Distribution-free Approach Research Articles

Tail variance and confidence of using tail conditional expectation: analytical representation, capital adequacy, and asymptotics

Abstract In this paper, we explore the applications of Tail Variance (TV) as a measure of tail riskiness and the confidence level of using Tail Conditional Expectation (TCE)-based risk capital. While TCE measures the expected loss of a risk that exceeds a certain threshold, TV measures the variability of risk along its tails. We first derive analytical formulas of TV and TCE for a large variety of probability distributions. These formulas are useful instruments for relevant research works on tail risk measures. We then propose a distribution-free approach utilizing TV to estimate the lower bounds of the confidence level of using TCE-based risk capital. In doing so, we introduce sharpened conditional probability inequalities, which halve the bounds of conventional Markov and Cantelli inequalities. Such an approach is easy to implement. We further investigate the characterization of tail risks by TV through an exploration of TV’s asymptotics. A distribution-free limit formula is derived for the asymptotics of TV. To further investigate the asymptotic properties, we consider two broad distribution families defined on tails, namely, the polynomial-tailed distributions and the exponential-tailed distributions. The two distribution families are found to exhibit an asymptotic equivalence between TV and the reciprocal square of the hazard rate. We also establish asymptotic relationships between TCE and VaR for the two families. Our asymptotic analysis contributes to the existing research by unifying the asymptotic expressions and the convergence rate of TV for Student-t distributions, exponential distributions, and normal distributions, which complements the discussion on the convergence rate of univariate cases in [28]. To show the usefulness of our results, we present two case studies based on real data from the industry. We first show how to use conditional inequalities to assess the confidence of using TCE-based risk capital for different types of insurance businesses. Then, for financial data, we provide alternative evidence for the relationship between the data frequency and the tail categorization by the asymptotics of TV.

Two-stage stochastic optimization of virtual power plant with wind power and uncertain demand

ABSTRACT The virtual power plant (VPP) has been recognized as an effective way to facilitate penetration of renewable and distributed energy resources in electricity markets. This paper introduces an adaptive curtailment strategy for a VPP comprising a wind power plant and uncertain demand, and explores the economic advantage of adaptively adjusting wind power curtailment. A two-stage stochastic model is proposed to deal with the uncertainties in wind power generation (WPG), load demand and market prices. In the model, the bid decision is made in the face of uncertainties in the first stage, while the control (curtailment) decision is made based on realized uncertain parameters in the second stage. This paper provides the closed-form optimal curtailment decision and characterizes the optimal bid decision. An efficient binary search algorithm is developed for optimizing the bid decision. By using a distribution-free approach, we show that as the prediction accuracy of WPG improves, the optimal bid decision converges toward the expected minimal power exchange, leading to a decrease in expected operational cost with diminishing marginal return. Numerical experiments based on real-world data demonstrate that compared with the existing greedy strategy and coordinated strategy, the proposed model can decrease the expected operational cost up to 16.9% and 11.0%, respectively.

Investigating a closed-loop dual-channel green supply chain with stochastic demand rate and collection ratio using the distribution-free approach

This paper investigates a dual-channel closed-loop supply chain in an uncertain market environment consisting of a manufacturer, a retailer, and a collector, where the manufacturer can produce green products by utilizing recycling components from the collector and raw materials from an outside source. According to our analysis, market demand is stochastic and strongly influenced by channels’ selling prices, the level of green innovation, and retail service. Additionally, the collection rate of used-obsolete products is also random and affected by green innovation levels and refund prices. We analyze the model by dividing it into three scenarios based on players’ collaborative and non-collaborative behavior and then solve the problem in centralized and decentralized cases through a distribution-free approach. Using vertical Nash and Stackelberg game approaches, the interaction between players is modeled in the decentralized scenario. The model’s findings are illustrated with a numerical example, and a sensitivity analysis is also performed to provide insights into management by determining how different parameters will impact profits. Results suggest that the higher expected value of demand randomness is linked to increased market scale but that random variables with higher variance levels are more uncertain, leading to lower channel profitability. Moreover, a collaborative effort between manufacturer and collector could improve both prices and greenness, ultimately boosting the fraction collected. In addition, results reveal that the retailer Stackelberg frameworks will most adversely affect players, whereas the manufacturer Stackelberg frameworks will generate profits almost every time.

Optimizing production-inventory replenishment and lead time decisions under a fill rate constraint in a two-echelon sustainable supply chain with quality issues

This paper considers a single vendor-single buyer coordinated supply chain under stochastic demand. GHG emissions arise from the vendor's production process. A carbon tax and penalties for exceeding multiple emissions limits are considered. A random number of defectives characterises the batch received by the buyer, who performs an inspection activity. Setup and transportation times are controllable through a capital investment, whose outcome is uncertain. An optimisation problem is formulated under the minimax distribution-free approach. The objective is to find the production-inventory replenishment policy and the length of setup and transportation times that minimise the long-run expected total system cost rate considering a fill rate constraint. Several properties of the cost function are demonstrated, and an optimisation method is developed. Finally, numerical experiments are carried out to draw insights on the effect of the model parameters, including different legislation schemes concerning GHG emissions, on the optimal system performance. The novelty of the paper consists into investigating the effect of a controllable lead time on coordinated supply chain decisions in presence of GHG emissions and quality issues, in a stochastic environment. A major finding is that controlling lead time enhances both the overall cost efficiency and the environmental sustainability of the system.

Promoting Education Quality in Curbing Business Crime Costs: A Quantile Analysis

Studies have concentrated on numerous strategies and measures to curb the crime rate. Out of these, interventions via education are most promising as teaching children has a multiplier effect on crime reduction. Empirically there is a debate on the effect of education on the incidence of crime, and it is reasoned that education with opportunities, relevance, and ethics plays a decisive role in reducing crime. This study has used more comprehensive indicators and a robust distribution-free approach to estimate the effect of education on the crime rate. The contour plot of Quantile on Quantile regression helps assess each level of education quality on each level of the business cost of crime. This model does not require the assumption of normality and linearity of variables. This study points out that primary and science education reduces crime rates at low levels, while management studies and research help reduce crime at high levels. This study provides a robust assessment of how education may influence the cost of crime at their different corresponding levels. These estimates can help determine how the marginal effects will affect and change in response to the change in the cost of crime. This study is an instrument in for assessing the role of education on crime at each quantile data position.

Process Flexibility: A Distribution-Free Approach to Long Chain Resilience

Process Flexibility: A Distribution-Free Approach to Long Chain Resilience

Distribution-Free Approaches for an Integrated Cargo Routing and Empty Container Repositioning Problem with Repacking Operations in Liner Shipping Networks

Cargo (laden container) routing and empty container repositioning are crucial components in liner shipping, which are closely relevant. However, most existing works focus on cargo routing and empty container repositioning separately, which cannot optimize the global shipping cost. In addition, (1) appropriate repacking operations for laden containers can curtail the total handling costs, and (2) the perfect information for uncertain laden and empty container demands may be difficult to obtain due to insufficient historical data. This work investigates an integrated cargo routing and empty container repositioning problem with repacking operations in which only partial information about uncertain laden and empty container demands is known. The objective is to minimize the total costs including repacking operation cost and laden and empty container transportation costs. For the problem, a new chance-constrained programming model based on moment-based ambiguous sets is formulated. Then, four distribution-free solution approaches are adopted to solve the investigated problem. Numerical experiments are conducted to compare the proposed methods.

A three-echelon green supply chain management for biodegradable products with three transportation modes

Currently, many industries are trying to save the environment by following government rules to reduce global warming issues. In this study, green supply chain management with less pollution for biodegradable products is developed and an outsourcing strategy is included only to control the quality of the finished bio-products which would be delivered to the multi-retailer. Three different transportation modes are proposed for products to identify the mode that causes the least carbon emissions through a single setup but an unequal number of shipments and lots. These products have specific properties that can be recycled after use and do not contain harmful chemicals that can negatively impact the environment. The demand pattern of this product is service level dependent; however, the demand during lead time duration is followed a normal distribution. The proposed model considers a distribution free approach with known mean and standard deviation. The model is analytically solved to obtain a quasi-closed-form optimum solution. A numerical example is illustrated to test the performance of the proposed model. Some special cases are considered in this study. In addition, several managerial insights are provided. A sensitivity analysis is conducted to help the industry move toward this fruitful strategy in the future. The numerical study proves that the proposed method is superior to existing research models in different aspects for the pollution reduction strategies. The transportation by road provides ten percentage reduced costs comparing with water and air mode transportation. For both cases, the service level is much better in the transportation by road than other two modes.

Bayesian discrete conditional transformation models

We propose a novel Bayesian model framework for discrete ordinal and count data based on conditional transformations of the responses. The conditional transformation function is estimated from the data in conjunction with an a priori chosen reference distribution. For count responses, the resulting transformation model is novel in the sense that it is a Bayesian fully parametric yet distribution-free approach that can additionally account for excess zeros with additive transformation function specifications. For ordinal categoric responses, our cumulative link transformation model allows the inclusion of linear and non-linear covariate effects that can additionally be made category-specific, resulting in (non-)proportional odds or hazards models and more, depending on the choice of the reference distribution. Inference is conducted by a generic modular Markov chain Monte Carlo algorithm where multivariate Gaussian priors enforce specific properties such as smoothness on the functional effects. To illustrate the versatility of Bayesian discrete conditional transformation models, applications to counts of patent citations in the presence of excess zeros and on treating forest health categories in a discrete partial proportional odds model are presented.

Distribution-Free Approach to the Design and Analysis of Randomized Stroke Trials With the Modified Rankin Scale.

Many methods have been suggested for analyzing the modified Rankin Scale (mRS). However, there lacks a unified approach to analysis and sample size determination that properly uses the ordinal nature of the data. We propose a simple method for CI estimation and corresponding sample size determination. We quantify treatment effect by the win probability (WinP) that a randomly selected patient in the treatment group has an equal or a better mRS score than a patient in the control group. Thus, a win probability of 0.5 means no effect, likened to a draw in competitive sports. We estimate the win probability and its SE based on the ranks of mRS scores, where tied scores are handled by average ranks. Corresponding methods for hypothesis testing, CI estimation, and sample size determination are derived. The methods are evaluated with a simulation study based on real data from 10 randomized stroke trials that used mRS as the outcome measure. Simulation results demonstrated that the methods performed very well in terms of CI coverage, tail errors, and assurance to achieving the prespecified precision. Because the methods are very simple, we implemented them in an Excel spreadsheet, requiring only user inputs on frequencies of mRS scores in 2 comparison groups. Sound statistical methods are important for the success of randomized stroke trials. The proposed methods and associated spreadsheet should prove useful for stroke researchers in the planning and analysis of randomized trials. Meta-analysis has also been made easy for trials with ordinal scores.

Transoral Laser Microsurgery in Recurrent Laryngeal Cancer: A Systematic Review and Meta-analysis.

To determine the oncological outcomes of salvage transoral laser microsurgery (TLM) in the treatment of patients suffering from recurrent laryngeal cancer. PubMed/MEDLINE, Cochrane Library, and Scopus databases were searched. English language, original studies investigating oncological outcomes of TLM in adult patients with recurrent laryngeal cancer were included. Data were pooled using a distribution-free approach for estimating summary local control (LC), disease-specific survival (DSS), and overall survival (OS) curves with random effects. Two hundred and thirty-five patients underwent salvage TLM after primary (chemo)radiotherapy. The mean follow-up time was 60.8 months (95% CI: 32.7-88.9). Estimated pooled LC rates (95% CI) at 1, 3 and 5 years were 74.2% (61.7-89.4), 53.9% (38.5-75.3), and 39.1% (25.2-60.8). Estimated pooled DSS rates (95% CI) at 1, 3 and 5 years were 88.4% (82.0-95.3), 67.8% (50.9-90.3), and 58.9% (42.7-81.1). Two hundred and seventy-one patients underwent TLM after primary laser treatment. The mean follow-up time was 70.9 months (95% CI: 36.9-104.9). Estimated pooled LC rates (95% CI) at 1, 3 and 5 years were 72.2% (64.7-80.6), 53.2% (42.2-66.9), and 40.4% (29.6-55.2). Estimated pooled DSS rates (95% CI) at 1, 3 and 5 years were 92.1% (85.5-99.1), 77.0% (64.4-92.0), and 67.1% (51.6-87.3). TLM is a valuable treatment option for the management of locally recurrent laryngeal carcinoma if performed by experienced surgeons and following rigorous patients' selection criteria. Further studies should be conducted to define stage-based clinical guidelines. NA Laryngoscope, 133:1425-1433, 2023.

A Distribution-Free Model for Longitudinal Metagenomic Count Data.

Longitudinal metagenomics has been widely studied in the recent decade to provide valuable insight for understanding microbial dynamics. The correlation within each subject can be observed across repeated measurements. However, previous methods that assume independent correlation may suffer from incorrect inferences. In addition, methods that do account for intra-sample correlation may not be applicable for count data. We proposed a distribution-free approach, namely CorrZIDF, which extends the current method to model correlated zero-inflated metagenomic count data, offering a powerful and accurate solution for detecting significance features. This method can handle different working correlation structures without specifying each margin distribution of the count data. Through simulation studies, we have shown the robustness of CorrZIDF when selecting a working correlation structure for repeated measures studies to enhance the efficiency of estimation. We also compared four methods using two real datasets, and the new proposed method identified more unique features that were reported previously on the relevant research.

Null-free False Discovery Rate Control Using Decoy Permutations.

The traditional approaches to false discovery rate (FDR) control in multiple hypothesis testing are usually based on the null distribution of a test statistic. However, all types of null distributions, including the theoretical, permutation-based and empirical ones, have some inherent drawbacks. For example, the theoretical null might fail because of improper assumptions on the sample distribution. Here, we propose a null distribution-free approach to FDR control for multiple hypothesis testing in the case-control study. This approach, named target-decoy procedure, simply builds on the ordering of tests by some statistic or score, the null distribution of which is not required to be known. Competitive decoy tests are constructed from permutations of original samples and are used to estimate the false target discoveries. We prove that this approach controls the FDR when the score function is symmetric and the scores are independent between different tests. Simulation demonstrates that it is more stable and powerful than two popular traditional approaches, even in the existence of dependency. Evaluation is also made on two real datasets, including an arabidopsis genomics dataset and a COVID-19 proteomics dataset.

An extended CAViaR model for early-warning of exceedances of the air pollution standards. The case of PM10 in the city of Madrid

The last decade has witnessed significant advances in air pollution control. However, the European Union standard for PM10 is still exceeded in large cities. Most of the literature on anticipating PM10 exceedances of the standard is based on PM10 forecasting. However, it fails when it comes to forecasting extreme values. This is the reason why we focus on a Value-at-Risk-based approach imported from finance: The Conditional Autoregressive Value-at-Risk (CAViaR) modelling, which we extend with a meteorological indicator. CAViaR has a core advantage with respect to traditional VaR-based methods used in finance: it is a direct distribution-free approach, naturally linked to the concept of VaR, thus overcoming the limitations of the typical indirect approaches used in financial literature. This is the first article using a CAViaR-based strategy in air pollution control. This methodology has been applied in the city of Madrid to forecast the risk of short-term PM10 exceedances of the standard. Backtesting reveals that the extended CAViaR outperforms the standard CAViaR model and other typical specifications in the financial literature on the topic, including the EWMA model proposed by RiskMetrics, the Gaussian GARCH(1,1) specification, and a hybrid strategy combining GARCH-type forecasting with the block-maxima approach of the Extreme Value Theory. Specifically, the extended CAViaR exhibits the highest p-values in the back-tests used. Our proposal provides citizens and authorities with a set of PM10 VaR-values, together with their respective probabilities which is key information to decide whether or not to take any remedial measure and, if so, what measures to take.

Simplifying Bias Correction for Selective Sampling: A Unified Distribution-Free Approach to Handling Endogenously Selected Samples

Exploiting informative endogenously selected samples while minimizing sample selection bias with minimum modeling assumptions and analytical burden.

A solution to the transportation hazard problem in a supply chain with an unreliable manufacturer

<abstract><p>The current study focuses on a two-echelon supply chain for a reliable retailer, an unreliable manufacturer, and selling price-dependent demand. Due to an unreliable manufacturer and transportation hazards, shortages arise, which negatively impact the reputation of the retailer. Moreover, customers are more conscious of the environment, as a result, most of the industry focuses on the production of green products. To reduce the holding cost of the retailer, a fuel consumption-based single-setup-multi-unequal-increasing-delivery policy was utilized in this current study. With this transportation policy, the number of shipments increases, which directly increases carbon emissions and transportation hazards. To protect the environment, the green level of the product is enhanced through some investments. The demand varies with the price of the product as well as with the level of the greenness of the product. Due to uncertain demand, the rate of the production is treated as controllable. A classical optimization technique and distribution-free approach have been utilized to obtain the optimum solution and the optimized system profit. To prove the applicability, the study is illustrated numerically and graphically via a well-explained analysis of sensitivity. The study proves that single-setup-multi-unequal-increasing delivery policy is $ 0.62 \% $ beneficial compared to single-setup-single-delivery policy and $ 0.35 \% $ better than the single-setup-multi-delivery policy.</p></abstract>

Semi-supervised multiple testing

An important limitation of standard multiple testing procedures is that the null distribution should be known. Here, we consider a null distribution-free approach for multiple testing in the following semi-supervised setting: the user does not know the null distribution, but has at hand a sample drawn from this null distribution. In practical situations, this null training sample (NTS) can come from previous experiments, from a part of the data under test, from specific simulations, or from a sampling process. In this work, we present theoretical results that handle such a framework, with a focus on the false discovery rate (FDR) control and the Benjamini-Hochberg (BH) procedure. First, we provide upper and lower bounds for the FDR of the BH procedure based on empirical p-values, called here the semi-supervised BH procedure. These bounds match when α(n+1)∕m is an integer, where n is the NTS sample size and m is the number of tests. Second, we give a power analysis for that procedure suggesting that it mimics an oracle power when n is sufficiently large in front of m; namely n≳m∕(max(1,k)), where k denotes the number of “detectable” alternatives. Third, to complete the picture, we also present a negative result that evidences an intrinsic transition phase to the general semi-supervised multiple testing problem and shows that the semi-supervised BH method is optimal in the sense that its performance boundary follows this transition phase. Our theoretical properties are supported by numerical experiments, which also show that the delineated boundary is of correct order without further tuning any constant. Finally, we demonstrate that our work provides a theoretical ground for standard practice in astronomical data analysis, and in particular for the procedure proposed in Mary et al. (2020) for galaxy detection.

Simulation experiments comparing nonstationary design-flood adjustments based on observed annual peak flows in the conterminous United States

While nonstationary flood frequency analysis (NSFFA) methods have proliferated, few studies have rigorously compared them for modeling changes in both the central tendency and variability of annual peak-flow series, also known as the annual maximum series (AMS), in hydrologically diverse areas. Through Monte Carlo experiments, we appraise five methods for updating estimates of 10- and 100-year floods at gauged sites using synthetic records based on sample moments and change trajectories of observed AMS in the conterminous United States (CONUS). We compare two methods that consider changes in both central tendency and variability - a Gamma generalized linear model estimated with weighted least squares and the Generalized Additive Model for Location, Scale, Shape (GAMLSS) - with a distribution-free approach (quantile regression), and baseline cases assuming stationarity or only changes in central tendency.‘Trend-space’ plots identify realistic AMS changes for which modeling trends in both central tendency and variability were warranted based on fractional root mean squared errors (fRMSE). They also reveal statistical properties of AMS under which NSFFA models perform especially well or poorly. For instance, quantile regression performed especially well (poorly) under strong negative (positive) skewness. Although the nonstationary LP3 distribution accommodates most AMS with trends well, the sensitivity of NSFFA model performance to different sample moments and trends suggests the need for more flexibility in prescribing design-flood adjustments in the CONUS. A follow-up comparison of regional NSFFA models pooling at-site AMS would further illuminate NSFFA guidance, especially for AMS with properties less conducive to NSFFA modeling, such as positive skewness and increasing variability.

Joint pricing and ordering decisions for a loss-averse retailer with quantity-oriented reference point effect and demand uncertainty: a distribution-free approach

PurposeThis work examines the joint pricing and ordering (JPO) decisions for a loss-averse retailer with quantity-oriented reference point (RP) effect under demand uncertainty.Design/methodology/approachThe demand is assumed to be uncertain with the mean and variance as the only known information. The prospect theory is used to model the retailer's expected utility. An expected utility maximization model in the distribution-free approach (DFA) is then developed. Using duality theory, the expected utility under the worst-case distribution is transformed into tractable piece-wise functions. To examine the effectiveness of the DFA in coping with the demand uncertainty, a stochastic programming model is developed and its solutions are used as benchmarks.FindingsThe proposed model and solution approach can effectively hedge against the demand uncertainty. The JPO decisions are significantly influenced by the LA coefficient and the reference level. The LA has a stronger influence than the reference level does on the expected utility. An excessive LA is detrimental while an appropriate reference level is beneficial to the retailer.Practical implicationsThe results of this work are applicable to loss-averse retailers with the quantity-oriented RP when making JPO decisions with difficulty in predicting the demands.Originality/valueThe demand is assumed to be uncertain in this work, but a certain demand distribution is usually assumed in the existing literature. The DFA is used to study JPO decisions for the loss-averse retailer with quantity-oriented RP effect under the uncertain demand.

Environmental and economic sustainability through innovative green products by remanufacturing

The rapid advancements in technology demand the massive development of innovative green products to save global warming issues. Older non-green products are quickly outdated with the expansion of technology. Consequently, their market demand declines sharply. It leads to shorter life cycles, causing a remarkably uncertain environment and high waste generation. This problem is more in innovative green products compared to primary non-green products, primarily characterized by swift technological advancement and low acceptance rate among consumers. This study’s focal point is to simultaneously consider the environmental risks associated with innovative green product manufacturing, while both the return rate and the demand are random with insufficient information. Three models are developed based on different production strategies for two different types of innovative green products. Moreover, a distribution-free approach is used rather than assuming any specific distribution for the random variables (demand and return rate). Models dealing with manufacturing strategies of innovative green products are rare in the literature; furthermore, a novel concept of green innovation by remanufacturing is used to improve waste management. The findings show that highly green innovative products perform better compared to less innovative products when uncertainties in demand and supply are high. Furthermore, new green products should be introduced only when the expected profit of new products is more than the loss of the existing products. The new policy innovation by remanufacturing is cost-effective compared to the traditional innovation policy. All three models are solved with numerical examples. Sensitivity analysis is performed to illustrate the robustness of models.