An asymptotic optimality theory for the estimation of cointegration regressions is developed in this paper. The theory applies to a reasonably wide class of estimators without making any specific assumptions about the probability distribution or short-run dynamics of the data-generating process. Due to the nonstandard nature of the estimation problem, the conventional minimum variance criterion does not provide a convenient measure of asymptotic efficiency. An alternative criterion, based on the concentration or peakedness of the limiting distribution of an estimator, is therefore adopted. The limiting distribution of estimators with maximum asymptotic efficiency is characterized in the paper and used to discuss the optimality of some known estimators. A new asymptotically efficient estimator is also introduced. This estimator is obtained from the ordinary least-squares estimator by a time domain correction which is nonparametric in the sense that no assumption of a finite parameter model is required. The estimator can be computed with least squares without any initial estimations.

Asymptotically Efficient Estimation of Cointegration Regressions

The problem of estimating state variables and parameters is considered for discrete-time systems in the presence of random disturbances and measurement noise. The solution of the linear problem is given and an approximation technique is developed for nonlinear systems. A dynamic programming formulation of the estimation problem is also developed.

On the estimation of state variables and parameters for noisy dynamic systems

Given subsets A and B of Euclidean d-space R a and 0 ~ 0, we set A + B -{x + Y l x E A, y E B} and OA = {Ox Ix 6 A }. Further given a convex subset g2 of R d we shall say that a set function F : 2 ~ \ {~} ~ [0, + ~ ] is convex (concave} if the inequality F(OA + (1 O)B) ~ Or(A} + (1 0 ) / ' ( B ) (>=.) holds for all It ~ A, B c_ D, and all 0 < 0 < 1. Here we shall s tudy such set functions of the special form given in the following

Convex set functions ind-space

/pdf/convex-measures-on-locally-convex-spaces-svo9pbqoxb.pdf

Convex measures on locally convex spaces

Let CN be a cube of volume one centered at the origin in RN and let Pκ be a iΓ-dimensional subspace of RN. We prove that CN Π Pκ has iΓ-dimensionai volume greater than or equal to one. As an application of this inequality we obtain a precise version of Minkowski's linear forms theorem. We also state a conjecture which would allow our method to be generalized. I* Introduction* Let CN — [ — 1/2, 1/2]^ be the iV-dimensional cube of volume one centered at the origin in RN and suppose that Pκ is a if-dimensional linear subspace of RN. Dr. Anton Good has conjectured that the iί-dimensional volume of Pκ Π CN is always greater than or equal to one. In case K = N — 1 this has recently been proved by Hensley [6], who also obtained upper bounds for this volume. Our purpose in this paper is to prove the conjecture for arbitrary K and to give some applications to Minkowski's theorem on linear forms. In fact we prove a more general inequality for the product of spheres of various dimensions which contains the conjecture as a special case. We write x for the column vector I I in Rn and

/pdf/a-geometric-inequality-with-applications-to-linear-forms-1vmybd7ekp.pdf

A geometric inequality with applications to linear forms

T. W. Anderson [1] has proved the following theorem and has given applications to probability and statistics. THEOREM 1. Let $E$ be a convex set in $n$-space, symmetric about the origin. Let $f(x) \geqq 0$ be a function such that i) $f(x) = f(-x)$, ii) $\{x |f(x) \geqq u\} = K_u$ is convex for every $u (0 \geqq u \geqq \infty)$ and iii) $\int_E f(x) dx < \infty$, then$ \begin{equation*}\tag{(1)}\int_E f(x + ky) dx \geqq \int_E f(x + y) dx\quad for 0 \leqq k \leqq 1.$ The purpose of this paper is to prove what can be considered a generalization of Anderson's Theorem and to give different statistical applications. Functions in $L_1$ satisfying the hypothesis were called unimodal by Anderson and he noted in [1] that if we let $\varphi(y)$ be equal to the right hand side of (1) then $\varphi$ is not unimodal in his sense insofar as it does not necessarily satisfy $ii$ (i.e., there exist $f, E$, and $u$ such that $\{x\mid \varphi(x) \geqq u\}$ is not convex). His example is the case where $n = 2$ and \begin{equation*}f(x) = \begin{cases}3,\quad\|x_1\| \leqq 1,\quad\|x_2\| \leqq 1, \\ 2,\quad\|x_1\| \leqq 1,\quad 1 < \|x_2\| \leqq 5, \\ 0,\quad {other} x,\end{cases}\end{equation*} where $x_1, x_2$ are the components of $x$ relative to rectangular cartesian coordinate system. Let $E$ be the set of vectors where $|x_1| \leqq 1, |x_2| \leqq 1$. The set $\{x \mid \varphi(x) \geqq 6\}$ is not convex since for $x = (.5,4)$ and $x = (1,0), \varphi(x) = 6$, while for $x = (.75.2), \varphi (x) < 6$. The point of departure of this paper is to see what can be said about $\varphi$. This is achieved in Theorem 2, giving a stronger and more symmetrical statement than Anderson makes (but one which does not yield more information for his applications). The main Lemma, presented below, proceeds along the line of his argument but squeezes out additional information (convexity of level lines) under a weaker hypothesis (no symmetry assumptions) than Anderson uses at the corresponding stage of his argument.

/pdf/a-theorem-on-convex-sets-with-applications-1vz3hg85gb.pdf

S. Sherman

Papers

A Theorem on Convex Sets with Applications