Estimation Of Conditional Expectation Research Articles

Stone's theorem for distributional regression in Wasserstein distance

We extend the celebrated Stone's theorem to the framework of distributional regression. More precisely, we prove that weighted empirical distributions with local probability weights satisfying the conditions of Stone's theorem provide universally consistent estimates of the conditional distributions, where the error is measured by the Wasserstein distance of order p ≥ 1 . Furthermore, for p = 1, we determine the minimax rates of convergence on specific classes of distributions. We finally provide some applications of these results, including the estimation of conditional tail expectation or probability weighted moments.

A comparative study of methods for estimating model-agnostic Shapley value explanations

Shapley values originated in cooperative game theory but are extensively used today as a model-agnostic explanation framework to explain predictions made by complex machine learning models in the industry and academia. There are several algorithmic approaches for computing different versions of Shapley value explanations. Here, we consider Shapley values incorporating feature dependencies, referred to as conditional Shapley values, for predictive models fitted to tabular data. Estimating precise conditional Shapley values is difficult as they require the estimation of non-trivial conditional expectations. In this article, we develop new methods, extend earlier proposed approaches, and systematize the new refined and existing methods into different method classes for comparison and evaluation. The method classes use either Monte Carlo integration or regression to model the conditional expectations. We conduct extensive simulation studies to evaluate how precisely the different method classes estimate the conditional expectations, and thereby the conditional Shapley values, for different setups. We also apply the methods to several real-world data experiments and provide recommendations for when to use the different method classes and approaches. Roughly speaking, we recommend using parametric methods when we can specify the data distribution almost correctly, as they generally produce the most accurate Shapley value explanations. When the distribution is unknown, both generative methods and regression models with a similar form as the underlying predictive model are good and stable options. Regression-based methods are often slow to train but quickly produce the Shapley value explanations once trained. The vice versa is true for Monte Carlo-based methods, making the different methods appropriate in different practical situations.

A nonstationary bivariate design flood estimation approach coupled with the most likely and expectation combination strategies

In this study, a framework is proposed to estimate the nonstationary bivariate design flood, which includes three key steps. First, a nonstationary copula model was constructed to analyze the temporal variation in the bivariate joint distribution. Second, the equivalent reliability method was used to calculate the design value of the dominant variable for a return period and the design lifespan length under non-stationarity. Third, the design value of the secondary variable that was conditioned on the design value of the dominant variable, was calculated using the conditional most likely combination and conditional expectation combination strategies. Through the above three steps, the typical design value combination of the nonstationary bivariate was obtained for a specific design standard. A case study, based on the nonstationary annual maximum 1-day (AM1) and 15-day (AM15) flood volume at the Yichang station, was conducted to illustrate the framework. The results indicate that the joint distribution of the AM1 and AM15 flood volumes varied over time. For a specific design standard, the design values of the AM15 and AM1 flood volumes decreased with an increase in the design lifespan length. Moreover, the conditional most likely estimation of the design value for the AM1 flood volume that was conditioned on the design value for the AM15 flood volume was greater than its conditional expectation estimation.

Probabilistic threshold analysis by pairwise stochastic approximation for decision-making under uncertainty

The concept of probabilistic parameter threshold analysis has recently been introduced as a way of probabilistic sensitivity analysis for decision-making under uncertainty, in particular, for health economic evaluations which compare two or more alternative treatments with consideration of uncertainty on outcomes and costs. In this paper we formulate the probabilistic threshold analysis as a root-finding problem involving the conditional expectations, and propose a pairwise stochastic approximation algorithm to search for the threshold value below and above which the choice of conditionally optimal decision options changes. Numerical experiments for both a simple synthetic testcase and a chemotherapy Markov model illustrate the effectiveness of our proposed algorithm, without any need for accurate estimation or approximation of conditional expectations which the existing approaches rely upon. Moreover we introduce a new measure called decision switching probability for probabilistic sensitivity analysis in this paper.

Robust Estimator of Conditional Tail Expectation of Pareto-Type Distribution

In this paper, we use the extreme value index estimator, called the t-Hill, to derive a robust estimator of conditional tail expectation (CTE) in the case of heavy-tailed losses. The CTE is rapidly turning into the favored measure for statutory assessment of the balance sheet at whatever point true stochastic techniques are utilized to fix the provisions for risks. Under the extreme value methodology, we prove the asymptotic normality of the robust nonparametric conditional tail expectation estimator when the loss variable follows any distribution with infinite second moment. In addition, the numerical performance of our new estimator is studied and compared to a well-established estimator, with favorable results.

Consistent regression using data-dependent coverings

We introduce a procedure to generate an estimator of the regression function based on a data-dependent quasi-covering of the feature space. A quasi-partition is generated from the quasi-covering and the estimator predicts the conditional empirical expectation over the cells of the quasi-partition. We provide sufficient conditions to ensure the consistency of the estimator. Each element of the quasi-covering is labeled as significant or insignificant. We avoid the condition of cell shrinkage commonly found in the literature for data-dependent partitioning estimators. This reduces the number of elements in the quasi-covering. An important feature of our estimator is that it is interpretable. The proof of the consistency is based on a control of the convergence rate of the empirical estimation of conditional expectations, which is interesting in itself.

Parametric g-formula implementations for causal survival analyses.

The g-formula can be used to estimate the survival curve under a sustained treatment strategy. Two available estimators of the g-formula are noniterative conditional expectation and iterative conditional expectation. We propose a version of the iterative conditional expectation estimator and describe its procedures for deterministic and random treatment strategies. Also, because little is known about the comparative performance of noniterative and iterative conditional expectation estimators, we explore their relative efficiency via simulation studies. Our simulations show that, in the absence of model misspecification and unmeasured confounding, our proposed iterative conditional expectation estimator and the noniterative conditional expectation estimator are similarly efficient, and that both are at least as efficient as the classical iterative conditional expectation estimator. We describe an application of both noniterative and iterative conditional expectation to answer "when to start" treatment questions using data from the HIV-CAUSALCollaboration.

Nonparametric Estimation of Conditional Expectation with Auxiliary Information and Dimension Reduction

Nonparametric estimation of the conditional expectation of an outcome Y given a covariate vector U is of primary importance in many statistical applications such as prediction and personalized medicine. In some problems, there is an additional auxiliary variable Z in the training dataset used to construct estimators, but Z is not available for future prediction or selecting patient treatment in personalized medicine. For example, in the training dataset longitudinal outcomes are observed, but only the last outcome Y is concerned in the future prediction or analysis. The longitudinal outcomes other than the last point is then the variable Z that is observed and related with both Y and U. Previous work on how to make use of Z in the estimation of mainly focused on using Z in the construction of a linear function of U to reduce covariate dimension for better estimation. Using , we propose a two-step estimation of inner and outer expectations, respectively, with sufficient dimension reduction for kernel estimation in both steps. The information from Z is utilized not only in dimension reduction, but also directly in the estimation. Because of the existence of different ways for dimension reduction, we construct two estimators that may improve the estimator without using Z. The improvements are shown in the convergence rate of estimators as the sample size increases to infinity as well as in the finite sample simulation performance. A real data analysis about the selection of mammography intervention is presented for illustration.

Conditional expectation estimation through attributable components

Abstract A general methodology is proposed for the explanation of variability in a quantity of interest x in terms of covariates z = (z1, …, zL). It provides the conditional mean $\bar{x}(z)$ as a sum of components, where each component is represented as a product of non-parametric one-dimensional functions of each covariate zl that are computed through an alternating projection procedure. Both x and the zl can be real or categorical variables; in addition, some or all values of each zl can be unknown, providing a general framework for multi-clustering, classification and covariate imputation in the presence of confounding factors. The procedure can be considered as a preconditioning step for the more general determination of the full conditional distribution $\boldsymbol{\rho}(x|z) $ through a data-driven optimal-transport barycenter problem. In particular, just iterating the procedure once yields the second order structure (i.e. the covariance) of $\boldsymbol{\rho}(x|z) $. The methodology is illustrated through examples that include the explanation of variability of ground temperature across the continental United States and the prediction of book preference among potential readers.

B-spline techniques for volatility modeling

This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility (SLV) models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines obtained by including basis functions with infinite support. We first discuss the application of shape-constrained B-splines to the estimation of conditional expectations, not merely from a scatter plot but also from the given marginal distributions. One application is the Monte Carlo calibration of SLV models by Markov projection. We then present a new technique for the calibration of an implied volatility surface to sparse option data. We use a B-spline parameterization of the Radon–Nikodym derivative of the underlying’s risk-neutral probability density with respect to a roughly calibrated base model. We show that this method provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch a Galerkin method with B-spline finite elements to the solution of the partial differential equation satisfied by the Radon–Nikodym derivative.

Convergence Analysis of Random Generators in Monte Carlo Simulation: Mersenne Twister and Sobol

We investigate further the random generators used in finance Monte Carlo simulation: Mersenne Twister and Sobol Quasi-Random Generator. We focus the analysis on the statistical properties of the random numbers generated across multiple dimensions in Monte Carlo finance simulation. We particularly focus on the distributions and processes used in the estimation of conditional expectation in a finance framework when using Mersenne Twister and Sobol generators.We characterize the convergence properties of the sample distributions by relying on empirical distribution theory. Then, we provide practical analysis on the convergence of distribution encountered in finance problems. Finally, we characterize the differences of the random generators when simulating such processes.

Nonparametric tests of conditional treatment effects with an application to single-sex schooling on academic achievements

Summary We develop a general class of nonparametric tests for treatment effects conditional on covariates. We consider a wide spectrum of null hypotheses regarding conditional treatment effects, including the following: (a) the null hypothesis of the conditional stochastic dominance between treatment and control groups; (b) the null hypothesis that the conditional average treatment effect is nonpositive for each value of covariates; (c) the null hypothesis of no distributional (or average) treatment effect conditional on covariates. The test statistics are based on L1-type functionals of uniformly consistent nonparametric kernel estimators of conditional expectations that characterize the null hypotheses. We show that our tests using the standard normal critical values have asymptotically correct size. We also show that the proposed nonparametric tests are consistent against general fixed alternatives and have non-negligible powers against some n−1/2 local alternatives to the null hypothesis with inequality constraints and n−1/2h−d/4 local alternatives to the null hypothesis with equality constraints, where h is a bandwidth, n is the sample size and d is the dimension of continuous covariates. We illustrate the usefulness of our tests by applying them to the effect of single-sex schooling on academic achievements using Korean data.

Estimation of a Semiparametric Natural Direct Effect Model Incorporating Baseline Covariates.

Establishing cause-effect relationships is a standard goal of empirical science. Once the presence of a causal relationship is established, the precise causal mechanism involved becomes a topic of interest. A particularly popular type of mechanism analysis concerns questions of mediation, that is to what extent an effect is direct, and to what extent it is mediated by a third variable. A semiparametric theory has recently been proposed which allows multiply robust estimation of direct and mediated marginal effect functionals in observational studies (Tchetgen Tchetgen & Shpitser, 2012). In this paper we extend the new theory to handle parametric models of natural direct and indirect effects within levels of pre-exposure variables with an identity or log link function, where the model for the observed data likelihood is otherwise unrestricted. We show that estimation is generally not feasible in this model because of the curse of dimensionality associated with the required estimation of auxiliary conditional densities or expectations, given high-dimensional covariates. Thus, we consider multiply robust estimation and propose a more general model which assumes that a subset but not all of several working models holds.

Strategies for Efficient Computation of the Expected Value of Partial Perfect Information

Expected value of information methods evaluate the potential health benefits that can be obtained from conducting new research to reduce uncertainty in the parameters of a cost-effectiveness analysis model, hence reducing decision uncertainty. Expected value of partial perfect information (EVPPI) provides an upper limit to the health gains that can be obtained from conducting a new study on a subset of parameters in the cost-effectiveness analysis and can therefore be used as a sensitivity analysis to identify parameters that most contribute to decision uncertainty and to help guide decisions around which types of study are of most value to prioritize for funding. A common general approach is to use nested Monte Carlo simulation to obtain an estimate of EVPPI. This approach is computationally intensive, can lead to significant sampling bias if an inadequate number of inner samples are obtained, and incorrect results can be obtained if correlations between parameters are not dealt with appropriately. In this article, we set out a range of methods for estimating EVPPI that avoid the need for nested simulation: reparameterization of the net benefit function, Taylor series approximations, and restricted cubic spline estimation of conditional expectations. For each method, we set out the generalized functional form that net benefit must take for the method to be valid. By specifying this functional form, our methods are able to focus on components of the model in which approximation is required, avoiding the complexities involved in developing statistical approximations for the model as a whole. Our methods also allow for any correlations that might exist between model parameters. We illustrate the methods using an example of fluid resuscitation in African children with severe malaria.

Fast Convergence of Regress-Later Estimates in Least Squares Monte Carlo

Many problems in financial engineering involve the estimation of unknown conditional expectations across a time interval. Often Least Squares Monte Carlo techniques are used for the estimation. One method that can be combined with Least Squares Monte Carlo is the "Regress-Later" method. Unlike conventional methods where the value function is regressed on a set of basis functions valued at the beginning of the interval, the "Regress-Later" method regresses the value function on a set of basis functions valued at the end of the interval. The conditional expectation across the interval is then computed exactly for each basis function. We provide sufficient conditions under which we derive the convergence rate of Regress-Later estimators. Importantly, our results hold on non-compact sets. We show that the Regress-Later method is capable of converging significantly faster than conventional methods and provide an explicit example. Achieving faster convergence speed provides a strong motivation for using Regress-Later methods in estimating conditional expectations across time.

On data-based optimal stopping under stationarity and ergodicity

The problem of optimal stopping with finite horizon in discrete time is considered in view of maximizing the expected gain. The algorithm proposed in this paper is completely nonparametric in the sense that it uses observed data from the past of the process up to time $-n+1$, $n\in\mathbb{N}$, not relying on any specific model assumption. Kernel regression estimation of conditional expectations and prediction theory of individual sequences are used as tools. It is shown that the algorithm is universally consistent: the achieved expected gain converges to the optimal value for $n\to\infty$ whenever the underlying process is stationary and ergodic. An application to exercising American options is given, and the algorithm is illustrated by simulated data.

The consequences of measurement error when estimating the impact of obesity on income

AbstractThis paper examines the consequences of using self-reported measures of BMI when estimating the effect of BMI on income for women using both Irish and US data. We find that self-reported BMI is subject to substantial measurement error and that this error deviates from classical measurement error. These errors cause the traditional least squares estimator to overestimate the relationship between BMI and income. We show that neither the conditional expectation estimator nor the instrumental variables approach adequately address the bias and briefly discuss alternative approaches that could be considered when faced with non-classical measurement error.JEL codesC13, C26, I14

Nonlinear Filtering of Stochastic Navier-Stokes Equation with Itô-Lévy Noise

In this article, we study the existence and uniqueness of the strong pathwise solution of stochastic Navier-Stokes equation with Itô-Lévy noise. Nonlinear filtering problem is formulated for the recursive estimation of conditional expectation of the flow field given back measurements of sensor output data. The corresponding Fujisaki-Kallianpur-Kunita and Zakai equations describing the time evolution of the nonlinear filter are derived. Existence and uniqueness of measure-valued solutions are proven for these filtering equations.

Reducing variance in the numerical solution of BSDEs

Numerical methods based on time discretization and estimation of conditional expectations for solving backward stochastic differential equations (BSDEs) have been the object of considerable research, particularly in view of the applications to finance. We introduce and implement a simple control variate technique to reduce the simulation error of the conditional expectation estimates in BSDE methods. These modifications increase the accuracy of the existing algorithms without additional computational cost.

Distributed Functional Scalar Quantization Simplified

Distributed functional scalar quantization (DFSQ) theory provides optimality conditions and predicts performance of data acquisition systems in which a computation on acquired data is desired. We address two limitations of previous works: prohibitively expensive decoder design and a restriction to sources with bounded distributions. We rigorously show that a much simpler decoder has equivalent asymptotic performance as the conditional expectation estimator previously explored, thus reducing decoder design complexity. The simpler decoder has the feature of decoupled communication and computation blocks. Moreover, we extend the DFSQ framework with the simpler decoder to acquire sources with infinite-support distributions such as Gaussian or exponential distributions. Finally, through simulation results we demonstrate that performance at moderate coding rates is well predicted by the asymptotic analysis, and we give new insight on the rate of convergence.