Empirical Bayesian Research Articles

Development of a prognostic model for early-stage gastric cancer-related DNA methylation-driven genes and analysis of immune landscape

Background and AimsThis study aimed to develop a prognostic model based on DNA methylation-driven genes for patients with early-stage gastric cancer and to examine immune infiltration and function across varying risk levels.MethodsWe analyzed data from stage I/II gastric cancer patients in The Cancer Genome Atlas which included clinical details, mRNA expression profiles, and level 3 DNA methylation array data. Using the empirical Bayes method of the limma package, we identified differentially expressed genes (DEGs), and the MethylMix package facilitated the identification of DNA methylation-driven genes (DMGs). Univariate Cox regression and LASSO (least absolute shrinkage and selector operation) analyses were utilized to pinpoint critical genes. A risk score prediction model was formulated using two genes that demonstrated the most significant hazard ratios (HRs). Model performance was evaluated within the initial cohort and verified in the GSE84437 cohort; a nomogram was also constructed based on these genes. We further examined 50 methylation sites associated with three CpG islands in C1orf35 and 14 methylation sites linked to one CpG island in FAAH. The CIBERSORT package was employed to identify immune cell clusters in the prediction model.ResultsA total of 176 DNA methylation-driven genes were refined down to a four-gene signature (ZC3H12A was hypermethylated; GATA3, C1orf35, and FAAH were hypomethylated), which exhibited a significant correlation with overall survival (OS), as evidenced by p-values below 0.05 following univariate Cox regression and LASSO analysis. Specifically, for the risk score prediction model, C1orf35, which had the highest hazard ratio (HR = 2.035, p = 0.028), and FAAH, with the lowest hazard ratio (HR = 0.656, p = 0.012), were selected. The Kaplan–Meier analysis demonstrated distinct survival outcomes between the high-risk and low-risk score groups. The model’s predictive accuracy was confirmed with an area under the curve (AUC) of 0.611 for 3-year survival and 0.564 for 5-year survival. Notably, the hypomethylation of the three CpG islands in C1orf35 and the single CpG island in FAAH was significantly different in stage I/II gastric cancer patients compared to normal tissues. Additionally, the high-risk score group showed a notable association with resting CD4 memory T cells.ConclusionPromoter hypomethylation of C1orf35 and FAAH in early-stage gastric cancer underscores their potential as biomarkers for accurate diagnosis and treatment. The developed predictive model employing genes affected by DNA methylation serves as a crucial independent prognostic factor in early-stage gastric cancer.

A Basket Trial Design Based on Power Priors

In basket trials a treatment is investigated in several subgroups. They are primarily used in oncology in early clinical phases as single-arm trials with a binary endpoint. For their analysis primarily Bayesian methods have been suggested, as they allow partial sharing of information based on the observed similarity between subgroups. Fujikawa et al. suggested an approach using empirical Bayes methods that allows flexible sharing based on easily interpretable weights derived from the Jensen-Shannon divergence between the subgroup-wise posterior distributions. We show that this design is closely related to the method of power priors and investigate several modifications of Fujikawa’s design using methods from the power prior literature. While in Fujikawa’s design, the amount of information that is shared between two baskets is only determined by their pairwise similarity, we also discuss extensions where the outcomes of all baskets are considered in the computation of the sharing weights. The results of our comparison study show that the power prior design has comparable performance to fully Bayesian designs in a range of different scenarios. At the same time, the power prior design is computationally cheap and even allows analytical computation of operating characteristics in some settings.

The application of Bayesian method in estimating heterogeneity of treatment effect

It is well recognized that the benefit of medical interventions may not be distributed evenly in the target population due to patient heterogeneity, and conclusions based on traditional randomized clinical trials do not apply to everyone. Considering the increasing cost of randomized trials and difficulties in recruiting patients, there is an urgent need to develop analytical methods to estimate treatment effects in subgroups. In particular, due to limited sample sizes of subgroups, standard analysis tends to yield wide confidence intervals for the treatment effect, which are often uninformative. In this article, the superiority of the empirical Bayes estimate of the conditional average treatment effect is demonstrated by using a linear regression model to specifically represent treatment-covariate interactions. To obtain more accurate prior distributions and posterior estimates of treatment effects in subgroups, we utilize the covariate-dependent precision parameter across subgroups. Based on subgroup data obtained from clinical trials, the covariate-dependent precision parameter is used to assess the similarity of treatment effects across subgroups, which is an approach to obtain more reasonable prior information. To verify the validity of the empirical Bayes method, we evaluate it by a real clinical trial comparing insulin and metformin.

Indirect Effects of Municipal Public Health Nurse Workforce on Cancer Standardized Mortality Ratios Mediated by Cancer Screening Rates.

This study examined the indirect effects of the number of Japanese municipal public health nurses (PHNs) on cancer standardized mortality ratios (SMRs), using cancer screening and diagnostic follow-up rates as mediators. Ecological study using municipalities as the unit of analysis MEASUREMENTS: Aggregate, municipal-level government data were analyzed using a linear model with empirical Bayes estimates of SMRs (EBSMRs) for gastric, colorectal, and lung cancers as the dependent variables, and the number of PHNs, cancer screening rate, diagnostic follow-up rate, and adjustment variables as independent variables. Structural equation modeling (SEM) was used to examine the indirect effects of PHNs. Cancer screening rates were significantly negatively associated with EBSMR, except for gastric cancer in women. No significant association was observed between the EBSMR and diagnostic follow-up rates. SEM revealed a significant indirect effect of the number of PHNs, most of which was due to the cancer screening rate. From a population-based public health perspective, increasing the number of PHNs and focusing on improving cancer screening rates may effectively reduce cancer SMRs.

Empirical Bayes Methods, Evidentialism, and the Inferential Roles They Play.

Empirical Bayes-based Methods (EBM) is an increasingly popular form of Objective Bayesianism (OB). It is identified in particular with the statistician Bradley Efron. The main aims of this paper are, first, to describe and illustrate its main features and, second, to locate its role by comparing it with two other statistical paradigms, Subjective Bayesianism (SB) and Evidentialism. EBM's main formal features are illustrated in some detail by schematic examples. The comparison between what Efron calls their underlying "philosophies" is by way of a distinction made between confirmation and evidence. Although this distinction is sometimes made in the statistical literature, it is relatively rare and never to the same point as here. That is, the distinction is invariably spelled out intra- and not inter-paradigmatically solely in terms of one or the other accounts. The distinction made in this paper between confirmation and evidence is illustrated by two well-known statistical paradoxes: the base-rate fallacy and Popper's paradox of ideal evidence. The general conclusion reached is that each of the paradigms has a basic role to play and all are required by an adequate account of statistical inference from a technically informed and fine-grained philosophical perspective.

A comparison of the renal function biomarkers serum creatinine, pro-enkephalin and cystatin C to predict clearance of pemetrexed.

For drugs with a narrow therapeutic window, there is a delicate balance between efficacy and toxicity, thus it is pivotal to administer the right dose from the first administration onwards. Exposure of pemetrexed, a cytotoxic drug used in lung cancer treatment, is dictated by kidney function. To facilitate optimized dosing of pemetrexed, accurate prediction of drug clearance is pivotal. Therefore, the aim of this study was to investigate the performance of the kidney function biomarkers serum creatinine, cystatin C and pro-enkephalin in terms of predicting the elimination of pemetrexed. We performed a population pharmacokinetic analysis using a dataset from two clinical trials containing pharmacokinetic data of pemetrexed and measurements of all three biomarkers. A three-compartment model without covariates was fitted to the data and the obtained individual empirical Bayes estimates for pemetrexed clearance were considered the "true" values (Cltrue). Subsequently, the following algorithms were tested as covariates for pemetrexed clearance: the Chronic Kidney Disease Epidemiology Collaboration equation using creatinine (CKD-EPICR), cystatin C (CKD-EPICYS), a combination of both (CKD-EPICR-CYS), pro-enkephalin as an absolute valueor in a combined algorithm with age and serum creatinine, and lastly, a combination of pro-enkephalin with cystatin C. The dataset consisted of 66 subjects with paired observations for all three kidney function biomarkers. Inclusion of CKD-EPICR-CYS as a covariate on pemetrexed clearance resulted in the best model fit, with the largest decrease in objective function (p < 0.00001) and explaining 35% of the total inter-individual variability in clearance. The predictive performance of the model to containing CKD-EPICR-CYS to predict pemetrexed clearance was good with a normalized root mean squared error and mean prediction error of 19.9% and 1.2%, respectively. In conclusion, this study showed that the combined CKD-EPICR-CYS performs best in terms predicting pharmacokinetics of pemetrexed. Despite the hypothesized disadvantages, creatinine remains to be a suitable and readily available marker to predict pemetrexed clearance in clinical practice.

Radiation image reconstruction and uncertainty quantification using a Gaussian process prior

We propose a complete framework for Bayesian image reconstruction and uncertainty quantification based on a Gaussian process prior (GPP) to overcome limitations of maximum likelihood expectation maximization (ML-EM) image reconstruction algorithm. The prior distribution is constructed with a zero-mean Gaussian process (GP) with a choice of a covariance function, and a link function is used to map the Gaussian process to an image. Unlike many other maximum a posteriori approaches, our method offers highly interpretable hyperparamters that are selected automatically with the empirical Bayes method. Furthermore, the GP covariance function can be modified to incorporate a priori structural priors, enabling multi-modality imaging or contextual data fusion. Lastly, we illustrate that our approach lends itself to Bayesian uncertainty quantification techniques, such as the preconditioned Crank–Nicolson method and the Laplace approximation. The proposed framework is general and can be employed in most radiation image reconstruction problems, and we demonstrate it with simulated free-moving single detector radiation source imaging scenarios. We compare the reconstruction results from GPP and ML-EM, and show that the proposed method can significantly improve the image quality over ML-EM, all the while providing greater understanding of the source distribution via the uncertainty quantification capability. Furthermore, significant improvement of the image quality by incorporating a structural prior is illustrated.

Population Pharmacokinetics and Exposure–Response Analysis of a Fixed‐Dose Combination of Ivermectin and Albendazole in Children, Adolescents, and Adults

Trichuris trichiura is a soil‐transmitted helminth causing intestinal disease. Albendazole is the standard treatment despite its moderate efficacy, which is improved when co‐administered with ivermectin. A fixed‐dose combination adds practical advantages mainly for mass drug administration. The aim of this article is to define the population pharmacokinetic models and exposure–response of an innovative albendazole/ivermectin combination. Data were obtained from a phase I clinical trial in healthy adults and from a phase II trial in children and adolescents infected with T. trichiura. Nonlinear mixed‐effects models were built for albendazole and ivermectin using NONMEM®. Area under the curve was calculated using the empirical Bayes estimates of the pharmacokinetic parameters of each individual and used for evaluation of exposure–response between cure rate and pharmacokinetic exposure. The pharmacokinetics of albendazole was described using a two‐compartmental model with first‐order absorption and the pharmacokinetics of ivermectin was described using a two‐compartmental model with zero‐order followed by first‐order absorption. Clearance and volume of distribution increased with body weight for both albendazole and ivermectin. Day 1 area under the curve of albendazole and ivermectin from the children and adolescents treated with the combination regimens were similar to the healthy adults treated with control drugs. A flat exposure–response relationship was observed between the cure rate and drug exposure. Population pharmacokinetic of a combination of albendazole and ivermectin in children, adolescents, and adults, either healthy or infected by T. trichiura was described. The dosage selected in the phase II trial was appropriate for the subsequent phase III.

Inflammation proteomic profiling of psychosis in young adults: Findings from the ALSPAC birth cohort

Psychotic disorder is associated with altered levels of various inflammatory markers in blood, but existing studies have typically focused on a few selected biomarkers, have not examined specific symptom domains notably negative symptoms, and are based on individuals with established/chronic illness. Based on data from young people aged 24 years from the Avon Longitudinal Study of Parents and Children (ALSPAC), a UK birth cohort, we have examined the associations of 67 plasma immune/inflammatory proteins assayed using the Olink Target 96 Inflammation panel with psychotic disorder, positive (any psychotic experiences and definite psychotic experiences) and negative symptoms, using linear models with empirical Bayes estimation. The analyses included between 2317 and 2854 individuals. After adjustment for age, sex, body mass index and smoking and correction for multiple testing, positive symptoms and psychotic disorder were consistently associated with upregulation of CDCP1 and IL-6, and psychotic disorder was additionally associated with upregulation of MMP-10. Negative symptoms were associated with upregulation of CDCP1 and TRAIL. CDCP1 and MMP-10 are novel markers of psychosis identified in this study, and are involved in immune regulation, immune cell activation/migration, blood-brain barrier disruption, and extracellular matrix abnormalities. Our findings highlight psychosis symptom domains have overlapping and distinct immune associations, and support a role of inflammation and immune dysfunction in the pathogenesis of psychosis.

Safety assessment of basiliximab using real-world adverse event data from the FDA Adverse Event Reporting System Database: A retrospective observational study.

This study analyzed adverse drug events (ADEs) associated with basiliximab, sourced from the Food and Drug Administration Adverse Event Reporting System (FAERS) database, spanning the first quarter of 2004 to the fourth quarter of 2023. We collected ADE data for basiliximab from 2004 Q1 to 2023 Q4. After standardization, we employed several signal quantification methods for analysis, such as the Reporting Odds Ratio (ROR), Proportional Reporting Ratio (PRR), Bayesian Confidence Propensity for Neural Networks (BCPNN), and empirical bayes geometric mean (EBGM). In this analysis of 1520 ADEs reports citing basiliximab as the primary suspect, we identified 295 preferred terms across 24 system organ classifications (SOCs). The 3 most prevalent SOCs were investigated (n = 1403, ROR 2.84, PRR 2.54, IC 1.34, EBGM 2.54), infections and infestations (n = 1198, ROR 2.85, PRR 2.59, IC 1.37, EBGM 2.59), and renal and urinary disorders (n = 903, ROR 6.01, PRR 5.48, IC 2.45, EBGM 5.47). Increased blood creatinine and pyrexia were the most frequently reported adverse events (AEs) associated with basiliximab, and cytomegalovirus infection also demonstrated significant signal intensity. Notably, this study revealed some adverse reactions beyond basiliximab drug instructions, such as mitral valve calcification, diastolic dysfunction, pelvic fluid collection, testicular swelling, soft tissue necrosis, and muscle necrosis. Although basiliximab offers therapeutic benefits, it carries the risk of several adverse reactions. Clinicians should monitor patients for signs of increased serum creatinine level, fever, cytomegalovirus infection, anaphylactic shock, mitral valve calcification, diastolic dysfunction, pelvic fluid collection, testicular swelling, soft tissue necrosis, muscle necrosis, and other events during clinical use.

Geographic Patterns of Youth Suicide in San Diego County

ObjectiveOur objectives were to evaluate for any cluster patterns of youth suicide deaths and characterize the level of child opportunity in the communities where suicide deaths occurred. MethodsDecedents <18 years were identified from San Diego County Medical Examiner death reports from 2000 to 2020. We mapped decedents’ residential Zone Improvement Plan (ZIP) codes and calculated suicide rates per 10,000 youths. ZIP codes identified in overlapping spatial statistical approaches — the spatial scan statistic and Local Moran with Empirical Bayes (EB) rates — were considered a cluster for the final analysis. We obtained Child Opportunity Index (COI) scores for each ZIP code to determine if there were differences in: 1) ZIP codes with suicide deaths compared to ZIPs with no deaths 2) differences in distribution of suicide death rates across quintiles of COI. ResultsScan statistic identified 25 ZIP codes within a cluster (RR 2.6, P = 0.00066). Local Moran with EB rates identified two ZIP codes as a high-high cluster (P < 0.05). The location identified as a cluster in both approaches was in Alpine. The median COI for ZIP codes with suicide deaths was higher at 63.5 (IQR 38–83) compared to 47 (IQR 22.5–75.5) for ZIP codes without suicide deaths. There was a significant difference in suicide rates between Very Low and Moderate levels of Overall opportunity (P = .013). ConclusionWe identified a cluster of youth suicides in one of the most populous counties in the country. These findings serve to inform policies and prevention programs that aim to mitigate youth suicide mortality.

Are Alpha-2 Adrenergic Agonists Being Used in Infants?

Purpose: To identify and quantify adverse events (AEs) associated with alpha-2 adrenergic agonists prescribed for the treatment of glaucoma in infants. Methods: We queried the Federal Adverse Event Reporting System (FAERS) from 2004-2023Q1 for AE reports related to brimonidine use in patients aged 12 months or younger. We then conducted a disproportionality analysis using data mining algorithms, including the reporting odds ratio, proportional reporting ratio, empirical bayes geometric mean, and information component to identify significant symptoms. Results: We identified 35 unique AE reports associated with brimonidine. Of these, 27 cases involved hospitalization, 13 cases involved life-threatening complications, 18 cases reported other complications, and 1 case involved a congenital anomaly. The most commonly reported AE was hypotonia, occurring in 20 cases. This was followed by other systemic symptoms, including hypothermia, depressed level of consciousness, lethargy, general toxicity, and pallor, among others. All symptoms were found to be significant in the disproportionality analysis. Notably, most cases were not known to involve an ophthalmic route of exposure. Conclusions: The use of alpha-2 adrenergic agonists in infants aged 1 year or younger has been associated with various systemic AEs, including hypotension, respiratory depression, and central nervous system depression. Ophthalmologists should be aware of these potential risks. Further, more rigorous warnings should be in place to prevent unintentional exposure of infants to brimonidine.

Developing Safety Performance Functions for Severe Distraction-Related Crashes along Kentucky’s Rural and Urban Two-Lane Roadways

This study develops safety performance functions (SPFs) for severe (“KA” or “fatal and suspected serious injury”) distraction-related crashes along Kentucky’s rural and urban two-lane undivided roadway segments using recent four-year (2018–2021) crash records. Additional efforts were made to meticulously scrutinize crash narratives categorized as non-distracted and correct those cases. To account for crash under-dispersion, the Conway–Maxwell–Poisson, heterogeneous Conway–Maxwell–Poisson, zero-inflated Conway–Maxwell–Poisson, and zero-inflated heterogeneous Conway–Maxwell–Poisson (ZI-HTCMP) models were fitted and compared. The ZI-HTCMP model outperformed the other models with respect to several goodness-of-fit measures (e.g., mean absolute deviance and mean square prediction error). From the developed SPFs for rural and urban two-lane roads, wider lanes and higher speed limits (55 mph) were associated with increased severe distraction-related crash frequencies. Furthermore, some variables were found to be significant in rural areas, but insignificant in urban areas, and vice versa. For example, major collector roads, minor collector/local roads, the presence of roadside guardrails, wider right-hand shoulders, the presence of horizontal curves, and the presence of vertical grades were associated with increased crash frequencies along rural two-lane roads. In addition, the proportion of heavy vehicles (&gt;5%) and the existence of paved shoulders were associated with increased crashes along urban two-lane roads. The empirical Bayes method was then used to rank the top 10 distraction-related high crash locations (HCLs) for both rural and urban two-lane segments. In-depth investigation of HCLs highlighted single-vehicle distraction-related crashes as the most common collision type. Countermeasures were proposed to help reduce severe distraction-related crashes, for example, installing chevron signs along rural two-lane roads.

Quantifying the effectiveness of an active treatment in improving highway-railway grade crossing safety in Canada: an empirical Bayes observational before–after study

Transportation engineers face challenging safety investment decisions, particularly for highway-railway grade crossings (HRGCs), where rare collision occurrences and incomplete historical records complicate the assessment of countermeasure cost-effectiveness. This study introduces an empirical Bayes (EB) observational before–after approach to address these challenges, specifically examining the impact of a widely used safety countermeasure: flashing lights, bells, and gates (FLBG). The research covers a total 171 FLBG crossings with pre-implementation conditions varying between standard reflective crossing signs and flashing lights and bells, utilizing collision data from 2000 to 2012 for analysis. This study is the first to substantiate quantitatively the effectiveness of FLBG treatment in improvising the safety of HRGCs and show that the effect is dependent of crossing characteristics, such as track angle, pavement conditions, and train speed limits. These findings can provide valuable support to decision-makers, enabling them to make more effective decisions for grade crossing safety improvement projects.

Benchmarking building energy consumption for space heating using an empirical Bayesian approach with urban-scale energy model

Calibration of an urban-scale building energy model is crucial for advancing building energy efficiency codes and policies to reduce carbon dioxide emissions whereas the accuracy of the model is challenged by performance gaps resulting from sparse data and uncertainties in modeling. An empirical Bayesian (EB) approach, incorporated a probability-based mixed-effects energy model and Monte Carlo Markov chain simulation, was developed to predict and benchmark building energy use intensity (EUI) for space heating, using the metering data of 2020 stochastically sampled district heating substations from a municipal utility database. The limitations, implementations and performances of different modeling strategy, including EB as well as Maximum Likelihood Estimation (MLE), Maximum A Posteriori (MAP) and Fully Bayes (FB) were compared. The results show that EB strategy achieved superior prediction performance than other strategies on the novel test dataset. The preparation, calibration, and validation efforts for implementing the EB approach were demonstrated. The proposed methodology is recommended as a robust approach for benchmarking urban-scale building EUI, distinguished from other machine learning approaches by requiring fewer training samples, better interpreting uncertainties in the data used, and exhibiting superior generalization capabilities. The proposed methodology was successfully demonstrated through the urban-scale benchmarking of space heating EUI in Beijing.

Estimating Urban Traffic Safety and Analyzing Spatial Patterns through the Integration of City-Wide Near-Miss Data: A New York City Case Study

City-wide near-miss data can be beneficial for traffic safety estimation. In this study, we evaluate urban traffic safety and examine spatial patterns by incorporating city-wide near-miss data (59,277 near-misses). Our methodology employs a grid-based method, the Empirical Bayes (EB) approach, and spatial analysis tools including global Moran’s I and local Moran’s I. The study findings reveal that near-misses have the strongest correlation with observed crash frequency among all the variables studied. Interestingly, the ratio of near-misses to crashes is roughly estimated to be 1957:1, providing a potentially useful benchmark for urban areas. For other variables, an increased number of intersections and bus stops, along with a greater road length, contribute to a higher crash frequency. Conversely, residential and open-space land use rates show a negative correlation with crash frequency. Through spatial analysis, potential risk hotspots including roads linking bridges and tunnels, and avenues bustling with pedestrian activity, are highlighted. The study also identified negative local spatial correlations in crash frequencies, suggesting significant safety risk variations within relatively short distances. By mapping the differences between observed and predicted crash frequencies, we identified specific grid areas with unexpectedly high or low crash frequencies. These findings highlight the crucial role of near-miss data in urban traffic safety policy and planning, particularly relevant with the imminent rise of autonomous and connected vehicles. By integrating near-miss data into safety estimations, we can develop a more comprehensive understanding of traffic safety and, thus, more effectively address urban traffic risks.

Speech and neuroimaging effects following HiCommunication: a randomized controlled group intervention trial in Parkinson's disease.

Speech, voice and communication changes are common in Parkinson's disease. HiCommunication is a novel group intervention for speech and communication in Parkinson's disease based on principles driving neuroplasticity. In a randomized controlled trial, 95 participants with Parkinson's disease were allocated to HiCommunication or an active control intervention. Acoustic analysis was performed pre-, post- and six months after intervention. Intention-to-treat analyses with missing values imputed in linear multilevel models and complimentary per-protocol analyses were performed. The proportion of participants with a clinically relevant increase in the primary outcome measure of voice sound level was calculated. Resting-state functional MRI was performed pre- and post-intervention. Spectral dynamic causal modelling and the parametric empirical Bayes methods were applied to resting-state functional MRI data to describe effective connectivity changes in a speech-motor-related network of brain regions. From pre- to post-intervention, there were significant group-by-time interaction effects for the measures voice sound level in text reading (unstandardized b = 2.3, P = 0.003), voice sound level in monologue (unstandardized b = 2.1, P = 0.009), Acoustic Voice Quality Index (unstandardized b = -0.5, P = 0.016) and Harmonics-to-Noise Ratio (unstandardized b = 1.3, P = 0.014) post-intervention. For 59% of the participants, the increase in voice sound level after HiCommunication was clinically relevant. There were no sustained effects at the six-month follow-up. In the effective connectivity analysis, there was a significant decrease in inhibitory self-connectivity in the left supplementary motor area and increased connectivity from the right supplementary motor area to the left paracentral gyrus after HiCommunication compared to after the active control intervention. In conclusion, the HiCommunication intervention showed promising effects on voice sound level and voice quality in people with Parkinson's disease, motivating investigations of barriers and facilitators for implementation of the intervention in healthcare settings. Resting-state brain effective connectivity was altered following the intervention in areas implicated, possibly due to reorganization in brain networks.

Constrained Bayes in multiplicative area‐level models under the precautionary loss function

AbstractConsider the problem of benchmarking small‐area estimates under multiplicative models with positive parameters. The goal is to propose a loss function that guarantees positive constrained estimates of small‐area parameters in this situation. The weighted precautionary loss function is introduced to solve the problem. Compared with the weighted Kullback–Leibler (KL) loss function, our proposed loss function penalizes underestimation of the small‐area parameters of interest more for small values of parameters. This property is appealing when we estimate disease rates. It tends to give larger estimates of small‐area parameters compared with those obtained under the KL loss function. The hierarchical empirical Bayes and constrained hierarchical empirical Bayes estimates of small‐area parameters and their corresponding risk functions under the new proposed loss function are obtained. The performance of the proposed methods is investigated using simulation studies and a real dataset.

The empirical Bayes estimators of the rate parameter of the gamma distribution with a conjugate gamma prior under Stein’s loss function

For the hierarchical gamma and gamma model, we calculate the Bayes estimator of the rate parameter of the gamma distribution under Stein’s loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein’s Loss (PESL). We also obtain the Bayes estimator of the rate parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the rate parameter of the gamma distribution with a conjugate gamma prior by the moment and MLE methods. In numerical simulations, we have illustrated five aspects: the two inequalities of the Bayes estimators and the PESLs for the oracle method; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data; the comparisons of the Bayes estimators and the PESLs of the oracle, moment, and MLE methods; and the marginal densities of the model for various hyperparameters. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, the model could potentially be used to fit right-skewed data, not left-skewed data.

High dimensional discriminant rules with shrinkage estimators of the covariance matrix and mean vector

Linear discriminant analysis (LDA) is a typical method for classification problems with large dimensions and small samples. There are various types of LDA methods that are based on the different types of estimators for the covariance matrices and mean vectors. In this paper, we consider shrinkage methods based on a non-parametric approach. For the precision matrix, methods based on the sparsity structure or data splitting are examined. Regarding the estimation of mean vectors, Non-parametric Empirical Bayes (NPEB) methods and Non-parametric Maximum Likelihood Estimation (NPMLE) methods, also known as f-modeling and g-modeling, respectively, are adopted. The performance of linear discriminant rules based on combined estimation strategies of the covariance matrix and mean vectors are analyzed in this study. Particularly, the study presents a theoretical result on the performance of the NPEB method and compares it with previous studies. Simulation studies with various covariance matrices and mean vector structures are conducted to evaluate the methods discussed in this paper. Furthermore, real data examples such as gene expressions and EEG data are also presented.