Conditional Distribution Function Research Articles

Added value of an atmospheric circulation pattern‐based statistical downscaling approach for daily precipitation distributions in complex terrain

AbstractReliable prediction of heavy precipitation events causing floods in a world of changing climate is crucial for the development of appropriate adaption strategies. Many attempts to provide such predictions have already been conducted but there is still much potential for improvement left. This is particularly true for statistical downscaling of heavy precipitation due to changes present in the corresponding atmospheric drivers. In this study, a circulation pattern (CP) conditional downscaling to the station level is proposed which considers occurring frequency changes of CPs. Following a strict circulation‐to‐environment approach we use atmospheric predictors to derive CPs. Subsequently, precipitation observations are used to derive CP conditional cumulative distribution functions (CDFs) of daily precipitation. Raw precipitation time series are sampled from these CDFs. Bias correction is applied to the sampled time series with quantile mapping (QM) and parametric transfer functions (PTFs) as methods being tested. The added value of this CP conditional downscaling approach is evaluated against the corresponding common non‐CP conditional approach. The performance evaluation is conducted by using Kling–Gupta Efficiency (KGE), root mean squared error (RMSE), and mean absolute error (MAE) metrics. In both cases the applied bias correction is identical. Potential added value can therefore only be attributed to the CP conditioning. It can be shown that the proposed CP conditional downscaling approach is capable of yielding more reliable and accurate downscaled daily precipitation time series in comparison to a non‐CP conditional approach. This can be seen in particular for the extreme parts of the distribution. Above the 95th percentile, an average performance gain of +0.24 and a maximum gain of +0.6 in terms of KGE is observed. These findings support the assumption of conserving and utilizing atmospheric information through CPs can be beneficial for more reliable statistical precipitation downscaling. Due to the availability of these atmospheric predictors in climate model output, the presented method is potentially suitable for downscaling precipitation projections.

The Automatic Solution of Macromolecular Crystal Structures via Molecular Replacement Techniques: REMO22 and Its Pipeline.

A description of REMO22, a new molecular replacement program for proteins and nucleic acids, is provided. This program, as with REMO09, can use various types of prior information through appropriate conditional distribution functions. Its efficacy in model searching has been validated through several test cases involving proteins and nucleic acids. Although REMO22 can be configured with different protocols according to user directives, it has been developed primarily as an automated tool for determining the crystal structures of macromolecules. To evaluate REMO22's utility in the current crystallographic environment, its experimental results must be compared favorably with those of the most widely used Molecular Replacement (MR) programs. To accomplish this, we chose two leading tools in the field, PHASER and MOLREP. REMO22, along with MOLREP and PHASER, were included in pipelines that contain two additional steps: phase refinement (SYNERGY) and automated model building (CAB). To evaluate the effectiveness of REMO22, SYNERGY and CAB, we conducted experimental tests on numerous macromolecular structures. The results indicate that REMO22, along with its pipeline REMO22 + SYNERGY + CAB, presents a viable alternative to currently used phasing tools.

ASYMPTOTIC NORMALITY OF L1-APPROACH A KERNEL ESTIMATOR OF CONDITIONAL CUMULATIVE DISTRIBUTION FUNCTION IN THE FUNCTIONAL SINGLE INDEX MODEL

The main of this paper is to treat the estimation of conditional distribution function for functional data. We defined the L1 norm estimator. Under some assumption in functional data analysis the asymptotic normality of estimator is established.

Estimation and bootstrap for stochastically monotone Markov processes

The Markov property is shared by several popular models for time series such as autoregressive or integer-valued autoregressive processes as well as integer-valued ARCH processes. A natural assumption which is fulfilled by corresponding parametric versions of these models is that the random variable at time t gets stochastically greater conditioned on the past, as the value of the random variable at time t-1 increases. Then the associated family of conditional distribution functions has a certain monotonicity property which allows us to employ a nonparametric antitonic estimator. This estimator does not involve any tuning parameter which controls the degree of smoothing and is therefore easy to apply. Nevertheless, it is shown that it attains a rate of convergence which is known to be optimal in similar cases. This estimator forms the basis for a new method of bootstrapping Markov chains which inherits the properties of simplicity and consistency from the underlying estimator of the conditional distribution function.

Two-stage conditional density estimation based on Bernstein polynomials

Two-stage conditional probability density function estimators are proposed and studied. Specifically, the Nadaraya-Watson (NW) and local linear (LL) conditional distribution function estimators have been smoothed using Bernstein polynomials in the first stage. Second, the proposed estimators are obtained by differentiating NW and LL estimators. The asymptotic properties of these estimators are established such as asymptotic bias, variance, and normality. Finally, a simulation study is carried out to assess the relative advantage of our estimators compared to other competitors. In addition, the well-known Old Faithful Geyser data is analyzed using the proposed estimators.

Deep quantile and deep composite triplet regression

A main difficulty in actuarial claim size modeling is that covariates may have different effects on the body of the conditional distribution and on its tail. To cope with this problem, we introduce a deep composite regression model whose splicing point is given in terms of a quantile of the conditional claim size distribution (rather than a constant). This allows us to simultaneously fit different regression models in the two different parts of the conditional distribution function. To facilitate M-estimation for such models, we introduce and characterize the class of strictly consistent scoring functions for the triplet consisting of a quantile, as well as the lower and upper expected shortfall beyond that quantile. In a second step, this elicitability result is applied to fit deep neural network regression models. We demonstrate the applicability of our approach and its superiority over classical approaches on a real data set from accident insurance.

Local likelihood of quantile difference under left-truncated, right-censored and dependent assumptions

We, in this paper, focus on the inference of conditional quantile difference (CQD) for left-truncated and right-censored model. Based on local conditional likelihood function of the observed data, local likelihood ratio function and smoothed local log-likelihood ratio (log-SLL) of the CQD are constructed, and the maximum local likelihood estimator of the CQD is further defined from the log-SLL. When the observations are assumed to be a sequence of stationary α-mixing random variables, we establish asymptotic normality of the defined estimator, and prove the Wilks' theorem of adjusted log-SLL. Besides, we define another estimator of the CQD based on product-limit estimator of conditional distribution function and give its asymptotic normality. Also, simulation study and real data analysis are conducted to investigate the finite sample behaviour of the proposed methods.

Single Functional Index Quantile Regression for Functional Data with Missing Data at Random

The primary goal of this research was to estimate the quantile of a conditional distribution using a semi-parametric approach in the presence of randomly missing data, where the predictor variable belongs to a semi-metric space. The authors assumed a single index structure to link the explanatory and response variable. First, a kernel estimator was proposed for the conditional distribution function, assuming that the data were selected from a stationary process with missing data at random (MAR). By imposing certain general conditions, the study established the model’s uniform almost complete consistencies with convergence rates.

ASYMPTOTIC NORMALITY SINGLE FUNCTIONAL INDEX QUANTILE REGRESSION UNDER RANDOMLY CENSORED DATA

The main objective of this paper is to estimate non-parametrically the quantiles of a conditional distribution based on the single-index model in the censorship model when the sample is considered as an independent and identically distributed (i.i.d.) random variables. First of all, a kernel type estimator for the conditional cumulative distribution function (cond-cdf) is introduced. Afterwards, we give an estimation of the quantiles by inverting this estimated cond-cdf, the asymptotic properties are stated when the observations are linked with a single-index structure. Finally, a simulation study is carried out to evaluate the performance of this estimate.

Modeling climate extremes using the four-parameter kappa distribution for [formula omitted]-largest order statistics

Accurate estimation of the T-year return levels of climate extremes using statistical distribution is a critical step in the projection of future climate and in engineering design for disaster response. We show how the estimation of such quantities can be improved by fitting the four-parameter kappa distribution for r-largest order statistics (rK4D), which was developed in this study. The rK4D is an extension of the generalized extreme value distribution for r-largest order statistics (rGEVD), similar to the four-parameter kappa distribution (K4D), which is an extension of the generalized extreme value distribution (GEVD). This new distribution (rK4D) can be useful not only for fitting data when three parameters in the GEVD are not sufficient to capture the variability of the extreme observations, but also in reducing the estimation uncertainty by making use of the r-largest extreme observations instead of only the block maxima. We derive a joint probability density function (PDF) of rK4D and the marginal and conditional cumulative distribution functions and PDFs. To estimate the parameters, the maximum likelihood estimation and the maximum penalized likelihood estimation methods were considered. The usefulness and practical effectiveness of the rK4D are illustrated by the Monte Carlo simulation and by an application to the Bangkok extreme rainfall data. A few new distributions for r-largest order statistics are also derived as special cases of the rK4D, such as the r-largest logistic, the r-largest generalized logistic, and the r-largest generalized Gumbel distributions. These distributions for r-largest order statistics would be useful in modeling extreme values for many research areas, including hydrology and climatology.

Asymptotic properties of nonparametric quantile estimation with spatial dependency

The purpose of this work is to nonparametrically estimate the conditional quantile for a locally stationary multivariate spatial process. The new kernel quantile estimate derived from the one of conditional distribution function (CDF). The originality in the paper is based on the ability to take into account some local spatial dependency in estimate CDF form. Consistency and asymptotic normality of the estimates are obtained under ‐mixing condition. Numerical study and application to real data are given in order to illustrate the performance of our methodology.

Consistent Estimation of Distribution Functions under Increasing Concave and Convex Stochastic Ordering

A random variable Y 1 is said to be smaller than Y 2 in the increasing concave stochastic order if for all increasing concave functions for which the expected values exist, and smaller than Y 2 in the increasing convex order if for all increasing convex ψ. This article develops nonparametric estimators for the conditional cumulative distribution functions of a response variable Y given a covariate X, solely under the assumption that the conditional distributions are increasing in x in the increasing concave or increasing convex order. Uniform consistency and rates of convergence are established both for the K-sample case and for continuously distributed X.

Self-similar behavior of the neutron fracture functions

In this paper, we have employed Fractal formalism to calculate the Fracture Functions of the Leading neutron produced in [Formula: see text] collisions. The Fractal concept describes the self-similar behavior of the proton structure at Leading neutron production of semi-inclusive Deep Inelastic Scattering at the low values of the fractional momentum variable [Formula: see text]. The Fracture Functions (FFs) parameterized the non-perturbative part of the fragmentation process at the initial scale of Q[Formula: see text]. In this analysis, we benefit from the Leading neutron (Ln) experimental data published by H1 Collaboration and in order to estimate the uncertainty of neutron FFs and corresponding observables, we used the Hessian method. As a consequence of this Next-to-Leading order QCD analysis, we achieve a nice agreement between the prediction of our model for neutron FFs and experimental data. It seems that the Fractal approach based on the self-similar behavior of these conditional parton distribution functions can nicely describe the experimental data at low [Formula: see text].

LOCAL LINEAR MODELLING OF THE CONDITIONAL DISTRIBUTION FUNCTION FOR FUNCTIONAL ERGODIC DATA

The focus of functional data analysis has been mostly on independent functional observations. It is therefore hoped that the present contribution will provide an informative account of a useful approach that merges the ideas of the ergodic theory and the functional data analysis by using the local linear approach. More precisely, we aim, in this paper, to estimate the conditional distribution function (CDF) of a scalar response variable given a random variable taking values in a semimetric space. Under the ergodicity assumption, we study the uniform almost complete convergence (with a rate), as well as the asymptotic normality of the constructed estimator. The relevance of the proposed estimator is verified through a simulation study.

Transfer learning based on sparse Gaussian process for regression

Transfer learning is to use the knowledge obtained from the source domain to improve the learning efficiency when the target domain has insufficient labeled data. For regression problems, when the conditional distribution function and the marginal distribution function of the source domain and the target domain are different, how to effectively extract similar knowledge for transfer learning is still a problem. In this paper, we propose a transfer learning method for regression problem based on the sparse Gaussian process (GP). GP models are very popular in regression modeling, as they have the capability to produce uncertainty estimation, however, they cannot be used directly for transfer learning. We propose an adaptive neural kernel network (ANKN) to ensure that the GP model can effectively transfer knowledge. Additionally, although many sparse GP methods are proposed to solve the time consumption problem of the GP models in large datasets, they cannot maintain the transfer performance. We propose a transfer inducing point (TIP) algorithm for data selection in large datasets to maintain the transfer performance. The experiments with transfer regression problems on both real-world small datasets and large datasets indicate that the our method significantly increases prediction accuracy and effectiveness.

A Mixture of Clayton, Gumbel, and Frank Copulas: A Complete Dependence Model

Knowledge of the dependence between random variables is necessary in the area of risk assessment and evaluation. Some of the existing Archimedean copulas, namely the Clayton and the Gumbel copulas, allow for higher correlations on the extreme left and right, respectively. In this study, we use the idea of convex combinations to build a hybrid Clayton–Gumbel–Frank copula that provides all dependence scenarios from existing Archimedean copulas. The corresponding density and conditional distribution functions of the derived models for two random variables, as well as an estimator for the proportion parameter associated with the proposed model, are also derived. The results show that the proposed model is able to show any case of dependence by providing coefficients for the upper tail and lower tail dependence.

Optimal Bayesian Estimation of a Regression Curve, a Conditional Density, and a Conditional Distribution

In this paper, several related estimation problems are addressed from a Bayesian point of view, and optimal estimators are obtained for each of them when some natural loss functions are considered. The problems considered are the estimation of a regression curve, a conditional distribution function, a conditional density, and even the conditional distribution itself. These problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, and parametric and nonparametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function commonly used in estimating a real function of the unknown parameter. The cornerstone of these Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate the results.

A multi-quantile regression time series model with interquantile lipschitz regularization for wind power probabilistic forecasting

Modern decision-making processes require uncertainty-aware models, especially those relying on non-symmetric costs and risk-averse profiles. The objective of this work is to propose a dynamic model for the conditional non-parametric distribution function (CDF) to generate probabilistic forecasts for a renewable generation time series. To do that, we propose an adaptive non-parametric time-series model driven by a regularized multiple-quantile-regression (MQR) framework. In our approach, all regression models are jointly estimated through a single linear optimization problem that finds the global-optimal parameters in polynomial time. An innovative feature of our work is the consideration of a Lipschitz regularization of the first derivative of coefficients in the quantile space, which imposes coefficient smoothness. The proposed regularization induces a coupling effect among quantiles creating a single non-parametric CDF model with improved out-of-sample performance. A case study with realistic wind-power generation data from the Brazilian system shows: 1) the regularization model is capable to improve the performance of MQR probabilistic forecasts, and 2) our MQR model outperforms five relevant benchmarks: two based on the MQR framework, and three based on parametric models, namely, SARIMA, and GAS with Beta and Weibull CDF.

A Method of Probability Distribution Modeling of Multi-Dimensional Conditions for Wind Power Forecast Error Based on MNSGA-II-Kmeans

How to consider both the influence of weather and wind power in the modeling process of probability distribution of wind power forecast error (WPFE), and to emphasize the application value of conditional modeling, is rarely studied at present. This paper proposes a novel method of conditional probability distribution modeling for WPFE. This method uses a proposed MNSGA-II-Kmeans algorithm to perform multi-objective clustering of multi-dimensional influencing factors (MDIF), including weather and wind power. It can maximize the difference between the probability distributions of each MDIF mode’s WPFE while clustering, thus ensuring the application value of the conditional modeling way. Based on the clustering results, by using the versatile distribution to simulate the probability distribution of WPFE and the support vector machine to realize the recognition of MDIF modes, the specific conditional probability distribution function of WPFE can be provided to stochastic economic dispatch by identifying the forecast MDIF data. A wind plant of north China with historical data is selected for calculation. The results verify the effectiveness of the proposed method, and by comparison with the non-conditional probability distribution of WPFE that does not consider MDIF, it can effectively increase the wind power consumption of the power system.

Statistical Inference for Multivariate Conditional Cumulative Distribution Function Estimation By Stochastic Approximation Method

This paper handles non-parametric estimation of a conditional cumulative distribution function (CCDF). Using a recursive approach, we set forward a multivariate recursive estimator defifined by stochastic approximation algorithm. Our basic objective is to investigate the statistical inference of our estimator and compare it with that of non-recursive Nadaraya-Watson’s estimator. From this perspective, we fifirst derive the asymptotic properties of the proposed estimator which highly depend on the choice of two parameters, the stepsize (γn) as well as the bandwidth (hn). The second generation plug-in method, a method of bandwidth selection minimizing the Mean Weighted Integrated Squared Error (MW ISE) of the estimator in reference, entails the optimal choice of the bandwidth and therefore maintains an appropriate choice of the stepsize parameter. Basically, we demonstrate that, under some conditions, the Mean Squared Error (MSE) of the proposed estimator can be smaller than the one of Nadaraya Watson’s estimator. We corroborate our theoretical results through simulation studies and two real dataset applications, namely the Insurance Company Benchmark (COIL 2000) dataset as well as the French Hospital Data of COVID-19 epidemic.