Representing a function of two variables as a superposition over angles of ridge functions with a given shape via linear programming

A ridge function with shape function g   in the horizontal direction is a function of the form g(x)h(y,0)$g(x)h(y,0)$. Along each horizontal line it has the shape g(x)$g(x)$, multiplied by a function h(y,0)$h(y,0)$ which depends on the y-value of the horizontal line. Similarly a ridge function with shape function g   in the vertical direction has the form g(y)h(x,π/2)$g(y)h(x,\pi /2)$. For a given shape function g it may or may not be possible to represent an arbitrary   function f(x,y)$f(x,y)$ as a superposition over all angles of a ridge function with shape g   in each direction, where h=_hf=_hf,g$h=h_f=h_{f,g}$ depends on the functions f and g   and also on the direction, θ:h=_hf,g(·,θ)$\theta :h=h_{f,g}(·,\theta )$. We show that if g   is Gaussian centered at zero then this is always possible and we give the function _hf,g$h_{f,g}$ for a given f(x,y)$f(x,y)$. For highpass or for odd shapes g  , we show it is impossible to represent an arbitrary f(x,y)$f(x,y)$, i.e. in general there is no _hf,g$h_{f,g}$. Note that our problem is similar to tomography, where the problem is to invert the Radon transform, except that the use of the word inversion is here somewhat “inverted”: in tomography f(x,y)$f(x,y)$ is unknown and we find it by inverting the projections of f  ; here, f(x,y)$f(x,y)$ is known, g(z)$g(z)$ is known, and _hf(·,θ)=_hf,g(·,θ)$h_f(·,\theta )=h_{f,g}(·,\theta )$ is the unknown.     

On asymptotic properties of the parameter estimator for a type of SPDE

In this paper, we study a random field _U?(t,x)$U_{?}(t,x)$ governed by some type of stochastic partial differential equations with an unknown parameter θ$\theta$ and a small noise ?$?$. We construct an estimator of θ$\theta$ based on the continuous observation of N   Fourier coefficients of _U?(t,x)$U_{?}(t,x)$, and prove the strong convergence and asymptotic normality of the estimator when the noise ?$?$ tends to zero.     

Regression models for functional data by reproducing kernel Hilbert spaces methods

Non-parametric regression models are developed when the predictor is a function-valued random variable X={_Xt}t∈T$X=\{X_t{\}}_{t\in T}$. Based on a representation of the regression function f(X)$f(X)$ in a reproducing kernel Hilbert space such models generalize the classical setting used in statistical learning theory. Two applications corresponding to scalar and categorical response random variable are performed on stock-exchange and medical data. The results of different regression models are compared.     

Nonparametric estimation of conditional expectation

Denote the integer lattice points in the N  -dimensional Euclidean space by Z^N${\mathbb{Z}}^N$ and assume that (_Xi,_Yi)$(X_{\mathbf{i}},Y_{\mathbf{i}})$, i∈Z^N$\mathbf{i}\in {\mathbb{Z}}^N$ is a mixing random field. Estimators of the conditional expectation r(x)=E[_Yi|_Xi=x]$r(\mathbf{x})=\mathrm{E}[Y_{\mathbf{i}}|X_{\mathbf{i}}=\mathbf{x}]$ by nearest neighbor methods are established and investigated. The main analytical result of this study is that, under general mixing assumptions, the estimators considered are asymptotically normal. Many difficulties arise since points in higher dimensional space N?2$N?2$ cannot be linearly ordered. Our result applies to many situations where parametric methods cannot be adopted with confidence.     

On the maximum of periodic integer-valued sequences with exponential type tails via max-semistable laws

In this work we study the limiting distribution of the maximum term of periodic integer-valued sequences with marginal distribution belonging to a particular class where the tail decays exponentially. This class does not belong to the domain of attraction of any max-stable distribution. Nevertheless, we prove that the limiting distribution is max-semistable when we consider the maximum of the first k_n observations, for a suitable sequence {_kn}$\{k_n\}$ increasing to infinity. We obtain an expression for calculating the extremal index of sequences satisfying certain local conditions similar to conditions D^(m)(_un)$D^{(m)}(u_n)$, m∈N$m\in \mathbb{N}$, defined by Chernick et al. (1991). We apply the results to a class of max-autoregressive sequences and a class of moving average models. The results generalize the ones obtained for the stationary case.     

The use of an identity in Anderson for multivariate multiple testing

Consider the model where there are I$I$ independent multivariate normal treatment populations with p×1$p\times 1$ mean vectors _μi${\mathit{\mu }}_i$, i=1,…,I$i=1,\dots ,I$, and covariance matrix Σ$\Sigma$. Independently the (I+1)$(I+1)$st population corresponds to a control and it too is multivariate normal with mean vector _μI+1${\mathit{\mu }}_{I+1}$ and covariance matrix Σ$\Sigma$. Now consider the following two multiple testing problems.     

An identity for the Fisher information and Mahalanobis distance

Consider a mixture problem consisting of k classes. Suppose we observe an s-dimensional random vector X   whose distribution is specified by the relations P(X∈A|Y=i)=_Pi(A)$P(X\in A|Y=i)=P_i(A)$, where Y   is an unobserved class identifier defined on {1,…,k}$\{1,\dots ,k\}$, having distribution P(Y=i)=_pi$P(Y=i)=p_i$. Assuming the distributions _Pi$P_i$ having a common covariance matrix, elegant identities are presented that connect the matrix of Fisher information in Y   on the parameters _p1,…,_pk$p_1,\dots ,p_k$, the matrix of linear information in X, and the Mahalanobis distances between the pairs of P  's. Since the parameters are not free, the information matrices are singular and the technique of generalized inverses is used. A matrix extension of the Mahalanobis distance and its invariant forms are introduced that are of interest in their own right. In terms of parameter estimation, the results provide an independent of the parameter upper bound for the loss of accuracy by esimating _p1,…,_pk$p_1,\dots ,p_k$ from a sample of X^′$X^{\prime }$s, as compared with the ideal estimator based on a random sample of Y^′$Y^{\prime }$s.     

Truncated linear estimation of a bounded multivariate normal mean

We consider the problem of estimating the mean θ$\theta$ of an _Np(θ,_Ip)${\mathrm{N}}_{\mathrm{p}}(\theta ,I_p)$ distribution with squared error loss ∥δ−θ∥²$\parallel \delta -\theta {\parallel }^2$ and under the constraint ∥θ∥≤m$\parallel \theta \parallel \le m$, for some constant m>0$m>0$. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator _δmle${\delta }_{\mathrm{mle}}$. We obtain for fixed (m,p)$(m,p)$ sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates _δmle${\delta }_{\mathrm{mle}}$, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.     

Characterizations based on record values and order statistics

Consider a sequence of independent and identically distributed random variables {_Xi,i?1}$\{X_i,i?1\}$ with a common absolutely continuous distribution function F  . Let _X1:n?_X2:n???_Xn:n$X_{1:n}?X_{2:n}???X_{n:n}$ be the order statistics of {_X1,_X2,…,_Xn}$\{X_1,X_2,\dots ,X_n\}$ and {_Yl,l?1}$\{Y_l,l?1\}$ be the sequence of record values generated by {_Xi,i?1}$\{X_i,i?1\}$. In this work, the conditional distribution of _Yl$Y_l$ given _Xn:n$X_{n:n}$ is established. Some characterizations of F   based on record values and _Xn:n$X_{n:n}$ are then given.     

Prediction in moving average processes

For the stationary invertible moving average process of order one with unknown innovation distribution F, we construct root-n   consistent plug-in estimators of conditional expectations E(h(_Xn+1)|_X1,…,_Xn)$E(h(X_{n+1})|X_1,\dots ,X_n)$. More specifically, we give weak conditions under which such estimators admit Bahadur-type representations, assuming some smoothness of h or of F. For fixed h it suffices that h   is locally of bounded variation and locally Lipschitz in _L2(F)$L_2(F)$, and that the convolution of h and F   is continuously differentiable. A uniform representation for the plug-in estimator of the conditional distribution function P(_Xn+1?·|_X1,…,_Xn)$P(X_{n+1}?·|X_1,\dots ,X_n)$ holds if F has a uniformly continuous density. For a smoothed version of our estimator, the Bahadur representation holds uniformly over each class of functions h that have an appropriate envelope and whose shifts are F-Donsker, assuming some smoothness of F. The proofs use empirical process arguments.     

Distributions associated with (k1,k2) events on semi-Markov binary trials

We derive neat expressions for the probability generating functions of relevant waiting times associated with (_k1,_k2)$(k_1,k_2)$ events on semi-Markov binary trials. These lead to evaluation of relevant probabilities associated with numbers of occurrence of such events on a string of a fixed length. Our methodology is general enough and provides a template for treating more general events than those of type (_k1,_k2)$(k_1,k_2)$. Also, the same template is extendable to semi-Markov trials with more than two outcomes.