On improved estimators of the generalized variance

Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p W_p(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ I_k) with ξ(p × k) unknown. Denote the columns of X by X₍₁₎ ,…, X_(k) and set ψ⁽⁰⁾(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ⁽ⁱ⁾(X, X) = min[ψ⁽ⁱ⁻¹⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |] and Ψ⁽ⁱ⁾(S, X) = min[ψ⁽⁰⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ⁽⁰⁾(S, X) is dominated by each of Ψ⁽ⁱ⁾(S, X), i = 1,…,k and for every i, ψ⁽ⁱ⁾(S, X) is better than ψ⁽ⁱ⁻¹⁾(S, X). In particular, ψ^(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X₍₁₎X′₍₁₎|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X₍₁₎X′₍₁₎ + + X_(k)X′_(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.     

A connection between a generalized Pascal matrix and the hypergeometric function

The n × n generalized Pascal matrix P(t) whose elements are related to the hypergeometric function ₂F₁(a, b; c; x) is presented and the Cholesky decomposition of P(t) is obtained. As a result, it is shown that

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is the solution of the Gauss's hypergeometric differential equation,

x(1 − x)y″ + [1 + (a + b − 1)x]y′ − ABY = 0

. where a and b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of P(t) is given.     

Asymptotic properties of Cramér-Smirnov statistics—A new approach

Let Y₁,…, Y_n be independent identically distributed random variables with distribution function F(x, θ), θ = (θ′₁, θ′₂), where θ_i (i = 1, 2) is a vector of p_i components, p = p₁ + p₂ and for θI, an open interval in ^p, F(x, θ) is continuous. In the present paper the author shows that the asymptotic distribution of modified Cramér-Smirnov statistic under H_n: θ₁ = θ₁₀ + n^−1/2γ, θ₂ unspecified, where γ is a given vector independent of n, is the distribution of a sum of weighted noncentral χ₁² variables whose weights are eigenvalues of a covariance function of a Gaussian process and noncentrality parameters are Fourier coefficients of the mean function of the Gaussian process. Further, the author exploits the special form of the covariance function by using perturbation theory to obtain the noncentrality parameters and the weights. The technique is applicable to other goodness-of-fit statistics such as U² [G. S. Watson, Biometrika 48 (1961), 109–114].     

Prophet regions and sharp inequalities for pth absolute moments of martingales

Exact comparisons are made relating E|Y₀|^p, E|Y_n−1|^p, and E(max_j≤n−1 |Y_j|^p), valid for all martingales Y₀,…,Y_n−1, for each p ≥ 1. Specifically, for p > 1, the set of ordered triples {(x, y, z) : X = E|Y₀|^p, Y = E |Y_n−1|^p, and Z = E(max_j≤n−1 |Y_j|^p) for some martingale Y₀,…,Y_n−1} is precisely the set {(x, y, z) : 0≤x≤y≤z≤Ψ_n,p(x, y)}, where Ψ_n,p(x, y) = xψ_n,p(y/x) if x > 0, and = a_n−1,py if x = 0; here ψ_n,p is a specific recursively defined function. The result yields families of sharp inequalities, such as E(max_j≤n−1 |Y_j|^p) + ψ_n,p^*(a) E |Y₀|^p ≤ aE |Y_n−1|^p, valid for all martingales Y₀,…,Y_n−1, where ψ_n,p^* is the concave conjugate function of ψ_n,p. Both the finite sequence and infinite sequence cases are developed. Proofs utilize moment theory, induction, conjugate function theory, and functional equation analysis.     

On computing the dispersion function

A two-dimensional analogue of the well-known bisection method for root finding is presented in order to solve the following problem, related to the dispersion function of a set of random variables: given distribution functionsF ₁,...,F _n and a probabilityp, find an interval [a,b] of minimum width such thatF _i(b)–F _i(a ^–)p, fori=1,...,n.The author wishes to thank Dr. I. D. Coope, for helpful advice offered during the preparation of this paper, and the referee, whose comments contributed to a clearer presentation.     

On the Strong Law of Large Numbers and the Law of the Logarithm for Weighted Sums of Independent Random Variables with Multidimensional Indices

Let (X, X ; ^d} be a field of independent identically distributed real random variables, 0 < p < 2, and {a _, ; ( , ) ^d × ^d, ≤ } a triangular array of real numbers, where ^d is the d-dimensional lattice. Under the minimal condition that sup _, |a _, | < ∞, we show that | |^{− 1/p} ∑ _≤ a _, X → 0 a.s. as | | → ∞ if and only if E(|X|^p(L|X|)^{d − 1}) < ∞ provided d ≥ 2. In the above, if 1 ≤ p < 2, the random variables are needed to be centered at the mean. By establishing a certain law of the logarithm, we show that the Law of the Iterated Logarithm fails for the weighted sums ∑ _≤ a _, X under the conditions that EX = 0, EX² < ∞, and E(X²(L|X|)^{d − 1}/L₂|X|) < ∞ for almost all bounded families {a _, ; ( , ) ^d × ^d, ≤ of numbers.     

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.     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

Extremal problems concerning Kneser graphs

Let and be two intersecting families of k-subsets of an n-element set. It is proven that | | ≤ (_k−1ⁿ⁻¹) + (_k−1ⁿ⁻¹) holds for , and equality holds only if there exist two points a, b such that {a, b} ∩ F ≠ for all F . For an example showing that in this case max | | = (1−o(1))(_kⁿ) is given. This disproves an old conjecture of Erdös [7]. In the second part we deal with several generalizations of Kneser's conjecture.     

A weak law for normed weighted sums of random elements in rademacher type p banach spaces

For weighted sums Σ_{j = 1}ⁿa_jV_j of independent random elements {V_n, n ≥ 1} in real separable, Rademacher type p (1 ≤ p ≤ 2) Banach spaces, a general weak law of large numbers of the form (Σ_{j = 1}ⁿa_jV_j − v_n)/b_n →^p 0 is established, where {v_n, n ≥ 1} and b_n → ∞ are suitable sequences. It is assumed that {V_n, n ≥ 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of |V| and the growth behaviors of the constants {a_n, n ≥ 1} and {b_n, n ≥ 1}. No assumption is made concerning the existence of expected values or absolute moments of the {V_n, n >- 1}.     

Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

For X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δ_a,r,C,D(X) = (I − (ar(|X|²)) D^−1/2CD^1/2 |X|⁻²)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X₁, X₂, …, X_n are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X₁, X₂, …, X_n), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.     

Two New Classes of Difference Families

A Z-cyclic triplewhist tournament for 4n+1 players, or briefly a TWh(4n+1), is equivalent to a n-set {(a_i, b_i, c_i, d_i) | i=1, …, n} of quadruples partitioning Z_4n+1−{0} with the property that ⁿ_i=1 {±(a_i−c_i), ±(b_i−d_i)}=ⁿ_i=1 {±(a_i−b_i), ±(c_i−d_i)}=ⁿ_i=1 {±(a_i−d_i), ±(b_i−c_i)}=Z_4n+1−{0}. The existence problem for Z-cyclic TWh(p)'s with p a prime has been solved for p1 (mod 16). I. Anderson et al. (1995, Discrete Math.138, 31–41) treated the case of p≡5 (mod 8) while Y. S. Liaw (1996, J. Combin. Des.4, 219–233) and G. McNay (1996, Utilitas Math.49, 191–201) treated the case of p≡9 (mod 16). In this paper, besides giving easier proofs of these authors' results, we solve the problem also for primes p≡1 (mod 16). The final result is the existence of a Z-cyclic TWh(v) for any v whose prime factors are all≡1 (mod 4) and distinct from 5, 13, and 17.     

Arbitrariness of the pilot estimator in adaptive kernel methods

Consider estimating a smooth p-variate density f at 0 using the classical kernel estimator f_n(0) = n⁻¹ Σ_ib_n^−pw(b_n⁻¹X_i) based on a sample {X_i} from f. Under familiar conditions, assigning b_n = bn^{−1/(4 + p)} gives the best MSE decay rate O(n^{−4/(4 + p)}, but the optimal b, b* say, depends on f through its second derivatives, raising a feasibility objection to its use. By prescribing a pilot estimate of b* based on the same sample, Woodroofe has shown that there need be asymptotically no loss as against knowing the constant exactly, but his proposal is critically dependent on achieving a certain consistency rate for b*. Admitting a minor change in the risk function, we show by a tightness argument applied to the error process that any consistent estimator of b* may be used to achieve the same performance.     

Relative -difference sets in p-subgroups of

In this note, we study relative (p^a,p^b,p^a,p^a−b)-relative difference sets in certain p-subgroups of SL(n,K), K=F_q, where q is a prime power.     

Characterization of generalized convex functions by their best approximation in sign-monotone norms

Let {u₀, u₁,… u_{n − 1}} and {u₀, u₁,…, u_n} be Tchebycheff-systems of continuous functions on [a, b] and let f ε C[a, b] be generalized convex with respect to {u₀, u₁,…, u_{n − 1}}. In a series of papers ([1], [2], [3]) D. Amir and Z. Ziegler discuss some properties of elements of best approximation to f from the linear spans of {u₀, u₁,…, u_{n − 1}} and {u₀, u₁,…, u_n} in the L_p-norms, 1 p ∞, and show (under different conditions for different values of p) that these properties, when valid for all subintervals of [a, b], can characterize generalized convex functions. Their methods of proof rely on characterizations of elements of best approximation in the L_p-norms, specific for each value of p. This work extends the above results to approximation in a wider class of norms, called “sign-monotone,” [6], which can be defined by the property: ¦ f(x)¦ ¦ g(x)¦,f(x)g(x) 0, a x b, imply f g . For sign-monotone norms in general, there is neither uniqueness of an element of best approximation, nor theorems characterizing it. Nevertheless, it is possible to derive many common properties of best approximants to generalized convex functions in these norms, by means of the necessary condition proved in [6]. For {u₀, u₁,…, u_n} an Extended-Complete Tchebycheff-system and f ε C⁽ⁿ⁾[a, b] it is shown that the validity of any of these properties on all subintervals of [a, b], implies that f is generalized convex. In the special case of f monotone with respect to a positive function u₀(x), a converse theorem is proved under less restrictive assumptions.     

Complex Roots of a Random Algebraic Polynomial

This paper, for any constantK, provides an exact formula for the average density of the distribution of the complex roots of equation η₀ + η₁z + η₂z² + ··· + η_{n − 1}z^{n − 1} = Kwhere η_j = a_j + ib_jand {a_j}^{n − 1}_{j = 0}and {b_j}^{n − 1}_{j = 0}are sequences of independent identically and normally distributed random variables andKis a complex number withKas its real and imaginary parts. The case of real roots of the above equation with real coefficients andK,z Ris well known. Further we obtain the limiting behaviour of this distribution function asntends to infinity.     

Simultaneous similarity of matrices

In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT⁻¹, TBT⁻¹). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set ⁰_n,2,r,π M_n × M_n (M_n = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø₁,…,ø_s in the entries of A and B whose values are constant on all pairs similar in _n,2,r,π to (A, B). The values of the functions ø_i(A, B), I = 1,…, s, determine a finite number (at most κ(n, 2, r)) of similarity classes in _n,2,r,π. Let S_n be the subspace of complex symmetric matrices in M_n. For (A, B) ε S_n × S_n we consider the similarity class (TAT^t, TBT^t), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial |λI − (A + xB)| determines a finite number of similarity classes for almost all pairs (A, B) ε S_n × S_n.     

Roots under convolution of sequences

The convolution a * b of the sequences a = a₀, a₁, a₂, and b is the sequence with elements ∑₀ⁿ a_kb_{n − k}. One sets 1, 1, 1, equal to σ. Given that a * a with a ≥ 0 is close to σ * σ, how close is a to σ? More generally, one asks how close a is to σ if the p-th convolution power, a*P with a ≥ 0, is close to σ*P. Power series and complex analysis form a natural tool to estimate the ‘summed deviation’ ρ = σ * (a — σ) in terms of b = a * a — σ * σ or b = a*P − σ*P. Optimal estimates are found under the condition ∑_k=0ⁿ b_k² = %plane1D;512;(n^{2β + 1}) whenever −½ < β < p − 1. It is not known what the optimal estimates are for the special case b_n = %plane1D;512;(n^β).     

Équations du type Monge-Ampère sur les variétés Riemanniennes compactes, I

Let (V_n, g) be a C^∞ compact Riemannian manifold. For a suitable function on V_n, let us consider the change of metric: g′ = g + Hess(), and the function, as a ratio of two determinants, M() = ¦g′¦ ¦g¦⁻¹. Using the method of continuity, we first solve in C^∞ the problem: Log M() = λ + ƒ, λ > 0, ƒ ε C^∞. Then, under weak hypothesis on F, we solve the general equation: Log M() = F(P, ), F in C^∞(V_n × ¦α, β¦), using a method of iteration. Our study gives rise to an interesting a priori estimate on ¦¦, which does not occur in the complex case. This estimate should enable us to solve the equation above when λ 0, providing we can overcome difficulties related to the invertibility of the linearised operator. This open question will be treated in our next article.     

On Rado numbers for

For all integers m3 and all natural numbers a₁,a₂,…,a_m−1, let n=R(a₁,a₂,…,a_m−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to

a₁x₁+a₂x₂++a_m−1x_m−1=x_m.