On Symmetrizing Transformation of the Sample Coefficient of Variation from a Normal Population

Variance-stabilizing transformation (VST) for the sample coefficient of variation is often used as a normalizing transformation and may be used for inference on the population coefficient of variation. However, for small samples, the VST may not be symmetric and hence there is a scope of improvement in its performance by seeking a symmetrizing transformation. This article investigates such a transformation that has been obtained by solving a differential equation. The solution may be complex; hence, a numerical strategy is employed in order to make the approximation practically useful. This transformation has been compared with explicitly available VST. The approach has been illustrated on real data from an agricultural experiment concentrating on inference on single samples; however, the method may be generally applicable to multiple samples when testing the homogeneity of coefficients of variation for many populations by following usual normal-theory-based methods applied on transformed statistics.     

Kurtosis modelling by means of the J-transformation

Summary: The H–family of distributions or H–distributions, introduced by Tukey (1960; 1977), are generated by a single transformation of the standard normal distribution and allow for leptokurtosis represented by the parameter h. Alternatively, Haynes et al. (1997) generated leptokurtic distributions by applying the K–transformation to the normal distribution. In this study we propose a third transformation, the so–called J–transformation, and derive some properties of this transformation. Moreover, so-called elongation generating functions (EGFs) are introduced. By means of EGFs we are able to visualize the strength of tail elongation and to construct new transformations. Finally, we compare the three transformations towards their goodness–of–fit in the context of financial return data.     

Normalization of the origin-shifted exponential distribution for control chart construction

This study demonstrates that a location parameter of an exponential distribution significantly influences normalization of the exponential. The Kullback–Leibler information number is shown to be an appropriate index for measuring data normality using a location parameter. Control charts based on probability limits and transformation are compared for known and estimated location parameters. The probabilities of type II error (β-risks) and average run length (ARL) without a location parameter indicate an ability to detect an out-of-control signal of an individual chart using a power transformation similar to using probability limits. The β-risks and ARL of control charts with an estimated location parameter deviate significantly from their theoretical values when a small sample size of n≤50 is used. Therefore, without taking into account of the existence of a location parameter, the control charts result in inaccurate detection of an out-of-control signal regardless of whether a power or natural logarithmic transformation is used. The effects of a location parameter should be eliminated before transformation. Two examples are presented to illustrate these findings.     

An Approximate Normalizing Power Transformation for the FDistribution

A power transformation of the Fdistribution is presented, yielding simple normal approximations for both probabilities and quantiles of the distribution. The transformation proposed is shown to produce the well-known Wilson-Hilferty cube root transformation (Wilson and Hilferty, 1931) for the chi-square distribution as a limiting case, as well as the Fisher logarithmic transformation (Fisher, 1924, 1925) for equal degrees of freedom. A numerical assessment of the accuracy achieved for approximating tail probabilities and a comparison with some of the existing approximate procedures are given.     

On the Relation between Edge and Vertex Modelling in Shape Analysis

Objects in the plane with no obvious landmarks can be described by either vertex transformation vectors or edge transformation vectors. In this paper we provide the relation between the two transformation vectors. Grenander & Miller (1994 ) use a multivariate normal distribution with a block circulant covariance matrix to model the edge transformation vector. This type of model is also feasible for the vertex transformation vector and in certain cases the free parameters of the two models match up in a simple way. A vertex model and an edge model are applied to a data set of sand particles to explore shape variability.     

On robustness of usual confidence region under transformation misspecification

Robustness of confidence region for linear model parameters following a misspecified transformation of dependent variable is studied. It is shown that when error standard deviation is moderate to large the usual confidence region is robust against transformation misspecification. When error standard deviation is small the usual confidence region could be very conservative for structured models and slightly liberal for unstructured models. However, the conservativeness in structured case can be controlled if the transformation is selected with the help of data rather than prior information since this is the case when data is able to provide a very accurate estimate of transformation.     

On a new family of shifted logarithmic transformations

Most parametric statistical methods are based on a set of assumptions: normality, linearity and homoscedasticity. Transformation of a metric response is a popular method to meet these assumptions. In particular, transformation of the response of a linear model is a popular method when attempting to satisfy the Gaussian assumptions on the error components in the model. A particular problem with common transformations such as the logarithm or the Box–Cox family is that negative and zero data values cannot be transformed. This paper proposes a new transformation which allows negative and zero data values. The method for estimating the transformation parameter consider an objective criteria based on kurtosis and skewness for achieving normality. Use of the new transformation and the method for estimating the transformation parameter are illustrated with three data sets.     

An alternative approximation to the variance of transformation score

An alternative approximation to the variance of transformation score is given, based on an asymptotic expansion of the transformation estimator. It is then compared with the variance approximation given by Lawrance (1987) in terms of standardized scores. Simulations show that the two standardized scores behave very similarly when model error standard deviation is small. However,when error standard deviation is not small, the new standardized score outperforms that of Lawrance,especially in the structured models.     

Semiparametric Estimation in Transformation Models with Cure Fraction

Semiparametric transformation model has been extensively investigated in the literature. The model, however, has little dealt with survival data with cure fraction. In this article, we consider a class of semi-parametric transformation models, where an unknown transformation of the survival times with cure fraction is assumed to be linearly related to the covariates and the error distributions are parametrically specified by an extreme value distribution with unknown parameters. Estimators for the coefficients of covariates are obtained from pseudo Z-estimator procedures allowing censored observations. We show that the estimators are consistent and asymptotically normal. The bootstrap estimation of the variances of the estimators is also investigated.     

Median estimation through a regression transformation

The author considers the problem of constructing confidence intervals for the median of a future observation at certain values of exogenous variables, following a normalizing transformation. He shows that when this transformation is estimated, the usual interval obtained through an inverse transformation needs to be corrected, even when the sample size is large. He then gives a simple analytical solution to this problem and provides simulation results confirming the good small‐sample properties of the corrected interval. He also presents two concrete illustrations.     

Predicting a Future Median Life through a Power Transformation

A simple and unified prediction interval (PI) for the median of a future lifetime can be obtained through a power transformation. This interval usually possesses the correct coverage, at least asymptotically, when the transformation is known. However, when the transformation is unknown and is estimated from the data, a correction is required. A simple correction factor is derived based on large sample theory. Simulation shows that the unified PI after correction performs well. When compared with the existing frequentist PI's, it shows an equivalent or a better performance in terms of coverage probability and average length of the interval. Its nonparametric aspect and the ease of usage make it very attractive to practitioners. Real data examples are provided for illustration.     

On Sensitivity of Inverse Response Plot Estimation and the Benefits of a Robust Estimation Approach

Abstract. Inverse response plots are a useful tool in determining a response transformation function for response linearization in regression. Under some mild conditions it is possible to seek such transformations by plotting ordinary least squares fits versus the responses. A common approach is then to use nonlinear least squares to estimate a transformation by modelling the fits on the transformed response where the transformation function depends on an unknown parameter to be estimated. We provide insight into this approach by considering sensitivity of the estimation via the influence function. For example, estimation is insensitive to the method chosen to estimate the fits in the initial step. Additionally, the inverse response plot does not provide direct information on how well the transformation parameter is being estimated and poor inverse response plots may still result in good estimates. We also introduce a simple robustified process that can vastly improve estimation.     

Consistency of the bootstrap for the ransformed two-sample t-test

A method of bootstrapping the two-sample t-test after a Box-Cox transformation is proposed. The procedure is shown to be consistent and asymptotically as efficient as the non-bootstrapped Box-Cox t-test. Because the bootstrap samples are drawn without the assumption of the same distributional shapes,the procedure may be more robust against violation of this assumption. Simulation results support this conjecture.     

Forecasting of some; transformed series

The delta method is proposed as a. retransformation approach for coramputing forecasts for a nonstationary process (Z$inf:t$einf t, = 1, 2, ...), A derived variance stabilizing transformation technique is also used to compute forecasts, The performance of the two methods is compared to other techniques. Numerical results show that forecasts based on the variance stabilizing transformation method (VSTM) can lead to forecasts with a lower mean square error (MSE) as compared to other transformation techniqnes.     

传统产业转型升级的政策变迁及实施效果——中国纺织产业的个案研究

从"政策变迁"和"实施效果"两个视角切入,在全面梳理中国纺织产业转型升级已有政策的基础上,基于文本计量分析法从四个维度研究中国纺织产业转型升级政策的历史变迁及主要特征,应用"目标达成"模式并结合模糊优选模型对政策效果进行评价。研究表明:中国纺织产业转型升级政策表现出比较明显的"周期性"、"专用性"、"治理性"和"约束性",而"行业整合"和"产业培育"类政策相对较少,并且指导性和可操作性亟待加强;中国纺织产业转型升级政策颁布的前五年实际效果一般,但是近年来稳步上升;不同政策工具对具体指标的影响程度、实际效果及持续性存在差异。     

Comparison of measuring devices

Although it is common to determine if two devices give the same results, we apply an affine transformation to make the two devices the same on the average. The device with a lower variability in its measurement results is more accurate. Repeated observations are required. The design is balanced if each object is measured the same number of times by each device. The properties this design yields justify this approach.     

Local influence in box-cox transformation for multivariate regression data

The Box-Cox power family of transformations for multivariate regression data is considered. The influence of cases on the maximum likelihood estimators of the transformation parameters is investigated using the local influence approach, An example is given to- illustrate the local influence method and to show the effectiveness of the method.     

Semiparametric Regression Analysis of Longitudinal Skewed Data

In this paper, we develop a semiparametric regression model for longitudinal skewed data. In the new model, we allow the transformation function and the baseline function to be unknown. The proposed model can provide a much broader class of models than the existing additive and multiplicative models. Our estimators for regression parameters, transformation function and baseline function are asymptotically normal. Particularly, the estimator for the transformation function converges to its true value at the rate n ^{? 1 ∕ 2}, the convergence rate that one could expect for a parametric model. In simulation studies, we demonstrate that the proposed semiparametric method is robust with little loss of efficiency. Finally, we apply the new method to a study on longitudinal health care costs.     

Semiparametric transformation models with random effects for clustered doubly censored data

For right-censored data, Zeng et al. [Semiparametirc transformation modes with random effects for clustered data. Statist Sin. 2008;18:355–377] proposed a class of semiparametric transformation models with random effects to formulate the effects of possibly time-dependent covariates on clustered failure times. In this article, we demonstrate that the approach of Zeng et al. can be extended to analyse clustered doubly censored data. The asymptotic properties of the nonparametric maximum likelihood estimators of the model parameters are derived. A simulation study is conducted to investigate the performance of the proposed estimators.     

Improved approximation for transformation diagnostics

The suitability of a normal linear regression model may require transformation of the original response, and transformation diagnostics are designed to detect the need for such transformation. A common approach to transformation diagnostics is to construct an artificial explanatory variable, which is then tested in the augmented linear regression model for the original response. This paper describes corresponding diagnostics based directly on score statistics with accurate approximations for their standard errors. Several transformation models are covered. Some numerical illustrations are given.