Heteroscedastic Variance Research Articles

Oracle-efficient estimation and global inferences for variance function of functional data

A new two-step reconstruction-based moment estimator and an asymptotically correct smooth simultaneous confidence band as a global inference tool are proposed for the heteroscedastic variance function of dense functional data. Step one involves spline smoothing for individual trajectory reconstructions and step two employs kernel regression on the individual squared residuals to estimate each trajectory variability. Then by the method of moment an estimator for the variance function of functional data is constructed. The estimation procedure is innovative by synthesizing spline smoothing and kernel regression together, which allows one not only to apply the fast computing speed of spline regression but also to employ the flexible local estimation and the extreme value theory of kernel smoothing. The resulting estimator for the variance function is shown to be oracle-efficient in the sense that it is uniformly as efficient as the ideal estimator when all trajectories were known by “oracle”. As a result, an asymptotically correct simultaneous confidence band for the variance function is established. Simulation results support our asymptotic theory with fast computation. As an illustration, the proposed method is applied to the analyses of two real data sets leading to a number of discoveries.

Indonesia Gold Price Forecasting Using ARIMA Model (0,1,1) - GARCH (1,0)

A frequently employed time series model is the Autoregressive Integrated Moving Average (ARIMA) model. In highly volatile data, ARIMA models sometimes produce residual variances that are heteroscedasticity. One method that can overcome the problem of residual variance heteroscedasticity is the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) method. The purpose of this study is to obtain the ARIMA-GARCH model for daily gold price data in Indonesia for the period 1 January 2022 to 31 December 2022, and to obtain daily gold price forecasting results in Indonesia. The daily gold price forecasting model obtained for Indonesia is ARIMA (0,1,1) - GARCH (1,0) with a MAPE value of 0.5745% which shows that the model is very good because the MAPE value is less than 10%. The results of Indonesia's daily gold price forecast from January 1st, 2023 to January 3rd, 2023 remain stable.

Nonparametric mean and variance adaptive classification rule for high‐dimensional data with heteroscedastic variances

AbstractIn this study, we introduce an innovative methodology aimed at enhancing Fisher's Linear Discriminant Analysis (LDA) in the context of high‐dimensional data classification scenarios, specifically addressing situations where each feature exhibits distinct variances. Our approach leverages Nonparametric Maximum Likelihood Estimation (NPMLE) techniques to estimate both the mean and variance parameters. By accommodating varying variances among features, our proposed method leads to notable improvements in classification performance. In particular, unlike numerous prior studies that assume the distribution of heterogeneous variances follows a right‐skewed inverse gamma distribution, our proposed method demonstrates excellent performance even when the distribution of heterogeneous variances takes on left‐skewed, symmetric, or right‐skewed forms. We conducted a series of rigorous experiments to empirically validate the effectiveness of our approach. The results of these experiments demonstrate that our proposed methodology excels in accurately classifying high‐dimensional data characterized by heterogeneous variances.

Compositional variable selection in quantile regression for microbiome data with false discovery rate control

AbstractAdvancement in high‐throughput sequencing technologies has stimulated intensive research interests to identify specific microbial taxa that are associated with disease conditions. Such knowledge is invaluable both from the perspective of understanding biology and from the biomedical perspective of therapeutic development, as the microbiome is inherently modifiable. Despite availability of massive data, analysis of microbiome compositional data remains difficult. The nature that relative abundances of all components of a microbial community sum to one poses challenges for statistical analysis, especially in high‐dimensional settings, where a common research theme is to select a small fraction of signals from amid many noisy features. Motivated by studies examining the role of microbiome in host transcriptomics, we propose a novel approach to identify microbial taxa that are associated with host gene expressions. Besides accommodating compositional nature of microbiome data, our method both achieves FDR‐controlled variable selection, and captures heterogeneity due to either heteroscedastic variance or non‐location‐scale covariate effects displayed in the motivating dataset. We demonstrate the superior performance of our method using extensive numerical simulation studies and then apply it to real‐world microbiome data analysis to gain novel biological insights that are missed by traditional mean‐based linear regression analysis.

Modeling price dynamics and risk Forecasting in Tehran Stock Exchange: Conditional Variance Heteroscedasticity Hidden Markov Models

Modeling price dynamics and risk Forecasting in Tehran Stock Exchange: Conditional Variance Heteroscedasticity Hidden Markov Models

Multiple comparisons of treatment against control under unequal variances using parametric bootstrap

ABSTRACT In one-way analysis of variance models, performing simultaneous multiple comparisons of treatment groups with a control group may be of interest. Dunnett's test is used to test such differences and assumes equal variances of the response variable for each group. This assumption is not always met even after transformation. A parametric bootstrap (PB) method is developed here for comparing multiple treatment group means against the control group with unequal variances and unbalanced data. In simulation studies, the proposed method outperformed Dunnett's test in controlling the type I error under various settings, particularly when data have heteroscedastic variance and unbalanced design. Simulations show that power is often lower for the PB method than for Dunnett's test under equal variance, balanced data, or smaller sample size, but similar to or higher than for Dunnett's test with unequal variance, unbalanced data and larger sample size. The method is applied to a dataset concerning isotope levels found in elephant tusks from various geographical areas. These data have very unbalanced group sizes and unequal variances. This example illustrates that the PB method is easy to implement and avoids the need for transforming data to meet the equal variance assumption, simplifying interpretation of results.

Distributed estimation for large-scale expectile regression

Analysis of large volume of data is very complex due to not only the high level of skewness and heteroscedasticity of variance but also the difficulty of data storage. Expectile regression is a common alternative method to analyze heterogeneous data. Distributed storage can reduce effectively the storage burden of a single machine. In this paper, we consider fitting linear expectile regression model to estimate conditional expectile based on large-scale data. We store the data in a distributed manner and construct a gradient-enhanced loss (GEL) function as a proxy for the global loss function. A distributed algorithm is proposed for the optimization of the GEL function. The asymptotic properties of the proposed estimator are established. Simulation studies are conducted to assess the finite-sample performance of our proposed estimator. Applications to an analysis of the National Health Interview Survey data set demonstrate the practicability of the proposed method.

Retire: Robust expectile regression in high dimensions

High-dimensional data can often display heterogeneity due to heteroscedastic variance or inhomogeneous covariate effects. Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. The former is computationally challenging due to the non-smooth nature of the check loss, and the latter is sensitive to heavy-tailed error distributions. In this paper, we propose and study (penalized) robust expectile regression (retire), with a focus on iteratively reweighted ℓ1-penalization which reduces the estimation bias from ℓ1-penalization and leads to oracle properties. Theoretically, we establish the statistical properties of the retire estimator under two regimes: (i) low-dimensional regime in which d≪n; (ii) high-dimensional regime in which s≪n≪d with s denoting the number of significant predictors. In the high-dimensional setting, we thoroughly analyze the statistical properties of the solution path of iteratively reweighted ℓ1-penalized retire estimation, adapted from the local linear approximation algorithm for folded-concave regularization. Under a mild minimum signal strength condition, we demonstrate that with as few as log(logd) iterations, the final iterate of our proposed approach achieves the oracle convergence rate. At each iteration, we solve the weighted ℓ1-penalized convex program using a semismooth Newton coordinate descent algorithm. Numerical studies demonstrate the promising performance of the proposed procedure in comparison to both non-robust and quantile regression based alternatives.

Injury severity analysis of time-of-day fluctuations and temporal volatility in reverse sideswipe collisions: A random parameter model with heterogeneous means and heteroscedastic variances

Problem: Sideswipe collisions in the opposite direction often result in more severe injuries than the typical same-direction crashes, especially when light trucks are involved. This study investigates the time-of-day fluctuations and temporal volatility of potential factors that affect the injury severity of reverse sideswipe collisions. Methods: A series of random parameters logit models with heterogeneous means and heteroscedastic variances are developed and utilized to explore unobserved heterogeneity inherent in variables and preclude biased parameter estimation. The segmentation of estimated results is also examined through temporal instability tests. Results: Based on crash data in North Carolina, a number of contributing factors are identified that have profound associations with obvious and moderate injuries. Meanwhile, significant temporal volatility is observed in the marginal effects of several factors such as driver restraint, alcohol or drugs impact, Sport Utility Vehicle (SUV) at fault, and adverse road surface across three different periods. Fluctuations in the time of day indicate that restraint with belts is more effective in mitigating the obvious injury in the nighttime, and high-class roadway sustains a higher probability of resulting in more serious injury compared to the daytime. Practical Applications: The findings of this study could help further guide the implementation of safety countermeasures related to atypical sideswipe collisions.

Genome-wide variance quantitative trait locus analysis suggests small interaction effects in blood pressure traits

Genome-wide variance quantitative trait loci (vQTL) analysis complements genome-wide association study (GWAS) and has the potential to identify novel variants associated with the trait, explain additional trait variance and lead to the identification of factors that modulate the genetic effects. I conducted genome-wide analysis of the UK Biobank data and identified 27 vQTLs associated with systolic blood pressure (SBP), diastolic blood pressure (DBP) and pulse pressure (PP). The top single-nucleotide polymorphisms (SNPs) are enriched for expression QTLs (eQTLs) or splicing QTLs (sQTLs) annotated by GTEx, suggesting their regulatory roles in mediating the associations with blood pressure (BP). Of the 27 vQTLs, 14 are known BP-associated QTLs discovered by GWASs. The heteroscedasticity effects of the 13 novel vQTLs are larger than their genetic main effects, which were not detected by existing GWASs. The total R-squared of the 27 top SNPs due to variance heteroscedasticity is 0.28%, compared with 0.50% owing to their main effects. The overall effect size of the variance heteroscedasticity is small in GWAS SNPs compared with their main effects. For the 411, 384 and 285 GWAS SNPs associated with SBP, DBP and PP, respectively, their heteroscedasticity effects were 0.52%, 0.43%, and 0.16%, and their main effects were 5.13%, 5.61%, and 3.75%, respectively. The number and effects of the vQTLs are small, which suggests that the effects of gene–environment and gene–gene interactions are small. The main effects of the SNPs remain the major source of genetic variance for BP, which would probably be true for other complex traits as well.

Robust PC with wild bootstrap estimation of linear model in the presence of outliers, multicollinearity and heteroscedasticity error variance

The regression model estimator is considered efficient if it is robust and resistant to the presence of heteroscedasticity variance, multicollinearity or unusual observations called outliers. However, in regard to these problems, the wild bootstrap and robust wild bootstrap are no longer efficient since they could not produce the smallest variance. Hence this research investigates the use of robust PC with wild bootstrap techniques on regression model as an estimator for real and simulation data in a situation where multicollinearity, heteroscedasticity and multiple outliers are present. This paper proposed a robust procedure based on the weighted residuals which combined the Tukey bisquare weighted function, principal component analysis (PCA) to remedy the multicollinearity problems, least trimmed squares (LTS) estimator, robust location and scale, and the wild bootstrap sampling procedure of Wu and Liu that remedy the heteroscedasticity error variance. RPCW Boot Wu and RPCW Boot Liu were obtained through a modified version of R Boot Wu and R Boot Liu. Finally, based on the real data and simulation study, the performance of the RPCW Boot Wu and RPCW Boot Liu is compared with the existing R Boot Wu, R Boot Liu and also with Boot Wu, Boot Liu using the biased, RMSE and standard error. The numerical example and simulation study shows that the RPCW Boot Wu and RPCW Boot Liu techniques have proven to be a good alternative estimator for regression model with lower standard error values.

A new nonparametric lack-of-fit test of nonlinear regression in presence of heteroscedastic variances

In this article, a nonparametric lack-of-fit test of nonlinear regression in presence of heteroscedastic variances is proposed. We consider regression models with a discrete or continuous response variable without distributional assumptions so that the test is widely applicable. The test statistic is developed using a k-nearest neighbor augmentation defined through the ranks of the predictor variable. The asymptotic distribution of the test statistic is derived under the null and local alternatives for the case of using fixed number of nearest neighbors. The parametric standardizing rate is achieved for the asymptotic distribution of the proposed test statistic. This allows the proposed test to have faster convergence rate than most of nonparametric methods. Numerical studies show that the proposed test has good power to detect both low and high frequency alternatives even for moderate sample size. The proposed test is applied to an engineering data example.

Arbitrage Pricing with Heterogeneous Spatial Effects and Heteroscedastic Disturbances

Abstract We develop a heterogeneous spatial arbit and regression coefficients, and heteroscedastic variances, and further establish identification of parameters and asymptotic normality of the conditional QML estimators under some mild conditions. We apply the proposed model to study a real data set of 11 eurozone stock index returns and extend the Fama–French five-factor model to regional stock indices, in which heterogeneous spatial effects and heteroscedastic disturbances are highly significant and they both play very important roles in explaining distinct endogenous effects and distinct risks of the 11 eurozone stock markets. Our empirical results reveal unique characteristics of each of 11 eurozone stock markets and their inner connections.

The Effects of Breeding Environment and Food Type on the Survival and Growth of Red Sea Cucumber (Apostichopus japonicus) Larvae

Jwa, M.-S.; Kim, S.-C., and Hong, C.-Y., 2022. The effects of breeding environment and food type on the survival and growth of red sea cucumber (Apostichopus japonicus) larvae. Journal of Coastal Research, 38(3), 578–584. Coconut Creek (Florida), ISSN 0749-0208. This study aimed to investigate the effects of water temperature, salinity, and food type on the survival and growth of red sea cucumber (Apostichopus japonicus) larvae, over 14 days. Early Auricularia larvae (average length 475 ± 10.3 µm) were exposed to four different water temperatures (18, 22, 26, and 30 °C), four different salinities (25, 30, 35, and 40 psu), and four food types (Chaetoceros calcitrans, Pavlova lutheri, Isochrysis galbana, Rhodotorula+effective microorganisms (EM)). Subsequently, the survival rate, larval developmental stage (early Auricularia–Doliolaria), and the required time (days) to arrive at each stage were observed. The experiment was conducted at a private sea cucumber aquaculture farm in Jeju Special Self-Governing Province, and the results were statistically analyzed using SPSS version 22 (SPSS Inc., USA). This research team evaluated the average significance difference between groups, which was determined in p < 0.05, by applying a Duncan test due to variance heteroscedasticity. The results indicate the following optimum rearing conditions of red sea cucumber larvae: water temperature of 22 °C, salinity of 30–31 psu, with Rhodotorula+EM as food. The findings of this study could facilitate increasing seed production of the red sea cucumber A. japonicus.

Near-source magnitude scaling of spectral accelerations: analysis and update of Kotha et al. (2020) model

Ground-motion models (GMMs) are often used to predict the random distribution of Spectral accelerations ( $$\mathrm{SAs}$$ ) at a site due to a nearby earthquake. In probabilistic seismic hazard and risk assessment, large earthquakes occurring close to a site are considered as critical scenarios. GMMs are expected to predict realistic $$\mathrm{SAs}$$ with low within-model uncertainty ( $${\upsigma }_{\upmu }$$ ) for such rare scenarios. However, the datasets used to regress GMMs are usually deficient of data from critical scenarios. The (Kotha et al., A Regionally Adaptable Ground-Motion Model for Shallow Crustal Earthquakes in Europe Bulletin of Earthquake Engineering 18:4091–4125, 2020) GMM developed from the Engineering strong motion (ESM) dataset was found to predict decreasing short-period $$\mathrm{SAs}$$ with increasing $${M}_{W}\ge {\mathrm{M}}_{\mathrm{h}}=6.2$$ , and with large $${\upsigma }_{\upmu }$$ at near-source distances $$\le 30\mathrm{km}$$ . In this study, we updated the parametrisation of the GMM based on analyses of ESM and the Near source strong motion (NESS) datasets. With $${\mathrm{M}}_{\mathrm{h}}=5.7$$ , we could rectify the $${M}_{W}$$ scaling issue, while also reducing $${\upsigma }_{\upmu }$$ at $${M}_{W}\ge {\mathrm{M}}_{\mathrm{h}}$$ . We then evaluated the GMM against NESS data, and found that the $$\mathrm{SAs}$$ from a few large, thrust-faulting events in California, New Zealand, Japan, and Mexico are significantly higher than GMM median predictions. However, recordings from these events were mostly made on soft-soil geology, and contain anisotropic pulse-like effects. A more thorough non-ergodic treatment of NESS was not possible because most sites sampled unique events in very diverse tectonic environments. We provide an updated set of GMM coefficients, $${\upsigma }_{\upmu }$$ , and heteroscedastic variance models; while also cautioning against its application for $${M}_{W}\le 4$$ in low-moderate seismicity regions without evaluating the homogeneity of $${M}_{W}$$ estimates between pan-European ESM and regional datasets.

Correlation between the Korean pork grade system and the amount of pork primal cut estimated with AutoFom III.

It is impossible to know the amount of pork primal cut by pig carcass grade which is determined only by carcass weight and backfat thickness in the Korean Pig Carcass System. The aim of this study was to investigate the correlation between the pig carcass grade and the amount of pork primal cut estimated with AutoFom III. A total of 419,321 Landrace, Yorkshire, and Duroc (LYD) pigs were graded with the Korean Pig Carcass Grade System. Amounts of belly, neck, loin, tenderloin, spare ribs, shoulder, and ham were estimated with AutoFom III. Regression equations for seven primal cuts according to each grade were derived. There were significant differences among the three carcass grades due to heteroscedasticity variance (p < 0.0001). Three regression equations were derived from AutoFom III estimation of primal cuts according to carcass grades. The coefficient of determination of the regression equation was 0.941 for grade 1+, 0.982 for grade 1, and 0.993 for grade 2. Regression equations obtained from this study are suitable for AutoFom III software, a useful tool for the analysis of each pig carcass grade in the Korean Pig Carcass Grade System. The high reliability of predicting the amount of primal cut with AutoFom III is advantageous for the management of slaughterhouses to optimize their product sorting in Korea.

Quantile composite-based path modeling: algorithms, properties and applications

Composite-based path modeling aims to study the relationships among a set of constructs, that is a representation of theoretical concepts. Such constructs are operationalized as composites (i.e. linear combinations of observed or manifest variables). The traditional partial least squares approach to composite-based path modeling focuses on the conditional means of the response distributions, being based on ordinary least squares regressions. Several are the cases where limiting to the mean could not reveal interesting effects at other locations of the outcome variables. Among these: when response variables are highly skewed, distributions have heavy tails and the analysis is concerned also about the tail part, heteroscedastic variances of the errors is present, distributions are characterized by outliers and other extreme data. In such cases, the quantile approach to path modeling is a valuable tool to complement the traditional approach, analyzing the entire distribution of outcome variables. Previous research has already shown the benefits of Quantile Composite-based Path Modeling but the methodological properties of the method have never been investigated. This paper offers a complete description of Quantile Composite-based Path Modeling, illustrating in details the method, the algorithms, the partial optimization criteria along with the machinery for validating and assessing the models. The asymptotic properties of the method are investigated through a simulation study. Moreover, an application on chronic kidney disease in diabetic patients is used to provide guidelines for the interpretation of results and to show the potentialities of the method to detect heterogeneity in the variable relationships.

Doubly generalized logit: A closed-form discrete choice model system with multivariate generalized extreme value distributed utilities

In this study, we generalize our previous q-generalized multinomial logit model (Nakayama & Chikaraishi, 2015), in which the heteroscedastic variance and flexible shape of the utility distribution are considered, by allowing for statistical dependency of alternatives. This is achieved by introducing the q-generalization of McFadden’s multivariate Gumbel distribution. Thus, the logit model is doubly generalized; 1) each utility follows the generalized extreme value distribution that includes the Gumbel, Weibull, and Fréchet distributions; and 2) the utility distribution is multivariate, and therefore, a nested or cross-nested structure and dependency of alternatives are allowed. The proposed doubly generalized logit model system allows for deriving new closed-form discrete choice models such as the q-generalized nested logit model and q-generalized cross-nested logit (CNL) model. Furthermore, the model system includes conventional logit models such as the multinomial logit, nested logit, and CNL models as special cases as well as the new generalized logit models, while retaining a closed-form expression. We empirically confirm that the goodness-of-fit of the proposed model could be substantially better compared to that of the conventional models in some cases, though the degree of improvements varies across cases.

A Nonparametric Lack-of-Fit Test of Constant Regression in the Presence of Heteroscedastic Variances

We consider a k-nearest neighbor-based nonparametric lack-of-fit test of constant regression in presence of heteroscedastic variances. The asymptotic distribution of the test statistic is derived under the null and local alternatives for a fixed number of nearest neighbors. Advantages of our test compared to classical methods include: (1) The response variable can be discrete or continuous regardless of whether the conditional distribution is symmetric or not and can have variations depending on the predictor. This allows our test to have broad applicability to data from many practical fields; (2) this approach does not need nonlinear regression function estimation that often affects the power for moderate sample sizes; (3) our test statistic achieves the parametric standardizing rate, which gives more power than smoothing-based nonparametric methods for moderate sample sizes. Our numerical simulation shows that the proposed test is powerful and has noticeably better performance than some well known tests when the data were generated from high frequency alternatives or binary data. The test is illustrated with an application to gene expression data and an assessment of Richards growth curve fit to COVID-19 data.

Time series analysis of the number of Covid-19 deaths in Iraq

In this paper, the time series data for the number of deaths from Coronavirus (COVID-19) patients in Iraq were analyzed for the period from 4/3/2020 to 18/2/2021. ARCH, GARCH, and TGARCH models were applied due to the changing volatility of the series leading to a heteroscedastic variance. The appropriate models for the series were diagnosed and the best model was chosen and used for forecasting by the exponential smoothing methods. The comparison criterion used was the Root Mean Squared Error and the Sum of Squared Residuals. The most appropriate model for modeling and forecasting the Coronavirus deaths series in Iraq was diagnosed as TGARCH (1,1). Finally, the method of Holt-winter-additive forecasting was the best method among the