Планування експерименту при розв’язанні оберненої задачі побудови толерантних (референсних) інтервалів

We examine the accuracy and precision of parameter estimates for both the exponential and normal distributions when using only a collection of sample extremes. That is, we consider a collection of random variables, where each of the random variables is either the minimum or maximum of a sample of nj independent, identically distributed random variables drawn from a normal or exponential distribution with unknown parameters. Previous work derived estimators for the population parameters assuming the nj sample sizes are constant. Since sample sizes are often not constant in applications, we derive new unbiased estimators that take into account the varying sample sizes. We also perform simulations to assess how the previously derived estimators perform when the constant sample size is simply replaced with the average sample size. We explore how varying the mean, standard deviation, and probability distribution of the sample sizes affects the estimation error. Overall, our results demonstrate that using the average sample size in place of the constant sample size still results in reliable estimates for the population parameters, especially when the average sample size is large. Our estimation framework is applied to a biological example involving plant pollination.

Some Thoughts About Data Type, Distribution, and Statistical Significance

Inference for the model of several treatments and a control

Предложена методика интервального вычисления энтропии случайной величины при замене оценок параметров распределения на их значения, определённые в интервальном виде в системе центр-радиус. Центром оценок параметров распределений принимали их значения, полученные по методу моментов. Радиусы оценок принимали равными полуширине их доверительных интервалов. Решение поставленной задачи рассмотрено для нормального распределения, логистического распределения, гамма-распределения, распределения Вейбулла, логарифмически нормального распределения, показательного распределения.

Thanks to the performance improvement introduced by micro sized functional fillers, application of epoxy composites for electrical insulation purposes has become popular. This paper investigates the dielectric properties of epoxy micro-composites filled with alumina (Al2O3). In particular, measurements of relative permittivity, dissipation factor, and electrical breakdown are performed, and a comprehensive statistical analysis on dielectric properties was conducted. AC breakdown strength (AC-BDS) was analyzed for normal distribution using four methods (Anderson–Darling, Shapiro–Wilk, Ryan–Joiner, and Kolmogorov–Smirnov). In addition, the AC-BDS was analyzed at risk probabilities of 1%, 5%, 10%, and 50% using Weibull distribution functions. Both normal and Weibull distributions were evaluated using the Anderson–Darling (A-D) statistic and p-value. Additionally, the log-normal, gamma, and exponential distributions of AC-BDS were examined by A-D goodness-of-fit test. The hypothesis test results of AC-BDS were fit by normal and Weibull distributions, and the compliance was evaluated by p-value and each method statistics. In addition, the experimental results of AC-BDS were fit by log-normal and gamma distributions, and the goodness-of-fit was evaluated by p-value and A-D testing. On the other hand, exponential distribution was not suitable for p-value and A-D testing. The results showed that the distributions of AC-BDS were the best using log-normal distribution. Meanwhile, statistical analysis results verified the apparent effect of temperature on dielectric properties using a paired t-test. The analysis results of this paper not only contribute to better characterization of epoxy/Al2O3 micro-composites but also introduce a comprehensive approach for performing statistical analysis for electrical insulation materials.

Is the cross-sectional distribution of house prices close to a (log) normal distribution, as is often assumed in empirical studies on house price indexes? How does the distribution evolve over time? To address these questions, we investigate the cross-sectional distribution of house prices in the Greater Tokyo Area for the period 1986 to 2009. We find that size-adjusted house prices follow a lognormal distribution except for the period of the housing bubble and its collapse in Tokyo, for which the price distribution has a substantially heavier right tail than that of a lognormal distribution. In addition, we find that, during the bubble era, the sharp price movements were concentrated in particular areas, and this spatial heterogeneity is the source of the fat upper tail. These findings suggest that the shape of the size-adjusted price distribution, especially the shape of the tail part, may contain information useful for the detection of housing bubbles. Specifically, the presence of a bubble can be safely ruled out if recent price observations are found to follow a lognormal distribution. On the other hand, if there are many outliers, especially near the upper tail, this may indicate the presence of a bubble, since such price observations are unlikely to occur if they follow a lognormal distribution. This method of identifying bubbles is quite different from conventional ones based on aggregate measures of housing prices, and therefore should be a useful tool to supplement existing methods.

The lognormal and Weibull distributions are the most popular distributions for modeling lifetime data. In practical applications, they usually fit the data at hand well. However, their predictions may lead to large differences. The main purpose of the present article is to investigate the impacts of mis-specification between the lognormal and Weibull distributions on the interval estimation of a pth quantile of the distributions for complete data. The coverage probabilities of the confidence intervals (CIs) with mis-specification are evaluated. The results indicate that for both the lognormal and the Weibull distributions, the coverage probabilities are significantly influenced by mis-specification, especially for a small or a large p on lower or upper tail of the distributions. In addition, based on the coverage probabilities with correct and mis-specification, a maxmin criterion is proposed to make a choice between these two distributions. The numerical results indicate that for p ≤ 0.05 and 0.6 ≤ p ≤ 0.8, Weibull distribution is suggested to evaluate CIs of a pth quantile of the distributions, while, for 0.2 ≤ p ≤ 0.5 and p = 0.99, lognormal distribution is suggested to evaluate CIs of a pth quantile of the distributions. Besides, for p = 0.9 and 0.95, lognormal distribution is suggested if the sample size is large enough, while, for p = 0.1, Weibull distribution is suggested if the sample size is large enough. Finally, a simulation study is conducted to evaluate the efficiency of the proposed method.

The article examines the approximation error of the chi-square distribution using a normal distribution, which is relevant when applying statistical tests to evaluate the quality of random number generators. The aim of the study is to determine under what conditions it is acceptable to replace chi-square statistics with their approximations (normal distributions) in order to simplify the calculation of p-values using the complementary error function. To achieve this goal, the research employed mathematical analysis methods, including analysis of the gamma distribution, which includes the chi-square distribution as a special case, and the application of the Berry — Esseen inequality for evaluating the accuracy of approximation. An analytical expression for the third absolute central moment of the distribution was obtained, allowing for an analytical estimation of the minimum length of a bit sequence necessary to achieve a specified level of approximation accuracy. The results showed that, in order to achieve the high level of accuracy required in cryptographic applications, there are significant practical limitations due to the required sample size. These limitations are related to computational complexity, memory requirements, and time costs. The issue of determining the optimal number of intervals in a test using chi-square statistics is considered to optimize the balance between the desired sensitivity with resistance to random fluctuations. The scientific novelty of this work lies in formalizing the conditions for using normal approximation to calculate p-values and developing recommendations for selecting the number of intervals and estimating the minimum sample size. The results obtained contribute to increasing the statistical significance and validity of statistical tests for verifying pseudo-random number generators. They also reduce the influence of heuristic considerations in determining the sample size, which increases the reliability of evaluating random number generator characteristics. In practical terms, this work aims to unify statistical analysis procedures in cryptography by formalizing conditions under which it is correct to replace the chi-square distribution with the normal distribution.

The cost of medical resources used is often recorded for each patient in clinical studies in order to inform decision-making. Although cost data are generally skewed to the right, interest is in making inferences about the population mean cost. Common methods for non-normal data, such as data transformation, assuming asymptotic normality of the sample mean or non-parametric bootstrapping, are not ideal. This paper describes possible parametric models for analysing cost data. Four example data sets are considered, which have different sample sizes and degrees of skewness. Normal, gamma, log-normal, and log-logistic distributions are fitted, together with three-parameter versions of the latter three distributions. Maximum likelihood estimates of the population mean are found; confidence intervals are derived by a parametric BC(a) bootstrap and checked by MCMC methods. Differences between model fits and inferences are explored.Skewed parametric distributions fit cost data better than the normal distribution, and should in principle be preferred for estimating the population mean cost. However for some data sets, we find that models that fit badly can give similar inferences to those that fit well. Conversely, particularly when sample sizes are not large, different parametric models that fit the data equally well can lead to substantially different inferences. We conclude that inferences are sensitive to choice of statistical model, which itself can remain uncertain unless there is enough data to model the tail of the distribution accurately. Investigating the sensitivity of conclusions to choice of model should thus be an essential component of analysing cost data in practice.

Simple fixed bed models such as the Bohart–Adams and Thomas equations are often used to describe the breakthrough characteristics of water and air contaminant adsorption in fixed bed adsorbers. However, these popular models are confined to correlating highly symmetric breakthrough curves. The present study explores the feasibility of using two probability distributions (normal and log-normal) to track asymmetric breakthrough curves for water and air contaminant adsorption in fixed bed columns (ciprofloxacin, ammonium, hydrogen chloride, and hydrogen sulfide). The normal and log-normal probability distributions provided accurate fits to the slightly asymmetric ciprofloxacin breakthrough curve. They also provided a good representation of the overall shape of the ammonium breakthrough curve, but failed to describe the leakage of ammonium during the initial period of column operation. The log-normal distribution was able to match the asymmetric HCI and H2S breakthrough curves. The normal distribution, by comparison, failed to describe these two asymmetric breakthrough curves. Because the log-normal distribution has a floating inflection point, it was very effective in describing the asymmetric HCI and H2S breakthrough curves that were sharp initially and then broadened significantly as the column approached saturation. The ability of the normal distribution to track such asymmetric breakthrough data was impeded by its invariant inflection point.

The available coefficient of friction (ACOF) for human locomotion is the maximum coefficient of friction that can be supported without a slip at the shoe and floor interface. A statistical model was introduced to estimate the probability of slip by comparing the ACOF with the required coefficient of friction, assuming that both coefficients have stochastic distributions. This paper presents an investigation of the stochastic distributions of the ACOF of quarry tiles under dry, water and glycerol conditions. One hundred friction measurements were performed on a walkway under the surface conditions of dry, water and 45% glycerol concentration. The Kolmogorov-Smirnov goodness-of-fit test was used to determine if the distribution of the ACOF was a good fit with the normal, log-normal and Weibull distributions. The results indicated that the ACOF appears to fit the normal and log-normal distributions better than the Weibull distribution for the water and glycerol conditions. However, no match was found between the distribution of ACOF under the dry condition and any of the three continuous distributions evaluated. Based on limited data, a normal distribution might be more appropriate due to its simplicity, practicality and familiarity among the three distributions evaluated.

Comparing computing formulas for estimating concentration ratios

A previous method for deciding if a set of time-to-fail data follows a lognormal distribution or a Weibull distribution is expanded upon. Pearson's s-correlation coefficient is calculated for lognormal and Weibull probability plots of the time-to-fail data. The test statistic is the ratio of the two s-correlation coefficients. When standardized, the lognormal and Weibull variables map into 1 of 2 gamma distributions with no dependence on the shape or scaling factors, confirming earlier observations. Using a set of Monte Carlo simulations, the test statistic was found to be s-normally distributed to good approximation. Formulas for estimating the mean and standard deviation of the test statistic were derived, allowing for an estimate of the probability of hypothesis test errors. As anticipated, the test capability increases with increasing sample size, but only if a substantial fraction of the parts actually fail. If less than 10% of the parts are stressed to failure, then it is almost impossible to distinguish between lognormal and Weibull distributions. If all parts are stressed to failure, the probability of making a correct choice is fair for sample sizes as small as 10, and becomes quite good if the sample size is at least 50. The statistical technique for distinguishing lognormal from Weibull distributions is presented. Its theoretical foundation is given at a qualitative level, and the range of useful application is explored. An approximate form for the distribution of the test statistic is inferred from Monte Carlo simulation.

SYNOPTIC ABSTRACTTwo tables of percentiles of the asymptotic test statistics of separate families of hypotheses proposed by D. R. Cox are presented. The symmetrical probability levels of α at 0.025, 0.25, 0.50, 0.75, and 0.975 are evaluated for various selected sample sizes. Three different methods for approximating these selected percentiles are compared. A comparison for small sample sizes is also presented. 95% confidence intervals are given for each of the percentile points estimated by the three methods. These results can be used for testing hypotheses involving lognormal and exponential distributions.

A return on investment analysis of applying health monitoring to LED lighting systems