Uniform in bandwidth consistency for various kernel estimators involving functional data

Nonparametric M-regression with free knot splines

A fast algorithm for computing least-squares cross-validations for nonparametric conditional kernel density functions

A conditional beta probability density function is employed to model time-delay measurement errors associated with sensors used in triangulation ranging applications. This type of density function is less restrictive than the beta density used in previous analyses in that end points can be extended out beyond the imposed range constraints and data editing is allowed before range calculations. The resulting conditional range density function is computed along with the mean and variance. Comparisons are made of the pertinent range statistics associated with the beta and conditional beta density functions. Finally, the conditional beta density function range statistics are compared with those obtained from a gated Gaussian density function for equivalent values of time-delay measurement error standard deviation. The Gaussian statistics are determined via Monte-Carlo simulation.

Consider the sample X1, X2, ..., XN of N independent and identically distributed (iid) random variables with common cumulative distribution function (cdf)F, and let Fu be their conditional excess distribution function F. We define the ordered sample by . Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions F , and large u ,Fu is well approximated by the Generalized Pareto Distribution.The mixed method is a method for determining thresholds. This method consists in minimizing the variance of a convex combination of other thresholds.The objective of the mixed method is to determine by which probability distribution one can approach this conditional distribution. In this article, we propose a theorem which specifies the conditional distribution of excesses when the deterministic threshold tends to the end point.

SUMMARY We extend the idea of crossvalidation to choose the smoothing parameters of the 'double-kernel' local linear regression for estimating a conditional density. Our selection rule optimises the estimated conditional density function by minimising the integrated squared error. We also discuss three other bandwidth selection rules, an ad hoc method used by Fan et al. (1996), a bootstrap method of Hall et al. (1999) for bandwidth selection in the estimation of conditional distribution functions, modified by Bashtannyk & Hyndman (2001) to cover conditional density functions, and finally a simple approach proposed by Hyndman & Yao (2002). The performance of the new approach is compared with these three methods by simulation studies, and our method performs outstandingly well. The method is illustrated by an application to estimating the transition density and the Value-at-Risk of treasury-bill data.

In this paper kernel estimates of the joint and conditional probability density functions are used to generate synthetic streamflow sequences. Streamflow is assumed to be a Markov process with time dependence characterized by a multivariate probability density function. Kernel methods are used to estimate this multivariate density function. Simulation proceeds by sequentially resampling from the conditional density function derived from the kernel estimate of the underlying multivariate probability density function. This is a nonparametric method for the synthesis of streamflow that is data‐driven and avoids prior assumptions as to the form of dependence (e.g., linear or nonlinear) and the form of the probability density functions (e.g., Gaussian). We show, using synthetic examples with known underlying models, that the nonparametric method presented is more flexible than the conventional models used in stochastic hydrology and is capable of reproducing both linear and nonlinear dependence. The effectiveness of this model is illustrated through its application to simulation of monthly streamflow from the Beaver River in Utah.

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.

Using the two-point Edgeworth series up to second order in the linear rms density fluctuation we construct the weakly non-linear conditional probability distribution function for the density field around an overdense region. This requires calculating the two-point analogues of the skewness parameter S3. We test the dependence of the two-point skewness on distance from the peak for scale-free power spectra and Gaussian smoothing. The statistical features of such a conditional distribution are given as the values obtained within linear theory corrected by the terms that arise as a result of weakly non-linear evolution. The expected density around the peak is found to be always below the linear prediction while its dispersion is always larger than in the linear case. For large enough overdensities the weakly non-linear corrections can be more significant than the peak constraint introduced by Bardeen et al. We apply these results to the spherical model of collapse as developed by Hoffman & Shaham and find that in general the effect of weakly non-linear interactions is to decrease the scale from which a peak gathers mass and therefore also the mass itself. In the case of an open universe this results in steepening of the final profile of the virialized proto-object.

A kernel-based parametric method for conditional density estimation

Pseudo-measurements are artificially generated estimates derived from historical data to represent power system variables that are not directly measured in real-time. In active distribution networks (ADNs), where the limited sensor coverage and high uncertainty from distributed energy resources (DERs) exist, pseudo-measurements are essential for ensuring the system observability. This paper presents a deep learning model for the pseudo-measurement generation in active distribution networks. The model uses a Quantized SoftMax Network (QSN) to estimate conditional probability distribution functions (CPDFs) without assuming any predefined statistical model. To capture spatial and temporal patterns, the QSN is combined with convolutional neural networks (CNNs), gated recurrent units (GRUs), and an attention mechanism. The proposed model is tested on the IEEE 123-bus system using real residential load and renewable generation data. Results show that the proposed model provides more accurate and reliable pseudo-measurements compared to existing conditional kernel density estimator (CKDE) and deep mixture density network (DMDN)-based models.

Saddlepoint approximations are derived for the conditional cumulative distribution function and density ofwhereis the sample mean ofni.i.d. bivariate random variables andg(x, y) is a non-linear function. The relative error of orderO(n–1) is retained. The results extend the important work of Skovgaard (1987), and are useful in conditional inference, especially in the case of small or moderate sample sizes. Generalizations to higher-dimensional random vectors are also discussed. Some examples are demonstrated.

SUMMARY The case-cohort design is a common means of reducing cost in large epidemiological cohort studies. Under this design, covariates are measured only on the cases and a subcohort randomly selected from the entire cohort. In this paper, we demonstrate how to use the case-cohort data to estimate the regression parameter of the additive hazards model, which specifies that the conditional hazard function given a set of covariates is the sum of an arbitrary baseline hazard function and a regression function of the covariates. The proposed estimator is shown to be consistent and asymptotically normal with an easily estimated variance. The subcohort may be selected by independent Bernoulli sampling with arbitrary selection probabilities or by stratified simple random sampling. The efficiencies of various sampling schemes are investigated both analytically and by simulation. A real example is provided.

Bayesian modeling of joint and conditional distributions

In this paper, the waiting time analysis for an OFDMA wireless cellular system with finite buffering and Adaptive Modulation and Coding (AMC) is addressed. When AMC is considered, several concentric regions within cells serving users with different data rates are differentiated. Phase type (PHT) distribution functions are used to model the conditional cumulative distribution function of the waiting time for users in the different coverage regions. The representation of the conditional waiting time distribution through PHT distributions allows expressing cumulative distribution function (CDF) as a linear combination of negative exponential distributions. The analytical method proposed here captures the exact behavior of the waiting time and the obtained solution is based on matrix analytic derivation, which allows evaluating system performance for any size of buffer and any number of coverage regions.

We present a study of the distribution of ozone in the lowermost stratosphere with the goal of characterizing the observed variability. The air in the lowermost stratosphere is divided into two population groups based on Ertel's potential vorticity at 300 hPa. High (low) potential vorticity at 300 hPa indicates that the tropopause is low (high), and the identification of these two groups is made to account for the dynamic variability. Conditional probability distribution functions are used to define the statistics of the ozone distribution from both observations and a three‐dimensional model simulation using winds from the Goddard Earth Observing System Data Assimilation System for transport. Ozone data sets include ozonesonde observations from northern midlatitude stations (1991–1996) and midlatitude observations made by the Halogen Occultation Experiment (HALOE) on the Upper Atmosphere Research Satellite (UARS) (1994–1998). The conditional probability distribution functions are calculated at a series of potential temperature surfaces spanning the domain from the midlatitude tropopause to surfaces higher than the mean tropical tropopause (∼380 K). The probability distribution functions are similar for the two data sources, despite differences in horizontal and vertical resolution and spatial and temporal sampling. Comparisons with the model demonstrate that the model maintains a mix of air in the lowermost stratosphere similar to the observations. The model also simulates a realistic annual cycle. Results show that during summer, much of the observed variability is explained by the height of the tropopause. During the winter and spring, when the tropopause fluctuations are larger, less of the variability is explained by tropopause height. This suggests that more mixing occurs during these seasons. During all seasons, there is a transition zone near the tropopause that contains air characteristic of both the troposphere and the stratosphere. The relevance of the results to the assessment of the environmental impact of aircraft effluence is also discussed.