Asymptotic Sample Research Articles

Sampling Distribution and Point and Interval Estimation of Kelley’s Percentile Coefficient of Kurtosis

There is a robust measure for estimating kurtosis based on quantiles, developed by the American psychologist Truman Lee Kelley, but it is seldom used. This underutilization is likely because it is not available in any statistical package, including R program. Additionally, Kelley reported its standard error when the random variable is drawn from a normal distribution but did not establish its sampling distribution. The objective of this study is to determine its asymptotic sampling distribution and to develop an R script to provide point and interval estimates for this measure. The R program was chosen because it is freely available, developed by the mathematical community, and considered one of the most comprehensive statistical packages currently available. To determine the asymptotic sampling distribution, samples of 100, 500, 1000, 5000, 10000, and 20000 data points were generated from three symmetric distributions: uniform (platykurtic), normal (mesokurtic), and Laplace (leptokurtic). From these 18 source samples, 1000 samples were drawn by resampling with replacement to obtain 18 bootstrap sampling distributions. Normality was then determined using Grubbs (outliers), D’Agostino (symmetry), Anscombe-Glynn (mesokurtosis), and Anderson-Darling, Lilliefors, Shapiro-Francia, and D’Agostino-Belanger-D’Agostino (normality) tests. The script included tests for several assumptions to decide which confidence interval to use: Wald type (normal asymptotic) or bootstrap (Gaussian, percentile, and percentile bias-corrected and accelerated). As an example, the script was applied to a random sample drawn from the raised cosine distribution on waiting time in a social service. It was concluded that the asymptotic sampling distribution of Kelley’s kurtosis measure is normal and that this measure appears less specific than classical Pearson and Fisher measures in the presence of a distribution very close to normal. It is suggested to use this script, which may have practical and academic utility.

Pollution concentration monitoring using a new Birnbaum–Saunders control chart

AbstractAir pollution monitoring is an important issue in environmental science. The Birnbaum–Saunders (BS) distribution, originally applied to describe product failure time distribution to fatigue failures and general random wear failures, is also well to describe the pollutant concentration data due to accumulations of various pollutants in the air over time. Sulfur dioxide (SO2) is a critical factor in air pollution. Hence, it is important to monitor its concentration variation for air pollution prevention. Due to the complexity of its distribution form, there is no reliable and easy‐to‐use control chart for monitoring pollutant concentrations based on the BS distribution. We found that the SO2 concentration data follows the BS distribution. In this study, we propose a new median control chart based on the exact sampling distribution of the monitoring statistic to detect shifts in the median of BS distribution. Thus, given the false alarm rate, the control limits for such control charts can be obtained precisely satisfying a preset in‐control average run length using Monte Carlo simulations. The out‐of‐control average run lengths are calculated by simulation to evaluate the detection performance of the proposed chart when the median shifts occur. We further compare the detection performance of the proposed chart and those of the existing control charts based on asymptotic sampling distributions. In order to improve the detection ability of the proposed chart for small median shifts, an exponentially weighted moving average (EWMA) control chart is constructed. The results of numerical analyses demonstrated that the proposed EWMA chart performs much better than all existing control charts for monitoring the median of BS distribution. Finally, the proposed control charts are applied to monitor the median of SO2 concentrations for air pollution control, showing that both charts can effectively detect a shift in the median of SO2 concentrations. The proposed EWMA control chart even detects out a small shift in the median of SO2 concentrations. The results provide a continuous monitoring solution for air pollution prevention.

No star is good news: A unified look at rerandomization based on [formula omitted]-values from covariate balance tests

Randomized experiments balance all covariates on average and are considered the gold standard for estimating treatment effects. Chance imbalances are nonetheless common in realized treatment allocations. To inform readers of the comparability of treatment groups at baseline, contemporary scientific publications often report covariate balance tables with not only covariate means by treatment group but also the associated p-values from significance tests of their differences. The practical need to avoid small p-values as indicators of poor balance motivates balance check and rerandomization based on these p-values from covariate balance tests (ReP) as an attractive tool for improving covariate balance in designing randomized experiments. Despite the intuitiveness of such strategy and its possibly already widespread use in practice, the literature lacks results about its implications on subsequent inference, subjecting many effectively rerandomized experiments to possibly inefficient analyses. To fill this gap, we examine a variety of potentially useful schemes for ReP and quantify their impact on subsequent inference. Specifically, we focus on three estimators of the average treatment effect from the unadjusted, additive, and interacted linear regressions of the outcome on treatment, respectively, and derive their asymptotic sampling properties under ReP. The main findings are threefold. First, the estimator from the interacted regression is asymptotically the most efficient under all ReP schemes examined, and permits convenient regression-assisted inference identical to that under complete randomization. Second, ReP, in contrast to complete randomization, improves the asymptotic efficiency of the estimators from the unadjusted and additive regressions. Standard regression analyses are accordingly still valid but in general overconservative. Third, ReP reduces the asymptotic conditional biases of the three estimators and improves their coherence in terms of mean squared difference. These results establish ReP as a convenient tool for improving covariate balance in designing randomized experiments, and we recommend using the interacted regression for analyzing data from ReP designs.

Asymptotic sampling distributions made easy: loose linkage in the ancestral recombination graph

Asymptotic sampling distributions made easy: loose linkage in the ancestral recombination graph

Asymptotic Sample Size for Common Test of Relative Risk Ratios in Stratified Bilateral Data

In medical clinical studies, various tests usually relate to the sample size. This paper proposes several methods to calculate sample sizes for a common test of relative risk ratios in stratified bilateral data. Under the prespecified significant level and power, we derive some explicit formulae and an algorithm of the sample size. The sample sizes of the stratified intra-class model are obtained from the likelihood ratio, score, and Wald-type tests. Under pooled data, we calculate sample size based on the Wald-type test and its log-transformation form. Numerical simulations show that the proposed sample sizes have empirical power close to the prespecified value for given significance levels. The sample sizes from the iterative method are more stable and effective.

A bootstrapping-based weighted average asymptotic sampling formulation for reliability estimation of highly safe structures

A bootstrapping-based weighted average asymptotic sampling formulation for reliability estimation of highly safe structures

Least Squares Polynomial Chaos Regression for Stochastic Analysis of Transmission Lines Without and With Noise

Stochastic transmission line (TL) analysis is often challenging due to the difficulty of fully identifying the probability distributions of all randomly varying parameters, especially in the presence of noise. These problems cannot be directly addressed by the existing polynomial chaos expansion (PCE) framework. To overcome the limitations, this paper proposes an improved PCE method for analyzing the stochastic TL model with noise. A regression-based non-intrusive approach termed least squares polynomial chaos regression (LSPCR) is employed to determine the PCE coefficients. Six algorithms for LSPCR are developed by respectively combining three sampling strategies, i.e., standard sampling (SS), asymptotic sampling (AS), and coherence-optimal sampling (COS), with two types of norm problems: the least-squares optimization problem (LSO) and the l1-minimum problem (l1-M). Numerical experiments are conducted on noiseless and noisy problems. The results reveal that the LSO-based algorithms (LSOs) are much faster than the l1-M-based algorithms (l1-Ms), regardless of the sampling strategy. In the absence of noise, these six algorithms require only a relatively small number of samples to achieve higher accuracy in solution moments compared to the Stochastic Galerkin (SG) method. When the sample size is large, the AS-based algorithms (ASs) and the COS-based algorithms (COSs) yield more accurate results than the SS-based algorithms (SSs), regardless of the norm problems. In the presence of noise, the LSOs outperform the l1-Ms for small sample sizes. However, no evident differences are observed among the six algorithms with large sample sizes.

A simple consistent Bayes factor for testing the Kendall rank correlation coefficient

In this paper, we propose a simple and easy-to-implement Bayesian hypothesis test for the presence of an association, described by Kendall's τ coefficient, between two variables measured on at least an ordinal scale. Owing to the absence of the likelihood functions for the data, we employ the asymptotic sampling distributions of the test statistic as the working likelihoods and then specify a truncated normal prior distribution on the noncentrality parameter of the alternative hypothesis, which results in the Bayes factor available in closed form in terms of the cumulative distribution function of the standard normal distribution. Investigating the asymptotic behaviour of the Bayes factor we find the conditions of the priors so that it is consistent to whichever the hypothesis is true. Simulation studies and a real-data application are used to illustrate the effectiveness of the proposed Bayes factor. It deserves mentioning that the proposed method can be easily covered in undergraduate and graduate courses in nonparametric statistics with an emphasis on students' Bayesian thinking for data analysis.

Probabilistic reliability analysis for active vibration control of piezoelectric laminated composite plates using mesh-free finite volume method

The implementation of deterministic design approaches in the presence of uncertainty is an adequate design task. Probabilistic methods are then used in structural design procedures, offering structural safety predictions. In this study, reliability-based analysis of a piezoelectric laminated composite plate subjected to the mechanical loads was investigated, accounting for the epistemic source of uncertainty. Material properties of the plate were considered temperature-dependent. The sophisticated mesh-free finite volume approach was employed to actively control the vibration of the temperature-dependent piezoelectric laminated composite plate through the first-order shear deformation principles. The induced vibration was treated as a non-stationary random process, and the generalized failure index was estimated using the first-passage probability concept by adopting the novel asymptotic sampling technique. The veracity of the suggested model was corroborated by alternate approaches in the literature, comprising the finite element approach. Furthermore, the temperature-dependent piezoelectric laminated composite plate with different ply orientations was considered to assess the safety index variation against changes in the ambient temperature and the laminate stacking orientation. The obtained results revealed that with increasing temperature, the reliability of the composite plate reduced. It was also realized that the reliability index is directly correlated with the piezoelectric voltage.

Sparse Multi-Reference Alignment: Phase Retrieval, Uniform Uncertainty Principles and the Beltway Problem

Motivated by cutting-edge applications like cryo-electron microscopy (cryo-EM), the Multi-Reference Alignment (MRA) model entails the learning of an unknown signal from repeated measurements of its images under the latent action of a group of isometries and additive noise of magnitude \(\sigma \). Despite significant interest, a clear picture for understanding rates of estimation in this model has emerged only recently, particularly in the high-noise regime \(\sigma \gg 1\) that is highly relevant in applications. Recent investigations have revealed a remarkable asymptotic sample complexity of order \(\sigma ^6\) for certain signals whose Fourier transforms have full support, in stark contrast to the traditional \(\sigma ^2\) that arise in regular models. Often prohibitively large in practice, these results have prompted the investigation of variations around the MRA model where better sample complexity may be achieved. In this paper, we show that sparse signals exhibit an intermediate \(\sigma ^4\) sample complexity even in the classical MRA model. Further, we characterize the dependence of the estimation rate on the support size s as \(O_p(1)\) and \(O_p(s^{3.5})\) in the dilute and moderate regimes of sparsity respectively. Our techniques have implications for the problem of crystallographic phase retrieval, indicating a certain local uniqueness for the recovery of sparse signals from their power spectrum. Our results explore and exploit connections of the MRA estimation problem with two classical topics in applied mathematics: the beltway problem from combinatorial optimization, and uniform uncertainty principles from harmonic analysis. Our techniques include a certain enhanced form of the probabilistic method, which might be of general interest in its own right.

Characterizing Sampling Variability for Item Response Theory Scale Scores in a Fixed-Parameter Calibrated Projection Design.

A common practice of linking uses estimated item parameters to calculate projected scores. This procedure fails to account for the carry-over sampling variability. Neglecting sampling variability could consequently lead to understated uncertainty for Item Response Theory (IRT) scale scores. To address the issue, we apply a Multiple Imputation (MI) approach to adjust the Posterior Standard Deviations of IRT scale scores. The MI procedure involves drawing multiple sets of plausible values from an approximate sampling distribution of the estimated item parameters. When two scales to be linked were previously calibrated, item parameters can be fixed at their original published scales, and the latent variable means and covariances of the two scales can then be estimated conditional on the fixed item parameters. The conditional estimation procedure is a special case of Restricted Recalibration (RR), in which the asymptotic sampling distribution of estimated parameters follows from the general theory of pseudo Maximum Likelihood (ML) estimation. We evaluate the combination of RR and MI by a simulation study to examine the impact of carry-over sampling variability under various simulation conditions. We also illustrate how to apply the proposed method to real data by revisiting Thissen et al. (2015).

Bayesian Modeling of Sequential Discoveries

We aim at modeling the appearance of distinct tags in a sequence of labeled objects. Common examples of this type of data include words in a corpus or distinct species in a sample. These sequential discoveries are often summarized via accumulation curves, which count the number of distinct entities observed in an increasingly large set of objects. We propose a novel Bayesian method for species sampling modeling by directly specifying the probability of a new discovery, therefore, allowing for flexible specifications. The asymptotic behavior and finite sample properties of such an approach are extensively studied. Interestingly, our enlarged class of sequential processes includes highly tractable special cases. We present a subclass of models characterized by appealing theoretical and computational properties, including one that shares the same discovery probability with the Dirichlet process. Moreover, due to strong connections with logistic regression models, the latter subclass can naturally account for covariates. We finally test our proposal on both synthetic and real data, with special emphasis on a large fungal biodiversity study in Finland. Supplementary materials for this article are available online.

Evaluation of alternative methods for estimating the precision of REML-based estimates of variance components and heritability.

Residual Maximum Likelihood (REML) analysis is the most widely used method to estimate variance components and heritability. This method is based on large sample theory under the assumption that the parameter estimates are asymptotically multivariate normally distributed with covariance matrix given by the inverse of the information matrix. Hence, these sampling variances could be biased if the assumption of asymptotic approximation is incorrect, especially when the sample size is small. Though it is difficult to assess the impact of sample size, an alternative option is to generate a full distribution of the parameters to determine the uncertainty of estimates. In this study, we compared the REML estimates of variance components, heritability and sampling variances of body-weight (BW), body-depth (BD), and condition-factor (K) with those obtained from four sampling-based methods viz., parametric and nonparametric bootstrap, asymptotic sampling and Bayesian estimation. The aim was to understand if a sample size of order 1413 was sufficient to contain adequate information for a reliable asymptotic approximation. The REML solution was close to values obtained from different sampling-based methods indicating that the present sample size was sufficient to estimate reliable genetic variation in different traits with varying heritability. The variance and heritability estimated by a nonparametric bootstrap estimate based on randomization of family effects gave comparable results as evaluated by REML for different traits. Hence, the nonparametric bootstrap estimate can be effectively used to understand whether the sample size is large enough to contain sufficient information under likelihood estimation assumptions.

Robustness of the Poverty Measures: Evidence from Farm Households in Akwa Ibom State, Nigeria

The use of a plethora of poverty indexes is sometimes fraught with difficulties. The purpose of this research was to quantitatively assess poverty and to examine the robustness of the poverty metrics. Selecting representative farm homes required a multistage sample technique, which was implemented. A total of 150 rural homes were surveyed using questionnaires. Stochastic dominance and the weighted poverty measures of Foster, Greer and Thorbecke were used in this work to examine the weighted poverty measures' resilience and sensitivity to changes in the poverty line. According to the findings, as people become older and their families get larger, the likelihood, severity, and depth of poverty increases. An asymptotic sampling distribution was utilized to infer whether poverty was larger across a variety of hypothetical poverty lines by stochastic dominance analysis. First-order stochastic dominance was found, with the Cumulative Distribution Function (CDF) of households headed by people over 60 years old lying totally above the other distribution functions (CDFs). The CDF of single families was lower than the CDF of married households, according to the findings. At any poverty level, the CDF of families with more than 10 household members stochastically dominated those with fewer family members. Many households will be lifted out of poverty if poverty-reduction initiatives are targeted at those over 60 and those with big families.

Robustness of the Poverty Measures: Evidence from Farm Households in Akwa Ibom State, Nigeria

The use of a plethora of poverty indexes is sometimes fraught with difficulties. The purpose of this research was to quantitatively assess poverty and to examine the robustness of the poverty metrics. Selecting representative farm homes required a multistage sample technique, which was implemented. A total of 150 rural homes were surveyed using questionnaires. Stochastic dominance and the weighted poverty measures of Foster, Greer and Thorbecke were used in this work to examine the weighted poverty measures' resilience and sensitivity to changes in the poverty line. According to the findings, as people become older and their families get larger, the likelihood, severity, and depth of poverty increases. An asymptotic sampling distribution was utilized to infer whether poverty was larger across a variety of hypothetical poverty lines by stochastic dominance analysis. First-order stochastic dominance was found, with the Cumulative Distribution Function (CDF) of households headed by people over 60 years old lying totally above the other distribution functions (CDFs). The CDF of single families was lower than the CDF of married households, according to the findings. At any poverty level, the CDF of families with more than 10 household members stochastically dominated those with fewer family members. Many households will be lifted out of poverty if poverty-reduction initiatives are targeted at those over 60 and those with big families.

A Machine Learning Approach to Assess Differential Item Functioning in Psychometric Questionnaires Using the Elastic Net Regularized Ordinal Logistic Regression in Small Sample Size Groups.

Assessing differential item functioning (DIF) using the ordinal logistic regression (OLR) model highly depends on the asymptotic sampling distribution of the maximum likelihood (ML) estimators. The ML estimation method, which is often used to estimate the parameters of the OLR model for DIF detection, may be substantially biased with small samples. This study is aimed at proposing a new application of the elastic net regularized OLR model, as a special type of machine learning method, for assessing DIF between two groups with small samples. Accordingly, a simulation study was conducted to compare the powers and type I error rates of the regularized and nonregularized OLR models in detecting DIF under various conditions including moderate and severe magnitudes of DIF (DIF = 0.4 and 0.8), sample size (N), sample size ratio (R), scale length (I), and weighting parameter (w). The simulation results revealed that for I = 5 and regardless of R, the elastic net regularized OLR model with w = 0.1, as compared with the nonregularized OLR model, increased the power of detecting moderate uniform DIF (DIF = 0.4) approximately 35% and 21% for N = 100 and 150, respectively. Moreover, for I = 10 and severe uniform DIF (DIF = 0.8), the average power of the elastic net regularized OLR model with 0.03 ≤ w ≤ 0.06, as compared with the nonregularized OLR model, increased approximately 29.3% and 11.2% for N = 100 and 150, respectively. In these cases, the type I error rates of the regularized and nonregularized OLR models were below or close to the nominal level of 0.05. In general, this simulation study showed that the elastic net regularized OLR model outperformed the nonregularized OLR model especially in extremely small sample size groups. Furthermore, the present research provided a guideline and some recommendations for researchers who conduct DIF studies with small sample sizes.

Conditional Inference in Small Sample Scenarios Using a Resampling Approach

This paper discusses a non-parametric resampling technique in the context of multidimensional or multiparameter hypothesis testing of assumptions of the Rasch model. It is based on conditional distributions and it is suggested in small sample size scenarios as an alternative to the application of asymptotic or large sample theory. The exact sampling distribution of various well-known chi-square test statistics like Wald, likelihood ratio, score, and gradient tests as well as others can be arbitrarily well approximated in this way. A procedure to compute the power function of the tests is also presented. A number of examples of scenarios are discussed in which the power function of the test does not converge to 1 with an increasing deviation of the true values of the parameters of interest from the values specified in the hypothesis to be tested. Finally, an attempt to modify the critical region of the tests is made aiming at improving the power and an R package is provided.

A critical evaluation of asymptotic sampling method for highly safe structures

A critical evaluation of asymptotic sampling method for highly safe structures

Rank-one multi-reference factor analysis

In recent years, there is a growing need for processing methods aimed at extracting useful information from large datasets. In many cases, the challenge is to discover a low-dimensional structure in the data, often concealed by the existence of nuisance parameters and noise. Motivated by such challenges, we consider the problem of estimating a signal from its scaled, cyclically shifted and noisy observations. We focus on the particularly challenging regime of low signal-to-noise ratio (SNR), where different observations cannot be shift-aligned. We show that an accurate estimation of the signal from its noisy observations is possible, and derive a procedure which is proved to consistently estimate the signal. The asymptotic sample complexity (the number of observations required to recover the signal) of the procedure is \(1{/}{\text {SNR}}^4\). Additionally, we propose a procedure which is experimentally shown to improve the sample complexity by a factor equal to the signal’s length. Finally, we present numerical experiments which demonstrate the performance of our algorithms and corroborate our theoretical findings.

Rank Estimation for Mean Residual Life Transformation Model

This paper proposes a new and important class of mean residual life regression model, which is called the mean residual life transformation model. The link function is assumed to be unknown and increasing in its second argument, but it is permitted to be not differentiable. The mean residual life transformation model encompasses the proportional mean residual life model, the additive mean residual life model, and so on. Under maximum rank correlation estimation, we present the estimation procedures, whose asymptotic and finite sample properties are established. The consistent variance can be estimated by a resampling method via perturbing the U -statistics objective function repeatedly which avoids the usual sandwich choice. Monte Carlo simulations reveal good finite sample performance and the estimators are illustrated with the Oscar data set.