Heteroscedastic Variance Research Articles

Global Optimization for Black-Box Simulation via Sequential Intrinsic Kriging

In this paper we investigate global optimization for black-box simulations using metamodels to guide this optimization. As a novel metamodel we introduce intrinsic Kriging, for either deterministic or random simulation. For deterministic simulation we study the famous `efficient global optimization' (EGO) method, substituting intrinsic Kriging for universal Kriging. For random simulation we investigate a state-of-the-art two-stage algorithm accounting for heteroscedastic variances of the simulation responses, and introduce a new variant with the following two features: (1) this variant uses intrinsic Kriging; (2) this variant uses a different procedure to allocate the total available number of replications over simulated points. We perform several numerical experiments with deterministic and random simulations, to compare (1) the classic EGO and our EGO with intrinsic Kriging; (2) the classic two-stage algorithm and our modified version. We conclude that in most experiments (1) EGO with intrinsic Kriging outperforms classic EGO; (2) there is no significant difference between the classic algorithm and our modifed two-stage algorithm.

Anomalías de calendario en los mercados accionarios latinoamericanos: una revisión mediante el procedimiento de Bonferroni

This article builds upon and corrects traditional calendar anomalies in the main Latin American stock markets for the period between 1991 and 2013. It analyzes stock indexes from Argentina, Brazil, Chile, Colombia, Mexico and Peru. For the study, we use econometric models for the analysis of heteroscedastic variance supplemented with significance tests, including the Bonferroni correction. Results indicate mixed evidence of these anomalies; but, after the introduction of Bonferroni corrections, evidence suggests that such anomalies disappear, in cases where return and volatility both are considered, for almost all countries during the last period of our sample.

Stochastic approximation Monte Carlo Gibbs sampling for structural change inference in a Bayesian heteroscedastic time series model

We consider a Bayesian deterministically trending dynamic time series model with heteroscedastic error variance, in which there exist multiple structural changes in level, trend and error variance, but the number of change-points and the timings are unknown. For a Bayesian analysis, a truncated Poisson prior and conjugate priors are used for the number of change-points and the distributional parameters, respectively. To identify the best model and estimate the model parameters simultaneously, we propose a new method by sequentially making use of the Gibbs sampler in conjunction with stochastic approximation Monte Carlo simulations, as an adaptive Monte Carlo algorithm. The numerical results are in favor of our method in terms of the quality of estimates.

Constructing Inverse Probability Weights for Continuous Exposures

Inverse probability-weighted marginal structural models with binary exposures are common in epidemiology. Constructing inverse probability weights for a continuous exposure can be complicated by the presence of outliers, and the need to identify a parametric form for the exposure and account for nonconstant exposure variance. We explored the performance of various methods to construct inverse probability weights for continuous exposures using Monte Carlo simulation. We generated two continuous exposures and binary outcomes using data sampled from a large empirical cohort. The first exposure followed a normal distribution with homoscedastic variance. The second exposure followed a contaminated Poisson distribution, with heteroscedastic variance equal to the conditional mean. We assessed six methods to construct inverse probability weights using: a normal distribution, a normal distribution with heteroscedastic variance, a truncated normal distribution with heteroscedastic variance, a gamma distribution, a t distribution (1, 3, and 5 degrees of freedom), and a quantile binning approach (based on 10, 15, and 20 exposure categories). We estimated the marginal odds ratio for a single-unit increase in each simulated exposure in a regression model weighted by the inverse probability weights constructed using each approach, and then computed the bias and mean squared error for each method. For the homoscedastic exposure, the standard normal, gamma, and quantile binning approaches performed best. For the heteroscedastic exposure, the quantile binning, gamma, and heteroscedastic normal approaches performed best. Our results suggest that the quantile binning approach is a simple and versatile way to construct inverse probability weights for continuous exposures.

Heteroscedasticity in a two-factors design model

This paper studies the two-factors design model when the heteroscedasticity of variance is present in errors. As can be observed, testing of hypothesis based on the main effects for this design model can be performed using Hotelling's T2 test. Simultaneous confidence intervals are also proposed. Finally, the proposed methodology is applied to a real-life example.

Weighted quantile regression for analyzing health care cost data with missing covariates

Analysis of health care cost data is often complicated by a high level of skewness, heteroscedastic variances and the presence of missing data. Most of the existing literature on cost data analysis have been focused on modeling the conditional mean. In this paper, we study a weighted quantile regression approach for estimating the conditional quantiles health care cost data with missing covariates. The weighted quantile regression estimator is consistent, unlike the naive estimator, and asymptotically normal. Furthermore, we propose a modified BIC for variable selection in quantile regression when the covariates are missing at random. The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost and is naturally adapted to the skewness and heterogeneity of the cost data. The method is semiparametric in the sense that it does not require to specify the likelihood function for the random error or the covariates. We investigate the weighted quantile regression procedure and the modified BIC via extensive simulations. We illustrate the application by analyzing a real data set from a health care cost study.

Q-Generalized Logit Route Choice an Network Equilibrium Model

The multinomial logit model is expressed as a closed-form equation, and plays an important role in the field of tra sportation. The Gumbel-distributed utility in the multinomial logit model is restrictive in certain applications, especially in ro te choice behavior and network equilibrium analysis, although it is mathematically convenient. The range of variation of utility in the logit model is unbounded. The Gumbel distribution is left-skewed and has a very thi tail to the left. In additionFor example, the utility in the multinomial logit has a the homogeneity of homosce astic variance. In this study, the multinomial logit model is extended by generalizing the Gumbel-distributed utility to allow heteroscedastic variance and flexible shape. Then, the generalized logit model with a generalized Gumbel distribution is incorporated into the transportation network equilibrium model. The network equilibrium model with a generalized logit route choice is formulated as an optimization problem under uncongested networks. The objective function includes Tsallis entropy, which is a type of generalized entropy. In this study, the generalization of the Gumbel distribution, logit model, and network equilibrium model has a unified framework with q-analysis or Tsallis statistics.

Tests of Linear Hypotheses in the ANOVA under Heteroscedasticity

It is often of interest to undertake a general linear hypothesis testing (GLHT) problem in the one-way ANOVA without assuming the equality of the group variances. When the equality of the group variances is valid, it is well known that the GLHT problem can be solved by the classical F-test. The classical Ftest, however, may lead to misleading conclusions when the variance homogeneity assumption is seriously violated since it does not take the group variance heteroscedasticity into account. To our knowledge, little work has been done for this heteroscedastic GLHT problem except for some special cases. In this paper, we propose a simple approximate Hotelling T 2 (AHT) test. We show that the AHT test is invariant under a‐ne-transformations, difierent choices of the coe‐cient matrix used to deflne the same hypothesis, and difierent labeling schemes of the group means. Simulations and real data applications indicate that the AHT test is comparable with or outperforms some well-known approximate solutions proposed for the k-sample Behrens-Fisher problem which is a special case of the heteroscedastic GLHT problem.

A Markov-chain-based regression model with random effects for the analysis of 18O-labelled mass spectra

The enzymatic 18O-labelling is a useful technique for reducing the influence of the between-spectra variability on the results of mass-spectrometry experiments. A difficulty in applying the technique lies in the quantification of the corresponding peptides due to the possibility of an incomplete labelling, which may result in biased estimates of the relative peptide abundance. To address the problem, Zhu et al. [A Markov-chain-based heteroscedastic regression model for the analysis of high-resolution enzymatically 18O-labeled mass spectra, J. Proteome Res. 9(5) (2010), pp. 2669–2677] proposed a Markov-chain-based regression model with heteroscedastic residual variance, which corrects for the possible bias. In this paper, we extend the model by allowing for the estimation of the technical and/or biological variability for the mass spectra data. To this aim, we use a mixed-effects version of the model. The performance of the model is evaluated based on results of an application to real-life mass spectra data and a simulation study.

Heteroscedastic variance covariance matrices for unbiased two groups linear classification methods

Heteroscedastic variance covariance matrices for unbiased two groups linear classification methods

Selecting traits that explain species–environment relationships: a generalized linear mixed model approach

AbstractQuestionQuantification of the effect of species traits on the assembly of communities is challenging from a statistical point of view. A key question is how species occurrence and abundance can be explained by the trait values of the species and the environmental values at the sites.MethodsUsing a sites × species abundance table, a site × environment data table and a species × trait data table, we address the above question using a novel generalized linear mixed model (GLMM) approach. TheGLMMovercomes problems of pseudo‐replication and heteroscedastic variance by including sites and species as random factors. The method is equally applicable to presence–absence data as to count and multinomial data. We present a tiered forward selection approach for obtaining a parsimonious model and compare the results with alternative methods (the fourth corner method andRLQordination).ResultsWe illustrate the approach on a presence–absence version on two data sets. In theDuneMeadow data, species presence is parsimoniously explained by moisture and manure on the meadows in combination with seed mass and specific leaf area (SLA). In theGrazedGrassland data, species presence is parsimoniously explained by the grazing intensity and soil phosphorus in combination with theC:Nratio and flowering mode.ConclusionsOurGLMMapproach can be used to identify which species traits and environmental variables best explain the species distribution, and which traits are significantly correlated with environmental variables. We argue that the method is better suited for providing an interpretable and predictive model than the fourth corner method andRLQ.

Stock index realized volatility forecasting in the presence of heterogeneous leverage effects and long range dependence in the volatility of realized volatility

In this article, we account for the presence of heterogeneous leverage effects and the persistence in the volatility of stock index realized volatility. The Heterogeneous Autoregressive (HAR) Realized Volatility (RV) model is extended in order to account for asymmetric responses to negative and positive shocks occurring at distinct frequencies, as well as, for the long range dependence in the heteroscedastic variance of the residuals. Compared with established HAR and Autoregressive Fractionally Integrated Moving Average (ARFIMA) realized volatility models, the proposed model exhibits superior in sample fitting, as well as, out of sample volatility forecasting performance. The latter is further improved when the Realized Power Variation (RPV) is used as a regressor, while we show that our analysis is also robust against microstructure noise.

Bayesian model averaging using particle filtering and Gaussian mixture modeling: Theory, concepts, and simulation experiments

Bayesian model averaging (BMA) is a standard method for combining predictive distributions from different models. In recent years, this method has enjoyed widespread application and use in many fields of study to improve the spread‐skill relationship of forecast ensembles. The BMA predictive probability density function (pdf) of any quantity of interest is a weighted average of pdfs centered around the individual (possibly bias‐corrected) forecasts, where the weights are equal to posterior probabilities of the models generating the forecasts, and reflect the individual models skill over a training (calibration) period. The original BMA approach presented by Raftery et al. (2005) assumes that the conditional pdf of each individual model is adequately described with a rather standard Gaussian or Gamma statistical distribution, possibly with a heteroscedastic variance. Here we analyze the advantages of using BMA with a flexible representation of the conditional pdf. A joint particle filtering and Gaussian mixture modeling framework is presented to derive analytically, as closely and consistently as possible, the evolving forecast density (conditional pdf) of each constituent ensemble member. The median forecasts and evolving conditional pdfs of the constituent models are subsequently combined using BMA to derive one overall predictive distribution. This paper introduces the theory and concepts of this new ensemble postprocessing method, and demonstrates its usefulness and applicability by numerical simulation of the rainfall‐runoff transformation using discharge data from three different catchments in the contiguous United States. The revised BMA method receives significantly lower‐prediction errors than the original default BMA method (due to filtering) with predictive uncertainty intervals that are substantially smaller but still statistically coherent (due to the use of a time‐variant conditional pdf).

Efficient Estimation for Rank‐Based Regression with Clustered Data

Rank-based inference is widely used because of its robustness. This article provides optimal rank-based estimating functions in analysis of clustered data with random cluster effects. The extensive simulation studies carried out to evaluate the performance of the proposed method demonstrate that it is robust to outliers and is highly efficient given the existence of strong cluster correlations. The performance of the proposed method is satisfactory even when the correlation structure is misspecified, or when heteroscedasticity in variance is present. Finally, a real dataset is analyzed for illustration.

Quantile Regression for Analyzing Heterogeneity in Ultra-High Dimension

Ultra-high dimensional data often display heterogeneity due to either heteroscedastic variance or other forms of non-location-scale covariate effects. To accommodate heterogeneity, we advocate a more general interpretation of sparsity, which assumes that only a small number of covariates influence the conditional distribution of the response variable, given all candidate covariates; however, the sets of relevant covariates may differ when we consider different segments of the conditional distribution. In this framework, we investigate the methodology and theory of nonconvex, penalized quantile regression in ultra-high dimension. The proposed approach has two distinctive features: (1) It enables us to explore the entire conditional distribution of the response variable, given the ultra-high-dimensional covariates, and provides a more realistic picture of the sparsity pattern; (2) it requires substantially weaker conditions compared with alternative methods in the literature; thus, it greatly alleviates the difficulty of model checking in the ultra-high dimension. In theoretic development, it is challenging to deal with both the nonsmooth loss function and the nonconvex penalty function in ultra-high-dimensional parameter space. We introduce a novel, sufficient optimality condition that relies on a convex differencing representation of the penalized loss function and the subdifferential calculus. Exploring this optimality condition enables us to establish the oracle property for sparse quantile regression in the ultra-high dimension under relaxed conditions. The proposed method greatly enhances existing tools for ultra-high-dimensional data analysis. Monte Carlo simulations demonstrate the usefulness of the proposed procedure. The real data example we analyzed demonstrates that the new approach reveals substantially more information as compared with alternative methods. This article has online supplementary material.

Firm debt structure, firm size and risk volatility in US industrial firms

US industrial-firm panel data on short-term and long-term borrowing (term debt structure) for annual and quarterly time periods over the years 1995 to 2008 are used to test an insulation hypothesis and a related volatility hypothesis. The former test uses a regression model relating the log of the ratio of accounts payable in trade to Long-Term Debt (LTD) to firm size and other variables. The focus is on the firm's response to the US Federal Reserve (FED)'s monetary policy, where the response is a micro perspective on the earlier macro debate over the existence of bank lending channels. The latter hypothesis uses the panel heteroscedastic variances from the first regression procedure to test for a quadratic-form risk function (either U-shaped or inverted U-shaped) using sigma squared and the Coefficient of Variation (CV) as risk indexes and firm size as a determinant. The findings suggest that there is some evidence that US industrial firms in their borrowing behaviour do insulate themselves from the effe...

Traditional and proposed tests of slope homogeneity for non‐normal and heteroscedastic data

The purpose of this study was to develop and compare tests of independent groups' slopes for non-normal distributions and heteroscedastic variances, including slope tests based on least squares, Theil-Sen, and trimmed estimation approaches. A slope test based on jackknife standard error estimates is proposed that can utilize each of the traditional estimation methods while also addressing problematic aspects of methods' standard error estimates. Simulations demonstrate that the proposed jackknife-based slope tests can improve standard error estimation, Type I error, and power relative to the traditional slope tests.

Generalized shrinkage F-like statistics for testing an interaction term in gene expression analysis in the presence of heteroscedasticity.

BackgroundMany analyses of gene expression data involve hypothesis tests of an interaction term between two fixed effects, typically tested using a residual variance. In expression studies, the issue of variance heteroscedasticity has received much attention, and previous work has focused on either between-gene or within-gene heteroscedasticity. However, in a single experiment, heteroscedasticity may exist both within and between genes. Here we develop flexible shrinkage error estimators considering both between-gene and within-gene heteroscedasticity and use them to construct F-like test statistics for testing interactions, with cutoff values obtained by permutation. These permutation tests are complicated, and several permutation tests are investigated here.ResultsOur proposed test statistics are compared with other existing shrinkage-type test statistics through extensive simulation studies and a real data example. The results show that the choice of permutation procedures has dramatically more influence on detection power than the choice of F or F-like test statistics. When both types of gene heteroscedasticity exist, our proposed test statistics can control preselected type-I errors and are more powerful. Raw data permutation is not valid in this setting. Whether unrestricted or restricted residual permutation should be used depends on the specific type of test statistic.ConclusionsThe F-like test statistic that uses the proposed flexible shrinkage error estimator considering both types of gene heteroscedasticity and unrestricted residual permutation can provide a statistically valid and powerful test. Therefore, we recommended that it should always applied in the analysis of real gene expression data analysis to test an interaction term.

Asymptotically distribution free tests in heteroscedastic unbalanced high dimensional ANOVA

In this paper, we develop the asymptotic theory for hypotheses testing in high-dimensional analysis of variance (HANOVA) when the distributions are completely unspecified. Most results in the literature have been restricted to obser- vations of no more than two-way designs for continuous data. Here we formulate the local alternatives in terms of departures from the null distribution so that the re- sponses can be either continuous or categorical. The asymptotic theory is presented for testing of main factor and interaction effects of up to order three in unbalanced designs with heteroscedastic variances and arbitrary number of factors. The test statistics are based on quadratic forms whose asymptotic theory is derived under non-classical settings where the number of variables is large while the number of replications may be limited. Simulation results show that the present test statistics perform well in both continuous and discrete HANOVA in type I error accuracy, power performance, and computing time. The proposed test is illustrated with a gene expression data analysis of Arabidopsis thaiana in response to multiple abiotic stresses.

Kriging metamodel with modified nugget-effect: The heteroscedastic variance case

Metamodels are commonly used to approximate and analyze simulation models. However, in cases where the simulation output variances are non-zero and not constant, many of the current metamodels which assume homogeneity, fail to provide satisfactory estimation. In this paper, we present a kriging model with modified nugget-effect adapted for simulations with heterogeneous variances. The new model improves the estimations of the sensitivity parameters by explicitly accounting for location dependent non-constant variances and smoothes the kriging predictor’s output accordingly. We look into the effects of stochastic noise on the parameter estimation for the classic kriging model that assumes deterministic outputs and note that the stochastic noise increases the variability of the classic parameter estimation. The nugget-effect and proposed modified nugget-effect stabilize the estimated parameters and decrease the erratic behavior of the predictor by penalizing the likelihood function affected by stochastic noise. Several numerical examples suggest that the kriging model with modified nugget-effect outperforms the kriging model with nugget-effect and the classic kriging model in heteroscedastic cases.