A general approach to optimal control of a regression experiment

Let $\text{θ}\text{θ}\text{?}$ = ($\text{θ}\text{θ}\text{?}$₁, $\text{θ}\text{θ}\text{?}$₂, …, $\text{θ}\text{θ}\text{?}$_n)′ be the least-squares estimator of θ = (θ₁, θ₂, …, θ_n)′ by the realization of the process y(t) = Σ_{k = 1}ⁿθ_kf_k(t) + ξ(t) on the interval T = [a, b] with f = (f₁, f₂, …, f_n)′ belonging to a certain set X. The process satisfies E(ξ(t))≡0 and has known continuous covariance r(s, t) = E(ξ(s)ξ(t)) on T × T. In this paper, A-, D-, and D_s-optimality are used as criteria for choosing f in X. A-, D-, and D_s-optimal models can be constructed explicitly by means of r.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

Increasing and decreasing operators on complete lattices

The (isotone) map f: X → X is an increasing (decreasing) operator on the poset X if f(x) ? f²(x) (f²(x) ? f(x), resp.) holds for each x ∈ X. Properties of increasing (decreasing) operators on complete lattices are studied and shown to extend and clarify those of closure (resp. anticlosure) operators. The notion of the decreasing closure, $\text{f}$, (the increasing anticlosure, $\text{f}$,) of the map f: X → X is introduced extending that of the transitive closure, $\text{f}\text{?}$, of f. $\text{f}\text{f}$, and $\text{f}$ are all shown to have the same set of fixed points. Our results enable us to solve some problems raised by H. Crapo. In particular, the order structure of H(X), the set of retraction operators on X is analyzed. For X a complete lattice H(X) is shown to be a complete lattice in the pointwise partial order. We conclude by claiming that it is the increasing-decreasing character of the identity maps which yields the peculiar properties of Galois connections. This is done by defining a u-v connection between the posets X and Y, where u: X → X (v: Y → Y) is an increasing (resp. decreasing) operator to be a pair f, g of maps f; X → Y, g: Y → X such that gf ? u, fg ? v. It is shown that the whole theory of Galois connections can be carried over to u-v connections.     

Definition and characterization of multivariate negative binomial distribution

The probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [β(s₁,…,s_n)]^?k where β is a polynomial of degree n being linear in each s_i and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x₁,…,x_n)$\text{exp}\text{(}\text{Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{θ}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}\text{)}\text{f(θ}{}_1\text{,…,θ}{}_{\mathrm{n}}\text{)}$ is negative binomial if and only if its univariate marginals are negative binomial. Let S_t, t = 1,…, m, be subsets of {s₁,…, s_n} with empty ∩_t=1^mS_t. Then an n-variate pgf is of a negative binomial if and only if for all s in S_t being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t.     

Positive dependence properties of elliptically symmetric distributions

Let X₁, …, X_p have p.d.f. g(x₁² + … + x_p²). It is shown that (a) X₁, …, X_p are positively lower orthant dependent or positively upper orthant dependent if, and only if, X₁,…, X_p are i.i.d. N(0, σ²); and (b) the p.d.f. of |X₁|,…, |X_p| is TP₂ in pairs if, and only if, In g(u) is convex. Let X₁, X₂ have p.d.f. $\text{f(x}{}_1\text{, x}{}_2\text{) =}\text{|Σ|}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{g((x}{}_1\text{, x}{}_2\text{)}\text{Σ}{}^{\mathrm{?1}}\text{(x}{}_1\text{, x}{}_2\text{)′)}$. Necessary and sufficient conditions are given for f(x₁, x₂) to be TP₂ for fixed correlation ?. It is shown that if f is TP₂ for all ? >0. then (X₁, X₂)′ ～ N(0, Σ). Related positive dependence results and applications are also considered.     

Hermite series estimates of a probability density and its derivatives

The following estimate of the pth derivative of a probability density function is examined: $\text{Σ}{}_{\mathrm{k\; =\; 0}}{}^{\mathrm{N}}\text{a}\text{?}{}_{\mathrm{k}}\text{h}{}_{\mathrm{k}}\text{(x)}$, where h_k is the kth Hermite function and $\text{a}\text{?}{}_{\mathrm{k}}\text{= (}\text{(?1)}{}^{\mathrm{p}}\text{n}\text{)}$Σ_{i = 1}ⁿh_k^(p)(X_i) is calculated from a sequence X₁,…, X_n of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n^?α) and O(n^?α log n), respectively, where $\text{α =}\text{2(r ? p)}\text{(2r + 1)}$. Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n^?β) and $\text{O(n}{}^{\mathrm{<mtext>?\beta </mtext><mtext>2</mtext>}}\text{log}\text{n)}$, respectively, where $\text{β =}\text{(2(r ? p) ? 1)}\text{(2r + 1)}$.     

A version of Chevet's theorem for stable processes

Let $\text{X}\text{}\text{?}\text{bo}\text{Y}$ be the injective tensor product of the separable Banach spaces X and Y and let S_X, S_Y and $\text{S}{}_{\mathrm{<mtext>X\; </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>bo\; Y</mtext>}}$ be the unit spheres of these spaces. The tensor product of two symmetric finite measures η₁ on S_X and η₂ on S_Y, η₁?η₂, is defined in a natural way as a measure on $\text{S}{}_{\mathrm{<mtext>X\; </mtext><mtext>\backslash </mtext><mtext>?</mtext><mtext>bo\; Y</mtext>}}$. It is shown that η₁? η₂ is the spectral measure of a p-stable random variable W on $\text{X}\text{}\text{?}\text{bo}\text{Y}$, 0 <p < 2, if and only if η₁ and η₂ are the spectral measures of p-stable random variables U and V on X and Y, respectively. Actually upper and lower bounds for $\text{(E∥ W∥}{}^{\mathrm{r}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}$ in terms of the random variables U and V are obtained. When X = C(S), Y = C(T) with S, T compact metric spaces, and η₁, and η₂ are discrete, our results imply that if θ_i, θ_ij are i.i.d. standard symmetric real valued p-stable random variables, 0 < p <2, x_i?C(S), and y_i?C(T), then the series ∑_ijθ_ijx_i(s) y_j(t) converges uniformly a.s. iff the series ∑_iθ_ix_i(s) and ∑_iθ_iy_i(t) both converge uniformly a.s. When p = 2 this follows from Chevet's theorem on Gaussian processes. Several examples are given. One of them requires an interesting upper bound on the probability distribution of the maximum of i.i.d. p-stable random variables taking values in a general Banach space.     

Probabilities of maximal deviations for nonparametric regression function estimates

Let (X, Y) have regression function m(x) = E(Y | X = x), and let X have a marginal density f₁(x). We consider two nonparameteric estimates of m(x): the Watson estimate when f₁ is known and the Yang estimate when f₁ is known or unknown. For both estimates the asymptotic distribution of the maximal deviation from m(x) is proved, thus extending results of Bickel and Rosenblatt for the estimation of density functions.     

The operator-valued Feynman-Kac formula with noncommutative operators

Let X(t) be a right-continuous Markov process with state space E whose expectation semigroup S(t), given by S(t) φ(x) = E_x[φ(X(t))] for functions φ mapping E into a Banach space L, has the infinitesimal generator A. For each x?E, let V(x) generate a strongly continuous semigroup T_x(t) on L. An operator-valued Feynman-Kac formula is developed and solutions of the initial value problem $\text{?u}\text{?t}\text{= Au + V(x)u, u(0) = φ}$ are obtained. Fewer conditions are assumed than in known results; in particular, the semigroups {T_x(t)} need not commute, nor must they be contractions. Evolution equation theory is used to develop a multiplicative operative functional and the corresponding expectation semigroup has the infinitesimal generator A + V(x) on a restriction of the domain of A.     

t-designs on hypergraphs

A $\text{t-}\text{design}\text{T=(X,}\text{B}\text{)}$, denoted by (λ; t, k, v), is a system $\text{B}$ of subsets of size k from a v-set X, such that each t-subset of X is contained in exactly λ elements of $\text{B}$. A hypergraph $\text{H=(Y,}\text{E}\text{)}$ is a finite set Y where $\text{E}\text{=(E}{}_{\mathrm{i}}\text{: i?I)}$ is a family of subsets (which we assume here are distinct) of Y such that E_i≠Ø, i?l, and ?E_i=Y. Let G be an automorphism group of $\text{H=(Y,}\text{E}\text{)}$ where O^l_i is the ith orbit of l-subsets of $\text{E}$. Let A(G; H; t, k)= (a_ij) be an m by n matrix, where a_ij is the number of copies of O^t_i that occur in the system of all t-subsets of all elements of O^k_j. Then there is a t-design $\text{T=(X,}\text{B}\text{)}$ with $\text{X=}\text{E}$, with parameters (λ; t, k, v), and with G an automorphism groupof T iff there is an m by s submatrix M of A(G; H; t, k) where M has uniform row sums λ. The calculus for applying this theorem is illustrated and numerous t-designs for 10?v?16 are found and presented. Using a theorem of Alltop on our (12; 4, 6, 13) and (60; 4, 7, 15) we obtain a (12; 5, 7, 14) and a (60; 5, 8, 16).     

Some properties of bivariate Gumbel Type A distributions with proportional hazard rates

We call a set of univariate distributions with the same mathematical form but different parameter values a family $\text{J}$. Consider a bivariate Gumbel Type A survival distribution, S₁₂(x₁, x₂), defined in (2.1), for which both marginal distributions, S₁(x₁), S₂(x₂), belong to the same family, $\text{J}$ of distributions. It is proved in this paper that subject to weak conditions, the crude hazard rates, h₁(t) and h₂(t), are proportional if and only if the marginal hazard rates, λ₁(t) and λ₂(t), are proportional (Theorem 1). It is also shown that the survival functions of W = min(X₁, X₂), and of the identified minimum, W_i = X_i, for X_i < X_j, j ≠ i, belong to the same family $\text{J}$ as do S₁(x₁), S₂(x₂) (Corollary 1). Counter-examples of distributions other than Gumbel Type A, for which these properties do not hold, are given. Some applications to the analysis of competing risks, using a family of Gompertz distributions, are discussed.     

Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments

The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) L_nx(t) + δq(t) f(x[g₁(t)],…, x[g_m(t)]) = 0, where δ = ± 1 and L₀x(t) = x(t), L_kx(t) = a_k(t)(L_{k ? 1}x(t))^., $\text{k = 1, 2,…, n (}{}^{\mathrm{.}}\text{=}\text{d}\text{dt}\text{)}$, is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y^.(t) + c₁(t) f(y[g₁(t)],…, y[g_m(t)]) = 0 and (a_{n ? 1}(t)z^.(t))^. ? c₂(t) f(z[g₁(t)],…, z[g_m(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos.     

On the sequence of Bayesian estimates of normal random walk process

Let (μ_t)^∞_t=0 be a k-variate (k?1) normal random walk process with successive increments being independently distributed as normal N(δ, R), and μ₀ being distributed as normal N(0, V₀). Let X_t have normal distribution N(μ_t, Σ) when μ_t is given, t = 1, 2,….Then the conditional distribution of μ_t given X₁, X₂,…, X_t is shown to be normal N(U_t, V_t) where U_t's and V_t's satisfy some recursive relations. It is found that there exists a positive definite matrix V and a constant θ, 0 < θ < 1, such that, for all t?1,

$\text{|R}\text{1}\text{2}\text{(V}{}^{\mathrm{?1}}{}_{\mathrm{t}}\text{?V}{}^{\mathrm{?1}}\text{R}\text{1}\text{2}\text{|<θ}{}^{\mathrm{t}}\text{|R}\text{1}\text{2}\text{(V}{}^{\mathrm{?1}}{}_0\text{?V}{}^{\mathrm{?1}}\text{)R}\text{1}\text{2}\text{|}$

where the norm |·| means that |A| is the largest eigenvalue of a positive definite matrix A. Thus, V_t approaches to V as t approaches to infinity. Under the quadratic loss, the Bayesian estimate of μ_t is U_t and the process {U_t}^∞_t=₀, U₀=0, is proved to have independent successive increments with normal N(θ, V_t?V_t+1+R) distribution. In particular, when V₀ =V then V_t = V for all t and {U_t}^∞_t=0 is the same as {μ_t}^∞_t=0 except that U₀ = 0 and μ₀ is random.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions

Let X be a p-dimensional random vector with density f(6X?θ6) where θ is an unknown location vector. For p ≥ 3, conditions on f are given for which there exist minimax estimators θ?(X) satisfying 6Xt6 · 6θ?(X) ? X6 ≤ C, where C is a known constant depending on f. (The positive part estimator is among them.) The loss function is a nondecreasing concave function of 6θ?? θ6². If θ is assumed likely to lie in a ball in $\text{R}$^p, then minimax estimators are given which shrink from the observation X outside the ball in the direction of P(X) the closest point on the surface of the ball. The amount of shrinkage depends on the distance of X from the ball.     

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on $\text{X}$ = $\text{X}$₁ × $\text{X}$₂ × … × $\text{X}$_n where each $\text{X}$_i is totally ordered satisfying f(x ∨ y) f(x ∧ y) ≥ f(x) f(y), where the lattice operations ∨ and ∧ refer to the usual ordering on $\text{X}$, is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies ?DΣ^?1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

A generalization of the Kruskal-Katona theorem

Let k_n ? k_n?1 ? … ? k₁ be positive integers and let ($\text{i}\text{j}$) denote the coefficient of xⁱ in $\text{Π}\text{r=1}\text{j}\text{(1 + x + x}{}^2\text{+ … + x}{}^{\mathrm{kr}}\text{)}$. For given integers l, m, where 1 ? l ? k_n + k_n?1 + … + k₁ and $\text{1 ? m ? (}\text{n}\text{n}\text{)}$, it is shown that there exist unique integers m(l), m(l ? 1),…, m(t), satisfying certain conditions, for which $\text{m = (}\text{m(l)}\text{l}\text{+ (}\text{m(l?1)}\text{l?1}\text{) + … + (}\text{m(t)}\text{t}\text{)}$. Moreover, any m l-subsets of a multiset with k_i elements of type i, i = 1, 2,…, n, will contain at least $\text{(}\text{m(l)}\text{l?1}\text{) + (}\text{m(l?1)}\text{l?2}\text{) + … + (}\text{m(t)}\text{t?1}$ different (l ? 1)-subsets. This result has been anticipated by Greene and Kleitman, but the formulation there is not completely correct. If k₁ = 1, the numbers ($\text{j}\text{i}$) are binomial coefficients and the result is the Kruskal-Katona theorem.     

Construction of translation planes from t-spread sets

A t-spread set [1] is a set $\text{C}$ of (t + 1) × (t + 1) matrices over GF(q) such that ∥C∥ = q^t+1, 0 ? C, I?C, and det(X ? Y) ≠ 0 if X and Y are distinct elements of $\text{C}$. The amount of computation involved in constructing t-spread sets is considerable, and the following construction technique reduces somewhat this computation. Construction: Let $\text{G}$ be a subgroup of GL(t + 1, q), (the non-singular (t + 1) × (t + 1) matrices over GF(q)), such that ∥G∥|a^t+1, and det (G ? H) ≠ 0 if G and H are distinct elements of $\text{G}$. Let A₁, A₂, …, A_n?GL(t + 1, q) such that det(A_i ? G) ≠ 0 for i = 1, …, n and all G?G, and det(A_i ? A_jG) ≠ 0 for i > j and all G?G. Let $\text{C = \&{0\&} ∪ G ∪ A}{}_1\text{G ∪ … ∪ A}{}_{\mathrm{n}}\text{G}$, and ∥C∥ = q^t+1. Then $\text{C}$ is a t-spread set. A t-spread set can be used to define a left V ? W system over V(t + 1, q) as follows: x + y is the vector sum; let e?V(t + 1, q), then xoy = yM(x) where M(x) is the unique element of $\text{C}$ with x = eM(x). Theorem: Let$\text{C}$be a t-spread set and F the associatedV ? Wsystem; the left nucleus = {y | CM(y) = C}, and the middle nucleus = }y | M(y)C = C}. Theorem: For$\text{C}$constructed as aboveG ? {M(x) | x?N_λ}. This construction technique has been applied to construct a V ? W system of order 25 with ∥N_λ∥ = 6, and ∥N_μ∥ = 4. This system coordinatizes a new projective plane.     

Reduction of variance for Gaussian densities via restriction to convex sets

Let X be a random vector with values in $\text{R}$ⁿ and a Gaussian density f. Let Y be a random vector whose density can be factored as k · f, where k is a logarithmically concave function on $\text{R}$ⁿ. We prove that the covariance matrix of X dominates the covariance matrix of Y by a positive semidefinite matrix. When k is the indicator function of a compact convex set A of positive measure the difference is positive definite. If A and X are both symmetric Var(a · X) is bounded above by an expression which is always strictly less than Var(a · X) for every a ∈ $\text{R}$ⁿ. Finally some counterexamples are given to show that these results cannot be extended to the general case where f is any logarithmically concave density.     

On a generalization of linear independence in finite-dimensional vector spaces

A set X in a vector space V is said to be k-independent (where k is a positive integer) if, for each x?X,X{x} admits a partition into k subsets {χ_gq}_θ=1,…,k such that x ? span χ_θ, θ = 1,…,k. It is proved that if dim V = n and X ? V is k-independent, then X cannot contain more than $\text{(}\text{n+k?1}\text{k}\text{)}$ elements; this bound is sharp for vector spaces over sufficiently large fields. A broader notion (k-independence in degrees) is considered, and similar results are obtained. Several unresolved problems are stated, some involving matroid generalizations of questions answered as yet only within vector space context.