Quantification of Variability and Uncertainty Using Mixture Distributions: Evaluation of Sample Size, Mixing Weights, and Separation Between Components

Variability is the heterogeneity of values within a population. Uncertainty refers to lack of knowledge regarding the true value of a quantity. Mixture distributions have the potential to improve the goodness of fit to data sets not adequately described by a single parametric distribution. Uncertainty due to random sampling error in statistics of interests can be estimated based upon bootstrap simulation. In order to evaluate the robustness of using mixture distribution as a basis for estimating both variability and uncertainty, 108 synthetic data sets generated from selected population mixture log-normal distributions were investigated, and properties of variability and uncertainty estimates were evaluated with respect to variation in sample size, mixing weight, and separation between components of mixtures. Furthermore, mixture distributions were compared with single-component distributions. Findings include: (1). mixing weight influences the stability of variability and uncertainty estimates; (2). bootstrap simulation results tend to be more stable for larger sample sizes; (3). when two components are well separated, the stability of bootstrap simulation is improved; however, a larger degree of uncertainty arises regarding the percentiles coinciding with the separated region; (4). when two components are not well separated, a single distribution may often be a better choice because it has fewer parameters and better numerical stability; and (5). dependencies exist in sampling distributions of parameters of mixtures and are influenced by the amount of separation between the components. An emission factor case study based upon NO(x) emissions from coal-fired tangential boilers is used to illustrate the application of the approach.     

Quantification of Variability and Uncertainty for Censored Data Sets and Application to Air Toxic Emission Factors

Many environmental data sets, such as for air toxic emission factors, contain several values reported only as below detection limit. Such data sets are referred to as "censored." Typical approaches to dealing with the censored data sets include replacing censored values with arbitrary values of zero, one-half of the detection limit, or the detection limit. Here, an approach to quantification of the variability and uncertainty of censored data sets is demonstrated. Empirical bootstrap simulation is used to simulate censored bootstrap samples from the original data. Maximum likelihood estimation (MLE) is used to fit parametric probability distributions to each bootstrap sample, thereby specifying alternative estimates of the unknown population distribution of the censored data sets. Sampling distributions for uncertainty in statistics such as the mean, median, and percentile are calculated. The robustness of the method was tested by application to different degrees of censoring, sample sizes, coefficients of variation, and numbers of detection limits. Lognormal, gamma, and Weibull distributions were evaluated. The reliability of using this method to estimate the mean is evaluated by averaging the best estimated means of 20 cases for small sample size of 20. The confidence intervals for distribution percentiles estimated with bootstrap/MLE method compared favorably to results obtained with the nonparametric Kaplan-Meier method. The bootstrap/MLE method is illustrated via an application to an empirical air toxic emission factor data set.     

Quantitative Analysis of Variability and Uncertainty with Known Measurement Error: Methodology and Case Study

The appearance of measurement error in exposure and risk factor data potentially affects any inferences regarding variability and uncertainty because the distribution representing the observed data set deviates from the distribution that represents an error-free data set. A methodology for improving the characterization of variability and uncertainty with known measurement errors in data is demonstrated in this article based on an observed data set, known measurement error, and a measurement-error model. A practical method for constructing an error-free data set is presented and a numerical method based upon bootstrap pairs, incorporating two-dimensional Monte Carlo simulation, is introduced to address uncertainty arising from measurement error in selected statistics. When measurement error is a large source of uncertainty, substantial differences between the distribution representing variability of the observed data set and the distribution representing variability of the error-free data set will occur. Furthermore, the shape and range of the probability bands for uncertainty differ between the observed and error-free data set. Failure to separately characterize contributions from random sampling error and measurement error will lead to bias in the variability and uncertainty estimates. However, a key finding is that total uncertainty in mean can be properly quantified even if measurement and random sampling errors cannot be separated. An empirical case study is used to illustrate the application of the methodology.     

Estimation of Error and Bias in Bayesian Monte Carlo Decision Analysis Using the Bootstrap

Bayesian Monte Carlo (BMC) decision analysis adopts a sampling procedure to estimate likelihoods and distributions of outcomes, and then uses that information to calculate the expected performance of alternative strategies, the value of information, and the value of including uncertainty. These decision analysis outputs are therefore subject to sample error. The standard error of each estimate and its bias, if any, can be estimated by the bootstrap procedure. The bootstrap operates by resampling (with replacement) from the original BMC sample, and redoing the decision analysis. Repeating this procedure yields a distribution of decision analysis outputs. The bootstrap approach to estimating the effect of sample error upon BMC analysis is illustrated with a simple value-of-information calculation along with an analysis of a proposed control structure for Lake Erie. The examples show that the outputs of BMC decision analysis can have high levels of sample error and bias.     

Measures of Compounding Conservatism in Probabilistic Risk Assessment

Concern about the degree of uncertainty and potential conservatism in deterministic point estimates of risk has prompted researchers to turn increasingly to probabilistic methods for risk assessment. With Monte Carlo simulation techniques, distributions of risk reflecting uncertainty and/or variability are generated as an alternative. In this paper the compounding of conservatism⁽¹⁾ between the level associated with point estimate inputs selected from probability distributions and the level associated with the deterministic value of risk calculated using these inputs is explored. Two measures of compounded conservatism are compared and contrasted. The first measure considered, F , is defined as the ratio of the risk value, R _d, calculated deterministically as a function of n inputs each at the j th percentile of its probability distribution, and the risk value, R _j that falls at the j th percentile of the simulated risk distribution (i.e., F=R_d/R_j). The percentile of the simulated risk distribution which corresponds to the deterministic value, R_d , serves as a second measure of compounded conservatism. Analytical results for simple products of lognormal distributions are presented. In addition, a numerical treatment of several complex cases is presented using five simulation analyses from the literature to illustrate. Overall, there are cases in which conservatism compounds dramatically for deterministic point estimates of risk constructed from upper percentiles of input parameters, as well as those for which the effect is less notable. The analytical and numerical techniques discussed are intended to help analysts explore the factors that influence the magnitude of compounding conservatism in specific cases.     

Estimating Upper Confidence Limits for Extra Risk in Quantal Multistage Models

Multistage models are frequently applied in carcinogenic risk assessment. In their simplest form, these models relate the probability of tumor presence to some measure of dose. These models are then used to project the excess risk of tumor occurrence at doses frequently well below the lowest experimental dose. Upper confidence limits on the excess risk associated with exposures at these doses are then determined. A likelihood-based method is commonly used to determine these limits. We compare this method to two computationally intensive "bootstrap" methods for determining the 95% upper confidence limit on extra risk. The coverage probabilities and bias of likelihood-based and bootstrap estimates are examined in a simulation study of carcinogenicity experiments. The coverage probabilities of the nonparametric bootstrap method fell below 95% more frequently and by wider margins than the better-performing parametric bootstrap and likelihood-based methods. The relative bias of all estimators are seen to be affected by the amount of curvature in the true underlying dose-response function. In general, the likelihood-based method has the best coverage probability properties while the parametric bootstrap is less biased and less variable than the likelihood-based method. Ultimately, neither method is entirely satisfactory for highly curved dose-response patterns.     

Integration of Probabilistic Exposure Assessment and Probabilistic Hazard Characterization

A method is proposed for integrated probabilistic risk assessment where exposure assessment and hazard characterization are both included in a probabilistic way. The aim is to specify the probability that a random individual from a defined (sub)population will have an exposure high enough to cause a particular health effect of a predefined magnitude, the critical effect size ( CES ). The exposure level that results in exactly that CES in a particular person is that person's individual critical effect dose ( ICED ). Individuals in a population typically show variation, both in their individual exposure ( IEXP ) and in their ICED . Both the variation in IEXP and the variation in ICED are quantified in the form of probability distributions. Assuming independence between both distributions, they are combined (by Monte Carlo) into a distribution of the individual margin of exposure ( IMoE ). The proportion of the IMoE distribution below unity is the probability of critical exposure ( PoCE ) in the particular (sub)population. Uncertainties involved in the overall risk assessment (i.e., both regarding exposure and effect assessment) are quantified using Monte Carlo and bootstrap methods. This results in an uncertainty distribution for any statistic of interest, such as the probability of critical exposure ( PoCE ). The method is illustrated based on data for the case of dietary exposure to the organophosphate acephate. We present plots that concisely summarize the probabilistic results, retaining the distinction between variability and uncertainty. We show how the relative contributions from the various sources of uncertainty involved may be quantified.     

Benchmark Dose Analysis via Nonparametric Regression Modeling

Estimation of benchmark doses (BMDs) in quantitative risk assessment traditionally is based upon parametric dose‐response modeling. It is a well‐known concern, however, that if the chosen parametric model is uncertain and/or misspecified, inaccurate and possibly unsafe low‐dose inferences can result. We describe a nonparametric approach for estimating BMDs with quantal‐response data based on an isotonic regression method, and also study use of corresponding, nonparametric, bootstrap‐based confidence limits for the BMD. We explore the confidence limits’ small‐sample properties via a simulation study, and illustrate the calculations with an example from cancer risk assessment. It is seen that this nonparametric approach can provide a useful alternative for BMD estimation when faced with the problem of parametric model uncertainty.     

Inference in Arch and Garch Models with Heavy–Tailed Errors

ARCH and GARCH models directly address the dependency of conditional second moments, and have proved particularly valuable in modelling processes where a relatively large degree of fluctuation is present. These include financial time series, which can be particularly heavy tailed. However, little is known about properties of ARCH or GARCH models in the heavy–tailed setting, and no methods are available for approximating the distributions of parameter estimators there. In this paper we show that, for heavy–tailed errors, the asymptotic distributions of quasi–maximum likelihood parameter estimators in ARCH and GARCH models are nonnormal, and are particularly difficult to estimate directly using standard parametric methods. Standard bootstrap methods also fail to produce consistent estimators. To overcome these problems we develop percentile–t, subsample bootstrap approximations to estimator distributions. Studentizing is employed to approximate scale, and the subsample bootstrap is used to estimate shape. The good performance of this approach is demonstrated both theoretically and numerically.     

Consumer-Phase Salmonella enterica serovar Enteritidis Risk Assessment for Egg-Containing Food Products

We describe a one-dimensional probabilistic model of the role of domestic food handling behaviors on salmonellosis risk associated with the consumption of eggs and egg-containing foods. Six categories of egg-containing foods were defined based on the amount of egg contained in the food, whether eggs are pooled, and the degree of cooking practiced by consumers. We used bootstrap simulation to quantify uncertainty in risk estimates due to sampling error, and sensitivity analysis to identify key sources of variability and uncertainty in the model. Because of typical model characteristics such as nonlinearity, interaction between inputs, thresholds, and saturation points, Sobol's method, a novel sensitivity analysis approach, was used to identify key sources of variability. Based on the mean probability of illness, examples of foods from the food categories ranked from most to least risk of illness were: (1) home-made salad dressings/ice cream; (2) fried eggs/boiled eggs; (3) omelettes; and (4) baked foods/breads. For food categories that may include uncooked eggs (e.g., home-made salad dressings/ice cream), consumer handling conditions such as storage time and temperature after food preparation were the key sources of variability. In contrast, for food categories associated with undercooked eggs (e.g., fried/soft-boiled eggs), the initial level of Salmonella contamination and the log10 reduction due to cooking were the key sources of variability. Important sources of uncertainty varied with both the risk percentile and the food category under consideration. This work adds to previous risk assessments focused on egg production and storage practices, and provides a science-based approach to inform consumer risk communications regarding safe egg handling practices.     

A Probabilistic Approach for Deriving Acceptable Human Intake Limits and Human Health Risks from Toxicological Studies: General Framework

The use of uncertainty factors in the standard method for deriving acceptable intake or exposure limits for humans, such as the Reference Dose (RfD), may be viewed as a conservative method of taking various uncertainties into account. As an obvious alternative, the use of uncertainty distributions instead of uncertainty factors is gaining attention. This paper presents a comprehensive discussion of a general framework that quantifies both the uncertainties in the no-adverse-effect level in the animal (using a benchmark-like approach) and the uncertainties in the various extrapolation steps involved (using uncertainty distributions). This approach results in an uncertainty distribution for the no-adverse-effect level in the sensitive human subpopulation, reflecting the overall scientific uncertainty associated with that level. A lower percentile of this distribution may be regarded as an acceptable exposure limit (e.g., RfD) that takes account of the various uncertainties in a nonconservative fashion. The same methodology may also be used as a tool to derive a distribution for possible human health effects at a given exposure level. We argue that in a probabilistic approach the uncertainty in the estimated no-adverse-effect-level in the animal should be explicitly taken into account. Not only is this source of uncertainty too large to be ignored, it also has repercussions for the quantification of the other uncertainty distributions.     

Probabilistic Framework for the Estimation of the Adult and Child Toxicokinetic Intraspecies Uncertainty Factors

Human health risk assessments use point values to develop risk estimates and thus impart a deterministic character to risk, which, by definition, is a probability phenomenon. The risk estimates are calculated based on individuals and then, using uncertainty factors (UFs), are extrapolated to the population that is characterized by variability. Regulatory agencies have recommended the quantification of the impact of variability in risk assessments through the application of probabilistic methods. In the present study, a framework that deals with the quantitative analysis of uncertainty (U) and variability (V) in target tissue dose in the population was developed by applying probabilistic analysis to physiologically-based toxicokinetic models. The mechanistic parameters that determine kinetics were described with probability density functions (PDFs). Since each PDF depicts the frequency of occurrence of all expected values of each parameter in the population, the combined effects of multiple sources of U/V were accounted for in the estimated distribution of tissue dose in the population, and a unified (adult and child) intraspecies toxicokinetic uncertainty factor UFH-TK was determined. The results show that the proposed framework accounts effectively for U/V in population toxicokinetics. The ratio of the 95th percentile to the 50th percentile of the annual average concentration of the chemical at the target tissue organ (i.e., the UFH-TK) varies with age. The ratio is equivalent to a unified intraspecies toxicokinetic UF, and it is one of the UFs by which the NOAEL can be divided to obtain the RfC/RfD. The 10-fold intraspecies UF is intended to account for uncertainty and variability in toxicokinetics (3.2x) and toxicodynamics (3.2x). This article deals exclusively with toxicokinetic component of UF. The framework provides an alternative to the default methodology and is advantageous in that the evaluation of toxicokinetic variability is based on the distribution of the effective target tissue dose, rather than applied dose. It allows for the replacement of the default adult and children intraspecies UF with toxicokinetic data-derived values and provides accurate chemical-specific estimates for their magnitude. It shows that proper application of probability and toxicokinetic theories can reduce uncertainties when establishing exposure limits for specific compounds and provide better assurance that established limits are adequately protective. It contributes to the development of a probabilistic noncancer risk assessment framework and will ultimately lead to the unification of cancer and noncancer risk assessment methodologies.     

Importance of Uncertainty and Variability to Predicted Risks from Trophic Transfer of PCBs in Dredged Sediments

Biomagnification of organochlorine and other persistent organic contaminants by higher trophic level organisms represents one of the most significant sources of uncertainty and variability in evaluating potential risks associated with disposal of dredged materials. While it is important to distinguish between population variability (e.g., true population heterogeneity in fish weight, and lipid content) and uncertainty (e.g., measurement error), they can be operationally difficult to define separately in probabilistic estimates of human health and ecological risk. We propose a disaggregation of uncertain and variable parameters based on: (1) availability of supporting data; (2) the specific management and regulatory context (in this case, of the U.S. Army Corps of Engineers/U.S. Environmental Protection Agency tiered approach to dredged material management); and (3) professional judgment and experience in conducting probabilistic risk assessments. We describe and quantitatively evaluate several sources of uncertainty and variability in estimating risk to human health from trophic transfer of polychlorinated biphenyls (PCBs) using a case study of sediments obtained from the New York-New Jersey Harbor and being evaluated for disposal at an open water off-shore disposal site within the northeast region. The estimates of PCB concentrations in fish and dietary doses of PCBs to humans ingesting fish are expressed as distributions of values, of which the arithmetic mean or mode represents a particular fractile. The distribution of risk values is obtained using a food chain biomagnification model developed by Gobas by specifying distributions for input parameters disaggregated to represent either uncertainty or variability. Only those sources of uncertainty that could be quantified were included in the analysis. Results for several different two-dimensional Latin Hypercube analyses are provided to evaluate the influence of the uncertain versus variable disaggregation of model parameters. The analysis suggests that variability in human exposure parameters is greater than the uncertainty bounds on any particular fractile, given the described assumptions.     

Percentiles of the Product of Uncertainty Factors for Establishing Probabilistic Reference Doses

Exposure guidelines for potentially toxic substances are often based on a reference dose (RfD) that is determined by dividing a no-observed-adverse-effect-level (NOAEL), lowest-observed-adverse-effect-level (LOAEL), or benchmark dose (BD) corresponding to a low level of risk, by a product of uncertainty factors. The uncertainty factors for animal to human extrapolation, variable sensitivities among humans, extrapolation from measured subchronic effects to unknown results for chronic exposures, and extrapolation from a LOAEL to a NOAEL can be thought of as random variables that vary from chemical to chemical. Selected databases are examined that provide distributions across chemicals of inter- and intraspecies effects, ratios of LOAELs to NOAELs, and differences in acute and chronic effects, to illustrate the determination of percentiles for uncertainty factors. The distributions of uncertainty factors tend to be approximately lognormally distributed. The logarithm of the product of independent uncertainty factors is approximately distributed as the sum of normally distributed variables, making it possible to estimate percentiles for the product. Hence, the size of the products of uncertainty factors can be selected to provide adequate safety for a large percentage (e.g., approximately 95%) of RfDs. For the databases used to describe the distributions of uncertainty factors, using values of 10 appear to be reasonable and conservative. For the databases examined the following simple "Rule of 3s" is suggested that exceeds the estimated 95th percentile of the product of uncertainty factors: If only a single uncertainty factor is required use 33, for any two uncertainty factors use 3 x 33 approximately 100, for any three uncertainty factors use a combined factor of 3 x 100 = 300, and if all four uncertainty factors are needed use a total factor of 3 x 300 = 900. If near the 99th percentile is desired use another factor of 3. An additional factor may be needed for inadequate data or a modifying factor for other uncertainties (e.g., different routes of exposure) not covered above.     

On the Effect of Probability Distributions of Input Variables in Public Health Risk Assessment

A central part of probabilistic public health risk assessment is the selection of probability distributions for the uncertain input variables. In this paper, we apply the first-order reliability method (FORM)^(1–3) as a probabilistic tool to assess the effect of probability distributions of the input random variables on the probability that risk exceeds a threshold level (termed the probability of failure) and on the relevant probabilistic sensitivities. The analysis was applied to a case study given by Thompson et al. ⁽⁴⁾ on cancer risk caused by the ingestion of benzene contaminated soil. Normal, lognormal, and uniform distributions were used in the analysis. The results show that the selection of a probability distribution function for the uncertain variables in this case study had a moderate impact on the probability that values would fall above a given threshold risk when the threshold risk is at the 50th percentile of the original distribution given by Thompson et al. ⁽⁴⁾ The impact was much greater when the threshold risk level was at the 95th percentile. The impact on uncertainty sensitivity, however, showed a reversed trend, where the impact was more appreciable for the 50th percentile of the original distribution of risk given by Thompson et al. ⁴ than for the 95th percentile. Nevertheless, the choice of distribution shape did not alter the order of probabilistic sensitivity of the basic uncertain variables.     

Bootstrap Estimation of Benchmark Doses and Confidence Limits with Clustered Quantal Data

The benchmark dose (BMD) is an exposure level that would induce a small risk increase (BMR level) above the background. The BMD approach to deriving a reference dose for risk assessment of noncancer effects is advantageous in that the estimate of BMD is not restricted to experimental doses and utilizes most available dose-response information. To quantify statistical uncertainty of a BMD estimate, we often calculate and report its lower confidence limit (i.e., BMDL), and may even consider it as a more conservative alternative to BMD itself. Computation of BMDL may involve normal confidence limits to BMD in conjunction with the delta method. Therefore, factors, such as small sample size and nonlinearity in model parameters, can affect the performance of the delta method BMDL, and alternative methods are useful. In this article, we propose a bootstrap method to estimate BMDL utilizing a scheme that consists of a resampling of residuals after model fitting and a one-step formula for parameter estimation. We illustrate the method with clustered binary data from developmental toxicity experiments. Our analysis shows that with moderately elevated dose-response data, the distribution of BMD estimator tends to be left-skewed and bootstrap BMDL s are smaller than the delta method BMDL s on average, hence quantifying risk more conservatively. Statistically, the bootstrap BMDL quantifies the uncertainty of the true BMD more honestly than the delta method BMDL as its coverage probability is closer to the nominal level than that of delta method BMDL. We find that BMD and BMDL estimates are generally insensitive to model choices provided that the models fit the data comparably well near the region of BMD. Our analysis also suggests that, in the presence of a significant and moderately strong dose-response relationship, the developmental toxicity experiments under the standard protocol support dose-response assessment at 5% BMR for BMD and 95% confidence level for BMDL.     

On the Bootstrap of the Maximum Score Estimator

This paper shows that the bootstrap does not consistently estimate the asymptotic distribution of the maximum score estimator. The theory developed also applies to other estimators within a cube‐root convergence class. For some single‐parameter estimators in this class, the results suggest a simple method for inference based upon the bootstrap.     

Uncertainty Distribution Associated with Estimating a Proportion in Microbial Risk Assessment

The uncertainty associated with estimates should be taken into account in quantitative risk assessment. Each input's uncertainty can be characterized through a probabilistic distribution for use under Monte Carlo simulations. In this study, the sampling uncertainty associated with estimating a low proportion on the basis of a small sample size was considered. A common application in microbial risk assessment is the estimation of a prevalence, proportion of contaminated food products, on the basis of few tested units. Three Bayesian approaches (based on beta(0, 0), beta(1/2, 1/2), and beta(l, 1)) and one frequentist approach (based on the frequentist confidence distribution) were compared and evaluated on the basis of simulations. For small samples, we demonstrated some differences between the four tested methods. We concluded that the better method depends on the true proportion of contaminated products, which is by definition unknown in common practice. When no prior information is available, we recommend the beta (1/2, 1/2) prior or the confidence distribution. To illustrate the importance of these differences, the four methods were used in an applied example. We performed two-dimensional Monte Carlo simulations to estimate the proportion of cold smoked salmon packs contaminated by Listeria monocytogenes, one dimension representing within-factory uncertainty, modeled by each of the four studied methods, and the other dimension representing variability between companies.     

Propagation of Uncertainty in Risk Assessments: The Need to Distinguish Between Uncertainty Due to Lack of Knowledge and Uncertainty Due to Variability

In quantitative uncertainty analysis, it is essential to define rigorously the endpoint or target of the assessment. Two distinctly different approaches using Monte Carlo methods are discussed: (1) the end point is a fixed but unknown value (e.g., the maximally exposed individual, the average individual, or a specific individual) or (2) the end point is an unknown distribution of values (e.g., the variability of exposures among unspecified individuals in the population). In the first case, values are sampled at random from distributions representing various "degrees of belief" about the unknown "fixed" values of the parameters to produce a distribution of model results. The distribution of model results represents a subjective confidence statement about the true but unknown assessment end point. The important input parameters are those that contribute most to the spread in the distribution of the model results. In the second case, Monte Carlo calculations are performed in two dimensions producing numerous alternative representations of the true but unknown distribution. These alternative distributions permit subject confidence statements to be made from two perspectives: (1) for the individual exposure occurring at a specified fractile of the distribution or (2) for the fractile of the distribution associated with a specified level of individual exposure. The relative importance of input parameters will depend on the fractile or exposure level of interest. The quantification of uncertainty for the simulation of a true but unknown distribution of values represents the state-of-the-art in assessment modeling.     

Inference on Counterfactual Distributions

Counterfactual distributions are important ingredients for policy analysis and decomposition analysis in empirical economics. In this article, we develop modeling and inference tools for counterfactual distributions based on regression methods. The counterfactual scenarios that we consider consist of ceteris paribus changes in either the distribution of covariates related to the outcome of interest or the conditional distribution of the outcome given covariates. For either of these scenarios, we derive joint functional central limit theorems and bootstrap validity results for regression‐based estimators of the status quo and counterfactual outcome distributions. These results allow us to construct simultaneous confidence sets for function‐valued effects of the counterfactual changes, including the effects on the entire distribution and quantile functions of the outcome as well as on related functionals. These confidence sets can be used to test functional hypotheses such as no‐effect, positive effect, or stochastic dominance. Our theory applies to general counterfactual changes and covers the main regression methods including classical, quantile, duration, and distribution regressions. We illustrate the results with an empirical application to wage decompositions using data for the United States. As a part of developing the main results, we introduce distribution regression as a comprehensive and flexible tool for modeling and estimating the entire conditional distribution. We show that distribution regression encompasses the Cox duration regression and represents a useful alternative to quantile regression. We establish functional central limit theorems and bootstrap validity results for the empirical distribution regression process and various related functionals.