This paper is concerned with inference about a function g that is identified by a conditional moment restriction involving instrumental variables. The function is nonparametric. It satisfies mild regularity conditions but is otherwise unknown. The paper presents test of the hypothesis that g is the mean of a random variable Y conditional on a covariate X . The need to test this hypothesis arise...

We study the simplest causal prediction algorithm that uses only conditional independences in purely observational data. A specific pattern of only four conditional independence relations amongst a quadruple of random variables already implies that one of these variables causes another without any confounding. As a consequence, it is possible to predict what would happen under an intervention o...

We consider a random-design regression model with vector-valued observations and develop nonparametric estimation of smooth conditional moment functions in the predictor variable. This includes estimation of higher order mixed moments and also functionals of the moments, such as conditional covariance, correlation, variance, and skewness functions. Our asymptotic analysis targets the limit dist...

In this paper we contribute two novel methods that simplify the demands of knowledge elicitation for particular types of Bayesian networks. The first method simplifies the task of experts providing conditional probabilities when the states that a random variable takes can be described by a fully ordered set. In this order, each state’s definition is inclusive of the preceding state’s definition...

Markov chains provided us with a way to model 1D objects such as contours probabilistically, in a way that led to nice, tractable computations. We now consider 2D Markov models. These are more powerful, but not as easy to compute with. In addition we will consider two additional issues. First, we will consider adding observations to our models. These observations are conditioned on the value of...

Suppose we are interested in the mean of an outcome variable missing not at random. Suppose however that one has available a fully observed shadow variable, which is associated with the outcome but independent of the missingness process conditional on covariates and the possibly unobserved outcome. Such a variable may be a proxy or a mismeasured version of the outcome and is available for all i...

Conditional distribution reflects the dependency link among random variables, but two-dimensional random variables Conditional Distribution has some limitations. In order to rich the content of conditional distribution this paper gives the extension of conditional distribution under discrete random variables and some examples. This article obtains the extension strictly in accordance with the d...

The paper deals with linear information inequalities valid for entropy functions induced by discrete random variables. Specifically, the so-called conditional Ingleton inequalities are in center of interest: these under conditional independence assumptions on inducing We discuss five this parti...

Our key aim is to propose effective estimators for the conditional probability density of a scalar response variable given functional co-variable, where considered have missing data at random. Such are constructed by combining approaches local linear method and kernel nearest neighborhood. The main feature this estimation possibility model phenomena. Under less restrictive conditions, we show s...