is obtained as a particular application. The advantage of our result is that we obtain in the degenerate case moderate deviations in non-Gaussian situations with non-convex rate functions. Received: 22 February 2000 / Revised version: 15 November 2002 / Published online: 28 March 2003 Research partially supported by the Swiss National Foundation, Contract No. 21-298333.90. Mathematics Subject Classification (2000): Primary 60F10; Secondary 62G20, 28A35 Key words or phrases: Rank-dependent moderate deviations – Empirical measures – Strong topology – U-statistics     

A Generalized φ-Divergence for Asymptotically Multivariate Normal Models

I. Csiszár's (Magyar. Tud. Akad. Mat. Kutató Int. Közl8 (1963), 85–108) -divergence, which was considered independently by M. S. Ali and S. D. Silvey (J. R. Statist. Soc. Ser. B28 (1966), 131–142) gives a goodness-of-fit statistic for multinomial distributed data. We define a generalized φ-divergence that unifies the -divergence approach with that of C. R. Rao and S. K. Mitra (“Generalized Inverse of Matrices and Its Applications,” Wiley, New York, 1971) and derive weak convergence to a χ² distribution under the assumption of asymptotically multivariate normal distributed data vectors. As an example we discuss the application to the frequency count in Markov chains and thereby give a goodness-of-fit test for observations from dependent processes with finite memory.     

Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach

We investigate the large-time behaviour of solutions to the nonlinear heat-conduction equation with absorption u_t = Δ(u^{σ + 1}) − u^β in Q = R^N × (0, ∞) (E) with N 1, σ > 0 and critical absorption exponent β = σ + 1 + 2/N; the initial function u(x, 0) = 0 is assumed to be integrable, nonnegative and compactly supported. We prove that u converges as t → ∞ to a unique self-similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem u_t = Δ(u^{σ + 1}), the same for any initial data. The cornerstone of the proof is a result about ω-limits of (infinite-dimensional) asymptotical dynamical systems. Combining this result with an asymptotic evaluation of the mass function as well as typical PDE estimates gives the behaviour of (E) for large times.Similar unusual asymptotic behaviour is obtained for the equation u_t = div(¦Du¦^σ Du) − u^β with same conditions on σ and u(x, 0) and critical value for β = σ + 1 + (σ + 2)/N.     

On the rate of convergence to normality for generalized linear rank statistics

Summary A generalized linear rank statistic is introduced to include, as special cases, both signed as well as unsigned linear rank statistics. For this statistic, the rate of convergence to asymptotic normality is investigated. It is shown that this rate is of orderO(N ^−1/2 logN) if the score generating function ϕ is twice differentiable, and it is of orderO(N ^−1/2) if the second derivative of ϕ satisfies Lipschitz's condition of order ≧1/2. The results obtained extend as well as generalize most of the earlier results obtained in this direction.     

Necessary and sufficient condition in Lyapunov robust control: Multi-input case

We consider a linear time-invariant finite-dimensional system x=Ax+Bu with multi-inputu, in which the matricesA andB are in canonical controller form. We assume that the system is controllable andB has rankm. We study the Lyapunov equationPA+A ^T P+Q=0, withQ>0, and investigate the properties thatP must satisfy in order that the canonical controller matrixA be Hurwitz. We show that, for the matrixA being Hurwitz, it is necessary and sufficient thatB ^T PB>0 and that the determinant ofB ^T PW be Hurwitz, whereW=block diag[w ₁,...,w _m ], with elementw _i =[s ^k _i –1,s ^k _i –2,...,s, 1] ^T ; here, the symbolsk _i ,i=1, 2, ...,m, denote the Kronecker invariants with respect to the pair {A, B}. This result has application in designing robust controllers for linear uncertain systems.     

Robust stabilization of uncertain linear dynamical systems with time-varying delay

In this paper, the problem of the robust stabilization for a class of uncertain linear dynamical systems with time-varying delay is considered. By making use of an algebraic Riccati equation, we derive some sufficient conditions for robust stability of time-varying delay dynamical systems with unstructured or structured uncertainties. In our approach, the only restriction on the delay functionh(t) is the knowledge of its upper boundh ^–. Some analytical methods are employed to investigate these stability conditions. Since these conditions are independent of the delay, our results are also applicable to systems with perturbed time delay. Finally, a numerical example is given to illustrate the use of the sufficient conditions developed in this paper.     

Robust tests based on dual divergence estimators and saddlepoint approximations

This paper is devoted to robust hypothesis testing based on saddlepoint approximations in the framework of general parametric models. As is known, two main problems can arise when using classical tests. First, the models are approximations of reality and slight deviations from them can lead to unreliable results when using classical tests based on these models. Then, even if a model is correctly chosen, the classical tests are based on first order asymptotic theory. This can lead to inaccurate p-values when the sample size is moderate or small. To overcome these problems, robust tests based on dual divergence estimators and saddlepoint approximations, with good performances in small samples, are proposed.     

Asymptotic Expansions for Large Deviation Probabilities of Noncentral Generalized Chi-Square Distributions

Asymptotic expansions for large deviation probabilities are used to approximate the cumulative distribution functions of noncentral generalized chi-square distributions, preferably in the far tails. The basic idea of how to deal with the tail probabilities consists in first rewriting these probabilities as large parameter values of the Laplace transform of a suitably defined function f_k; second making a series expansion of this function, and third applying a certain modification of Watson's lemma. The function f_k is deduced by applying a geometric representation formula for spherical measures to the multivariate domain of large deviations under consideration. At the so-called dominating point, the largest main curvature of the boundary of this domain tends to one as the large deviation parameter approaches infinity. Therefore, the dominating point degenerates asymptotically. For this reason the recent multivariate asymptotic expansion for large deviations in Breitung and Richter (1996, J. Multivariate Anal.58, 1–20) does not apply. Assuming a suitably parametrized expansion for the inverse g⁻¹ of the negative logarithm of the density-generating function, we derive a series expansion for the function f_k. Note that low-order coefficients from the expansion of g⁻¹ influence practically all coefficients in the expansion of the tail probabilities. As an application, classification probabilities when using the quadratic discriminant function are discussed.     

Finite-sample inference with monotone incomplete multivariate normal data, I

We consider problems in finite-sample inference with two-step, monotone incomplete data drawn from , a multivariate normal population with mean and covariance matrix . We derive a stochastic representation for the exact distribution of , the maximum likelihood estimator of . We obtain ellipsoidal confidence regions for through T², a generalization of Hotelling’s statistic. We derive the asymptotic distribution of, and probability inequalities for, T² under various assumptions on the sizes of the complete and incomplete samples. Further, we establish an upper bound for the supremum distance between the probability density functions of and , a normal approximation to .     

Tests for the mean direction of the Langevin distribution with large concentration parameter

In this paper we study the asymptotic behaviors of the likelihood ratio criterion (T_L^(s)), Watson statistic (T_W^(s)) and Rao statistic (T_R^(s)) for testing H_0s: μ (a given subspace) against H_1s: μ , based on a sample of size n from a p-variate Langevin distribution M_p(μ, κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ⁻¹. For the case when κ is unknown, it is shown that T_R^(s) T_L^(s) T_W^(s) in their powers up to the order κ⁻¹.