Nonparametric estimation of averages and errors for small data-sets in isotope geoscience: a proposal

Some Thoughts About Data Type, Distribution, and Statistical Significance

Введение. Исследование цитокинового профиля пациентов, страдающих крапивницей, является определяющим в понимании механизмов развития различных форм заболевания. Цель. Определение показателей уровня цитокинов (IL-9, IL-17, TNFβ) у детей с острой крапивницей. Материалы и методы. Обследовано 80 детей обоего пола, поступивших под наблюдение с диагнозом «острая крапивница». Количество мальчиков составило 43 (53,7%), девочек – 37 (46,3%). В контрольную группу вошли 20 практически здоровых детей. Средний возраст детей составил (7,48±0,39) года. Уровень содержания интерлейкинов IL-9, IL-17 и TNFβ в периферической крови был определен у 70 пациентов с острой крапивницей и 20 здоровых детей контрольной группы методом твердофазного иммуноферментного анализа из единой пробы сыворотки крови. Межгрупповые показатели сравнивали с применением критерия χ2 Пирсона (Pearson Chi-Square – рχ2), с дополнительным уточнением результатов критерием U Манна – Уитни (U-Mann – Whitney – pU). Внутригрупповые сравнения проводили критериями W Вилкоксона (Wilcoxon test – pW) и Z-знаков (Sign test – pZ). Результаты. Острая крапивница у детей сопровождалась гиперпродукцией в крови провоспалительных цитокинов. Отмечалось достоверное повышение уровня IL-9 (р < 0,000) и IL-17 (р < 0,000) в крови у детей с острой крапивницей и отеком Квинке. Установленные межсистемные связи выявили достоверную корреляцию между клиническими симптомами острой крапивницы и повышенными уровнями содержания провоспалительных цитокинов IL-9 и IL-17, а также достоверную корреляционную связь между повышенными уровнями содержания в крови IL-9, IL-17 и представителями кишечной микробиоты. Заключение. Результаты исследования свидетельствуют о важной роли цитокинов в патогенезе острой крапивницы и отека Квинке, а также об участии кишечной микрофлоры в развитии данной патологии у детей. Introduction. The study of the cytokine profile of patients suffering from urticaria is decisive in understanding mechanisms of development of various forms of the disease. Purpose. To determine cytokines (IL-9, IL-17, TNFβ) levels in children with acute urticaria. Materials and methods. We examined 80 children of both genders admitted under observation with diagnosis of acute urticaria. The number of boys was 43 (53.7%), girls – 37 (46.3%). The control group consisted of 20 practically healthy children. The average age of the children was 7.48±0.39 years. The level of interleukins IL-9, IL-17 and TNFβ in peripheral blood was determined in 70 patients with acute urticaria and 20 healthy children of the control group by enzyme-linked immunosorbent assay from a single blood serum sample. Intergroup indicators were compared using the χ2-Pearson test (Pearson Chi-Square – pχ2), with additional refinement of the results by the U-Wilcoxon test (U-Mann-Whitney – pU). Intragroup comparisons were performed using W-Wilcoxon test (Wilcoxon test – pW) and Z-signs (Sign test – pZ). Results. Acute urticaria in children was accompanied by hyperproduction of pro- inflammatory cytokines in the blood. There was a significant increase in IL-9 (p < 0.000) and IL-17 (p < 0.000) levels in the blood of children with acute urticaria and Quincke’s edema. The established intersystem relationships revealed a significant correlation between clinical symptoms of acute urticaria and elevated levels of pro-inflammatory cytokines IL-9 and IL-17, as well as a significant correlation between elevated blood levels of IL-9, IL-17 and representatives of intestinal microbiota. Conclusion. The results of the study indicate the important role of cytokines in the pathogenesis of acute urticaria and Quincke’s edema, as well as the participation of intestinal microflora in the development of this pathology in children.

The robust properties of the simultaneous estimators of location and scale parameters (μ*, σ*) proposed by Brown and Hwang are studied. As a pair of simultaneous M estimators of location and scale, their asymptotic efficiencies (0.650 for μ* and 0.541 for σ*) are higher than those for median (0.637) and median absolute deviation (0.368) under the normal distribution. Simulation indicates that the distributions of and are much flatter than those based on the sample mean and the sample standard deviation under the normal distribution when the sample size is small.

SummaryThe standard maximum likelihood estimator of censored regression models is constructed under the assumption of the normal distribution and it yields inefficient results in case of non‐normal asymmetric data or/and presence of outlier(s). In the analysis of censored real‐life data encountered in many disciplines such as finance, medical sciences, and engineering, there is often a violation of the assumption of normality. In this paper, we propose a generalization of the censored normal regression based on the extended normal distribution (EGT). The generalized model, with two additional shape parameters as a and b, includes classical Tobit model for a = 0 and b = 0 and Alpha‐Power Tobit model for a = 1. It provides very flexible estimation in cases symmetric‐asymmetric data and especially in case non‐normal asymmetric data or/and presence of outlier(s). In addition, Lehmann type II‐G Tobit model (LTII‐GT) is also proposed for b = 1 as a special case of generalization. The performance of proposed two new generalizations of Tobit, namely EGT and LTII‐GT has been compared with Tobit, Alpha‐Power Tobit, and classical estimators by means of a comprehensive Monte Carlo simulation study. The performance of proposed models' ML estimators is illustrated on a real data set. The superiority of the proposed estimators is shown.

Control charts are one of the most powerful tools used to detect and control industrial process deviations in statistical process control. In this paper, a moving average control chart based on a robust scale estimator of standard deviation, namely, the sample median absolute deviation (MAD) statistic, for monitoring process dispersion, is proposed. A simulation study is conducted to evaluate the performance of the proposed moving average median absolute deviation (MA‐MAD) chart, in terms of average run length for various distributions. The results show that the moving average MAD chart performs well in detecting small and moderate shifts in process dispersion, especially when the normality assumption is violated. In addition, this chart is very efficient, especially when the quality characteristic follows a skewed distribution. Numerical and simulated examples are given at the end of the paper.

Modeling uncertain data using Monte Carlo integration method for clustering

ROBUST: an interactive FORTRAN-77 package for exploratory data analysis using parametric, ROBUST and nonparametric location and scale estimates, data transformations, normality tests, and outlier assessment

Sign, Wilcoxon and Mann-Whitney tests are nonparametric methods in one or two-sample problems. The nonparametric methods are alternatives used for testing hypothesis when the standard methods based on the Gaussianity assumption are not suitable to be applied. Recently, the functional data analysis (FDA) has gained relevance in statistical modeling. In FDA, each observation is a curve or function which usually is a realization of a stochastic process. In the literature of FDA, several methods have been proposed for testing hypothesis with samples coming from Gaussian processes. However, when this assumption is not realistic, it is necessary to utilize other approaches. Clustering and regression methods, among others, for non-Gaussian functional data have been proposed recently. In this paper, we propose extensions of the sign, Wilcoxon and Mann-Whitney tests to the functional data context as methods for testing hypothesis when we have one or two samples of non-Gaussian functional data. We use random projections to transform the functional problem into a scalar one, and then we proceed as in the standard case. Based on a simulation study, we show that the proposed tests have a good performance. We illustrate the methodology by applying it to a real data set.

The progressively first failure censored (PFFC) data have become very popular in the past decade due to its usefulness in life testing experiments and in reliability theory. The PFFC data record the number of failures and increase the efficiency of the estimators. The inverse Pareto distribution (IPD) is useful when empirical data suggest a decreasing or upside-down bathtub-shaped failure rate functions. In this article, we consider the classical and Bayesian estimation of the model parameter and the reliability characteristics of the IPD using the PFFC data. The maximum likelihood estimators, asymptotic confidence, and bootstrap confidence intervals are considered in the classical estimation. Under the Bayesian paradigm, Bayes estimators based on non-informative and the gamma informative priors under the squared error loss function using Tierney-Kadane approximation, importance sampling, and Metropolis-Hasting (M-H) algorithm are assessed. In addition, the highest probability density (HPD) intervals based on the M-H algorithm are also constructed. To investigate the efficacy of each of the estimation procedures, numerical computations are performed based on a simulation study. Finally, a real data set is re-analyzed to show the applicability of the IPD model under a censoring scheme.

We propose a novel and unified framework for distribution-free testing under multivariate symmetry (that includes central symmetry, sign symmetry, spherical symmetry, etc.) based on the theory of optimal transport. Our approach leads to notions of distribution-free generalized multivariate signs, absolute ranks and signed-ranks. As a consequence, we develop analogues of the sign and Wilcoxon signed-rank tests that share many of the appealing properties of their one-dimensional counterparts. In particular, the proposed tests are exactly distribution-free in finite samples with an asymptotic normal limit, and adapt to various notions of multivariate symmetry. We study the consistency of the proposed tests and their behavior under local alternatives, and show that the proposed generalized Wilcoxon signed-rank (GWSR) test is particularly powerful against location shift alternatives. We show that in a large class of such models, our GWSR test suffers from no loss in (asymptotic) efficiency, when compared to Hotelling’s T 2 test, despite being nonparametric and exactly distribution-free. An appropriately score transformed version of the GWSR statistic leads to a locally asymptotically optimal test. Further, our method can be readily used to construct distribution-free confidence sets for the center of symmetry.

Adaptive nonparametric tests for a single sample location problem

In this paper, a robust statistical method called projection statistics (PS) is developed into an TV-dimensional complex form required to detect strong nonhomogeneities, or TV-dimensional outlier vectors (i.e., outliers), in space-time adaptive processing (STAP) radar training data, where N is the number of degrees of freedom (DOF) associated with the STAP scenario. The PS technique does not require the estimation of a covariance matrix, as is the case with the generalized inner product (GIP) test. Rather, it uses robust estimators of location (the median) and scale (the median absolute deviation from the median (MAD)) to identify and then excise strong nonhomogeneities in STAP training data that are capable of rendering traditional nonhomogeneity detection (NHD) techniques, such as the GIP, ineffective. Shown in the results section of this paper is sample matrix inversion (SMI) STAP performance using three separate NHD techniques: 1) a power test for high power outlier snapshots, 2) a GIP test, and 3) a PS-based test. The performance when not using any form of NHD is shown for comparison. It is clear from the results that the PS NHD method provides an SMI-STAP processor the best training data selection in the multi channel airborne radar measurements (MCARM) [B.N.S Babu et al., 1996] based radar data scenario studied here, and does so across all range cells of the scenario.

In a randomized block design, there are, in addition to the experimental factor or factors of interest, one or more nuisance factors. The role of blocking is to reduce or eliminate that part of the experimental error attributable to these nuisance factors. Standard parametric ANOVA can be used to analyze such designs, but it requires the normality assumption, and may be misleading if the errors are not normally distributed or if there are outliers. In this article, we consider some distribution‐free tests, such as the sign test, the Wilcoxon signed rank test, Friedman's test, aligned rank tests, Durbin's test, and the row mean score test. None of these methods require the normality assumption, and they can all be used to analyze such designs.

This paper investigates recent developments in the literature on nonparametric instrumental variables estimation and considers the practical importance of the features of these estimators in the context of typically applied econometric models. Our primary focus is on the estimation of econometric models with endogenous regressors, and their marginal effects, without a known functional form. We develop an estimator for the marginal effects and investigate its finite sample performance. We show that when instruments are weak, in the classic sense, the nonparametric estimates of the marginal effect outperforms the classic two-stage least squares estimator, even when the model is correctly specified. When the instruments are strong, we show that the nonparametric estimator for the partial effects is still effective compared to the two-stage least squares estimator even as the number of IVs increases. We also investigate bandwidth choice and find that a rule-of-thumb bandwidth performs relatively well. Whereas cross-validation leads to a better fit when the number of instruments is small, as the number of instruments increases the rule-of-thumb standard actually results in better model fit. In an empirical application we estimate the work-horse aggregate logit demand model, discuss the required nonparametric identification properties, and document the differences between nonparametric and parametric specifications on the estimation of demand elasticities.

In the case of two small toy data sets, we found out that subsampling has a much weaker behavior in the finding of the true number of clusters K than bootstrapping (Mucha and Bartel, Soft bootstrapping in cluster analysis and its comparison with other resampling methods. In: M. Spiliopoulou, L. Schmidt-Thieme, R. Janning (eds.) Data analysis, machine learning and knowledge discovery. Springer, Cham, 2014). In contradiction, Möller and Dörte (Intell Data Anal 10:139–162, 2006) pointed out that “subsampling … clearly outperformed the bootstrapping technique in the detection of correct clustering consensus results.” Obviously, there is a need for further investigations. Therefore here we compare these two resampling techniques based on real and artificial data sets by means of different indices: ARI or Jaccard. We consider hierarchical cluster analysis methods because they find all partitions into K = 2, 3, … clusters in one run only, and, moreover, these results are (usually) unique (Spaeth, Cluster analysis algorithms for data reduction and classification of objects. Ellis Horwood, Chichester, 1982). The methods are tested on two synthetic data sets and two real data sets. Obviously, bootstrapping is better than subsampling in finding the true number of clusters.