Parameter Evaluation and Prediction in Information Poor System Using Fuzzy-Set Theory

Mixed Model Variance Analysis with Normal Error and Possibly Non-Normal Other Random Effects: Part I: The Univariate Case

The robustness of confidence intervals for a scale parameter based on M-esimators is studied, especially in small size samples. The coverage probablity is used as measure of robustness. A theorem for a lower bound of the minimum coverage probability of M-estimators is presented and it is applied in order to examine the behavior of the standard deviation and the median absolute deviation, as interval estimators. This bound can confirm the robustness of any other scale M-estimator in interval estimation. The idea of stretching is used to formulate the family of distributions that are considered as underlying. Critical values for the confidence interval are computed where it is needed, that is for the median absolute deviation in the Normal, Uniform and Cauchy distribution and for the standard deviation in the Uniform and Cauchy distribution. Simulation results have been achieved for the estimation of the coverage probabilities and the critical values.

Consider a finite sample from an unknown distribution over a countable alphabet. Unobserved events are alphabet symbols which do not appear in the sample. Estimating the probabilities of unobserved events is a basic problem in statistics and related fields, which was extensively studied in the context of point estimation. In this work we introduce a novel interval estimation scheme for unobserved events. Our proposed framework applies selective inference, as we construct confidence intervals (CIs) for the desired set of parameters. Interestingly, we show that obtained CIs are dimension-free, as they do not grow with the alphabet size. Further, we show that these CIs are (almost) tight, in the sense that they cannot be further improved without violating the prescribed coverage rate. We demonstrate the performance of our proposed scheme in synthetic and real-world experiments, showing a significant improvement over the alternatives. Finally, we apply our proposed scheme to large alphabet modeling. We introduce a novel simultaneous CI scheme for large alphabet distributions which outperforms currently known methods while maintaining the prescribed coverage rate. Supplementary materials for this article are available online including a standardized description of the materials available for reproducing the work.

Summary This paper discusses interval estimation of the simple difference (SD) between the proportions of the primary infection and the secondary infection, given the primary infection, by developing three asymptotic interval estimators using Wald’s test statistic, the likelihood-ratio test, and the basic principle of Fieller’s theorem. This paper further evaluates and compares the performance of these interval estimators with respect to the coverage probability and the expected length of the resulting confidence intervals. This paper finds that the asymptotic confidence interval using the likelihood ratio test consistently performs well in all situations considered here. When the underlying SD is within 0.10 and the total number of subjects is not large (say, 50), this paper further finds that the interval estimators using Fieller’s theorem would be preferable to the estimator using the Wald’s test statistic if the primary infection probability were moderate (say, 0.30), but the latter is preferable to the former if this probability were large (say, 0.80). When the total number of subjects is large (say, 200), all the three interval estimators perform well in almost all situations considered in this paper. In these cases, for simplicity, we may apply either of the two interval estimators using Wald’s test statistic or Fieller’s theorem without losing much accuracy and efficiency as compared with the interval estimator using the asymptotic likelihood ratio test.

A new research frontier in AI and data mining seeks to develop methods to automatically discover relevant variables among many irrelevant ones. In this paper, we present four algorithms that output such crucial variables in PAC model with membership queries. The first algorithm executes the task under any unknown distribution by measuring the distance between virtual and real targets. The second algorithm exhausts virtual version space under an arbitrary distribution. The third algorithm exhausts universal set under the uniform distribution. The fourth algorithm measures influence of variables under the uniform distribution. Knowing the number r of relevant variables, the first algorithm runs in almost linear time for r. The second and the third ones use less membership queries than the first one, but runs in time exponential for r. The fourth one enumerates highly influential variables in quadratic time for r.

Accurate confidence interval (CI) estimation is fundamental to statistical inference, yet traditional methods particularly the classical t-distribution-based intervals are often unreliable in small-sample contexts and under non-normal conditions. This study investigates the performance of alternative CI estimation approaches, including bootstrap methods (percentile and bias-corrected and accelerated [BCa]) and Bayesian credible intervals, with the aim of identifying robust methods suitable for small samples. A simulation-based framework was employed, generating synthetic data from three distinct distributions: normal, exponential (skewed), and t-distribution with 3 degrees of freedom (heavy-tailed). For each distribution, sample sizes of n=10,15,20 and 30 were examined, with 10,000 replications per condition. Each CI method was evaluated based on coverage probability, average interval width, bias, and root mean squared error (RMSE). The results reveal that classical t-intervals perform adequately only under normality and larger sample sizes but exhibit significant under-coverage and inflated bias under skewed or heavy-tailed distributions. Bootstrap percentile intervals, while simple to implement, fail to correct for bias and skewness. In contrast, the BCa bootstrap method consistently achieved closer-to-nominal coverage and improved accuracy across all conditions. Bayesian credible intervals demonstrated the most robust performance overall, providing reliable coverage, minimal bias, and the lowest RMSE across all distributions and sample sizes. The study concludes that researchers working with small samples should reconsider their reliance on classical CI methods. Bootstrap BCa and Bayesian credible intervals are recommended as more robust alternatives for reliable inference. These findings have important implications for statistical methodology, applied research practice, and the reform of statistical training and reporting standards. Keywords: Confidence Intervals, Small Samples, Bootstrap, Bca, Bayesian Inference, Simulation, Coverage Probability, Interval Estimation.

Standardized effect sizes and confidence intervals are useful statistical assessments for comparing results across different studies when measurement units are not directly comparable. This paper aims to describe and compare confidence interval estimation methods for the standardized contrasts of treatment effects in ANCOVA designs. Sample size procedures are also presented to assure that the resulting confidence intervals yield informative estimation with adequate precision. Exact interval estimation approach has theoretical and empirical advantages in coverage probability and interval width over the approximate interval procedures. Numerical investigations of the existing method reveal that the omission of covariate variables has a negative impact on sample size calculations for precise interval estimation, especially when there is disparity in influential covariate variables. The proposed approaches and developed computer programs fully utilize covariate properties in interval estimation and provide accurate sample size determinations under the precision considerations of the expected interval width and the assurance probability of interval width.

A conditional approach to robustness is described and applied to interval estimation for a location parameter. Two new types of confidence interval estimators using two-dimensional numerical integrations within location-and-scale configurations emerge. Strong confidence intervals have robust conditional coverage probabilities. Bioptimal confidence intervals are as short as possible and have robust overall coverage probabilities. Robust intervals compromising the Gaussian and the slash distributions serve as examples. A comparison with a variety of other confidence intervals shows the nonrobustness of rank-based interval estimators. For samples of size 10, robust intervals based on M-estimators are clearly superior. Tuning constants must be chosen carefully for interval estimation. An interval based on Tukey's biweight, for example, should be used with tuning constants of about 7, 8, and 11 for samples of size 20, 10, and 5, respectively.

Interval estimates – estimates of parameters that include an allowance for sampling uncertainty – have long been touted as a key component of statistical analyses. There are several kinds of interval estimates, but the most popular are confidence intervals (CIs): intervals that contain the true parameter value in some known proportion of repeated samples, on average. The width of confidence intervals is thought to index the precision of an estimate; CIs are thought to be a guide to which parameter values are plausible or reasonable; and the confidence coefficient of the interval (e.g., 95 %) is thought to index the plausibility that the true parameter is included in the interval. We show in a number of examples that CIs do not necessarily have any of these properties, and can lead to unjustified or arbitrary inferences. For this reason, we caution against relying upon confidence interval theory to justify interval estimates, and suggest that other theories of interval estimation should be used instead. Electronic supplementary material The online version of this article (doi:10.3758/s13423-015-0947-8) contains supplementary material, which is available to authorized users.

Interval estimation in multivariate curve resolution by exploiting the principles of error propagation in linear least squares

Finding a new drug which is both safe and efficient is an expensive and time-consuming endeavour. In particular, establishing the ‘dose-effect relationship’ – how beneficial a drug is at different dosages – can be challenging. Predicting this curve requires gathering experimental data by exposing and recording how cells respond to various levels of the drug. However, extreme values are often observed at low and high dosages, potentially introducing errors that are hard to correct in the prediction process. Yet, these extreme observations are sometimes genuine so researchers cannot just ignore them. To improve dose-effect estimation, Zhou, Liu, Fang et al. developed a new general-purpose approach. It uses advanced statistical modelling to account for extremes in lab data. This strategy outperformed other methods when dealing with these observations while also providing higher efficiency in data analysis with more uniform data in experiments. To facilitate implementation, Zhou, Liu, Fang et al. set up a user-friendly tool baptized ‘REAP’; this free online resource allows scientists without advanced statistical experience to harness the new approach and to perform dose-effect analysis more easily and accurately. This could boost research across many different disciplines that examine the effects of chemicals on cells.

The enhanced interval approach (EIA) is one of the most important approaches for constructing interval type-2 fuzzy set (IT2 FS) from data intervals that are collected from a survey. However, the uniform distribution used in EIA is rough. And the shape (Left-shoulder, Right-shoulder or Interior) of the fuzzy set (FS) affects the value of the membership function (MF) of the final IT2 FS a lot. To guarantee that the final IT2 FSs are consistent with fuzzistics characteristic of the original data and improve robustness, this paper proposes a normal distribution associated with free parameter (FP) as the supplement of uniform distribution in the data part of EIA. Furthermore, a general frame for encoding words from data intervals, called general interval approach (GIA), is built. GIA includes a data part, fuzzy set (FS) part and footprint of uncertainty (FOU) part. The data part maps data intervals to probability distributions, in which normal distribution with FP and uniform distribution are included. The FS part encodes the probability distributions produced by the data part to fuzzy MFs. Gaussian MF is discussed, and the parameter transformation table is obtained. In the FOU part, the FOU of IT2 FS is built by collecting the obtained T1 FSs. The way to construct a Gaussian FOU is, for the first time, proposed in this paper. The validity of GIA is verified by experiments. Compared with EIA, the IT2 FSs built by GIA can keep the statistic characteristic of the original data intervals in the greatest degree and improve the robustness owing to the FP.

A problem with the estimation of quantiles occurs when the sample comes from an unknown distribution. The estimation uses the bootstrap method in the version that the literature refers to as exact. Three bootstrap estimators were used: two of them based on one order statistic, and the third on a linear combination of two order statistics (for an integer). The distribution of the exact bootstrap estimator based on a single order statistic is known. It has been shown that there is no general form of the distribution of the exact bootstrap estimator based on two order statistics. However, it is possible to calculate such a distribution – the article presents the algorithm that performs such a task. The bootstrap confidence intervals were constructed using the exact percentile method. It has been shown that if the estimator is based on a single order statistic, it is known in advance which elements of the primary sample are the limits of the confidence intervals, so there is no need to resample. The intervals determined by the exact percentile method were compared with those constructed using other methods. It has been shown that the information on the direction of the asymmetry of the distribution that the sample comes from is worth considering when selecting the rank of the order statistic used as an estimator. Attention is paid to the influence of the quality of the pseudorandom number generators on the results of the Monte Carlo simulation.

We consider the problem of distribution-free testing of the class of monotone monomials and the class of monomials over n variables.0 While there are very efficient algorithms for testing a variety of functions classes when the underlying distribution is uniform, designing distribution-free algorithms (which must work under any arbitrary and unknown distribution), tends to be a more challenging task. When the underlying distribution is uniform, Parnas et al. (SIAM Journal on Discrete Math, 2002) give an algorithm for testing (monotone) monomials whose query complexity does not depends on n, and whose dependence on the distance parameter is (inverse) linear. In contrast, Glasner and Servedio (in Proceedings of RANDOM, 2007) prove that every distribution-free testing algorithm for monotone monomials as well as for general monomials must have query complexity Ω(n1/5) (for a constant distance parameter e). In this paper we present distribution-free testing algorithms for these classes where the query complexity of the algorithms is O(n1/2/e). We note that as opposed to previous results for distribution-free testing, our algorithms do not build on the algorithms that work under the uniform distribution. Rather, we define and exploit certain structural properties of monomials (and functions that differ from them in a nonnegligible manner), which were not used in previous work on property testing.

ABSTRACTControl charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having non-normal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used in such circumstances. In this paper, we propose a new variance chart based on a simple statistic to monitor process variance shifts. We explore the sampling properties of the new monitoring statistic and calculate the average run lengths (ARLs) of the proposed variance chart. Furthermore, an arcsine transformed exponentially weighted moving average (EWMA) chart is proposed because the ARLs of this modified chart are more intuitive and reasonable than those of the variance chart. We compare the out-of-control variance detection performance of the proposed variance chart with that of the non-parametric Mood variance (NP-M) chart with runs rules, developed by Zombade and Ghute [Nonparametric control chart for variability using runs rules. Experiment. 2014;24(4):1683–1691], and the nonparametric likelihood ratio-based distribution-free exponential weighted moving average (NLE) chart and the combination of traditional exponential weighted moving average (EWMA) mean and EWMA variance (CEW) control chart proposed by Zou and Tsung [Likelihood ratio-based distribution-free EWMA control charts. J Qual Technol. 2010;42(2):174–196] by considering cases in which the critical quality characteristic has a normal, a double exponential or a uniform distribution. Comparison results showed that the proposed chart performs better than the NP-M with runs rules, and the NLE and CEW control charts. A numerical example of service times with a right-skewed distribution from a service system of a bank branch in Taiwan is used to illustrate the application of the proposed variance chart and of the arcsine transformed EWMA chart and to compare them with three existing variance (or standard deviation) charts. The proposed charts show better detection performance than those three existing variance charts in monitoring and detecting shifts in the process variance.