Moderate deviations for the random weighted sums of END random variables with consistently varying tails

In this paper, we study moderate deviations for random weighted sums of extended negative dependent (END) random variables, which are consistently-varying tailed and not necessarily identically distributed. When these END random variables are independent of their weights, and the weights are positive random variables with two-sided bounds, the results shows END structure and the dependence between the weights have no effects on the asymptotic behavior of moderate deviations of partial sums and random sums.     

A Dirichlet random coefficient regression model for quality indicators

We present a random coefficient regression model in which a response is linearly related to some explanatory variables with random coefficients following a Dirichlet distribution. These coefficients can be interpreted as weights because they are nonnegative and add up to one. The proposed estimation procedure combines iteratively reweighted least squares and the maximization on an approximated likelihood function. We also present a diagnostic tool based on a residual Q–Q plot and two procedures for estimating individual weights. The model is used to construct an index for measuring the quality of the railroad system in Spain.     

Second order tail behaviour of randomly weighted heavy-tailed sums and their maxima

Abstract

In this paper the second order asymptotics of the tail probabilities of randomly weighted sums and their maxima are established in the case that the underlying primary random variables are subexponential. No any assumption is made on the dependence structure between the random weights, but we assume these weights are bounded away from zero and infinity.     

Densities and distribution functions of the ratio of two linear functions and the product of Gamma variates: A saddlepoint approach

Saddlepoint approximations for the densities and the distribution functions of the ratio of two linear functions of gamma random variables and the product of gamma random variables are derived. Ratios of linear functions with positive and negative weights and non identical gamma variables are considered. The saddlepoint approximations are very accurate in the tails as in the center of the distribution. Extensive simulation studies are used to evaluate the accuracy of the proposed methods.     

Unconditional empirical likelihood approach for analytic use of public survey data

Modeling survey data often requires having the knowledge of design and weighting variables. With public-use survey data, some of these variables may not be available for confidentiality reasons. The proposed approach can be used in this situation, as long as calibrated weights and variables specifying the strata and primary sampling units are available. It gives consistent point estimation and a pivotal statistics for testing and confidence intervals. The proposed approach does not rely on with-replacement sampling, single-stage, negligible sampling fractions, or noninformative sampling. Adjustments based on design effects, eigenvalues, joint-inclusion probabilities or bootstrap, are not needed. The inclusion probabilities and auxiliary variables do not have to be known. Multistage designs with unequal selection of primary sampling units are considered. Nonresponse can be easily accommodated if the calibrated weights include reweighting adjustment for nonresponse. We use an unconditional approach, where the variables and sample are random variables. The design can be informative.     

Bayesian predictive inference of a finite population proportion under selection bias

We show how to infer about a finite population proportion using data from a possibly biased sample. In the absence of any selection bias or survey weights, a simple ignorable selection model, which assumes that the binary responses are independent and identically distributed Bernoulli random variables, is not unreasonable. However, this ignorable selection model is inappropriate when there is a selection bias in the sample. We assume that the survey weights (or their reciprocals which we call ‘selection’ probabilities) are available, but there is no simple relation between the binary responses and the selection probabilities. To capture the selection bias, we assume that there is some correlation between the binary responses and the selection probabilities (e.g., there may be a somewhat higher/lower proportion of positive responses among the sampled units than among the nonsampled units). We use a Bayesian nonignorable selection model to accommodate the selection mechanism. We use Markov chain Monte Carlo methods to fit the nonignorable selection model. We illustrate our method using numerical examples obtained from NHIS 1995 data.     

Combined Density Nowcasting in an Uncertain Economic Environment

We introduce a combined density nowcasting (CDN) approach to dynamic factor models (DFM) that in a coherent way accounts for time-varying uncertainty of several model and data features to provide more accurate and complete density nowcasts. The combination weights are latent random variables that depend on past nowcasting performance and other learning mechanisms. The combined density scheme is incorporated in a Bayesian sequential Monte Carlo method which rebalances the set of nowcasted densities in each period using updated information on the time-varying weights. Experiments with simulated data show that CDN works particularly well in a situation of early data releases with relatively large data uncertainty and model incompleteness. Empirical results, based on U.S. real-time data of 120 monthly variables, indicate that CDN gives more accurate density nowcasts of U.S. GDP growth than a model selection strategy and other combination strategies throughout the quarter with relatively large gains for the two first months of the quarter. CDN also provides informative signals on model incompleteness during recent recessions. Focusing on the tails, CDN delivers probabilities of negative growth, that provide good signals for calling recessions and ending economic slumps in real time.     

Asymptotic properties of kernel density estimation with complex survey data

Kernel density estimation has been used with great success with data that may be assumed to be generated from independent and identically distributed (iid) random variables. The methods and theoretical results for iid data, however, do not directly apply to data from stratified multistage samples. We present finite-sample and asymptotic properties of a modified density estimator introduced in Buskirk (Proceedings of the Survey Research Methods Section, American Statistical Association (1998), pp. 799–801) and Bellhouse and Stafford (Statist. Sin. 9 (1999) 407–424); this estimator incorporates both the sampling weights and the kernel weights. We present regularity conditions which lead the sample estimator to be consistent and asymptotically normal under various modes of inference used with sample survey data. We also introduce a superpopulation structure for model-based inference that allows the population model to reflect naturally occurring clustering. The estimator, and confidence bands derived from the sampling design, are illustrated using data from the US National Crime Victimization Survey and the US National Health and Nutrition Examination Survey.     

On a characteristic property of generalized Pareto distributions, extreme value distributions and their max domains of attraction

Using the independence of an arbitrary random variable Y and the weighted minima of independent, identically distributed random variables with weights depending on Y, we characterize extreme value distributions and generalized Pareto distributions. A discussion is made about an analogous characterization for distributions in the max domains of attraction of extreme value limit laws.     

Nonparametric estimation of the conditional distribution function in a semiparametric censorship model

In this paper we propose a new nonparametric estimator of the conditional distribution function under a semiparametric censorship model. We establish an asymptotic representation of the estimator as a sum of iid random variables, balanced by some kernel weights. This representation is used for obtaining large sample results such as the rate of uniform convergence of the estimator, or its limit distributional law. We prove that the new estimator outperforms the conditional Kaplan–Meier estimator for censored data, in the sense that it exhibits lower asymptotic variance. Illustration through real data analysis is provided.     

Convergence results for sequences of quadratic forms

Let (XI,)be a sequence of independent random variables, and let Q_n= where for each N,(an:,k)is a doubly indexed sequence of weights. The convergence and the rate of convergence of the sequence of quadratic forms {Q_n} are studied. These quadratic forms are linear sums of dependent variables; however, their convergence properties are similar to those of linear sums of independent variables provided the variables have finite rth absolute moments with 0 < r 2.while the rate of convergence has not been obtained for r< 2, it is shown to be different from that of linear sums.     

Estimating the error variance in nonparametric regression by a covariate-matched u-statistic

For nonparametric regression models with fixed and random design, two classes of estimators for the error variance have been introduced: second sample moments based on residuals from a nonparametric fit, and difference-based estimators. The former are asymptotically optimal but require estimating the regression function; the latter are simple but have larger asymptotic variance. For nonparametric regression models with random covariates, we introduce a class of estimators for the error variance that are related to difference-based estimators: covariate-matched U-statistics. We give conditions on the random weights involved that lead to asymptotically optimal estimators of the error variance. Our explicit construction of the weights uses a kernel estimator for the covariate density.     

Bayesian temporal density estimation with autoregressive species sampling models

We propose a novel Bayesian nonparametric (BNP) model, which is built on a class of species sampling models, for estimating density functions of temporal data. In particular, we introduce species sampling mixture models with temporal dependence. To accommodate temporal dependence, we define dependent species sampling models by modeling random support points and weights through an autoregressive model, and then we construct the mixture models based on the collection of these dependent species sampling models. We propose an algorithm to generate posterior samples and present simulation studies to compare the performance of the proposed models with competitors that are based on Dirichlet process mixture models. We apply our method to the estimation of densities for the price of apartment in Seoul, the closing price in Korea Composite Stock Price Index (KOSPI), and climate variables (daily maximum temperature and precipitation) of around the Korean peninsula.     

Simulation of Constrained Variables in Engineering Risk Analyses

The problem of sampling random variables with overlapping pdfs subject to inequality constraints is addressed. Often, the values of physical variables in an engineering model are interrelated. This mutual dependence imposes inequality constraints on the random variables representing these parameters. Ignoring the interdependencies and sampling the variables independently can lead to inconsistency/bias. We propose an algorithm to generate samples of constrained random variables that are characterized by typical continuous probability distributions and are subject to different kinds of inequality constraints. The sampling procedure is illustrated for various representative cases and one realistic application to simulation of structural natural frequencies.     

On exact strong laws of large numbers for ratios of random variables and their applications

Abstract

We study the almost sure convergence of weighted sums of ratios of independent random variables satisfying some general, mild conditions. The obtained results are applied to exact laws for order statistics. An exact law for independent random variables which are nonidentically distributed is also proved and applied to ratios of adjacent order statistics for a sample of uniformly distributed random variables.     

An iterative algorithm for optimal variable weighting in K-means clustering

The K-means clustering method is a widely adopted clustering algorithm in data mining and pattern recognition, where the partitions are made by minimizing the total within group sum of squares based on a given set of variables. Weighted K-means clustering is an extension of the K-means method by assigning nonnegative weights to the set of variables. In this paper, we aim to obtain more meaningful and interpretable clusters by deriving the optimal variable weights for weighted K-means clustering. Specifically, we improve the weighted k-means clustering method by introducing a new algorithm to obtain the globally optimal variable weights based on the Karush-Kuhn-Tucker conditions. We present the mathematical formulation for the clustering problem, derive the structural properties of the optimal weights, and implement an recursive algorithm to calculate the optimal weights. Numerical examples on simulated and real data indicate that our method is superior in both clustering accuracy and computational efficiency.     

Computation of the asymptotic null distribution of goodness-of-fit tests for multi-state models

We develop an improved approximation to the asymptotic null distribution of the goodness-of-fit tests for panel observed multi-state Markov models (Aguirre-Hernandez and Farewell, Stat Med 21:1899–1911, 2002) and hidden Markov models (Titman and Sharples, Stat Med 27:2177–2195, 2008). By considering the joint distribution of the grouped observed transition counts and the maximum likelihood estimate of the parameter vector it is shown that the distribution can be expressed as a weighted sum of independent c²₁{\chi^2_1} random variables, where the weights are dependent on the true parameters. The performance of this approximation for finite sample sizes and where the weights are calculated using the maximum likelihood estimates of the parameters is considered through simulation. In the scenarios considered, the approximation performs well and is a substantial improvement over the simple χ ² approximation.     

Some orthogonal expansions for the logistic distribution

We expand a continuous random variable as a sum of a sequence of un-correlated random variables. These variables are principal components of a Bernoulli process, as well as principal dimensions in continuous metric scaling on a particular distance function. We obtain expansions for the uniform, exponential and logistic distributions. A goodness-of-fit application is given.     

Positive dependence of random variables with a common marginal distribution

In this paper we review some notions of positive dependence of random variables with a common univariate marginal distribution and describe the related moment and probability inequalities. We first present a comparison between i.i.d. random variables and exchangeable random variables via an application of de Finetti's theorem, then describe some useful probability inequalities via partial orderings of the strength of their positive dependence. Finally, we state a result for random variables which are not necessarily exchangeable. Special applications to the multivariate normal distribution will be discussed, and the results involve only the correlation matrix of the distribution.     

Root-n-consistent semiparametric estimation of partially linear models for weakly dependent observations

We consider estimation of the linear part in a partially linear model for absolutely regular observations. The estimator using random weights are proposed and the asymptotic normality of the estimator is established without compact support assumption.