Effect of individual observations on the Box–Cox transformation

In this paper, we consider the influence of individual observations on inferences about the Box–Cox power transformation parameter from a Bayesian point of view. We compare Bayesian diagnostic measures with the ‘forward’ method of analysis due to Riani and Atkinson. In particular, we look at the effect of omitting observations on the inference by comparing particular choices of transformation using the conditional predictive ordinate and the k _d measure of Pettit and Young. We illustrate the methods using a designed experiment. We show that a group of masked outliers can be detected using these single deletion diagnostics. Also, we show that Bayesian diagnostic measures are simpler to use to investigate the effect of observations on transformations than the forward search method.     

Improving TAIEX forecasting using fuzzy time series with Box–Cox power transformation

Box–Cox together with our newly proposed transformation were implemented in three different real world empirical problems to alleviate noisy and the volatility effect of them. Consequently, a new domain was constructed. Subsequently, universe of discourse for transformed data was established and an approach for calculating effective length of the intervals was then proposed. Considering the steps above, the initial forecasts were performed using frequently used fuzzy time series (FTS) methods on transformed data. Final forecasts were retrieved from initial forecasted values by proper inverse operation. Comparisons of the results demonstrate that the proposed method produced more accurate forecasts compared with existing FTS on original data.     

Variable selection in joint mean and variance models of Box–Cox transformation

In many applications, a single Box–Cox transformation cannot necessarily produce the normality, constancy of variance and linearity of systematic effects. In this paper, by establishing a heterogeneous linear regression model for the Box–Cox transformed response, we propose a hybrid strategy, in which variable selection is employed to reduce the dimension of the explanatory variables in joint mean and variance models, and Box–Cox transformation is made to remedy the response. We propose a unified procedure which can simultaneously select significant variables in the joint mean and variance models of Box–Cox transformation which provide a useful extension of the ordinary normal linear regression models. With appropriate choice of the tuning parameters, we establish the consistency of this procedure and the oracle property of the obtained estimators. Moreover, we also consider the maximum profile likelihood estimator of the Box–Cox transformation parameter. Simulation studies and a real example are used to illustrate the application of the proposed methods.     

A generalised Box–Cox transformation for the parametric estimation of clinical reference intervals

Parametric methods for the calculation of reference intervals in clinical studies often rely on the identification of a suitable transformation so that the transformed data can be assumed to be drawn from a Gaussian distribution. In this paper, the two-stage transformation recommended by the International Federation for Clinical Chemistry is compared with a novel generalised Box–Cox family of transformations. Investigation is also made of sample sizes needed to achieve certain criteria of reliability in the calculated reference interval. Simulations are used to show that the generalised Box–Cox family achieves a lower bias than the two-stage transformation. It was found that there is a possibility that the two-stage transformation will result in percentile estimates that cannot be back-transformed to obtain the required reference intervals, a difficulty not observed when using the generalised Box–Cox family introduced in this paper.     

Box–Cox transformation for spatial linear models: a study on lattice data

In this article, we extended the classic Box–Cox transformation to spatial linear models. For a comparative study, the proposed models were applied to a real data set of Chinese population growth and economic development with three different structures: no spatial correction, conditional autoregressive and simultaneous autoregressive. Maximal likelihood method was used to estimate the Box–Cox parameter λ and other parameters in the models. The residuals of the models were analyzed through Moran’s I and Geary’s c.     

Local Box–Cox transformation on time-varying parametric models for smoothing estimation of conditional CDF with longitudinal data

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies, such as epidemiological studies and longitudinal clinical trials. Estimation approaches without any structural assumptions may lead to inadequate and numerically unstable estimators in practice. We propose in this paper a nonparametric approach based on time-varying parametric models for estimating the conditional distribution functions with a longitudinal sample. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model after local Box–Cox transformation. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Applications of our two-step estimation method have been demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through a simulation study. Application and simulation results show that smoothing estimation from time-variant parametric models outperforms the existing kernel smoothing estimator by producing narrower pointwise bootstrap confidence band and smaller root mean squared error.     

Imposing and Testing Curvature Conditions on a Box–Cox Cost Function

We present a new method for imposing and testing concavity of cost functions using asymptotic least squares, which can be easily implemented even for nonlinear cost functions. We provide an illustration for a (generalized) Box–Cox cost function with six inputs: capital, labor disaggregated in three skill levels, energy, and intermediate materials. We present a parametric concavity test and compare price elasticities when curvature conditions are imposed versus when they are not. Although concavity is statistically rejected, estimates are not very sensitive to its imposition. We find stronger substitution between the different type of labor than between any other two inputs.     

Assessing Type I error and power of multistate Markov models for panel data—A simulation study

Ordinal outcomes collected at multiple follow-up visits are common in clinical trials. Sometimes, one visit is chosen for the primary analysis and the scale is dichotomized amounting to loss of information. Multistate Markov models describe how a process moves between states over time. Here, simulation studies are performed to investigate the Type I error and power characteristics of multistate Markov models for panel data with limited non-adjacent state transitions. The results suggest that the multistate Markov models preserve the Type I error and adequate power is achieved with modest sample sizes for panel data with limited non-adjacent state transitions.     

Effect of measurement error and autocorrelation on the X¯ chart

Measurement error and autocorrelation often exist in quality control applications. Both have an adverse effect on the X¯ chart's performance. To counteract the undesired effect of autocorrelation, we build-up the samples with non-neighbouring items, according to the time they were produced. To counteract the undesired effect of measurement error, we measure the quality characteristic of each item of the sample several times. The chart's performance is assessed when multiple measurements are applied and the samples are built by taking one item from the production line and skipping one, two or more before selecting the next.     

The Likelihood Ratio Test with the Box–Cox Transformation for the Normal Mixture Problem: Power and Sample Size Study

Abstract

Through simulation and regression, we study the alternative distribution of the likelihood ratio test in which the null hypothesis postulates that the data are from a normal distribution after a restricted Box–Cox transformation and the alternative hypothesis postulates that they are from a mixture of two normals after a restricted (possibly different) Box–Cox transformation. The number of observations in the sample is called N. The standardized distance between components (after transformation) is D = (μ₂ ? μ₁)/σ, where μ₁ and μ₂ are the component means and σ² is their common variance. One component contains the fraction π of observed, and the other 1 ? π. The simulation results demonstrate a dependence of power on the mixing proportion, with power decreasing as the mixing proportion differs from 0.5. The alternative distribution appears to be a non-central chi-squared with approximately 2.48 + 10N ^?0.75 degrees of freedom and non-centrality parameter 0.174N(D ? 1.4)² × [π(1 ? π)]. At least 900 observations are needed to have power 95% for a 5% test when D = 2. For fixed values of D, power, and significance level, substantially more observations are necessary when π ≥ 0.90 or π ≤ 0.10. We give the estimated powers for the alternatives studied and a table of sample sizes needed for 50%, 80%, 90%, and 95% power.     

A fractional version of the Cox–Ingersoll–Ross interest rate model and pricing double barrier option with Hurst index H∈(23,1)

In this work, we study the existence and uniqueness of the solution to a fractional version of the Cox–Ingersoll–Ross (fCIR) stochastic differential equation. The strong convergence of this equation is analyzed and according to it’s framework, we obtain the price of the double barrier option under transaction cost. Finally, we verify the effect of the parameters of the model on the value of the option.     

Likelihood Function and Canonical Correlation Analysis of the Peña–Box Model

The Peña–Box model is a type of dynamic factor model whose factors try to capture the time-effect movements of a multiple time series. The Peña–Box model can be expressed as a vector autoregressive (VAR) model with constraints. This article derives the maximum likelihood estimates and the likelihood ratio test of the VAR model for Gaussian processes. Then a test statistic constructed by canonical correlation coefficients is presented and adjusted for conditional heteroscedasticity. Simulations confirm the validity of adjustments for conditional heteroscedasticity, and show that the proposed statistics perform better than the statistics used in the existing literature.     

Duality of the Multivariate Distributions of Marshall–Olkin Type and Tail Dependence

A dual class of the multivariate distributions of Marshall–Olkin type is introduced, and their copulas are presented and utilized to derive explicit expressions of the distributional tail dependencies, which describe the amount of dependence in the upper-orthant tail or lower-orthant tail of a multivariate distribution and can be used in the study of dependence among extreme values. A sufficient condition under which tail dependencies of two such distributions can be compared are obtained. Some examples are also presented to illustrate our results.     

A Cautionary Note on the Use of the Grønnesby and Borgan Goodness-of-Fit Test for the Cox Proportional Hazards Model

Grønnesby and Borgan (1996, Lifetime Data Analysis 2, 315–328) propose an omnibus goodness-of-fit test for the Cox proportional hazards model. The test is based on grouping the subjects by their estimated risk score and comparing the number of observed and a model based estimated number of expected events within each group. We show, using extensive simulations, that even for moderate sample sizes the choice of number of groups is critical for the test to attain the specified size. In light of these results we suggest a grouping strategy under which the test attains the correct size even for small samples. The power of the test statistic seems to be acceptable when compared to other goodness-of-fit tests.     

The asymptotic size and power of the augmented Dickey–Fuller test for a unit root

It is shown that the limiting distribution of the augmented Dickey–Fuller (ADF) test under the null hypothesis of a unit root is valid under a very general set of assumptions that goes far beyond the linear AR(∞) process assumption typically imposed. In essence, all that is required is that the error process driving the random walk possesses a continuous spectral density that is strictly positive. Furthermore, under the same weak assumptions, the limiting distribution of the ADF test is derived under the alternative of stationarity, and a theoretical explanation is given for the well-known empirical fact that the test's power is a decreasing function of the chosen autoregressive order p. The intuitive reason for the reduced power of the ADF test is that, as p tends to infinity, the p regressors become asymptotically collinear.     

Stress–strength reliability of a non-identical-component-strengths system based on upper record values from the family of Kumaraswamy generalized distributions

In an attempt to produce more realistic stress–strength models, this article considers the estimation of stress–strength reliability in a multi-component system with non-identical component strengths based on upper record values from the family of Kumaraswamy generalized distributions. The maximum likelihood estimator of the reliability, its asymptotic distribution and asymptotic confidence intervals are constructed. Bayes estimates under symmetric squared error loss function using conjugate prior distributions are computed and corresponding highest probability density credible intervals are also constructed. In Bayesian estimation, Lindley approximation and the Markov Chain Monte Carlo method are employed due to lack of explicit forms. For the first time using records, the uniformly minimum variance unbiased estimator and the closed form of Bayes estimator using conjugate and non-informative priors are derived for a common and known shape parameter of the stress and strength variates distributions. Comparisons of the performance of the estimators are carried out using Monte Carlo simulations, the mean squared error, bias and coverage probabilities. Finally, a demonstration is presented on how the proposed model may be utilized in materials science and engineering with the analysis of high-strength steel fatigue life data.     

Comparison of Statistical Methods for Pretest–Posttest Designs in Terms of Type I Error Probability and Statistical Power

The pretest–posttest design is widely used to investigate the effect of an experimental treatment in biomedical research. The treatment effect may be assessed using analysis of variance (ANOVA) or analysis of covariance (ANCOVA). The normality assumption for parametric ANOVA and ANCOVA may be violated due to outliers and skewness of data. Nonparametric methods, robust statistics, and data transformation may be used to address the nonnormality issue. However, there is no simultaneous comparison for the four statistical approaches in terms of empirical type I error probability and statistical power. We studied 13 ANOVA and ANCOVA models based on parametric approach, rank and normal score-based nonparametric approach, Huber M-estimation, and Box–Cox transformation using normal data with and without outliers and lognormal data. We found that ANCOVA models preserve the nominal significance level better and are more powerful than their ANOVA counterparts when the dependent variable and covariate are correlated. Huber M-estimation is the most liberal method. Nonparametric ANCOVA, especially ANCOVA based on normal score transformation, preserves the nominal significance level, has good statistical power, and is robust for data distribution.     

A note on the equality of the OLSE and the BLUE of the parametric function in the general Gauss–Markov model

In this note we consider the equality of the ordinary least squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of the estimable parametric function in the general Gauss–Markov model. Especially we consider the structures of the covariance matrix V for which the OLSE equals the BLUE. Our results are based on the properties of a particular reparametrized version of the original Gauss–Markov model.      

A powerful and interpretable alternative to the Jarque–Bera test of normality based on 2nd-power skewness and kurtosis,using the Rao's score test on the APD family

We introduce the 2nd-power skewness and kurtosis, which are interesting alternatives to the classical Pearson's skewness and kurtosis, called 3rd-power skewness and 4th-power kurtosis in our terminology. We use the sample 2nd-power skewness and kurtosis to build a powerful test of normality. This test can also be derived as Rao's score test on the asymmetric power distribution, which combines the large range of exponential tail behavior provided by the exponential power distribution family with various levels of asymmetry. We find that our test statistic is asymptotically chi-squared distributed. We also propose a modified test statistic, for which we show numerically that the distribution can be approximated for finite sample sizes with very high precision by a chi-square. Similarly, we propose a directional test based on sample 2nd-power kurtosis only, for the situations where the true distribution is known to be symmetric. Our tests are very similar in spirit to the famous Jarque–Bera test, and as such are also locally optimal. They offer the same nice interpretation, with in addition the gold standard power of the regression and correlation tests. An extensive empirical power analysis is performed, which shows that our tests are among the most powerful normality tests. Our test is implemented in an R package called PoweR.