Sample Quantiles Research Articles

On the Bahadur representation of sample quantiles for score functionals

On the Bahadur representation of sample quantiles for score functionals

Influence of physical activity measurement on the association between Life’s Essential 8 and incident cardiovascular disease in older women

ObjectiveThe American Heart Association’s Life’s Essential 8 (LE8) metric includes self-reported physical activity as one of the metrics for assessing cardiovascular health. Self-reported physical activity is prone to misclassification, whereas accelerometer measures are less biased. We examined associations of LE8 and incident cardiovascular disease (CVD) using self-reported and accelerometer-measured physical activity. MethodsParticipants in the Women’s Health Initiative (WHI) Objective Physical Activity and Cardiovascular Health Study (n = 4,243; mean age = 79 ± 7 years) with no CVD history completed the WHI physical activity questionnaire and the Community Healthy Activities Model Program for Seniors (CHAMPS) questionnaire prior to wearing a hip-worn accelerometer for up to seven days in 2012–2014. LE8 components (physical activity, diet, sleep, body mass index, smoking, blood pressure, non-HDL cholesterol, and glucose) were scored according to guidelines. Scores were created using five physical activity measures: WHI questionnaire (LE8WHI), CHAMPS (LE8CHAMPS), accelerometer-measured physical activity (LE8A), and sample quantiles of accelerometer-measured physical activity (LE8AQ) and daily steps (LE8STEPS). Hazard ratios (HR) for physician-adjudicated CVD were estimated using Cox regression. Results707 incident CVD events occurred over an average 7.5 years. Multivariable HRs (95 % CI) comparing women in the highest vs. lowest quartiles of LE8 scores were: LE8WHI = 0.53 (0.43–0.67), LE8CHAMPS = 0.47 (0.38–0.60), LE8A = 0.44 (0.36–0.56), LE8AQ = 0.44 (0.35–0.55), and LE8STEPS = 0.45 (0.35–0.57). ConclusionsThe LE8-incident CVD association varies by physical activity measurement, however all methods showed reduced risk. Device-measures of physical activity may be more accurate in the LE8, but when impractical to implement, also support use of self-reported measures.

이산확률분포의 표본 분위수 계산 방법에 관한 연구

In this paper, we studied the methods of computing sample quantiles for the case of discrete probability distributions. There are two main methods which are introduced in statistics textbooks. We compared these two methods by simulation from practical and educational necessity. We considered some cases of Poisson, binomial, geometric, negative binomial, and discrete uniform distributions by setting up the parameters, and for each distribution we performed 10,000 times of simulation of drawing random samples of size 20 and 50. At each time of simulation, we computed the difference between the population quantile and the sample quantile obtained by each method. We compared the two methods by using two criteria: one is the mean square error over 10,000 times of simulation, and the other is the frequency of obtaining closer sample quantile to the population quantile than the competing method. We also obtained the estimated probabilities of exact estimation of population quantiles for the two methods, and performed the tests of homogeneity on the probability distribution of under-estimation, exact estimation, and over-estimation of the population quantiles.

Sharp moderate and large deviations for sample quantiles

In this article, we discuss the sharp moderate and large deviations between the quantiles of population and the quantiles of samples. Cramér type moderate deviations and Bahadur-Rao type large deviations are established with some mild conditions. The results refine the moderate and large deviation principles of Xu and Miao (2011).

Kernel quantile estimators for nested simulation with application to portfolio value-at-risk measurement

Nested simulation has been widely used in portfolio risk measurement in recent years. We focus on one risk measure, value at risk (VaR), and study a kernel quantile estimator (KQE) for nested simulation to estimate this risk. We analyze the bias, variance, and mean squared error (MSE), based on which we show that the variance is reduced in the lower-order terms, while in some cases bias could be reduced in the dominant term. For practical implementation, we propose an efficient bootstrap-based algorithm to guide kernel bandwidth selection and budget allocation in nested simulation. We also conduct numerical experiments to show that KQE works quite well at different significance levels compared with the widely used sample quantile.

A Berry–Esseen theorem for sample quantiles under martingale difference sequences

In this paper, we establish the uniformly asymptotic normality for sample quantiles based on martingale difference sequences under some suitable conditions. We obtain the rate of normality approximation of O ( n − 1 / 4 log ⁡ n ) by using some classical methods such as Bernstein type inequality, and so on. Finally, we verify asymptotic normality for the fixed quantile of the martingale difference sequences and present some numerical simulations to demonstrate the finite sample performances of the theoretical results.

Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions

Analytic and asymptotic properties of the generalized Student and generalized Lomax distributions are discussed, with the main focus on the representation of these distributions as scale mixtures of the laws that appear as limit distributions in classical limit theorems of probability theory, such as the normal, folded normal, exponential, Weibull, and Fréchet distributions. These representations result in the possibility of proving some limit theorems for statistics constructed from samples with random sizes in which the generalized Student and generalized Lomax distributions are limit laws. An overview of known properties of the generalized Student distribution is given, and some simple bounds for its tail probabilities are presented. An analog of the ‘multiplication theorem’ is proved, and the identifiability of scale mixtures of generalized Student distributions is considered. The normal scale mixture representation for the generalized Student distribution is discussed, and the properties of the mixing distribution in this representation are studied. Some simple general inequalities are proved that relate the tails of the scale mixture with that of the mixing distribution. It is proved that for some values of the parameters, the generalized Student distribution is infinitely divisible and admits a representation as a scale mixture of Laplace distributions. Necessary and sufficient conditions are presented that provide the convergence of the distributions of sums of a random number of independent random variables with finite variances and other statistics constructed from samples with random sizes to the generalized Student distribution. As an example, the convergence of the distributions of sample quantiles in samples with random sizes is considered. The generalized Lomax distribution is defined as the distribution of the absolute value of the random variable with the generalized Student distribution. It is shown that the generalized Lomax distribution can be represented as a scale mixture of folded normal distributions. The convergence of the distributions of maximum and minimum random sums to the generalized Lomax distribution is considered. It is demonstrated that the generalized Lomax distribution can be represented as a scale mixture of Weibull distributions or that of Fréchet distributions. As a consequence, it is demonstrated that the generalized Lomax distribution can be limiting for extreme statistics in samples with random size. The convergence of the distributions of mixed geometric random sums to the generalized Lomax distribution is considered, and the corresponding extension of the famous Rényi theorem is proved. The law of large numbers for mixed Poisson random sums is presented, in which the limit random variable has a generalized Lomax distribution.

Some improved results on Berry–Esséen bounds for strong mixing random variables and applications

In this paper, we mainly establish a general form of Berry–Esséen bound for α-mixing random variables. With different choices of the parameters, the rates are shown as , , and approximately . These results improved some corresponding ones in the literature. An application to the Berry–Esséen bound of sample quantiles is further provided. Moreover, some simulations are also carried out to support the theoretical results.

Pointwise norm-based clustering of data in arbitrary dimensional space

ABSTRACT In this paper, a novel nonparametric norm-based clustering algorithm is proposed to classify real-valued continuous data sets given in arbitrary dimensional space. For data univariate, multivariate or high-dimensional, with the number of study variables close to or larger than the data size, our straightforward algorithm is implemented throughout under an univariate set-up, where we make use of the observation-wise (or pointwise) norms which quantify the distances of the observations from the origin zero or the null vector. The method begins with determination of the sample quantile using nonparamteric bootstrapping on the computed norms and always converges independently. By its design, the suggested algorithm is fast enough to detect the number of existing clusters itself and to form well-defined groups. Data study demonstrates its competitiveness in comparison to 2 popular clustering algorithms K-means and K-medoids.

연속형 분포의 표본 분위수 계산 방법에 관한 고찰

When sample data is given, we estimate the population quantiles by computing the quantiles of the data. In statistics textbooks there are two methods of computing sample quantiles. Method 1 uses a single order statistic or a simple mean of two adjacent order statistics as a sample quantile, and method 2 uses a weighted mean of two adjacent order statistics as a sample quantile. In this paper, we compared these two methods for the case of several continuous distributions by simulation. We considered some cases of uniform, normal, exponential, chi-square, and beta distributions, and for each distribution we performed 10,000 times of simulation of drawing a random sample of sizes 20, 30 and 50. In each time of simulation, we computed the difference between the population quantile and the sample quantile obtained by each method. We compared the two methods by two criteria: one is the mean square error over 10,000 times of simulation, and the other is the frequency of giving smaller difference from the population quantile than the competing method. As a result of simulation, method 1 turned out to be much more frequently superior to method 2 by criterion 2.

On asymptotic correlation coefficient for some order statistics

<abstract><p>By the relationship between a continuous population $ X $ and the uniform distribution $ U[0, 1] $, we gain for a sample quantile an equivalent expression of its variance and for two different sample quantiles the asymptotic correlation coefficient. As the population of interest can have no expectation, the obtained conclusions are applicable to the location estimating problem of a Cauchy distribution. On that occasion, we finally obtained a quick and effective estimator established by a linear function of some sample quantiles. For similar problems, the presented approach is worthy of reference.</p></abstract>

Robust CFAR Detector Based on KLQ Estimator for Multiple-Target Scenario

The order statistic (OS) constant false alarm rate (CFAR) detector is designed from a perspective of quantile estimation and uses the sample quantile (SQ) estimator to estimate background level. Following this idea, we find a more efficient quantile estimator, i.e. Kaigh-Lachenbruch quantile (KLQ) estimator, to construct a robust CFAR detector, which is referred to as KLQ-CFAR. We prove that the KLQ estimator is an asymptotic unbiased estimator to the quantile function and derive the asymptotic variance of KLQ estimator in Exponential case. It is shown that the KLQ estimator has more efficient estimation efficiency than SQ estimator. Subsequently, we explore the performance of KLQ-CFAR in Exponential clutter. In homogeneous background, the KLQ-CFAR conquers the shortcoming of large detection loss of OS-CFAR, and exhibits comparable performance with trimmed mean (TM) CFAR. The superiority of KLQ-CFAR is also reflected in multiple-target scenario, especially in the case where the number of interferences exceeds the preset parameters of KLQ- and TM-CFAR. In this case, the detection performance and false alarm control ability of KLQ-CFAR have significant improvement compared with TM-CFAR. Then, we extend the KLQ-CFAR from Exponential background to non-Gaussian clutter via transformation approach, and obtain the transformed KLQ (TKLQ) CFAR. We take Weibull clutter as a specific example. It is found that the behavior of TKLQ-CFAR in Weibull background is very similar to that of the KLQ-CFAR in Exponential clutter, but the difference lies in that the performance of TKLQ-CFAR is related to Weibull shape parameter. The TKLQ-CFAR tends to have better detection performance with the increase of Weibull shape parameter. Finally, the validity of TKLQ-CFAR is verified in skywave over-the-horizon radar data.

On block accelerations of quantile randomized Kaczmarz for corrupted systems of linear equations

With the growth of large data as well as large-scale learning tasks, the need for efficient and robust linear system solvers is greater than ever. The randomized Kaczmarz method (RK) and similar stochastic iterative methods have received considerable recent attention due to their efficient implementation and memory footprint. These methods can tolerate streaming data, accessing only part of the data at a time, and can also approximate the least squares solution even if the system is affected by noise. However, when data is instead affected by large (possibly adversarial) corruptions, these methods fail to converge, as corrupted data points draw iterates far from the true solution. A recently proposed solution to this is the quantileRK method, which avoids harmful corrupted data by exploring the space carefully as the method iterates. The exploration component requires the computation of quantiles of large samples from the system and is computationally much heavier than the subsequent iteration update. In this paper, we propose an approach that better uses the information obtained during exploration by incorporating an averaged version of the block Kaczmarz method. This significantly speeds up convergence, while still allowing for a constant fraction of the equations to be arbitrarily corrupted. We provide theoretical convergence guarantees as well as experimental supporting evidence. We also demonstrate that the classical projection-based block Kaczmarz method cannot be robust to sparse adversarial corruptions, but rather the blocking has to be carried out by averaging one-dimensional projections.

The shortest confidence interval for the ratio of quantiles of the Dagum distribution

Jȩdrzejczak et al. (REVSTAT-Statistical Journal 19(1), 87–97, 2021) constructed a confidence interval for a ratio of quantiles coming from the Dagum distribution, which is frequently applied as a theoretical model in numerous income distribution analyses. The proposed interval is symmetric with respect to the ratio of sample quantiles, which result may be unsatisfactory in many practical applications. The search for a confidence interval with a smaller length resulted in the derivation of the shortest interval with the ends being asymmetric relative to the ratio of sample quantiles. In the paper, the existence of the shortest confidence interval is shown and the method of obtaining such an interval is presented. The results of the calculations show a reduction in the length of the proposed confidence interval by several percent compared to the symmetrical confidence interval.

Improving linear quantile regression for replicated data

ABSTRACT This paper deals with improvement of linear quantile regression, when there are a few distinct values of the covariates but many replicates. One can improve asymptotic efficiency of the estimated regression coefficients by using suitable weights in quantile regression, or simply by using weighted least squares regression on the conditional sample quantiles. The asymptotic variances of the unweighted and weighted estimators coincide only in some restrictive special cases, e.g., when the density of the conditional response has identical values at the quantile of interest over the support of the covariate. The dominance of the weighted estimators is demonstrated in a simulation study, and through the analysis of a data set on tropical cyclones.

Precise Asymptotics for the Uniform Empirical Process and the Uniform Sample Quantile Process

One of the sources of “invariance principle” is that the limit properties of the uniform empirical process coincide with that of a Brownian bridge. The deep discussion of limit theorem of the uniform empirical process gathered wild interest of the researchers. In this paper, the precise convergence rate of the uniform empirical process is considered. As is well-known, when ε tends to 0, the precise asymptotic theorems can be demonstrated by referring to the classical method of Gut and Spǎtaru, by using some nice probability inequalities and so on. However, if ε tends to a positive constant, other powerful methods and tools are needed. The method of strong approximation is used in this paper. The main theorems are proved by using the Brownian bridge B t to approximate the uniform empirical process α n t . The relevant results for the uniform sample quantile process are also presented.

A Generic Online Nonparametric Monitoring and Sampling Strategy for High-Dimensional Heterogeneous Processes

With the rapid advancement of in-process measurements and sensor technology driven by zero-defect manufacturing applications, high-dimensional heterogeneous processes that continuously collect distinct physical characteristics frequently appear in modern industries. Such large-volume high-dimensional data place a heavy demand on data collection, transmission, and analysis in practice. Thus, practitioners often need to decide which informative data streams to observe given the resource constraints at each data acquisition time, which poses significant challenges for multivariate statistical process control and quality improvement. In this article, we propose a generic online nonparametric monitoring and sampling scheme to quickly detect mean shifts occurring in heterogeneous processes when only partial observations are available at each acquisition time. Our innovative idea is to seamlessly integrate the Thompson sampling (TS) algorithm with a quantile-based nonparametric cumulative sum (CUSUM) procedure to construct local statistics of all data streams based on the partially observed data. Furthermore, we develop a global monitoring scheme of using the sum of top- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${r}$ </tex-math></inline-formula> local statistics, which can quickly detect a wide range of possible mean shifts. Tailored to monitoring the heterogeneous data streams, the proposed method balances between exploration that searches unobserved data streams for possible mean shifts and exploitation that focuses on highly suspicious data streams for quick shift detection. Both simulations and a case study are comprehensively conducted to evaluate the performance and demonstrate the superiority of the proposed method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper is motivated by the critical challenge of online process monitoring by considering the cost-effectiveness and resource constraints in practice (e.g., limited number of sensors, limited transmission bandwidth or energy constraint, and limited processing time). Unlike the existing methodologies which rely on the restrictive assumptions (e.g., normally distributed, exchangeable data streams) or require historical full observations of all data streams to be available offline for training, this paper proposes a novel monitoring and sampling strategy that allows the practitioners to cost-effectively monitor high-dimensional heterogeneous data streams that contain distinct physical characteristics and follow different distributions. To implement the proposed methodology, it is necessary: (i) to identify sample quantiles for each data stream based on historical in-control data offline; (ii) to determine which data streams to observe at each acquisition time based on the resource constraints; and (iii) to automatically screen out the suspicious data streams to form the global monitoring statistic. Experimental results through simulations and a case study have shown that the proposed method has much better performance than the existing methods in reducing detection delay and effectively dealing with heterogeneous data streams.

Content‐adjusted tolerance intervals via bootstrap calibration

Tolerance intervals (TIs) are commonly employed in numerous industries, ranging from engineering to pharmaceuticals. However, closed‐form TIs are unavailable for most distributions. Although some approximate methods can be used to obtain TIs, coverage probabilities (CPs) of these TIs cannot achieve the nominal level, or can be even far different from the nominal level. In this study, we propose two content‐adjusted procedures for TIs based on bootstrap. The first one is based on the bootstrap sample quantile, while the second one is based on the asymptotic normality of empirical distribution. The simulation results show that the two calibration procedures can improve CPs of TIs for some non‐normal distributions according to extensive numerical simulations, and they are both proved to be effective through real data examples.

Construction and Simulation of Market Risk Warning Model Based on Deep Learning

The market is intricate and complicated, and the existing risk warning models have problems of low efficiency and poor generalization in predicting market risk data. Aiming at the problems, this study takes stock market risk warning as the research object and proposes a market risk warning model based on LSTM-VaR. 15 variables in the three categories of basic transaction data, statistical technical indicators, and moving interval data are selected as the stock market characteristic indicators, the LSTM(Long Short Term Memory) prediction model is constructed and the standard deviation of stock returns is predicted. Based on the predicted results, the probability distribution of return rate under the conditional distribution is obtained, and the VaR(Value at Risk) is calculated. 1% and 5% sample quantiles are taken as the warning line, and the LSTM-VaR warning model is obtained. The results show that the RMSE value of the model is the smallest, which is 0.013762, when the activation function of the LSTM-VaR model is the Leaky ReLu function, the training periods epochs are 10, the time window length N is 9, the batch size is 8, the number of neurons in each layer is 50, the dropout probability is 0.1, and adam is used as the optimizer. Compared with traditional prediction models such as MLP, the proposed model has better performance and can well realize market risk warning.

Post-Strata Based on Sample Quantiles

AbstractThe standard method of creating post-strata is to define the boundaries of the strata on the basis of population characteristics of auxiliary variables. Estimation treats the post-strata as strata for standard stratified-sample estimation. Samples often contain empty post-strata requiring adjustment to the estimation procedure. To avoid empty post-strata, we propose using the sample distribution function of an auxiliary variable to define the post-strata. We show that the large-sample efficiency of the sample-based post-stratification procedure is the same as that of the equivalently defined population-based procedure. In the simulation, the sample-based procedure was slightly more efficient than the classical procedure. The Monte Carlo coverage of a nominal 95% interval was approximately 95% for the sample-based procedure and approximately 94% for the classical procedure.