Second-order linearity of the general signed-rank statistic

Let X₁,…, X_n be i.i.d. random variables symmetric about zero. Let R_i(t) be the rank of |X_i − tn^−1/2| among |X₁ − tn^−1/2|,…, |X_n − tn^−1/2| and T_n(t) = Σ_{i = 1}ⁿφ((n + 1)⁻¹R_i(t))sign(X_i − tn^−1/2). We show that there exists a sequence of random variables V_n such that sup_{0 ≤ t ≤ 1} |T_n(t) − T_n(0) − tV_n| → 0 in probability, as n → ∞. V_n is asymptotically normal.     

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.     

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.     

L1-optimal estimates for a regression type function in R

Let X₁, X₂, …, X_n be random vectors that take values in a compact set in R^d, d ≥ 1. Let Y₁, Y₂, …, Y_n be random variables (“the responses”) which conditionally on X₁ = x₁, …, X_n = x_n are independent with densities f(y | x_i, θ(x_i)), i = 1, …, n. Assuming that θ lives in a sup-norm compact space Θ_q,d of real valued functions, an optimal L₁-consistent estimator of θ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter θ depends on Kolmogorov's entropy of Θ_q,d.     

The Pointwise Densities of the Cantor Measure

Let be the classical middle-third Cantor set and let μ be the Cantor measure. Set s = log 2/log 3. We will determine by an explicit formula for every point x the upper and lower s-densities Θ*^s(μ , x), Θ*^s(μ , x) of the Cantor measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F such that 9(Θ*^s(μ , x))^{− 1/s} + (Θ*^s(μ , x))^{− 1/s} = 16 holds for x \F. Furthermore, for μ_C almost all x, Θ*^s(μ , X) − 2 · 4^{− s} and Θ*^s(μ , x) = 4^{− s}. As an application, we will show that the s-dimensional packing measure of the middle-third Cantor set is 4^s.     

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.     

Multifunctions of faces for conditional expectations of selectors and Jensen's inequality

Let (T, , P) be a probability space, a P-complete sub-δ-algebra of and X a Banach space. Let multifunction t → Γ(t), t T, have a (X)-measurable graph and closed convex subsets of X for values. If x(t) ε Γ(t) P-a.e. and y(·) ε E_p x(·), then y(t) ε Γ(t) P-a.e. Conversely, x(t) ε F(Γ(t), y(t)) P-a.e., where F(Γ(t), y(t)) is the face of point y(t) in Γ(t). If X = , then the same holds true if Γ(t) is Borel and convex, only. These results imply, in particular, extensions of Jensen's inequality for conditional expectations of random convex functions and provide a complete characterization of the cases when the equality holds in the extended Jensen inequality.     

On deviations between empirical and quantile processes for mixing random variables

Let {X_n} be a strictly stationary φ-mixing process with Σ_j=1^∞ φ^1/2(j) < ∞. It is shown in the paper that if X₁ is uniformly distributed on the unit interval, then, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = (O(n^−3/4(log n)^1/2(log log n)^1/4) a.s., where F_n and F_n⁻¹(t) denote the sample distribution function and tth sample quantile, respectively. In case {X_n} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)^1/2(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)(log log n)^1/4) a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.     

Rates for approximation of unbounded functions by positive linear operators

Let g_a(t) and g_b(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {L_n} be a sequence of linear positive operators, each with domain containing 1, t, g_a(t), and g_b(t). If L_n(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, g_a(t), g_b(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(g_a(t)) (t → a⁺) and ƒ(t) = O(g_b(t)) (t → b⁻). We estimate the convergence rate of L_nƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′.     

Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.     

Exponential bounds of mean error for the kernel estimates of regression functions

Let (X, Y), (X₁, Y₁), …, (X_n, Y_n) be i.d.d. R^r × R-valued random vectors with E|Y| < ∞, and let Q_n(x) be a kernel estimate of the regression function Q(x) = E(Y|X = x). In this paper, we establish an exponential bound of the mean deviation between Q_n(x) and Q(x) given the training sample Zⁿ = (X₁, Y₁, …, X_n, Y_n), under conditions as weak as possible.     

A uniform bound for the deviation of empirical distribution functions

If X₁, …, X_n are independent R^d-valued random vectors with common distribution function F, and if F_n is the empirical distribution function for X₁, …, X_n, then, among other things, it is shown that P{sup_x F_n(x) ε} 2e²(2n)^de^−2nε2 for all nε² ≥ d². The inequality remains valid if the X_i are not identically distributed and F(x) is replaced by Σ_iP{X_i ≤ x}/n.     

A note on asymptotic expansions for Markov chains using operator theory

We consider asymptotic expansions for sums S_n on the form S_n = ƒ₀(X₀) + ƒ(X₁, X₀) + … + ƒ(X_n, X_n−1), where X_i is a Markov chain. Under different ergodicity conditions on the Markov chain and certain conditional moment conditions on ƒ(X_i, X_i−1), a simple representation of the characteristic function of S_n is obtained. The representation is in term of the maximal eigenvalue of the linear operator sending a function g(x) into the function x → E(g(X_i)exp[itƒ(X_i, x)]|X_i−1 = x).     

Admissibility and complete class results for the multinomial estimation problem with entropy and squared error loss

Let X ≡ (X₁, …, X_t) have a multinomial distribution based on N trials with unknown vector of cell probabilities p ≡ (p₁, …, p_t). This paper derives admissibility and complete class results for the problem of simultaneously estimating p under entropy loss (EL) and squared error loss (SEL). Let and f(x¦p) denote the (t − 1)-dimensional simplex, the support of X and the probability mass function of X, respectively. First it is shown that δ is Bayes w.r.t. EL for prior P if and only if δ is Bayes w.r.t. SEL for P. The admissible rules under EL are proved to be Bayes, a result known for the case of SEL. Let Q denote the class of subsets of of the form T = _j=1^kF_j where k ≥ 1 and each F_j is a facet of which satisfies: F a facet of such that F naF_jF ncT. The minimal complete class of rules w.r.t. EL when N ≥ t − 1 is characterized as the class of Bayes rules with respect to priors P which satisfy P( ⁰) = 1, ξ(x) ≡ ∫ f(x¦p) P(dp) > 0 for all x in {x : sup ^₀ f(x¦p) > 0} for some ⁰ in Q containing all the vertices of . As an application, the maximum likelihood estimator is proved to be admissible w.r.t. EL when the estimation problem has parameter space Θ = but it is shown to be inadmissible for the problem with parameter space Θ = ( minus its vertices). This is a severe form of “tyranny of boundary.” Finally it is shown that when N ≥ t − 1 any estimator δ which satisfies δ(x) > 0 x is admissible under EL if and only if it is admissible under SEL. Examples are given of nonpositive estimators which are admissible under SEL but not under EL and vice versa.     

Estimates of Freud-Christoffel functions for some weights with the whole real line as support

Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λ_n,p(W,j,x) = infPW_Lp(R)/|P^(j)(X)| where the infimum is taken over all polynomials P(x) of degree at most n − 1. The upper and lower bounds for λ_n,p(W,j,x) are obtained for all 0 < p ∞ and J = 0, 1, 2, 3,… for weights W(x) = exp(−Q(x)), where, among other things, Q(x) is bounded in [− A, A], and Q″ is continuous in β(−A, A) for some A > 0. For p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For p = 2, the results remove some restrictions on Q in Freud's work. The weights considered include W(x) = exp(− ¦x¦^α/2), α > 0, and W(x) = exp(− exp(¦x¦)), > 0.     

Theorems of large deviations in the multivariate invariance principle

Let B be a real separable Banach space with norm |ß|_B, X, X₁, X₂, … be a sequence of centered independent identically distributed random variables taking values in B. Let s_n = s_n(t), 0 ≤ t ≤ 1 be the random broken line such that s_n(0) = 0, s_n(k/n) = n^−1/2 Σ_i=1^k X_i for n = 1, 2, … and k = 1, …, n. Denote |s_n|_B = sup_{0 ≤ t ≤ 1} |s_n(t)|_B and assume that w(t), 0 ≤ t ≤ 1 is the Wiener process such that covariances of w(1) and X are equal. We show that under appropriate conditions P(|s_n|_B > r) = P(|w|_B > r)(1 + o(1)) and give estimates of the remainder term. The results are new already in the case of B having finite dimension.     

Prophet inequalities for cost of observation stopping problems

Comparisons are made between the expected gain of a prophet (an observer with complete foresight) and the maximal expected gain of a gambler (using only non-anticipating stopping times) observing a sequence of independent, uniformly bounded random variables where a non-negative fixed cost is charged for each observation. Sharp universal bounds are obtained under various restrictions on the cost and the length of the sequence. For example, it is shown for X₁, X₂, … independent, [0, 1]-valued random variables that for all c ≥ 0 and all n ≥ 1 that E(max_{1 ≤ j ≤ n}(X_j − jc)) − sup_{t Tn} E(X_t − tc) ≤ 1/e, where T_n is the collection of all stopping times t which are less than or equal to n almost surely.     

Limit distributions and one-parameter groups of linear operators on Banach spaces

Let = {U_t: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set (X) of all probability measures on X. By (Q; ) we denote the class of all limit measures of {U_tn(μ₁ * μ₂*…*μ_n)*δ_xn}, where {μ_n}Q, {x_n}X and measures U_tnμ_j (j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes L_m( ) as follows: L₀( )= ( (X); ), L_m( )= (L_m−1( ); ) for m=1, 2,… and L_∞( )=_m=0^∞L_m( ). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from L_m( ), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators.     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.