Non‐parametric Quantile Regression with Censored Data

Distribution-free prediction regions of multivariate response PLS models with applications to NIR datasets

The theory of large deviations is applied to the study of the asymptotic properties of the stochastic approximation algorithms (1.1) and (1.2). The method provides a useful alternative to the currently used technique of obtaining rate of convergence results by studying the sequence {(Xn-.)/θan} (for (1.1)), where θ is a 'stable' point of the algorithm. Let G be a bounded neighborhood of θ, which is in the domain of attraction of θ for the 'limit ODE'. The process xn(θ) is defined as a 'natural interpolation' of {Xj,j≥n} with xn(0) = Xn, and interpolation intervals {aj,j≥n}. Define τG n = min{t:xn(t)τG}. Then it is shown (among other things) that Px{τG n ≥ T} ~ exp-nqV, where q depends on {an,cn}, and V depends on the b(.) cov τn, and G. Such estimates imply that the asymptotic behavior is much better than suggested by the 'local linearization methods', and they yield much new insight into the asymptotic behavior. The technique is applicable to related problems in the asymptotic analysis of recursive algorithms, and requires weaker conditions on the dynamics than do the 'linearization methods'. The necessary basic background is provided and the optimal control problems associated with getting the V above are derived.

There has been an intense recent activity in embedding of very high-dimensional and nonlinear data structures, much of it in the data science and machine learning literature. We survey this activity in four parts. In the first part, we cover nonlinear methods such as principal curves, multidimensional scaling, local linear methods, ISOMAP, graph-based methods and diffusion mapping, kernel based methods and random projections. The second part is concerned with topological embedding methods, in particular mapping topological properties into persistence diagrams and the Mapper algorithm. Another type of data sets with a tremendous growth is very high-dimensional network data. The task considered in part three is how to embed such data in a vector space of moderate dimension to make the data amenable to traditional techniques such as cluster and classification techniques. Arguably, this is the part where the contrast between algorithmic machine learning methods and statistical modeling, represented by the so-called stochastic block model, is at its greatest. In the paper, we discuss the pros and cons for the two approaches. The final part of the survey deals with embedding in R2, that is, visualization. Three methods are presented: t-SNE, UMAP and LargeVis based on methods in parts one, two and three, respectively. The methods are illustrated and compared on two simulated data sets; one consisting of a triplet of noisy Ranunculoid curves, and one consisting of networks of increasing complexity generated with stochastic block models and with two types of nodes.

As a maritime country strategically located along the world's leading transportation routes, Indonesia often faces increased ship accidents. Based on the Basarnas Statistics Book, ship accidents handled by Basarnas from 2021 to 2023 increased by 3%. This condition requires an effective forecasting method to carry out SAR operations to predict ship accidents in the Indonesian region in the future and assess the readiness and needs of Basarnas resources. This study compares the forecasting results obtained using the Singular Spectrum Analysis (SSA) and the Local Linear methods. Both methods do not require parametric assumptions. The data used in this study are divided into training data and test data. This data is secondary data obtained from the Basarnas Statistics Book. The training data in this study is the number of SAR operations from January 2021 to December 2022, while the testing data is from January 2023 to December 2023. From the analysis results, it is known that the method with the smallest MAPE is the Local Linear method with a MAPE of test data of 18.67% (good forecasting category), optimal bandwidth (h) = 4.299, and CV (h) = 231.39 where bandwidth is used to determine the level of smoothness of the estimate, while the CV (h) value is used to select the optimal bandwidth that minimizes the estimation error. At the same time, the SSA method has a MAPE of 40.27% (fair forecasting category). This shows that the Local Linear method provides a more accurate forecast of the number of SAR operations related to ship accidents in Indonesia. This research contributes to the SDGs to make Basarnas an effective and accountable institution and improve the planning and decision-making process in SAR operations through accurate forecasting research is relevant to accurate forecasting.

Estimation of conditional quantiles at very high or low tails is of interest in numerous applications. Quantile regression provides a convenient and natural way of quantifying the impact of covariates at different quantiles of a response distribution. However, high tails are often associated with data sparsity, so quantile regression estimation can suffer from high variability at tails especially for heavy-tailed distributions. In this article, we develop new estimation methods for high conditional quantiles by first estimating the intermediate conditional quantiles in a conventional quantile regression framework and then extrapolating these estimates to the high tails based on reasonable assumptions on tail behaviors. We establish the asymptotic properties of the proposed estimators and demonstrate through simulation studies that the proposed methods enjoy higher accuracy than the conventional quantile regression estimates. In a real application involving statistical downscaling of daily precipitation in the Chicago area, the proposed methods provide more stable results quantifying the chance of heavy precipitation in the area. Supplementary materials for this article are available online.

This study deals with the transfer of mass and heat of nanofluid flow over three different geometries of the non‐Darcy permeable vertical cone/wedge/vertical plate. Influence of the Brownian motion and thermophoresis takes place due to the nanofluid. Boundary condition on the temperature is introduced at the surface where the thermal conductivity of the fluid obeys a linear relation with the temperature. The local linearization method is introduced for solving the governing equations, and is based on spectral discretization. To verify the numerical scheme, we compared our results with those in the existing literature. The impact of the governing parameters on the fluid velocity, temperature distribution, and concentration distribution of nanoparticles along with the Nusselt number and Sherwood number is discussed. Some important outcomes of the present study are that the Nusselt number is higher for the plane plate than that for the vertical cone and it significantly decreases with introduction of the radiation parameter. The nanofluid Lewis number decreases the diffusivity of mass of the nanofluid, and as a result it helps enhance the Sherwood number.

The nexus between access to electricity and CO2 damage in Asian Countries: The evidence from quantile regression models

In this article, we focus on the efficient estimators of the derivative of the nonparametric function in the nonparametric quantile regression model. We develop two ways of combining quantile regression information to derive the estimators. One is the weighted composite quantile regression estimator based on the quantile weighted loss function; the other is the weighted quantile average estimator based on the weighted average of quantile regression estimators at a single quantile. Furthermore, by minimizing the asymptotic variance, the optimal weight vector is computed, and consequently, the optimal estimator is obtained. Furthermore, we conduct some simulations to evaluate the performance of our proposed estimators under different symmetric error distributions. Simulation studies further illustrate that both estimators work better than the local linear least square estimator for all the symmetric errors considered except the normal error, and the weighted quantile average estimator performs better than the weighted composite quantile regression estimator in most situations.

ABSTRACTA number of nonstationary models have been developed to estimate extreme events as function of covariates. A quantile regression (QR) model is a statistical approach intended to estimate and conduct inference about the conditional quantile functions. In this article, we focus on the simultaneous variable selection and parameter estimation through penalized quantile regression. We conducted a comparison of regularized Quantile Regression model with B-Splines in Bayesian framework. Regularization is based on penalty and aims to favor parsimonious model, especially in the case of large dimension space. The prior distributions related to the penalties are detailed. Five penalties (Lasso, Ridge, SCAD0, SCAD1 and SCAD2) are considered with their equivalent expressions in Bayesian framework. The regularized quantile estimates are then compared to the maximum likelihood estimates with respect to the sample size. A Markov Chain Monte Carlo (MCMC) algorithms are developed for each hierarchical model to simulate the conditional posterior distribution of the quantiles. Results indicate that the SCAD0 and Lasso have the best performance for quantile estimation according to Relative Mean Biais (RMB) and the Relative Mean-Error (RME) criteria, especially in the case of heavy distributed errors. A case study of the annual maximum precipitation at Charlo, Eastern Canada, with the Pacific North Atlantic climate index as covariate is presented.

The ‐quantile regression generalizes both quantile regression and expectile regression, and has become popular for its robustness and effectiveness especially when . In this paper, we consider the data that are inherently distributed and propose two distributed ‐quantile regression estimators for a preconceived low‐dimensional parameter in the presence of high‐dimensional extraneous covariates. To handle the impact of high‐dimensional nuisance parameters, we first investigate regularized projection score for estimating low‐dimensional parameter of main interest in ‐quantile regression. To deal with the distributed data, we further propose two communication‐efficient surrogate projection score estimators and establish their theoretical properties. The finite‐sample performance of the proposed estimators is studied through simulations and an application to Communities and Crime data set is also presented.

Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. There is a great amount of work about linear and nonlinear QR models. Specifically, nonparametric estimation of the conditional quantiles received particular attention, due to its model flexibility. However, nonparametric QR techniques are limited in the number of covariates. Dimension reduction offers a solution to this problem by considering low-dimensional smoothing without specifying any parametric or nonparametric regression relation. Existing dimension reduction techniques focus on the entire conditional distribution. We, on the other hand, turn our attention to dimension reduction techniques for conditional quantiles and introduce a new method for reducing the dimension of the predictor X. The novelty of this paper is threefold. We start by considering a single index quantile regression model, which assumes that the conditional quantile depends on X through a single linear combination of the predictors, then extend to a multi index quantile regression model, and finally, generalize the proposed methodology to any statistical functional of the conditional distribution. The performance of the methodology is demonstrated through simulation examples and a real data application. Our results suggest that this method has a good finite sample performance and often outperforms existing methods.

This study addresses the problem of accurately estimating P(X<Y), particularly in the presence of weakly dependent data. Traditional nonparametric methods often assume independence between observations, which can lead to biased results in dependent settings. To overcome this limitation, we propose the use of local linear methods for the nonparametric estimation of P(X<Y), focusing on weakly dependent data modeled using an autoregressive (AR(1)) process. The primary objective is to assess the performance of the local linear estimator under both independent and weakly dependent conditions, comparing bias, variance, and mean squared error (MSE). A Monte Carlo simulation with 1000 repetitions was conducted, utilizing large sample sizes to ensure robust asymptotic properties. Results show that while the estimator performs well for independent data, with low bias and variance, its accuracy decreases significantly in the presence of weak dependence, leading to increased prejudice and MSE. These findings highlight the need for adjustments in nonparametric methods when dealing with dependent data. In conclusion, while effective for independent cases, the local linear method demonstrates limitations when applied to weakly dependent data, emphasizing the importance of considering dependence structures in future nonparametric estimations.

ABSTRACTStatistical analyses that physiologists use to test hypotheses predominantly centre on means, but the tail ends of the response distribution can behave quite differently and underpin important scientific phenomena. We demonstrate that quantile regression (QR) offers a way to bypass some limitations of least squares regression (LSR) by building a picture of independent variable effects across the whole distribution of a dependent variable. We used LSR and QR with simulated and real datasets. With simulated data, LSR showed no change in the mean response but missed significant effects in the tails of the distribution found using QR. With real data, LSR showed a significant change in the mean response but missed a lack of response in the upper quantiles which was biologically revealing. Together, this highlights that QR can help to ask and answer more questions about variation in nature.

Sea level forecasting is an essential task for coastal engineering, geodetic application, navigation, and recreational activities. Predicting the behavior of future sea level is important for monitoring and forecasting of changes in fishery and marine ecosystems as well as for protection of coastal. This research is focused on the analysis and prediction of hourly sea level time series data at the benchmark station located in Penang by using chaotic approach. The purpose of this research was to identify the presence of chaotic behavior by the phase space reconstruction and Cao methods, and also a local linear approximation method is applied for prediction purposes. The results notified that the value of correlation coefficient between the observed and predicted time series is 0.8838 which is near to one. This reveals that the local linear approximation method can be used to predict the sea level time series in Malaysia. Certainly, the result of this research is expected to help stakeholder such as Department of Survey and Mapping Malaysia (JUPEM) in having a better sea level management.

BackgroundSample size is one of the critical factors affecting the accuracy of the estimation of population genetic diversity parameters. Small sample sizes often lead to significant errors in determining the allelic richness, which is one of the most important and commonly used estimators of genetic diversity in populations. Correct estimation of allelic richness in natural populations is challenging since they often do not conform to model assumptions. Here, we introduce a simple and robust approach to estimate the genetic diversity in large natural populations based on the empirical data for finite sample sizes.ResultsWe developed a non-linear regression model to infer genetic diversity estimates in large natural populations from finite sample sizes. The allelic richness values predicted by our model were in good agreement with those observed in the simulated data sets and the true allelic richness observed in the source populations. The model has been validated using simulated population genetic data sets with different evolutionary scenarios implied in the simulated populations, as well as large microsatellite and allozyme experimental data sets for four conifer species with contrasting patterns of inherent genetic diversity and mating systems. Our model was a better predictor for allelic richness in natural populations than the widely-used Ewens sampling formula, coalescent approach, and rarefaction algorithm.ConclusionsOur regression model was capable of accurately estimating allelic richness in natural populations regardless of the species and marker system. This regression modeling approach is free from assumptions and can be widely used for population genetic and conservation applications.