General Set Of Sufficient Conditions Research Articles

On Semiparametric Instrumental Variable Estimation of Average Treatment Effects through Data Fusion

Suppose one is interested in estimating causal effects in the presence of potentially unmeasured confounding with the aid of a valid instrumental variable. This paper investigates the problem of making inferences about the average treatment effect when data are fused from two separate sources, one of which contains information on the treatment and the other contains information on the outcome, while values for the instrument and a vector of baseline covariates are recorded in both. We provide a general set of sufficient conditions under which the average treatment effect is nonparametrically identified from the observed data law induced by data fusion, even when the data are from two heterogeneous populations, and derive the efficiency bound for estimating this causal parameter. For inference, we develop both parametric and semiparametric methods, including a multiply robust and locally efficient estimator that is consistent even under partial misspecification of the observed data model. We illustrate the methods through simulations and an application on public housing projects.

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: I. The Heisenberg group

This paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the Hörmander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that generate the Heisenberg group, giving explicit conditions on the sign and size of the first and zero-th order terms in the equation. The optimality of the conditions is shown via several examples. A sequel of this paper applies the methods to other Carnot groups and to Grushin geometries.

Optimal Scaling of the Random Walk Metropolis: General Criteria for the 0.234 Acceptance Rule

Scaling of proposals for Metropolis algorithms is an important practical problem in Markov chain Monte Carlo implementation. Analyses of the random walk Metropolis for high-dimensional targets with specific functional forms have shown that in many cases the optimal scaling is achieved when the acceptance rate is approximately 0.234, but that there are exceptions. We present a general set of sufficient conditions which are invariant to orthonormal transformation of the coordinate axes and which ensure that the limiting optimal acceptance rate is 0.234. The criteria are shown to hold for the joint distribution of successive elements of a stationary pth-order multivariate Markov process.

Optimal Scaling of the Random Walk Metropolis: General Criteria for the 0.234 Acceptance Rule

Scaling of proposals for Metropolis algorithms is an important practical problem in Markov chain Monte Carlo implementation. Analyses of the random walk Metropolis for high-dimensional targets with specific functional forms have shown that in many cases the optimal scaling is achieved when the acceptance rate is approximately 0.234, but that there are exceptions. We present a general set of sufficient conditions which are invariant to orthonormal transformation of the coordinate axes and which ensure that the limiting optimal acceptance rate is 0.234. The criteria are shown to hold for the joint distribution of successive elements of a stationary pth-order multivariate Markov process.

Adaptive Alternating Minimization Algorithms

The classical alternating minimization (or projection) algorithm has been successful in the context of solving optimization problems over two variables. The iterative nature and simplicity of the algorithm has led to its application in many areas such as signal processing, information theory, control, and finance. A general set of sufficient conditions for the convergence and correctness of the algorithm are known when the underlying problem parameters are fixed. In many practical situations, however, the underlying problem parameters are changing over time, and the use of an adaptive algorithm is more appropriate. In this paper, we study such an adaptive version of the alternating minimization algorithm. More precisely, we consider the impact of having a slowly time-varying domain over which the minimization takes place. As a main result of this paper, we provide a general set of sufficient conditions for the convergence and correctness of the adaptive algorithm. Perhaps somewhat surprisingly, these conditions seem to be the minimal ones one would expect in such an adaptive setting. We present applications of our results to adaptive decomposition of mixtures, adaptive log-optimal portfolio selection, and adaptive filter design.

Continuum and Finite-Player Noncooperative Models of Competition

The anonymous interaction of large numbers of economic agents is a kind of noncooperative situation which is markedly different from small-numbers strategic conflict. The mathematical model of a nonatomic game, or a game with a continuum of players, has been introduced as a model for these many-agent situations on the basis that its equilibria should closely approximate those of games with large finite numbers of players. This paper contains a precise definition of what it means for a nonatomic game to be the limit of a sequence of finite-player games, and a theorem which states when the limit of equilibria of finite-player games will be an equilibrium of the nonatomic limit game. This is analogous to theorems prompted by Edgeworth's conjecture in core theory. It is derived from a general set of sufficient conditions for the graph of a noncooperative equilibrium correspondence to be closed.

A Critical Note on Portfolio Selection under Stochastic Taxation

Since Domar and Musgrave (1944), the analysis of taxation and risktaking has been developed by Mossin (1968), Penner (1964), Richter (1960), Stiglitz (1969) and others, but all of these have treated the tax rate as a certain parameter. Ekern (1971), on the other hand, concentrated precisely on this point. He termed uncertainty about the rate of (proportional income) tax and, by defining a change in it in terms of a corresponding change in a dispersion measure of the probability distribution of the tax rate, examined the qualitative effect of a change in political risk on risk-taking (the demand for a risky asset) as well as several related problems. Unfortunately, however, his curiosity could not be satisfied in full generality, and he had to use the quadratic utility function (which is well known to be very anomalous in the theory of portfolio selection) as an individual's objective function in order to make his analysis determinate. However, if we recall the analyses of Rothschild and Stiglitz (1970, 1971), Hahn (1970) and Diamond and Stiglitz (1974), we can refine Ekern's analysis concerning the qualitative effect on risk-taking of a change in political risk in the following two points: (a) When the tax rate and the rate of return on the risky asset are assumed to be stochastically independent, a simple reinterpretation of the rate of return on the risky asset allows us to apply the analysis of Rothschild and Stiglitz (R-S hereafter) (1971, pp. 70-74) to the present problem in a straightforward manner. Consequently, we can enumerate a quite general set of sufficient conditions for the qualitative effect on risk-taking of a mean preserving change in political risk to become determinate, and for that purpose we need neither to use the special utility function nor to define a change in political risk in a specific form. Or more conclusively, we may assert from this that at least the part of Ekern's analysis concerning the qualitative effect on risk-taking of a mean preserving change in political risk is included in the case where uncertainty about the tax rate forms a part of uncertainty about the rate of return on the risky asset. (b) We can also sign the response of optimal risk-taking to an alternative, mean utility-preserving change in political risk according to Theorem 2 of Diamond and Stiglitz (1974, p. 343). (A mean utility-