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

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.

Stochastic order characterization of uniform integrability and tightness

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

We study the problem of estimating conditional distribution functions from data containing additional errors. The only assumption on these errors is that a weighted sum of the absolute errors tends to zero with probability one for sample size tending to infinity. We prove sufficient conditions on the weights (e.g. fulfilled by kernel weights) of a local averaging estimate of the codf, based on data with errors, which ensure strong pointwise consistency. We show that two of the three sufficient conditions on the weights and a weaker version of the third one are also necessary for the spc. We also give sufficient conditions on the weights, which ensure a certain rate of convergence. As an application we estimate the codf of the number of cycles until failure based on data from experimental fatigue tests and use it as objective function in a shape optimization of a component.

The Use of Conditional Probability Functions in Range Data Analysis and Simulation

An objective of longitudinal analysis is to estimate the conditional distributions of an outcome variable through a regression model. The approaches based on modeling the conditional means are not appropriate for this task when the conditional distributions are skewed or cannot be approximated by a normal distribution through a known transformation. We study a class of time-varying transformation models and a two-step smoothing method for the estimation of the conditional distribution functions. Based on our models, we propose a rank-tracking probability and a rank-tracking probability ratio to measure the strength of tracking ability of an outcome variable at two different time points. Our models and estimation method can be applied to a wide range of scientific objectives that cannot be evaluated by the conditional mean-based models. We derive the asymptotic properties for the two-step local polynomial estimators of the conditional distribution functions. Finite sample properties of our procedures are investigated through a simulation study. Application of our models and estimation method is demonstrated through an epidemiological study of childhood growth and blood pressure. Supplementary materials for this article are available online.

ABSTRACTIn this paper we consider the uniform strong consistency, along with a rate, of the cumulative distribution function (CDF) estimator. We extend the extended Glivenko–Cantelli lemma (for empirical distribution function) in Fabian and Hannan (1985, pp. 80–83) to the kernel estimator of the CDF.

We consider the high order moments estimator of the frontier of a random pair introduced by Girard, S., Guillou, A., Stupfler, G. (2012). {\it Frontier estimation with kernel regression on high order moments}. In the present paper, we show that this estimator is strongly uniformly consistent on compact sets and its rate of convergence is given when the conditional cumulative distribution function belongs to the Hall class of distribution functions.

Saddlepoint approximations to densities of sums of random variables were introduced by Daniels (1954). These methods were further developed by Barndorff-Nielsen & Cox (1979) to approximate joint and conditional densities for canonical sufficient statistics in exponential family settings. Approximations for the cumulative distributions of sums were developed by Lugannani & Rice (1980) and Robinson (1982), while the approximation of conditional cumulative distribution functions was addressed by Skovgaard (1987). The modifications necessary to apply the Lugannani & Rice and Skovgaard approximations to conditional distributions of canonical sufficient statistics in exponential family settings have been summarized by Davison (1988). Reid (1988) provides an overall review of the saddlepoint literature. In the present paper the Lugannani & Rice, and Skovgaard approximations are used to reproduce the distributions of the generalized variance and Wilks' likelihood ratio statistic in a multivatiate analysis of variance. Representing the degree of freedom in a gamma regression model as an appropriate linear function, the generalized variance distribution arises as the marginal distribution of a canonical sufficient statistic to which the Lugannani & Rice approximation may be applied. The more recent approximations of Fraser (1990) and Barndorff-Nielsen (1990) are exactly the same for this setting. Comparison of its accuracy with the approximation of Hoel (1937) as well as a normal approximation (Johnson & Kotz, 1972, p. 198) shows the Lugannani & Rice approximation to be highly accurate and uniformly better than its competitors. The distribution of Wilks' likelihood ratio statistic has two distinct characterizations in terms of products of independent beta variates (Anderson, 1958, Ch. 8). In conjunction with these characterizations we use Skovgaard's approximation to construct two distinct

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.

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.

Three methods for the probability distribution function F(x) estimation, based on the traditional Monte Carlo method, the Chebyshev inequality, and the theorems of Smirnov and Kolmogorov, are developed and compared. The methods are based on the functional dependence of the sample size n on the estimated value of F(x) with an accuracy factor as a parameter. Expressions for this dependence are derived for all methods and compared. It is shown that the accuracy of the estimation can be expressed and controlled by two parameters: the confidence and the confidence limits factor. Based on the methods developed, two techniques for F(x) estimation are proposed and demonstrated: a technique with prespecified accuracy and a technique with controlled accuracy. The technique with controlled accuracy allows the processing of a random sample starting with a limited initial sample of observations and then proceeding with the processing on an observation-by-observation basis. This way, the time that is needed to sort the sample has been minimized, and the estimation of distribution functions in real-time has become possible. This technique is demonstrated by estimating the distribution functions of chaotic and Gaussian sequences that are applied in code-division multiple-access (CDMA) systems.

On the boundary properties of Bernstein polynomial estimators of density and distribution functions

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.

In this study we investigate a covariate-based stochastic approach to parameterize unresolved turbulent processes within a standard model of the idealised, wind-driven ocean circulation. We focus on vertical instead of horizontal coarse-graining, such that we avoid the subtle difficulties of horizontal coarsegraining. The corresponding eddy forcing is uniquely defined and has a clear physical interpretation related to baroclinic instability.We propose to emulate the baroclinic eddy forcing by sampling from the conditional probability distribution functions of the eddy forcing obtained from the baroclinic reference model data. These conditional probability distribution functions are approximated here by sampling uniformly from discrete reference values. We analyze in detail the different performances of the stochastic parameterization dependent on whether the eddy forcing is conditioned on a suitable flow-dependent covariate or on a timelagged covariate or on both. The results demonstrate that our non-Gaussian, non-linear methodology is able to accurately reproduce the first four statistical moments and spatial/temporal correlations of the stream function, energetics, and enstrophy of the reference baroclinic model.