The effect of rounding on the significance level and power of certain test statistics for non-normal data

In a previously published study, the effects of rounding on the significance and power of four test statistics were considered when the parent population was normal. Here we investigate how these tests will perform for rounded non-normal data. Guidelines are given on how the degree of precision recommended for normal populations can be applied when the population is non-normal.     

Symmetric and asymmetric rounding: a review and some new results

Using rounded data to estimate moments and regression coefficients typically biases the estimates. We explore the bias-inducing effects of rounding, thereby reviewing widely dispersed and often half forgotten results in the literature. Under appropriate conditions, these effects can be approximately rectified by versions of Sheppard’s correction formula. We discuss the conditions under which these approximations are valid and also investigate the efficiency loss caused by rounding. The rounding error, which corresponds to the measurement error of a measurement error model, has a marginal distribution, which can be approximated by the uniform distribution, but is not independent of the true value. In order to take account of rounding preferences (heaping), we generalize the concept of simple rounding to that of asymmetric rounding and consider its effect on the mean and variance of a distribution.     

On Generalized Linear Model Method for Detecting Dispersion Effects in Unreplicated Factorial Designs

There are numerous approaches to screen location effects for unreplicated experiments, but only a handful to screen dispersion effects. Generalized linear models, popular in analyses of non-normal data, were recently proposed to screen both location and dispersion effects simultaneously. This paper illustrates and explains the impact of unidentified location effects on dispersion effects identification for such procedures. A remedy is proposed to recover the loss of power of the GLM method due to such impact.     

Probabilistic rounding and Sheppard’s correction

When rounded data are used in place of the true values to compute the variance of a variable or a regression line, the results will be distorted. Under suitable smoothness conditions on the distribution of the variable(s) involved, this bias, however, can be corrected with very high precision by using the well-known Sheppard’s correction. In this paper, Sheppard’s correction is generalized to cover more general forms of rounding procedures than just simple rounding, viz., probabilistic rounding, which includes asymmetric rounding and mixture rounding.     

Effects of Rounding Data Sampled from the Exponential Distribution

This paper looks at the effects of rounding data sampled from the exponential distribution. It examines the nature of the rounded distribution, together with the resulting error distribution. The influence of these distributions on estimates and tests of hypothesis is investigated. The results indicate that even a moderate degree of rounding can cause the bias in an estimator to increase, whereas in hypothesis tests level of significance is altered.     

BAYESIAN ANALYSIS OF THE LINEAR REGRESSION MODEL WITH NON-NORMAL DISTURBANCES

This paper considers the Bayesian analysis of a linear regression model with identically independently distributed non-normal disturbances. The distribution of disturbances is approximated by an Edgeworth series distribution with cumulants, of order higher than fourth, negligible. The posterior distribution of the regression coefficients vector is obtained under the assumption of a g-prior distribution for the parameters of the model. The Bayes estimator and its Bayes risk of the estimator are derived under a quadratic loss structure.     

JACKKNIFE TESTS FOR HETEROSCEDASTICITY IN THE GENERAL LINEAR MODEL

In this paper, tests based on the Jackknife technique are proposed to test for heteroscedasticity in the linear regression model when the errors are non-normal. These are the Jackknifed Goldfeld-Quandt (GQ), and jack-knife related variations of White (H), Lagrange multiplier (LM), Glejser (GL) and Bickel (B) tests. The power of the proposed tests is compared with that of GQ, H, LM, GL and B tests; and the robustness to the error distribution is analyzed under several heteroscedastic assumptions. The GQ test is by far the best test if the error distribution is close to normal, however, GQ test is not robust against non-normal errors. By applying the jackknife technique to the regression a more robust statistic (GQJRG) is produced but the cost is a loss in power. The GQJRG statistic generally is not M powerful as the Bickel (BlOLS) and Glejser (GLlOLS) statistics.     

Robustness of Bayes Prediction Under Error-in-Variables Superpopulation Model

In this article, we consider Bayes prediction in a finite population under the simple location error-in-variables superpopulation model. Bayes predictor of the finite population mean under Zellner's balanced loss function and the corresponding relative losses and relative savings loss are derived. The prior distribution of the unknown location parameter of the model is assumed to have a non-normal distribution belonging to the class of Edgeworth series distributions. Effects of non normality of the “true” prior distribution and that of a possible misspecification of the loss function on the Bayes predictor are illustrated for a hypothetical population.     

面板数据的分位回归方法及其模拟研究

文章讨论了含有固定效应的面板数据模型,给出了3种估计未知参数的分位回归方法,蒙特卡洛模拟结果显示这些分位回归方法是处理面板数据的有效手段,且在误差非正态时优于均值回归方法。文章最后给出了一个真实数据的建模案例,得到了有利于决策的有用参考信息。     

Robust 2k factorial design with Weibull error distributions

It is well known that the least squares method is optimal only if the error distributions are normally distributed. However, in practice, non-normal distributions are more prevalent. If the error terms have a non-normal distribution, then the efficiency of least squares estimates and tests is very low. In this paper, we consider the 2^k factorial design when the distribution of error terms are Weibull W(p,σ). From the methodology of modified likelihood, we develop robust and efficient estimators for the parameters in 2^k factorial design. F statistics based on modified maximum likelihood estimators (MMLE) for testing the main effects and interaction are defined. They are shown to have high powers and better robustness properties as compared to the normal theory solutions. A real data set is analysed.     

Economic-statistical design of X ¥ charts for non-normal data by considering quality loss

When the X ¥ control chart is used to monitor a process, three parameters should be determined: the sample size, the sampling interval between successive samples, and the control limits of the chart. Duncan presented a cost model to determine the three parameters for an X ¥ chart. Alexander et al. combined Duncan's cost model with the Taguchi loss function to present a loss model for determining the three parameters. In this paper, the Burr distribution is employed to conduct the economic-statistical design of X ¥ charts for non-normal data. Alexander's loss model is used as the objective function, and the cumulative function of the Burr distribution is applied to derive the statistical constraints of the design. An example is presented to illustrate the solution procedure. From the results of the sensitivity analyses, we find that small values of the skewness coefficient have no significant effect on the optimal design; however, a larger value of skewness coefficient leads to a slightly larger sample size and sampling interval, as well as wider control limits. Meanwhile, an increase on the kurtosis coefficient results in an increase on the sample size and wider control limits.     

On Z-estimation by rounded data

It is often assumed in statistics that the random variables under consideration come from a continuous distribution. However, real data is always given in a rounded (discretized) form. The rounding errors become serious when the sample size is large. In this paper, we consider the situation where the mesh of discretization tends to zero as the sample size tends to infinity, and give some sets of sufficient conditions under which the rounding errors can be asymptotically ignored, in the context of Z-estimation. It is theoretically proved that the mid-point discretization is preferable.     

On the umvu estimators after using a normalizing transformation

The conditional distribution given complete sufficient statistics is used along with the Rao-Blackwell theorem to obtain uniformly minimum variance unbiased (UMVU) estimators after a transformation to normality has been applied to data. The estimators considered are for the mean, the variance and the cumulative distribution of the original non-normal data. Previous procedures to obtain UMVU estimators have used Laplace transforms, Taylor expansions and the jackknife. An integration method developed in this paper requires only integrability of the normalizing transformation function. This method is easy to employ and it is always possible to obtain a numerical result.     

Economic design of VSI X̄ control chart with correlated non-normal data under multiple assignable causes

It has been recently revealed that the Shewhart control charts with variable sampling interval (VSI) perform better than the traditional Shewhart chart with the fixed sampling interval in detecting shifts in the process. In most of these research works, the normality and independency of the process data or measurements are assumed and that the process is subjected to only one assignable cause. While, in practice, these assumptions usually do not hold, some recent studies are focused on working with only one or two of these violations. In this paper, the situation in which the process data are correlated and follow a non-normal distribution and that there is multiplicity of assignable causes in the process is considered. For this case, a cost model for the economic design of the VSI X? control chart is developed, where the Burr distribution is employed to represent the non-normal distribution of the process data. To obtain the optimal values of the design parameters, a genetic algorithm is employed in which the response surface methodology is applied. A numerical example is presented to show the applicability and effectiveness of the proposed methodology. Sensitivity analysis is also carried out to evaluate the effects of cost and input parameters on the performance of the chart.     

Applications of a new power normal family

The main purpose of this paper is to give an algorithm to attain joint normality of non-normal multivariate observations through a new power normal family introduced by the author (Isogai, 1999). The algorithm tries to transform each marginal variable simultaneously to joint normality, but due to a large number of parameters it repeats a maximization process with respect to the conditional normal density of one transformed variable given the other transformed variables. A non-normal data set is used to examine performance of the algorithm, and the degree of achievement of joint normality is evaluated by measures of multivariate skewness and kurtosis. Besides the above topic, making use of properties of our power normal family, we discuss not only a normal approximation formula of non-central F distributions in the frame of regression analysis but also some decomposition formulas of a power parameter, which appear in a Wilson-Hilferty power transformation setting.     

REDUCED DISTANCE MEASURES AND TRANSFORMATIONS IN PROCESSING MULTIVARIATE OUTLIERS

The study of multivariate outliers raises many problems of definition, principle and manipulation. Well-authenticated tests of discordancy exist only for the multivariate normal distribution. Detection of outliers in non-normal distributions involves the adoption of appropriate criteria to represent 'extremeness' of observations in a sample; corresponding tests of discordancy usually require tedious, or even intractable, distributional and computational manipulations. A class of transformations of the data is considered with a view of transferring some of the familiar and desirable features of discordancy tests for normal samples to non-normal situations.     

Effect on Bayesian inference of the degree of precision of recorded data

In a previous study, the effect of rounding on classical statistical techniques was considered. Here, we consider how rounded data may affect the posterior distribution and, thus, any Bayesian inferences made. The results in this paper indicate that Bayesian inferences can be sensitive to the roundingprocess.     

Median rankit control chart by the quantile approach

It is desirable that the data for a statistical control chart be normally distributed. However, if the data are not normal, then a transformation can be used, e.g. Box-Cox transformations, to produce a suitable control chart. In this paper we will discuss a quantile approach to produce a control chart and to estimate median rankit for various non-normal distributions. We will also provide examples of logistic data to indicate how a quantile approach could be used to construct a control chart for a non-normal distribution using a median rankit.     

A comparison of bootstrap method and Edgeworth expansion in approximating the distribution of sample variance — one sample and two sample cases

The performance of the bootstrap method and the Edgeworth expansion in approximating the distribution of sample variance are compared when the data are from a non-normal population. Both approximations are very good. so long as the parent population is close to normal.     

Multiple imputation using multivariate gh transformations

Multiple imputation has emerged as a popular approach to handling data sets with missing values. For incomplete continuous variables, imputations are usually produced using multivariate normal models. However, this approach might be problematic for variables with a strong non-normal shape, as it would generate imputations incoherent with actual distributions and thus lead to incorrect inferences. For non-normal data, we consider a multivariate extension of Tukey's gh distribution/transformation [38] to accommodate skewness and/or kurtosis and capture the correlation among the variables. We propose an algorithm to fit the incomplete data with the model and generate imputations. We apply the method to a national data set for hospital performance on several standard quality measures, which are highly skewed to the left and substantially correlated with each other. We use Monte Carlo studies to assess the performance of the proposed approach. We discuss possible generalizations and give some advices to practitioners on how to handle non-normal incomplete data.