Canonical row-column-exchangeable arrays

Consider a standard row-column-exchangeable array X = (X_ij : i,j ≥ 1), i.e., X_ij = f(a, ξ_i, η_j, λ_ij) is a function of i.i.d. random variables. It is shown that there is a canonical version of X, X′, such that X′, and α′, ξ′₁, ξ′₂,…, η′₁, η′₂,…, are conditionally independent given ∩_{n ≥ 1}σ(X′_ij : max(i,j) ≥ n). This result is quite a bit simpler to prove than the analogous result for the original array X, which is due to Aldous.     

On the unlinking conjecture of independent polynomial functions

The celebrated U-conjecture states that under the N_n(0,I_n) distribution of the random vector X=(X₁,…,X_n) in Rⁿ, two polynomials P(X) and Q(X) are unlinkable if they are independent [see Kagan et al., Characterization Problems in Mathematical Statistics, Wiley, New York, 1973]. Some results have been established in this direction, although the original conjecture is yet to be proved in generality. Here, we demonstrate that the conjecture is true in an important special case of the above, where P and Q are convex nonnegative polynomials with P(0)=0.     

Adaptive confidence region for the direction in semiparametric regressions

In this paper we aim to construct adaptive confidence region for the direction of ξ in semiparametric models of the form Y=G(ξ^TX,ε) where G(⋅) is an unknown link function, ε is an independent error, and ξ is a p_n×1 vector. To recover the direction of ξ, we first propose an inverse regression approach regardless of the link function G(⋅); to construct a data-driven confidence region for the direction of ξ, we implement the empirical likelihood method. Unlike many existing literature, we need not estimate the link function G(⋅) or its derivative. When p_n remains fixed, the empirical likelihood ratio without bias correlation can be asymptotically standard chi-square. Moreover, the asymptotic normality of the empirical likelihood ratio holds true even when the dimension p_n follows the rate of p_n=o(n^1/4) where n is the sample size. Simulation studies are carried out to assess the performance of our proposal, and a real data set is analyzed for further illustration.     

Optimal designs for nonparametric kernel regression

We consider the fixed design regression model Y_i = g(t_i) + ξ_i, i = 1, …, n, where ξ_i are (not necessarily i.i.d.) no variables, t_i constitute the design points where nonrepeatable measurements are to be taken and Y_i are the observations from which g and its derivatives are to be estimated. The dependency of the Integrated Mean Squared Error of two different types of kernel estimates on the design {t₁, …, t_n} is established. This allows the derivation of asymptotically optimal designs.     

Robustness of A-optimal designs

Suppose that Y=(Y_i) is a normal random vector with mean Xb and covariance σ²I_n, where b is a p-dimensional vector (b_j),X=(X_ij) is an n×p matrix. A-optimal designs X are chosen from the traditional set D of A-optimal designs for ρ=0 such that X is still A-optimal in D when the components Y_i are dependent, i.e., for i≠i′, the covariance of Y_i,Y_i^′ is ρ with ρ≠0. Such designs depend on the sign of ρ. The general results are applied to X=(X_ij), where X_ij∈{-1,1}; this corresponds to a factorial design with -1,1 representing low level or high level respectively, or corresponds to a weighing design with -1,1 representing an object j with weight b_j being weighed on the left and right of a chemical balance respectively.     

A test of location for exchangeable multivariate normal data with unknown correlation

We consider the problem of testing whether the common mean of a single n-vector of multivariate normal random variables with known variance and unknown common correlation ρ is zero. We derive the standardized likelihood ratio test for known ρ and explore different ways of proceeding with ρ unknown. We evaluate the performance of the standardized statistic where ρ is replaced with an estimate of ρ and determine the critical value c_n that controls the type I error rate for the least favorable ρ in [0,1]. The constant c_n increases with n and this procedure has pathological behavior if ρ depends on n and ρ_n converges to zero at a certain rate. As an alternate approach, we replace ρ with the upper limit of a (1−β_n) confidence interval chosen so that c_n=c for all n. We determine β_n so that the type I error rate is exactly controlled for all ρ in [0,1]. We also investigate a simpler approach where we bound the type I error rate. The former method performs well for all n while the less powerful bound method may be a useful in some settings as a simple approach. The proposed tests can be used in different applications, including within-cluster resampling and combining exchangeable p-values.     

On the Laczkovich-Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supn∈HAn is uncountable for every H∈ω[N] then _?n∈GAn is uncountable for some G∈ω[N]. This fact, by our definition, means that the σ-ideal [X]^?ω has property (LK). We prove that every σ-ideal generated by X/E has property (LK), for an equivalence relation E⊂X² of type Fσ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.     

Empirical likelihood confidence region for parameter in the errors-in-variables models

This paper proposes a constrained empirical likelihood confidence region for a parameter β₀ in the linear errors-in-variables model: Y_i=x_i^τβ₀+ε_i,X_i=x_i+u_i,(1?i?n), which is constructed by combining the score function corresponding to the squared orthogonal distance with a constrained region of β₀. It is shown that the coverage error of the confidence region is of order n⁻¹, and Bartlett corrections can reduce the coverage errors to n⁻². An empirical Bartlett correction is given for practical implementation. Simulations show that the proposed confidence region has satisfactory coverage not only for large samples, but also for small to medium samples.     

Weak regularity and consecutive topologizations and regularizations of pretopologies

L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ-compact pretopology. On the other hand, it is proved that for each n<ω there is a (regular) pretopology ρ (on a set of cardinality c) such that k(RT)ρ>n(RT)ρ for each k<n and n(RT)ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT-order ?ω₀. Moreover, all these pretopologies have the property that all the points except one are topological and regular.     

Characterizations and closure under convolution of two classes of multivariate life distributions

Let ?(η) be the class of positive random vectors $\text{T}$ for which min_1?i?nα_iT_i is IFRA (NBU) for all α_i > 0, i = 1,…,n where n is an arbitrary positive integer. Characterizations of the classes ? and η are obtained and utilized to show that η is closed under convolution and that ? is closed under convolution provided one of the two convoluted vectors has independent components.     

Decompositions of Borel bimeasurable mappings between complete metric spaces

We prove that every Borel bimeasurable mapping can be decomposed to a σ-discrete family of extended Borel isomorphisms and a mapping with a σ-discrete range. We get a new proof of a result containing the Purves and the Luzin-Novikov theorems as a by-product. Assuming an extra assumption on f, or that Fleissner's axiom (SCω₂) holds, we characterize extended Borel bimeasurable mappings as those extended Borel measurable ones which may be decomposed to countably many extended Borel isomorphisms and a mapping with a σ-discrete range.     

Densities for random balanced sampling

A random balanced sample (RBS) is a multivariate distribution with n components X_k, each uniformly distributed on [-1,1], such that the sum of these components is precisely 0. The corresponding vectors lie in an (n-1)-dimensional polytope M(n). We present new methods for the construction of such RBS via densities over M(n) and these apply for arbitrary n. While simple densities had been known previously for small values of n (namely 2,3, and 4), for larger n the known distributions with large support were fractal distributions (with fractal dimension asymptotic to n as n→∞). Applications of RBS distributions include sampling with antithetic coupling to reduce variance, and the isolation of nonlinearities. We also show that the previously known densities (for n?4) are in fact the only solutions in a natural and very large class of potential RBS densities. This finding clarifies the need for new methods, such as those presented here.     

Some local approximation properties of simple point processes

For simple point processes ξ on a Borel space S, we prove some approximations involving conditional distributions, given that ξ hits a small set B. Beginning with general versions of some classical limit theorems, going back to the pioneering work of Palm and Khinchin, we proceed to prove that, under suitable regularity conditions, the contributions to B and B ^c are asymptotically conditionally independent. We further derive approximations in total variation of reduced Palm distributions and show that, when ξ hits some small sets B ₁,..., B _n , the corresponding restrictions are asymptotically independent. Next we give general versions of the asymptotic relations P{ξ B > 0} ~ Eξ B and prove some ratio limit theorems for conditional expectations E[η | ξ B > 0], valid even when Eξ is not σ-finite and the Palm distributions may fail to exist.     

Asymptotic distribution of the OLS estimator for a mixed spatial model

We find the asymptotic distribution of the OLS estimator of the parameters β and ρ in the mixed spatial model with exogenous regressors Y_n=X_nβ+ρW_nY_n+V_n. The exogenous regressors may be bounded or growing, like polynomial trends. The assumption about the spatial matrix W_n is appropriate for the situation when each economic agent is influenced by many others. The error term is a short-memory linear process. The key finding is that in general the asymptotic distribution contains both linear and quadratic forms in standard normal variables and is not normal.     

Adaptive estimation of the transition density of a particular hidden Markov chain

We study the following model of hidden Markov chain: with (X_i) a real-valued positive recurrent and stationary Markov chain, and (?_i)_1?i?n+1 a noise independent of the sequence (X_i) having a known distribution. We present an adaptive estimator of the transition density based on the quotient of a deconvolution estimator of the density of X_i and an estimator of the density of (X_i,X_i+1). These estimators are obtained by contrast minimization and model selection. We evaluate the L² risk and its rate of convergence for ordinary smooth and supersmooth noise with regard to ordinary smooth and supersmooth chains. Some examples are also detailed.     

Weak consistency of least-squares estimators in linear models

Let Y_n, n≥1, be a sequence of integrable random variables with EY_n = x_n1β₁ + x_n2β₂ + … + x_npβ_p, where the x_ij's are known and β^T = (β₁, β₂,…, β_p) unknown. Let b_n be the least-squares estimator of β based on Y₁, Y₂,…, Y_n. Weak consistency of b_n, n≥1, has been considered in the literature under the assumption that each Y_n is square integrable. In this paper, we study weak consistency of b_n, n≥1, and associated rates of convergence under the minimal assumption that each Y_n is integrable.     

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f_n^(p) of f^(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R^m, p = (p₁,…,p_m), p_j ≥ 0, fixed integers, and for x = (x₁,…,x_m) in R^m, f^(p)(x) = ?^p₁+…+p_m f(x)/(?^p₁x₁ … ?^p_mx_m). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f_n^(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f_n^(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.     

Deformations of the generalised Picard bundle

Let X be a nonsingular algebraic curve of genus g?3, and let M_ξ denote the moduli space of stable vector bundles of rank n?2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g=3 then n?4 and if g=4 then n?3, and suppose further that n₀, d₀ are integers such that n₀?1 and nd₀+n₀d>nn₀(2g-2). Let E be a semistable vector bundle over X of rank n₀ and degree d₀. The generalised Picard bundle W_ξ(E) is by definition the vector bundle over M_ξ defined by the direct image where U_ξ is a universal vector bundle over X×M_ξ. We obtain an inversion formula allowing us to recover E from W_ξ(E) and show that the space of infinitesimal deformations of W_ξ(E) is isomorphic to H¹(X,End(E)). This construction gives a locally complete family of vector bundles over M_ξ parametrised by the moduli space M(n₀,d₀) of stable bundles of rank n₀ and degree d₀ over X. If (n₀,d₀)=1 and W_ξ(E) is stable for all E∈M(n₀,d₀), the construction determines an isomorphism from M(n₀,d₀) to a connected component M⁰ of a moduli space of stable sheaves over M_ξ. This applies in particular when n₀=1, in which case M⁰ is isomorphic to the Jacobian J of X as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over J, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles.     

Characterization of a class of bivariate distribution functions

Let F_X,Y(x,y) be a bivariate distribution function and P_n(x), Q_m(y), n, m = 0, 1, 2,…, the orthonormal polynomials of the two marginal distributions F_X(x) and F_Y(y), respectively. Some necessary conditions are derived for the co-efficients c_n, n = 0, 1, 2,…, if the conditional expectation E[P_n(X) ∥ Y] = c_nQ_n(Y) holds for n = 0, 1, 2,…. Several examples are given to show the application of these necessary conditions.