Conditional Density Function Research Articles

First-stage analysis for instrumental-variables quantile regression

In this article, we develop a first-stage linear regression command, fsivqreg, for an instrumental-variables quantile regression (QR) model. The quantile first stage is analogous to the least-squares case, that is, a linear projection of the endogenous variables on the instruments and other exogenous covariates, with the difference that the QR case is a weighted projection. The weights are given by the conditional density function of the innovation term in the QR structural model, at a given quantile. An empirical application illustrates its implementation.

Efficient data integration under prior probability shift.

Conventional supervised learning usually operates under the premise that data are collected from the same underlying population. However, challenges may arise when integrating new data from different populations, resulting in a phenomenon known as dataset shift. This paper focuses on prior probability shift, where the distribution of the outcome varies across datasets but the conditional distribution of features given the outcome remains the same. To tackle the challenges posed by such shift, we propose an estimation algorithm that can efficiently combine information from multiple sources. Unlike existing methods that are restricted to discrete outcomes, the proposed approach accommodates both discrete and continuous outcomes. It also handles high-dimensional covariate vectors through variable selection using an adaptive least absolute shrinkage and selection operator penalty, producing efficient estimates that possess the oracle property. Moreover, a novel semiparametric likelihood ratio test is proposed to check the validity of prior probability shift assumptions by embedding the null conditional density function into Neyman's smooth alternatives (Neyman, 1937) and testing study-specific parameters. We demonstrate the effectiveness of our proposed method through extensive simulations and a real data example. The proposed methods serve as a useful addition to the repertoire of tools for dealing dataset shifts.

Modeling forecast errors for microgrid operation using Gaussian process regression

Microgrids, denoting small-scale and self-sustaining grids, constitute a pivotal component in future power systems with a high penetration of renewable generators. The inherent uncertainty tied to renewable power generation, typified by photovoltaic and wind turbine systems, necessitates counterbalancing mechanisms. These mechanisms encompass Energy storage systems or conventional thermal fossil-fuel generators imbued with heightened flexibility. Addressing the uncertainty stemming from renewable generators mandates a cost-effective assessment and operational strategy for said compensatory devices. To this end, myriad uncertainty factors warrant scrutiny, conceivably concretized into a unified probability distribution function (PDF) that takes into account their temporal inter-dependencies. Diverse uncertainty factors, characterized by varying marginal distributions and scales, can be assimilated into a multivariate probability distribution through a conversion to normal distributions via rank correlation. However, with the escalation in the number of uncertainty factors embraced within a microgrid context, the endeavour becomes notably intricate when aiming to define conditional probability distributions originating from joint PDFs. This paper presents a method proposing the modelling of net-load forecast error distribution, considering the interplay among uncertainty factors. The approach introduces a data-driven Gaussian process regression technique for training and validating conditional PDFs among these uncertainty factors. Notably, this approach facilitates the transformation of said factors into normal distributions while preserving their inherent marginal characteristics. The resultant conditional density function, as per the proposed methodology, exhibits enhanced suitability for estimating net-load error distribution. Consequently, the conditional density function stemming from this proposed approach demonstrates superior aptitude in approximating the distribution of net load error.

Central Limit Theorem for Conditional Mode in the Single Functional Index Model with Data Missing at Random

This paper concentrates on nonparametrically estimating the conditional density function and conditional mode within the single functional index model for independent data, particularly when the variable of interest is affected by randomly missing data. This involves a semi-parametric single model structure and a censoring process on the variables. The estimator’s consistency (with rates) in a variety of situations, such as the framework of the single functional index model (SFIM) under the assumption of independent and identically distributed (i.i.d) data with randomly missing entries, as well as its performance under the assumption that the covariate is functional, are the main areas of focus. For this model, the nearly almost complete uniform convergence and rate of convergence are established. The rates of convergence highlight the critical part that the probability of concentration play in the law of the explanatory functional variable. Additionally, we establish the asymptotic normality of the derived estimators proposed under specific mild conditions, relying on standard assumptions in Functional Data Analysis (FDA) for the proofs. Finally, we explore the practical application of our findings in constructing confidence intervals for our estimators. The rates of convergence highlight the critical part that the probability of concentration play in the law of the explanatory functional variable.

Testing for the Markov property in time series via deep conditional generative learning.

The Markov property is widely imposed in analysis of time series data. Correspondingly, testing the Markov property, and relatedly, inferring the order of a Markov model, are of paramount importance. In this article, we propose a nonparametric test for the Markov property in high-dimensional time series via deep conditional generative learning. We also apply the test sequentially to determine the order of the Markov model. We show that the test controls the type-I error asymptotically, and has the power approaching one. Our proposal makes novel contributions in several ways. We utilise and extend state-of-the-art deep generative learning to estimate the conditional density functions, and establish a sharp upper bound on the approximation error of the estimators. We derive a doubly robust test statistic, which employs a nonparametric estimation but achieves a parametric convergence rate. We further adopt sample splitting and cross-fitting to minimise the conditions required to ensure the consistency of the test. We demonstrate the efficacy of the test through both simulations and the three data applications.

Conditional density function for surrogate scalar response

his paper presents the estimator of the conditional density function of surrogated scalar response variable given a functional random one. We construct a conditional density function by using the available (true) response data and the surrogate data. Then, we build up some asymptotic properties of the constructed estimator in terms of the almost complete convergences. As a result, we compare our estimator with the classical estimator through the Relatif Mean Square Errors (RMSE). Finally, we end this analysis by displaying the superiority of our estimator in terms of prediction when we are lacking complete data.

A first-stage representation for instrumental variables quantile regression

Summary This paper develops a first-stage linear regression representation for an instrumental variables (IV) quantile regression (QR) model. The quantile first stage is analogous to the least-squares case, i.e., a linear projection of the endogenous variables on the instruments and other exogenous covariates, with the difference that the QR case is a weighted projection. The weights are given by the conditional density function of the innovation term in the QR structural model, at a given quantile. We also show that the required Jacobian identification conditions for IVQR models are embedded in the quantile first stage. We then suggest procedures to evaluate the validity of instruments by evaluating their statistical significance using the first-stage representation. Monte Carlo experiments provide numerical evidence that the proposed tests work as expected in terms of empirical size and power. An empirical application illustrates the methods.

On the central limit theorem for a conditional mode estimator in the single functional index modeling for functional time series data under random censorship

The main objective of this paper is to investigate the estimation of conditional density function based on the single-index model in the censorship model when the sample is considered as an dependent random variables. First of all, a kernel type estimator for the conditional density function (cond-df) is introduced. Afterwards, the asymptotic properties are stated when the observations are linked with a single-index structure. The pointwise almost complete convergence and the uniform almost complete convergence (with rate) of the kernel estimate of this model are established. As an application the conditional mode in functional single-index model is presented. Under general conditions, the asymptotic normality of the conditional density estimator is established. Simulation study is also presented to illustrate the validity and finite sample performance of the considered estimator. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is discussed, but not tackled.

Nonparametric Estimation of the Expected Shortfall Regression for Quasi-Associated Functional Data

In this paper, we study the nonparametric estimation of the expected shortfall regression when the exogenous observation is functional. The constructed estimator is obtained by combining the double kernels estimator of both conditional value at risk and conditional density function. The asymptotic proprieties of this estimator are established under weak dependency condition. Precisely, we assume that the observations are generated from quasi-associated functional time series and we prove the almost complete convergence of the constructed estimator. This asymptotic result is obtained under a standard condition of functional time series analysis. The finite sample performance of this estimator is evaluated using artificial data.

On the singular gamma, Wishart, and beta matrix‐variate density functions

AbstractWhen a real positive definite matrix follows a Wishart or, more generally, a matrix‐variate gamma distribution with shape parameter and positive definite scale parameter matrix , one can represent as for some matrix of dimension . When , has a singular distribution whose properties can be studied via the density function of . It will be shown that when follows a matrix‐variate extended Gaussian distribution, the density function of the resulting singular gamma distribution can be obtained by making use of successive transformations and their associated Jacobians. The singular Wishart distribution will then be obtained as a particular case. The marginal and conditional density functions resulting from an arbitrary partitioning of will be considered as well. The same technique will also be applied to the derivation of the density functions of real and complex singular type‐1 and type‐2 beta‐distributed matrices. It so happens that the proposed approach, which is based on manipulations involving the wedge products of certain differential elements, generally proves more efficient than the intricate procedures that have hitherto been employed in the literature.

THE CONSISTENCY OF THE KERNEL ESTIMATION OF THE FUNCTION CONDITIONAL DENSITY FOR ASSOCIATED DATA IN HIGH-DIMENSIONAL STATISTICS

The purpose of the present paper is to investigate by the kernel method a nonparametric estimate of the conditional density function of a scalar response variable given a random variable taking values in a separable real Hilbert space when the observations are quasi-associated dependent. Under some general conditions, we establish the pointwise almost complete consistencies with rates of this estimator. The principal aim is to investigate of the convergence rate of the proposed estimator.

Comment on “Modeling Purchasing Behavior with Sudden ‘Death’: A Flexible Customer Lifetime Model”

In their 2012 paper, Bemmaor and Glady introduced the gamma/Gompertz/negative binomial distribution model for customer base analysis. Their model uses exponentially distributed interpurchase times and a Gompertz distributed customer lifetime, where the latter distribution is nonmemoryless. This comment corrects an error in their expression for the conditional expected number of individual future purchases [Formula: see text] in a forecasting interval of length [Formula: see text]. Contrary to their approach, the correct derivation of the conditional expectation must be based on the conditional survival and density functions of the lifetime distribution. Using the wrong formula leads managers to overestimate the expected future customer purchases. Further, the comment corrects the erroneous expressions for the conditional variance [Formula: see text] and the conditional mean residual lifetime [Formula: see text]. This paper was accepted by Raphael Thomadsen, marketing. Supplemental Material: The e-companion is available at https://doi.org/10.1287/mnsc.2022.4422 .

Nonparametric Estimation of the Density Function of the Distribution of the Noise in CHARN Models

This work is concerned with multivariate conditional heteroscedastic autoregressive nonlinear (CHARN) models with an unknown conditional mean function, conditional variance matrix function and density function of the distribution of noise. We study the kernel estimator of the latter function when the former are either parametric or nonparametric. The consistency, bias and asymptotic normality of the estimator are investigated. Confidence bound curves are given. A simulation experiment is performed to evaluate the performance of the results.

Delta family approach for the stochastic control problems of utility maximization

In this paper, we propose a new approach for stochastic control problems arising from utility maximization. The main idea is to directly start from the dynamical programming equation and compute the conditional expectation using a novel representation of the conditional density function through the Dirac Delta function and the corresponding series representation. We obtain an explicit series representation of the value function, whose coefficients are expressed through integration of the value function at a later time point against a chosen basis function. Thus we are able to set up a recursive integration time-stepping scheme to compute the optimal value function given the known terminal condition, e.g. utility function. Due to tensor decomposition property of the Dirac Delta function in high dimensions, it is straightforward to extend our approach to solving high-dimensional stochastic control problems. The backward recursive nature of the method also allows for solving stochastic control and stopping problems, i.e. mixed control problems. We illustrate the method through solving some two-dimensional stochastic control (and stopping) problems, including the case under the classical and rough Heston stochastic volatility models, and stochastic local volatility models such as the stochastic alpha beta rho (SABR) model.

Asymptotic properties of the estimator of the conditional distribution for associated functional data

The purpose of the paper was to investigate by the kernel method a nonparametric estimate of the conditional density function of a scalar response variable given a random variable taking values in a separable real Hilbert space when the observations are quasi-associated dependent. Under some general conditions, the authors established the pointwise almost complete consistencies with rates of this estimator. The principal aim is the investigate the convergence rate of the proposed estimator.

Non parametric estimations of the conditional density and mode when the regressor and the response are curves

We develop new estimation results for the functional relationship between a regressor and a response which are functions indexed by time or by spatial locations. The regressor is assumed to belong to a semi-metric space whereas the responses belong to a Hilbert space First, we build a double-kernel estimator of the conditional density function, via a Nadaraya-Watson method. Then, we deduce a conditional mode estimator as the value that maximizes the conditional density estimator. Then, we establish the strong uniform consistencies, with rates, of the two constructed estimators. In this context, we wished to set up these preliminary results which will certainly motivate several works on this same subject.

Daily activity pattern of pumas (Puma concolor) and their potential prey in a tropical cloud forest of Colombia

Ecosystems in the northern Andes face unprecedented habitat loss. Pumas are the top predators in the region and exert key ecological functions, such as population control and resource facilitation. However, little is known about the temporal niche of the species or its effects on behaviour of prey in the tropics. We hypothesized that there is a link between the activity patterns of pumas and their prey in a cloud forest of the Central Andes of Colombia. We installed 61 camera traps to estimate the degree of overlap between the daily activity curves of pumas and seven potential prey species, using conditional kernel density functions. Pumas, armadillos, mountain pacas, and white–eared opossums were mainly nocturnal, with little crepuscular activity and high temporal overlap. Central American agouti, mountain coati, little red brocket deer, and Cauca guan displayed a predominantly diurnal activity and temporal partitioning with pumas. As opportunistic predators, pumas were able to maximize foraging efficiency by preying on the crepuscular and nocturnal species. Conservation of this highland predator will largely depend on the suitable management of its native prey.

Nonparametric conditional density estimation in a deep learning framework for short-term forecasting

Short-term forecasting is an important tool in understanding environmental processes. In this paper, we incorporate machine learning algorithms into a conditional distribution estimator for the purposes of forecasting tropical cyclone intensity. Many machine learning techniques give a single-point prediction of the conditional distribution of the target variable, which does not give a full accounting of the prediction variability. Conditional distribution estimation can provide extra insight on predicted response behavior, which could influence decision-making and policy. We propose a technique that simultaneously estimates the entire conditional distribution and flexibly allows for machine learning techniques to be incorporated. A smooth model is fit over both the target variable and covariates, and a logistic transformation is applied on the model output layer to produce an expression of the conditional density function. We provide two examples of machine learning models that can be used, polynomial regression and deep learning models. To achieve computational efficiency, we propose a case–control sampling approximation to the conditional distribution. A simulation study for four different data distributions highlights the effectiveness of our method compared to other machine learning-based conditional distribution estimation techniques. We then demonstrate the utility of our approach for forecasting purposes using tropical cyclone data from the Atlantic Seaboard. This paper gives a proof of concept for the promise of our method, further computational developments can fully unlock its insights in more complex forecasting and other applications.

Asymptotic normality of conditional density and conditional mode in the functional single index model

The main objective of this paper is to investigate the nonparametric estimation of the conditional density of a scalar response variable Y, given the explanatory variable X taking value in a Hilbert space when the sample of observations is considered as an independent random variables with identical distribution (i.i.d) and are linked with a single functional index structure. First of all, a kernel type estimator for the conditional density function (cond-df) is introduced. Afterwards, the asymptotic properties are stated for a conditional density estimator when the observations are linked with a single-index structure from which one derives a central limit theorem (CLT) of the conditional density estimator to show the asymptotic normality of the kernel estimate of this model. As an application the conditional mode in functional single-index model is presented, and the asymptotic (1 – ζ) confidence interval of the conditional mode function is given for 0

Transition matrix and stochastic kernel for repeatability assessment of performance of Polish open pension funds

This study focuses on the issue of repeatability and reversal of performance achieved by Polish open pension funds. For the choice of the fund investing the contributions of the future pensioner, the phenomenon of repeatability and reversal of its performance is essential. Otherwise, historical rates of return cannot be used to predict future ones, and in that case the only rational method of choosing a pension fund is a pure random selection. Contrarily to the other studies taking into account numerous variables describing open pension funds, the author analyzes the only one variable interesting from a future pensioner’s viewpoint: rate of return. As a consequence the formulated conclusions are not flawed due to considering factors having no influence on future pensioners’ wealth. The analysis of these phenomena was carried out by adopting the dynamics of distribution testing methods. The estimated Markovian transition matrix and the conditional density function allowed us to formulate the conclusion about the weak performance repeatability of pension funds.