Quantile Estimation Research Articles

SQSTS: A sequential procedure for estimating steady-state quantiles using standardized time series

ABSTRACT We develop and evaluate SQSTS, an automated sequential procedure for computing confidence intervals (CIs) for steady-state quantiles based on the simulation analysis methods of standardized time series (STS), batching, and sectioning. Using recent theoretical developments for STS-based quantile estimation in dependent sequences, we formulate the key steps in SQSTS for controlling the growth of the batch size on successive iterations of the procedure. The variance parameter associated with the full-sample quantile estimator is estimated by a combination of estimators that are asymptotically independent of each other and the full-sample quantile estimator with increasing batch size and a fixed number of batches. Extensive experimentation revealed that SQSTS performed well compared to its competitors in terms of estimated CI coverage probabilities; and it outperformed those competitors with regard to average sample-size requirements. Finally, we outline an extension of SQSTS for computing individual CIs for a set of selected quantiles.

Quantile adaptive lasso: variable selection for quantile treatment effect estimation

Quantile treatment effects (QTEs) characterize different causal effects of treatment on outcome distribution. Propensity score (PS) methods are commonly employed for estimating QTEs in non-randomized studies. Previous studies have shown that insufficient and unnecessary adjustment for covariates in PS models leads to bias and efficiency loss of treatment effects. Confounders may vary across different quantiles of outcome distribution. To allow for the selection of covariates related to outcome at interesting quantiles corresponding to QTE, we proposed quantile adaptive lasso (QAL) to select covariates that can provide unbiased and efficient QTE estimation. QAL utilized linear quantile regression models to construct penalty weights. Theoretical analyses showed that QAL enjoys several asymptotic properties. Simulations showed the superiority of QAL over existing methods in variable selection and QTE estimation. We applied QAL to datasets from the China Health and Retirement Longitudinal Study to explore the impact of smoking on the severity of depression.

A new reading of the relationship between financial development, trade openness, vulnerability and economic growth in Africa: New perspectives from method of moment’s quantile regression

One of the most frequently debated subjects in international forums is economic growth, which is regarded as a global priority. Consequently, researchers have turned their attention from conventional economic growth at a single average coefficient to divisible economic growth at levels of its value. Although the existing literature has discussed several determinants of economic growth, our article contributes to examining the sources of economic growth in African countries during the generations of reforms from 1990 to 2019 and in the context of economic vulnerability. The variables used in the analysis are gross domestic product, trade openness, financial development, and economic vulnerability. The study uses a quantile regression econometric model to examine these variables at different stages of reform. Quantile regression (QR) estimates for quantiles 0.05 to 0.95 showed mixed results: financial development is favorable to African economic growth at all quantile levels. However, economic vulnerability is a major impediment to economic growth at all quantile levels. In addition, it was found that a high degree of trade openness has a detrimental effect on African economic growth from quantile 0.5 of the dependent variable. Finally, another important result proves that financial development is a remedy for decision-makers against economic vulnerability.

The joint effect of changes in urbanization and climate on trends in floods: A comparison of panel and single-station quantile regression approaches

Estimates of annual maximum (peak) flow quantiles are needed for basins undergoing changes in both urbanization and climate. Most previous work on the effect of urbanization on peak flows has considered urbanization alone and only the spatial variation in flood quantiles or its mean temporal effect, and most work on the effect of nonstationarity in climate has focused on single-station analyses, which give uncertain results for extreme quantiles. To address these gaps, three approaches to the statistical estimation of the joint effects of changes in impervious cover and climate on the estimation of peak-flow quantiles were compared: single-station quantile regression; a fixed effect panel-quantile regression (pQR) method using a location (mean) shift to homogenize the panel; and a location-scale panel regression model (pQRmom), which accounts for both scale (variance) and location effects. The different approaches were applied to a dataset consisting of instantaneous annual peak flows from 127 minimally nested basins in the midwestern United States with at least 4 % change in imperviousness. The annual maximum daily discharge from a water-balance model was selected as the primary climate predictor; in addition, to provide a comparison of climate predictors, precipitation was also considered. The coefficients from single-station regressions were usually sufficiently certain to determine the effects of climate variation but usually too uncertain to estimate the effects of urbanization. The panel-quantile regression approaches give much more certain results, but their estimates of quantile dependence differ: although both indicate urbanization effects decreasing with decreasing annual exceedance probability (AEP), the pQRmom urbanization coefficients are insignificantly different from zero for AEPs less than 0.10, whereas the pQR coefficients remain positive and are significant except for AEP = 0.01, the smallest AEP value considered. Although the location-scale structure of the pQRmom approach has less flexible quantile dependence than the pQR approach, the pQRmom approach has somewhat lower overall error, and it is found that by subsetting the dataset to homogenize the scale effects, the pQR and pQRmom results become similar, indicating the insignificant urbanization coefficients for small AEPs of the pQRmom results are likely correct for the study dataset.

US Funds’ returns-based ESG extraction and implementation: a multifaceted quantile regression approach

ABSTRACT This study introduces a novel ESG intrinsic-based return factor and its application in asset pricing. This factor is extracted using a parallelized rolling window estimation and extreme value-weighted quantile portfolios. It carries a positive risk premium, indicating that investors are willing to assess its risk exposure. We further show that higher returns can be obtained in the top 30% quantiles using a long-only trading strategy. We apply a Monotone Composite Quantile Regression Neural Network (MCQRNN) model to explain US fund returns and address the needs of investors seeking to optimize their investment strategies. This model surpasses traditional benchmark models by performing deep quantile estimation and considering the nonlinear relationships between fund returns and six firm-based characteristics. This approach empowers investors by explaining the core principles of impact investing and highlighting how our constructed ESG risk factor can generate competitive returns even in volatile markets when its risk is well assessed. Highlights A sustainable smart beta factor construction is ensured ESG Long-only trading strategy grants higher returns in the top 30% quantiles Using ML, we further assessed the need to implement ESG factors into asset pricing Positive ESG risk premium reveals the investors’ interest in impact investing US market remains promising for positive ESG investment signals

Regional frequency analysis of extreme wind in Pakistan using robust estimation methods

The quantile estimates of extreme wind speed are needed for various areas of interest using regional frequency analysis (RFA) and extreme value theory. These calculations are crucial for the coding of wind speed. The data was taken from the NASA official website at a 10-meter distance and measured in meters per second (m/s). RFA of annual maximum wind speed (AMWS) using L-moments is performed utilizing annual maximum wind speed data from sixteen sites (16) in Pakistan’s Khyber Pakhtunkhwa province. There are no sites that are found to be discordant. The wards method is used to construct a homogenous region and make two homogenous regions from 16 sites. The heterogeneity test justifies that both clusters are homogeneous. The most appropriate probability distribution from the Generalized Normal (GNO), Generalized Logistic (GLO), Pearson Type-3 (P3), Generalized Pareto (GPA), and Generalized Extreme Value (GEV) distributions is chosen to calculate regional quantiles. According to the L-moments diagram and Z statistics, the GEV for Cluster- Ι and GLO for Cluster- ΙΙ are the best suggestions from the others. Both clusters’ robustness is measured utilizing relative bias (RB) and relative root mean square error (RRMSE). Overall, GEV distribution is fit for cluster-Ι, and the GLO distribution is fit for cluster-ΙΙ. Utilizing the site mean and median as index parameters, we can also find at-site quantiles from regional quantiles. The study’s quantile estimates can be employed in codified structural designs with policy consequences.

Modeling extreme events: Univariate and multivariate data-driven approaches

AbstractThis article summarizes the contribution of team genEVA to the EVA (2023) Conference Data Challenge. The challenge comprises four individual tasks, with two focused on univariate extremes and two related to multivariate extremes. In the first univariate assignment, we estimate a conditional extremal quantile using a quantile regression approach with neural networks. For the second, we develop a fine-tuning procedure for improved extremal quantile estimation with a given conservative loss function. In the first multivariate sub-challenge, we approximate the data-generating process with a copula model. In the remaining task, we use clustering to separate a high-dimensional problem into approximately independent components. Overall, competitive results were achieved for all challenges, and our approaches for the univariate tasks yielded the most accurate quantile estimates in the competition.

Inferring Surface NO2 Over Western Europe: A Machine Learning Approach With Uncertainty Quantification

AbstractNitrogen oxides (NOx = NO + NO2) are of great concern due to their impact on human health and the environment. In recent years, machine learning (ML) techniques have been widely used for surface NO2 estimation with rapid developments in computational power and big data. However, the uncertainties inherent to such retrievals are rarely studied. In this study, a novel ML framework has been developed, enhanced with uncertainty quantification techniques, to estimate surface NO2 and provide corresponding data‐induced uncertainty. We apply the Boosting Ensemble Conformal Quantile Estimator (BEnCQE) model to infer surface NO2 concentrations over Western Europe at the daily scale and 1 km spatial resolution from May 2018 to December 2021. High NO2 mainly appears in urban areas, industrial areas, and roads. The space‐based cross‐validation shows that our model achieves accurate point estimates (r = 0.8, R2 = 0.64, root mean square error = 8.08 μg/m3) and reliable prediction intervals (coverage probability, PI‐50%: 51.0%, PI‐90%: 90.5%). Also, the model result agrees with the Copernicus Atmosphere Monitoring Service (CAMS) model. The quantile regression in our model enables us to understand the importance of predictors for different NO2 level estimations. Additionally, the uncertainty information reveals the extra potential exceedance of the World Health Organization (WHO) 2021 limit in some locations, which is undetectable by only point estimates. Meanwhile, the uncertainty quantification allows assessment of the model's robustness outside existing in‐situ station measurements. It reveals challenges of NO2 estimation over urban and mountainous areas where NO2 is highly variable and heterogeneously distributed.

Role of environmental, social, and governance (ESG) investment and natural capital stocks in achieving net-zero carbon emission

This study investigates the role of environmental, social, and governance (ESG) investment and natural capital stocks in achieving net-zero carbon emissions, focusing on the Chinese economy. This study uses the data from 2013 to 2022 and applies correlation analysis, quantile process estimation, quantile-factor augmented vector autoregressive, quantile dimension regression, and robustness analysis techniques. The results confirmed that ESG investment has a significant role in net-zero carbon emission and fosters natural capital stocks in the Chinese ecosystem. The results further established that natural capital stocks significantly benefit the Chinese ecosystem and reduce greenhouse gas emissions, CO2 emission – gaseous fuel, CO2 emission – solid fuel, and carbon intensity. Moreover, natural capital stocks have a significant role in achieving net-zero carbon emissions and it accelerates gross tree cover and green agricultural yield. The study is of great interest to corporations, Chinese banks, and the government in planning and achieving net-zero carbon emissions as per Prudential Financial regulation and accounting reporting systems. Achieving net-zero carbon emission is a process-based approach; therefore, the study recommends that policymakers customize investment mechanisms and natural capital accounting standards as per the territorial requirements and execute an ecological control system to make it sustainable.

Multivariate copula-based conditional quantiles: analytic higher-order moments and ratio estimation approaches

ABSTRACT In this paper, we aim to modify multivariate copula-based conditional quantiles for a targeted random variable, given multiple random variables attaining their respective quantiles. Specifically, we propose two modifications through the Cornish–Fisher expansion, one of which involves its analytic higher-order unconditional moments. The second one accounts for its higher-order conditional moments estimated using a ratio estimation method. By considering multivariate elliptical and Archimedean copulas with Johnson's SU margins, our simulation study shows that this method provides more accurate conditional moment estimators compared to the naive method. They result in expanded conditional quantile estimators, whose efficiency is relatively better than their unexpanded versions, particularly at lower and higher quantile levels under a stronger (tail) dependence assumption. A much higher efficiency is gained when the first modification is performed. These findings are validated using an empirical study based on cryptocurrency return data and a comparative analysis against the existing estimation approaches.

The Bahadur representations of quantile estimators in general unequal probability sampling

In this study, we establish the Bahadur representations of quantile estimators based on the Hájek and the Horvitz–Thompson estimators for the population distribution function in unequal probability sampling designs under general conditions. We also present the asymptotic normality of the above quantile estimators. The simulation studies using real data confirm the validity of the asymptotic normality.

Modeling longitudinal skewed functional data.

This paper introduces a model for longitudinal functional data analysis that accounts for pointwise skewness. The proposed procedure decouples the marginal pointwise variation from the complex longitudinal and functional dependence using copula methodology. Pointwise variation is described through parametric distribution functions that capture varying skewness and change smoothly both in time and over the functional argument. Joint dependence is quantified through a Gaussian copula with a low-rank approximation-based covariance. The introduced class of models provides a unifying platform for both pointwise quantile estimation and prediction of complete trajectories at new times. We investigate the methods numerically in simulations and discuss their application to a diffusion tensor imaging study of multiple sclerosis patients. This approach is implemented in the R package sLFDA that is publicly available on GitHub.

A novel hybrid model based on evolving multi-quantile long and short-term memory neural network for ultra-short-term probabilistic forecasting of photovoltaic power

Probabilistic forecasting is extremely crucial in eliminating uncertainty in photovoltaic (PV) power generation. Quantile regression long and short-term memory neural network (QRLSTM) is widely recognized as promising methods for PV power probabilistic forecasting due to their strong generalization ability. However, these models train the model for each quantile individually, which lacks consideration of the correlation and monotonicity between quantiles, and multiple training leads to excessive computational complexity. Furthermore, the non-differentiable pinball loss function generated by QR places significant demands on the optimization algorithms. To address these issues, this paper proposes an evolutive distributed chaotic particle swarm optimization (EDCPSO)-optimized multi-quantile LSTM (MQLSTM) to achieve high-quality probabilistic PV power prediction. MQLSTM is a multi-output network structure that simultaneously outputs all quantile estimates and adopts a loss function with all quantile scores and non-crossing constraints to guide the training of the model. This approach not only improves the quality and reasonableness of quantile estimations, but also reduces computational difficulty. Then, from the perspective of evolutionary computation, considering the weight parameters of each connection layer in MQLSTM as decision variables, we convert the probabilistic forecasting into an optimization problem and propose a EDCPSO to solve the training difficulty. It implements a targeted distributed chaos strategy based on the evolutionary state to improve convergence speed and search capability. The proposed model is tested to be superior in real cases.

Practical non-stationary extreme value analysis of peaks over threshold using the generalised Pareto distribution: Estimating uncertainties in return values

Choice of tuning parameters influences the performance of non-stationary extreme value modelling for peaks over threshold using the generalised Pareto (GP) distribution. We examine the effect of tuning parameter choice on maximum roughness-penalised likelihood estimation of GP models, the shape and scale parameters of which are assumed to vary smoothly on a one-dimensional “directional” covariate domain, under a B-spline representation. We examine the effect of (a) extreme value model parameterisation, (b) relative roughness penalty of GP parameters as a function of covariate, (c) cross-validation strategy for roughness parameter tuning, and (d) estimator for return value, on the estimation of return values corresponding to return periods 1000× the period of a sample of size 1000. Bootstrap resampling is used for thorough uncertainty quantification. We also compare results with those from stationary inference.Results from a large simulation study of 16 cases broadly representative of North Sea conditions for significant wave height with direction, indicate that (i) multiple two-group cross-validation yields lower return value estimates that ten-group cross-validation (leading to negative bias on average, for the case studies considered), (ii) the quantile of the bootstrap predictive estimator yields larger values than the mean over bootstraps of the quantile estimate (leading to reduced omni-directional bias for the case studies considered). Further, (iii) the use of stationary models for non-stationary tails is only reasonable when a high extreme value threshold is set for the stationary analysis. However, (iv) the relative performance of different modelling strategies is sensitive to the specific characteristics of the case study.

Willingness to pay for sheep traits and their heterogeneous effects on prices: Evidence from primary livestock markets in Ethiopia

We test the hypothesis whether levels of key traits of sheep heterogeneously affect market prices of sheep in a rural setting. Feasible generalized least squares and (un)conditional quantile regression estimations were made on a dataset of 1153 sheep transactions in two primary small ruminant markets in the Amhara region of Ethiopia. The empirical results show that animal traits affect the observed prices of sheep differently, but only partly explain the sheep price differences. Our results also reveal that in addition to animal traits, the type of buyers and seasonality of sheep marketing cause heterogeneity in the observed prices. These findings imply that targeting the animal traits demanded by the market and access to price information that enables farmers to respond to the seasonal changes in livestock markets are essential to increase the income of sheep keepers.

Foreign Direct Investment and Military Expenditure in the Face of Terrorism: Evidence from Sub-Saharan Africa

This study examines the relationship between military expenditure and foreign direct investment (FDI) amid terrorism. The study used Mean Group, Pooled Mean Group, Dynamic Fixed Effect, and Panel Quantile estimations techniques on 23 terrorized countries in Sub-Sharan Africa from 1999 to 2019. The outcome of the estimate discovered that military expenditure in the absence of terrorism negates FDI flow. But, amid terrorism, military expenditure has a significant positive impact on FDI inflow. Also, the results show that the effect varies at different quantiles. The results are robust using death from terrorism attacks to measure terrorism on a different model. The study suggests that countries should desist from high military spending in the absence of terrorism to facilitate FDI flow. Countries in mid-life terrorism should enhance military spending and augment military strategy with other options to boost investors' confidence in the economy's high influx of FDI.

Composite quantile estimation in partially functional linear regression model with randomly censored responses

Composite quantile estimation in partially functional linear regression model with randomly censored responses

Principal quantile treatment effect estimation using principal scores.

Intercurrent events and estimands play a key role in defining the treatment effects of interest precisely. Sometimes the median or other quantiles of outcomes in a principal stratum according to potential occurrence of intercurrent events are of interest in randomized clinical trials. Naïve analyses such as those based on the observed occurrence of the intercurrent events lead to biased results. Therefore, we propose principal quantile treatment effect estimators that can nonparametrically estimate the distribution of potential outcomes by principal score weighting without relying on the exclusion restriction assumption. Our simulation studies show that the proposed method works in situations where the median or quantiles may be regarded as the preferred population-level summary over the mean. We illustrate our proposed method by using data from a randomized controlled trial conducted on patients with nonerosive reflux disease.

Quantile regressions as a tool to evaluate how an exposure shifts and reshapes the outcome distribution: A primer for epidemiologists.

Quantifying how an exposure affects the entire outcome distribution is often important, e.g., for outcomes such as blood pressure which have non-linear effects on long-term morbidity and mortality. Quantile regressions offer a powerful way of estimating an exposure's relationship with the outcome distribution but remain underused in epidemiology. We introduce quantile regressions with a focus on distinguishing estimators for quantiles of the conditional and unconditional outcome distributions. We also present an empirical example in which we fit mean and quantile regressions to investigate educational attainment's association with later-life systolic blood pressure (SBP). We use data on 8,875 US-born respondents aged 50+ years from the US Health and Retirement Study. More education was negatively associated with mean SBP. Conditional and unconditional quantile regressions both suggested a negative association between education and SBP at all levels of SBP, but the absolute magnitudes of these associations were higher at higher SBP quantiles relative to lower quantiles. In addition to showing that educational attainment shifted the SBP distribution left-wards, quantile regression results revealed that education may have reshaped the SBP distribution through larger protective associations in the right tail, thus benefiting those at highest risk of cardiovascular diseases.

A new Kolmogorov-Smirnov test based on representative points in the exponential distribution family

Hypothesis testing for the exponential distribution family consistently gains significant attentions from the statistical community, especially in situations characterized by small sample sizes. In this paper, we propose a novel approach for constructing a Kolmogorov-Smirnov type test based on representative points. This testing statistic is derived through a comparison between two empirical functions: one from the observed data and another from chosen representative points. A distribution-free quantile estimator is used for sample standardization. We also discuss how to select an optimal number of representative points based on the loss function, which is recommended for constructing the empirical function from the exponential distribution family. To evaluate the efficiency of this new test, a comprehensive Monte Carlo simulation is conducted to compare this new test with tailor-made tests. Simulation results show that our test statistic will be more powerful especially for small sample sizes, and it is competitive in the comparison with tailor-made tests. Some datasets are analyzed to further illustrate the efficiency of the proposed method.