Covariance Terms Research Articles

Multivariate test applied to single Pu isotopic composition measurement

The determination of the plutonium isotopic composition by gamma spectrometry is one of the tasks performed by nuclear inspectors. It is typically measured using planar germanium detectors. The spectra are analysed with deconvolution codes, which calculate all isotopic abundances. The experimental results are then compared with the declared values and the acceptance criterion is often based on one unique isotope, 240Pu, to distinguish between reactor and weapon grade Pu.If the measurement is a part of a measurement series, it is possible to calculate covariance terms and carry out a more comprehensive comparison using Hotelling's test as recently presented in Berndt and Mortreau (2023) [1]. In this study, it was shown that the T2 test often rejects simulated false cases which are accepted if only the 240Pu abundance is considered. Hence, the T2 test should be preferred to the 240Pu criterion.In case of a single measurement, which is typical for inspections, the co-variance cannot be determined and the comprehensive calculation of the T2 test as performed in Berndt and Mortreau (2023) [1] cannot be applied. The purpose of the present paper is to consider approximations of the variance-covariance matrix to allow the application of the T2 test to isolated measurements. The results were then applied to more than 400 single measurements of Pu samples covering many different experimental conditions.

Nonparametric estimation of the relative error regression for twice censored and dependent data

Let ( Y i ) 1 ≤ i ≤ n be a sequence of dependent random variables (r.v.) of interest distributed as Y and ( X i ) 1 ≤ i ≤ n be a corresponding d−dimensional vector of covariates taking values on ℝ d . The r.v. Y is twice censored and satisfies the α−mixing property. In this article, we are interested in the relative error regression estimation in the case of twice-censored data under strong mixing conditions. Under suitable assumptions, the estimator’s strong uniform almost sure consistency with rate and asymptotic normality are established. We have highlighted the covariance term, which is relatively uncommon in this context. Furthermore, we give a result about the α-mixing rate in the case of twice-censoring. As far as we know, there is no analogous result with proof in the context of twice-censoring. A simulation study is conducted in both one and two dimensions for the covariate to highlight the performance of our estimator.

MANOCCA: a robust and computationally efficient test of covariance in high-dimension multivariate omics data.

Multivariate analysis is becoming central in studies investigating high-throughput molecular data, yet, some important features of these data are seldom explored. Here, we present MANOCCA (Multivariate Analysis of Conditional CovAriance), a powerful method to test for the effect of a predictor on the covariance matrix of a multivariate outcome. The proposed test is by construction orthogonal to tests based on the mean and variance and is able to capture effects that are missed by both approaches. We first compare the performances of MANOCCA with existing correlation-based methods and show that MANOCCA is the only test correctly calibrated in simulation mimicking omics data. We then investigate the impact of reducing the dimensionality of the data using principal component analysis when the sample size is smaller than the number of pairwise covariance terms analysed. We show that, in many realistic scenarios, the maximum power can be achieved with a limited number of components. Finally, we apply MANOCCA to 1000 healthy individuals from the Milieu Interieur cohort, to assess the effect of health, lifestyle and genetic factors on the covariance of two sets of phenotypes, blood biomarkers and flow cytometry-based immune phenotypes. Our analyses identify significant associations between multiple factors and the covariance of both omics data.

Contribution of Surface Radiative Effects, Heat Fluxes and Their Interactions to Land Surface Temperature Variability

AbstractLand surface temperature anomalies can be linked to changes in local surface energy balance, although the relationship between surface temperature variability and individual radiative processes remains unclear. In this paper, we quantify the contributions of surface radiative effects and non‐radiative heat fluxes to the variance of monthly land surface temperature using European Centre for Medium‐Range Weather Forecasts Reanalysis v5 data and Coupled Model Intercomparison Project Phase 6 simulations. The surface energy budget equation is used to link changes in surface radiation, surface heat fluxes and land surface temperature. Subsequently, surface radiation is decomposed into the radiative effects of clouds, air temperature, surface albedo and relative humidity using radiative kernels. The contributions of these radiative processes, including their coupling effects, are quantified using covariance matrices. The results reveal the air temperature radiative effect to be the most significant contributor to the variability of land surface temperature. In addition, the covariance terms reveal important coupling effects. For example, the contribution from the cloud radiative effect is found to be substantially dampened by its coupling with surface heat fluxes. The air temperature radiative effect is further decomposed into a forcing component and a feedback component using different regression methods, in an attempt to separate the air temperature radiative effect as the driver of the surface temperature variability. The cloud radiative effect becomes the primary contributor to the variance of surface temperature after separating the air temperature feedback, while the contribution of the air temperature radiative forcing remains important.

History matching for geological carbon storage using data-space inversion with spatio-temporal data parameterization

History matching based on monitoring data will enable uncertainty reduction, and thus improved aquifer management, in industrial-scale carbon storage operations. In traditional model-based data assimilation, geomodel parameters are modified to force agreement between flow simulation results and observations. In data-space inversion (DSI), history-matched quantities of interest, e.g., posterior pressure and saturation fields conditioned to observations, are inferred directly, without constructing posterior geomodels. This is accomplished efficiently using a set of O(1000) prior simulation results, data parameterization, and posterior sampling within a Bayesian setting. In this study, we develop and implement (in DSI) a deep-learning-based parameterization to represent spatio-temporal pressure and CO2 saturation fields at a set of time steps. The new parameterization uses an adversarial autoencoder (AAE) for dimension reduction and a convolutional long short-term memory (convLSTM) network to represent the spatial distribution and temporal evolution of the pressure and saturation fields. This parameterization is used with an ensemble smoother with multiple data assimilation (ESMDA) in the DSI framework to enable posterior predictions. A realistic 3D system characterized by prior geological realizations drawn from a range of geological scenarios is considered. A local grid refinement procedure is introduced to estimate the error covariance term that appears in the history matching formulation. Extensive history matching results are presented for various quantities, for multiple synthetic true models, using the new DSI framework. Substantial uncertainty reduction in posterior pressure and saturation fields is achieved in all cases. The framework is also applied to efficiently provide posterior predictions for a range of error covariance specifications. Such an assessment would be very expensive using a traditional model-based approach.

Statistical Perspective on the Petrological Utility of Polyphase Groundmass Compositions Inferred via Defocused Beam Electron Probe Microanalysis

Polyphase groundmasses (micro‐scale minerals with or without glass) are generated from silicate liquids during the cooling of natural lavas often alongside larger minerals formed long before eruption. Many researchers have posited that compositions gleaned from the analysis of groundmasses closely approximate the compositions of the melts they were derived from, and these have been used frequently to model pre‐eruptive magma conditions. However, it is difficult to confidently identify and sample these groundmasses once they are formed. Using a sample of lava that exhibits a wide degree of textural variation (ranging from holocrystalline to hypohyaline) we show that compositions of groundmasses sampled using defocused electron beams are significantly different from glass compositions in terms of mean composition and covariance. Despite this, several groundmass compositions qualify as ‘in equilibrium’ with matrix/rim olivine. When processed using available thermometers, however, modelled equilibrium temperatures are significantly higher than those produced using glass data, on average. Because of this, we prescribe caution in using polyphase groundmass data generated using defocused beam analysis even as a rudimentary approach.

From a point to a range of optimum estimates for maize plant density and nitrogen rate recommendations

AbstractThe interaction between nitrogen (N) rate, plant density, and hybrid on net return to seed and fertilizer cost remains unknown but important to be addressed given the multi‐input decision process corn growers make every year. We collected grain yield from a factorial experiment with five N rates (0–291 kg N ha−1), five plant densities (3.7–11.4 plants m−2), and four hybrids (old vs. new) over 2 years in Iowa. Data were sufficiently modeled with a quadratic plateau model with continuous covariate terms to include N × plant density interactions (R2 = 0.925, n = 387). We found a wide range of plant densities (6.7–8.4 plants m−2) and N rate (150–212 kg N ha−1) combinations to achieve 99% of the maximum net revenue. The economic optimum N rate increased with increasing plant density from 4 to 7 plants m−2. The fundamental relationships between grain yield response to N fertilizer and plant density were similar across old and new hybrids; however, the absolute values were different, with newer hybrids having 20% higher economic optimum N rate and 23% higher yield than the older hybrids. We concluded that by accounting for two inputs into the revenue equation, a larger range of combinations was found to achieve 99% of the maximum net revenue. Our study offers new insights into N rate by plant density by hybrid interaction to assist future research.

Clustering blood donors via mixtures of product partition models with covariates.

Motivated by the problem of accurately predicting gap times between successive blood donations, we present here a general class of Bayesian nonparametric models for clustering. These models allow for the prediction of new recurrences, accommodating covariate information that describes the personal characteristics of the sample individuals. We introduce a prior for the random partition of the sample individuals, which encourages two individuals to be co-clustered if they have similar covariate values. Our prior generalizes product partition models with covariates (PPMx) models in the literature, which are defined in terms of cohesion and similarity functions. We assume cohesion functions that yield mixtures of PPMx models, while our similarity functions represent the denseness of a cluster. We show that including covariate information in the prior specification improves the posterior predictive performance and helps interpret the estimated clusters in terms of covariates in the blood donation application.

A nonparametric mixed exponentially weighted moving average-moving average control chart with an application to gas turbines.

This study aims to develop a nonparametric mixed exponentially weighted moving average-moving average (NPEWMA-MA) sign control chart for monitoring shifts in process location, particularly when the distribution of a critical quality characteristic is either unknown or non-normal. In literature, the variance expression of the mixed exponentially weighted moving average-moving average (EWMA-MA) statistic is calculated by allowing sequential moving averages to be independent, and thus the exclusion of covariance terms results in an inaccurate variance expression. Furthermore, the effectiveness of the EWMA-MA control chart deteriorates when the distribution of a critical quality characteristic deviates from normality. The proposed NPEWMA-MA sign control chart addresses these by utilizing the corrected variance of the EWMA-MA statistic and incorporating the nonparametric sign test into the EWMA-MA charting structure. The chart integrates the moving average (MA) statistic into the exponentially weighted moving average (EWMA) statistic. The EWMA-MA charting statistic assigns more weight to recent w samples, with weights for previous observations decling exponentially. Monte Carlo simulations assess the chart's performance using various run length (RL) characteristics such as average run length (ARL), standard deviation of run length (SDRL), and median run length (MRL). Additional measures for overall performance include the average extra quadratic loss (AEQL) and relative mean index (RMI). The proposed NPEWMA-MA sign control chart demonstrates superior performance compared to existing nonparametric control charts across different symmetrical and asymmetric distributions. It efficiently detects process shifts, as validated through both a simulated study and a real-life example from a combined cycle power plant.

Trajectories of self-kindness, common humanity, and mindfulness during the COVID-19 pandemic: A person-oriented multi-trajectory approach.

The COVID-19 pandemic has produced unprecedented changes in the lives of many people. Although research has documented associations between concerns related to COVID-19 and poor mental health indicators, fewer studies have focused on positive factors that could help people better cope with this stressful social context. To fill this gap, the present research investigated the trajectories of self-compassion facets in times of dramatic social change. Using a longitudinal research design, we described the trajectories of self-kindness, common humanity, and mindfulness during the first eight months of the COVID-19 pandemic, in a representative sample of Canadian adults (N = 3617). Relying on a multi-trajectory group-based approach, we identified clusters of individuals following persistently low (4.0%), moderate-low (39.3%), moderate-high (46.7%), and high (10.0%) levels of self-kindness, common humanity, and mindfulness. Interestingly, we found that compassionate self-responding trajectories were mainly stable over time with minor fluctuations for some groups of individuals, in line with the epidemiological situation. In terms of covariates, we observed that older women were more likely to follow trajectories of high compassionate self-responding, as compared to the other age and gender groups. In terms of mental health indicators, we demonstrated that trajectory groups with high levels of compassionate self-responding were associated with greater life satisfaction, more happiness, better sleep quality, higher sleep quantity, and fewer negative emotions, as compared to lower trajectory groups. The results supported the idea that self-compassion during the COVID-19 pandemic could have favored better mental health indicators and could possibly be promoted as a psychological intervention in the general population.

Performance-based inference of selection on stomatal length and specific leaf area varies with climate-of-origin of the forest tree, Eucalyptus ovata

Understanding how functional traits affect plant performance and fitness is a key step in unravelling the role of natural selection in shaping the evolutionary trajectory of populations. We examined early-age selection acting on leaf traits via their effects on growth performance and fitness, measured in Eucalyptus ovata trees planted in a common-garden field trial embedded in a reforestation planting in Tasmania, Australia. We focused on two important leaf traits - stomatal length and specific leaf area (SLA) - measured two years after planting, and compared interplanted E. ovata groups originating from dry and wet home-site climates, with the trial site having intermediate long-term mean annual rainfall. Two-year height growth was used as the performance attribute, and the time-averaged tree survival over the subsequent six years as the fitness component. There was evidence for performance-based selection on the leaf traits, with the strength and form of selection depending on the trait and climate group being considered. In this sense, selection in the dry group operated mainly on stomatal length where a combination of directional (favouring longer stomata) and stabilizing selection was detected, whereas selection in the wet group acted only on SLA and was purely stabilizing. Estimates of performance-based correlational selection were not statistically significant. For both climate groups, estimates of fitness-based selection gradients provided evidence for significant directional (but not quadratic) selection on height performance, favouring individuals with faster growth, but did not indicate statistical support for direct effects of the leaf traits on tree survival, conditional on measured performance. These results validated qualitative inferences of selection from the performance-based analysis, and suggested that selection on the leaf traits appeared to be mediated by their effects on early-age height performance, which in turn directly influenced later-age survival. We discuss the mechanisms by which the focal traits may have affected height performance, and likely factors contributing to the different patterns of phenotypic selection observed in the two groups experiencing the same environment. We also provide expressions of analytical derivatives that were developed for the estimation of selection gradients based on a logistic regression model relating a binary fitness response to linear and nonlinear covariate terms for the target regressor variables.

Propensity score-incorporated adaptive design approaches when incorporating real-world data.

The propensity score-integrated composite likelihood (PSCL) method is one method that can be utilized to design and analyze an application when real-world data (RWD) are leveraged to augment a prospectively designed clinical study. In the PSCL, strata are formed based on propensity scores (PS) such that similar subjects in terms of the baseline covariates from both the current study and RWD sources are placed in the same stratum, and then composite likelihood method is applied to down-weight the information from the RWD. While PSCL was originally proposed for a fixed design, it can be extended to be applied under an adaptive design framework with the purpose to either potentially claim an early success or to re-estimate the sample size. In this paper, a general strategy is proposed due to the feature of PSCL. For the possibility of claiming early success, Fisher's combination test is utilized. When the purpose is to re-estimate the sample size, the proposed procedure is based on the test proposed by Cui, Hung, and Wang. The implementation of these two procedures is demonstrated via an example.

The Hubble tension survey: A statistical analysis of the 2012–2022 measurements

ABSTRACT In order to investigate the potential Hubble tension, we compile a catalogue of 216 measurements of the Hubble–Lemaître constant H0 between 2012 and 2022, which includes 109 model-independent measurements and 107 ΛCDM model-based measurements. Statistical analyses of these measurements show that the deviations of the results with respect to the average H0 are far larger than expected from their error bars if they follow a Gaussian distribution. We find that xσ deviation is indeed equivalent in a Gaussian distribution to xeqσ deviation in the frequency of values, where xeq = 0.72x0.88. Hence, a tension of 5σ, estimated between the Cepheid-calibrated type Ia supernovae and cosmic microwave background (CMB) data, is indeed a 3σ tension in equivalent terms of a Gaussian distribution of frequencies. However, this recalibration should be independent of the data whose tension we want to test. If we adopt the previous analysis of data of 1976–2019, the equivalent tension is reduced to 2.25σ. Covariance terms due to correlations of measurements do not significantly change the results. None the less, the separation of the data into two blocks with H0 < 71 and H0 ≥ 71 km s−1 Mpc−1 finds compatibility with a Gaussian distribution for each of them without removing any outlier. These statistical results indicate that the underestimation of error bars for H0 remain prevalent over the past decade, dominated by systematic errors in the methodologies of CMB and local distance ladder analyses.

Virtual Patient Simulation Using Copula Modeling.

Virtual patient simulation is increasingly performed to support model-based optimization of clinical trial designs or individualized dosing strategies. Quantitative pharmacological models typically incorporate individual-level patient characteristics, or covariates, which enable the generation of virtual patient cohorts. The individual-level patient characteristics, or covariates, used as input for such simulations should accurately reflect the values seen in real patient populations. Current methods often make unrealistic assumptions about the correlation between patient's covariates or require direct access to actual data sets with individual-level patient data, which may often be limited by data sharing limitations. We propose and evaluate the use of copulas to address current shortcomings in simulation of patient-associated covariates for virtual patient simulations for model-based dose and trial optimization in clinical pharmacology. Copulas are multivariate distribution functions that can capture joint distributions, including the correlation, of covariate sets. We compare the performance of copulas to alternative simulation strategies, and we demonstrate their utility in several case studies. Our work demonstrates that copulas can reproduce realistic patient characteristics, both in terms of individual covariates and the dependence structure between different covariates, outperforming alternative methods, in particular when aiming to reproduce high-dimensional covariate sets. In conclusion, copulas represent a versatile and generalizable approach for virtual patient simulation which preserve relationships between covariates, and offer an open science strategy to facilitate re-use of patient data sets.

Limit theorems for multifractal products of random fields

This paper investigates asymptotic properties of multifractal products of random fields and multidimensional multifractal measures. The obtained limit theorems provide sufficient conditions for the convergence of cumulative fields in the spaces Lq. New results on the rate of convergence of cumulative fields are presented. Simple unified conditions for the limit theorems and the calculation of the Rényi function are given. They are less restrictive than those in the known one-dimensional results. The developed methodology is also applied to multidimensional multifractal measures. Finally, an application to a new class of examples of geometric φ-sub-Gaussian random fields is presented. In this case, the general assumptions have a simple form and can be expressed in terms of covariance functions only.

Optimal sampling using Conditioned Latin Hypercube for digital soil mapping: An approach using Bhattacharyya distance

Soil properties are important because they determine the soil’s suitability for different types of plant growth, ecosystems and biota functioning. Soil properties influence nutrient cycling, carbon sequestration and soil management. Digital Soil Mapping (DSM) is a procedure to map soil properties. Soil sampling for DSM is a foundational step in building prediction accuracy and essential for incorporating variability in terms of environmental covariates (ancillary variables). Conditioned Latin Hypercube (CLH) sampling is a method for generating a sample of points from a multivariate distribution that has been conditioned on one or more covariates. It is an extension of Latin Hypercube sampling, which is a popular technique for generating samples from a multivariate distribution in a way that ensures that each dimension is sampled uniformly. CLH sampling carries the benefit of selecting sampling locations covering the feature space and forming a Hypercube of the original sample. However, determining the optimum sample size is crucial in soil survey exercises constrained by budget and time limits. For this purpose, a study was carried out on Scotland's Finzean Estate (44.8 km2) location. A dataset of 21 independent features (16 continuous and five categorical) and 17,932 sampling locations was created using Digital Elevation Model (DEM) derivatives, soil classes map, land cover map, the peat depth map and parent material map to further generate sub-samples and compare the generated sub-samples with the original population. Two hundred CLH sampling datasets were extracted from the original population (17932 data points) with different sizes (5, 10, 15, 20, …, 100) and each size was given 10 repetitions e.g. (5_1, 5_2, …, 5_10). The sample datasets were analysed by comparing the mean, standard deviation, boxplot and estimates of the probability density function (pdf) for all the 16 continuous independent features. All the mentioned comparisons suggest that the impact of increasing sample size on the distribution of covariates can be observed up to a certain point, beyond which further increases in sample size may not yield noticeable differences. Bhattacharyya distance, a statistical measurement that quantifies the similarity between two probability distributions, was calculated between every quantitative and qualitative element of respective sampling size and for the original population. In contrast, as the CLH sample dataset size increased, the Bhattacharyya distance value decreased and became constant.. The optimum number of samples based on the study was determined for the spatial extent of the Finzean Estate in Scotland and a range of 25–50 CLH samples was suggested based on the study. This work, therefore, achieved both reductions in sampling location numbers compared to classical approaches and identification of the precise location of these sample location to achieve optimal DSM.

Decoding the enigma of 100-year record-breaking rainfall over Tamil Nadu using wavelet analysis

There have been increasing trends of short-intensive rainstorm events, which lead to urban flood disasters in many coastal areas of India. The relationship between regional rainfall and global large-scale climate indices has shown high teleconnections in terms of recurrence and covariance. We have examined the teleconnection between the November and December 2015 rainstorm events (100-year record breaking rainfall) with the Oceanic Nino Index using the dataset from the Indian Meteorological Organisation (NEMR) and National Oceanic and Atmospheric Administration (ONI). The study has found out not only the existence of the cause-and-effect relationship but also the recurrences of these high rainfall events. The climatological trend of northeast monsoon rainfall (NEMR) and Oceanic Nino Index (ONI) is calculated, and the correlation for a period of 71 years (1951–2021) is quantified using Pearson's correlation method. We observed that Chennai, Nagapattinam, Cuddalore, Kancheepuram, Thiruvarur, and adjoining coastal districts are more prone to torrential rains during El Niño years as compared to La-Niña years which is unusual but not exceptional. Wavelet analysis is performed on the NEMR and ONI signals to decipher the teleconnections at different temporal scales. Wavelet coherence shows an in-phase relationship between the ONI and NEMR signal and high covariance (0.8–0.9 at 5% significance level using Monte Carlo generated noise), between the two in 2015 the year which broke the 100 years rainfall record for Chennai. The study highlights the crucial linkage (cause and effect relationship) between the regional rainstorms and the large-scale global index (ONI), and wavelet analysis can explain the distant relationship in the northeast monsoon. The study provides quantified insights especially in the present era of climate change, where the hydrological extremes are persistent. The study does not only connect ONI with NEMR but also quantifies the in-phase strength of the relationship which can guide the scientific community towards accurate predictions and develop disaster mitigation strategies.

Nonstationary frequency analysis and uncertainty quantification for extreme low lake levels in a large river-lake-catchment system

Extreme hydrological events have become increasingly frequent on a global scale. The middle Yangtze River also faces a substantial challenge in dealing with extreme flooding and drought. However, the long-term characteristics of the extreme hydrological regime have not yet been adequately recognized. Moreover, there is uncertainty in the extreme value estimation, and this uncertainty needs to be distinguished and quantified. In this study, we investigated the nonstationary frequency characteristics of extreme low lake levels (ELLLs), taking the Poyang Lake as an example. Daily lake levels from 1960 to 2022 were utilized to estimate the return level using the generalized Pareto distribution (GPD). The uncertainty from three sources, i.e., the parameter estimator, threshold selection, and covariate, was quantified via variance decomposition. The results indicate that (1) the parameter estimator is the predominant source of uncertainty, with a contribution rate of approximately 87 %. The total uncertainty of the covariate, threshold, and interaction term is only 13 %. (2) Two indexes, namely the annual minimum water level (WLmin) and the days with peak over the 90 % threshold per year (DPOT90), decreased (0.01–0.03 m/year) and increased (0.17–1.39 days/year), respectively, indicating a progressively severe drought trend for Poyang Lake. (3) The return level with return period of 5 to 100 years significantly decreased after the early 21st century. A large spatial heterogeneity was identified for the variation in the return level, and the change rate of the return level with a 100-year return period ranged from 5 % to 40 % for the whole lake. (4) The ELLLs had a stronger correlation with the catchment discharge than with the Yangtze River discharge and the large-scale atmospheric circulation indices. This study provides a methodology with reduced uncertainty for nonstationary frequency analysis (NFA) of ELLLs exemplified in large river–lake systems.

Approximate Gaussian variance inference for state‐space models

SummaryState‐space models require an accurate knowledge of the process error () and measurement error () covariance matrices for exact state estimation. Even though the matrix can be, in many situations, considered to be known from the measuring instrument specifications, it is still a challenge to infer the matrix online while providing reliable estimates along with a low computational cost. In this article, we propose an analytically tractable online Bayesian inference method for inferring the matrix in state‐space models. We refer to this method as approximate Gaussian variance inference (AGVI) using which we are able to treat the error variance and covariance terms in the full matrix as Gaussian hidden states and infer them simultaneously with the other hidden states in a closed‐form manner. The two case studies show that the method is able to provide statistically consistent estimates for the mean and uncertainties of the error variance terms for univariate and multivariate cases. The method also exceeds the performance of the existing adaptive Kalman filter methods both in terms of accuracy and computational efficiency.

Modeling clustered binary data with nonparametric unobserved heterogeneity: An application to stock crash analysis

AbstractVarious random effects models have been developed for clustered binary data; however, traditional approaches to these models generally rely heavily on the specification of a continuous random effect distribution such as Gaussian or beta distribution. In this article, we introduce a new model that incorporates nonparametric unobserved random effects on unit interval (0,1) into logistic regression multiplicatively with fixed effects. This new multiplicative model setup facilitates prediction of our nonparametric random effects and corresponding model interpretations. A distinctive feature of our approach is that a closed‐form expression has been derived for the predictor of nonparametric random effects on unit interval (0,1) in terms of known covariates and responses. A quasi‐likelihood approach has been developed in the estimation of our model. Our results are robust against random effects distributions from very discrete binary to continuous beta distributions. We illustrate our method by analyzing recent large stock crash data in China. The performance of our method is also evaluated through simulation studies.