Estimation of Slope for Linear Regression Model with Uncertain Prior Information and Student-t Error

This article considers estimation of the slope parameter of the linear regression model with Student-t errors in the presence of uncertain prior information on the value of the unknown slope. Incorporating uncertain non sample prior information with the sample data the unrestricted, restricted, preliminary test, and shrinkage estimators are defined. The performances of the estimators are compared based on the criteria of unbiasedness and mean squared errors. Both analytical and graphical methods are explored. Although none of the estimators is uniformly superior to the others, if the non sample information is close to its true value, the shrinkage estimator over performs the rest of the estimators.     

Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Abstract. We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor (BF), requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper‐parameter, which can be set to its minimal value. We show that our approach produces genuine BFs. The implied prior on the concentration matrix of any complete graph is a data‐dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models and show that in this case they coincide with those recently obtained using limiting versions of hyper‐inverse Wishart distributions as priors on the graph‐constrained covariance matrices.     

Asymptotic properties of the partition function and applications in tail index inference of heavy-tailed data

The so-called partition function is a sample moment statistic based on blocks of data and it is often used in the context of multifractal processes. It will be shown that its behaviour is strongly influenced by the tail of the distribution underlying the data both in independent identically distributed and weakly dependent cases. These results will be used to develop graphical and estimation methods for the tail index of a distribution. The performance of the tools proposed is analysed and compared with other methods by means of simulations and examples.     

Graphical displays for clarifying how allocation ratio affects total sample size for the two sample logrank test

For time‐to‐event data, the power of the two sample logrank test for the comparison of two treatment groups can be greatly influenced by the ratio of the number of patients in each of the treatment groups. Despite the possible loss of power, unequal allocations may be of interest due to a need to collect more data on one of the groups or to considerations related to the acceptability of the treatments to patients. Investigators pursuing such designs may be interested in the cost of the unbalanced design relative to a balanced design with respect to the total number of patients required for the study. We present graphical displays to illustrate the sample size adjustment factor, or ratio of the sample size required by an unequal allocation compared to the sample size required by a balanced allocation, for various survival rates, treatment hazards ratios, and sample size allocation ratios. These graphical displays conveniently summarize information in the literature and provide a useful tool for planning sample sizes for the two sample logrank test. Copyright © 2010 John Wiley & Sons, Ltd.     

SiZer analysis for the comparison of time series

SiZer (SIgnificant ZERo crossing of the derivatives) is a scale-space visualization tool for statistical inferences. In this paper we introduce a graphical device, which is based on SiZer, for the test of the equality of the mean of two time series. The estimation of the quantile in a confidence interval is theoretically justified by advanced distribution theory. The extension of the proposed method to the comparison of more than two time series is also done using residual analysis. A broad numerical study is conducted to demonstrate the sample performance of the proposed tool. In addition, asymptotic properties of SiZer for the comparison of two time series are investigated.     

Methods to Distinguish Between Polynomial and Exponential Tails

Two methods to distinguish between polynomial and exponential tails are introduced. The methods are based on the properties of the residual coefficient of variation for the exponential and non‐exponential distributions. A graphical method, called a CV‐plot, shows departures from exponentiality in the tails. The plot is applied to the daily log‐returns of exchange rates of US dollar and Japanese yen. New statistics are introduced for testing the exponentiality of tails using multiple thresholds. They give better control of the significance level than previous tests. The powers of the new tests are compared with those of some others for various sample sizes.     

Normal probability plots with confidence for the residuals in linear regression

Normal probability plots for a simple random sample and normal probability plots for residuals from linear regression are not treated differently in statistical text books. In the statistical literature, 1 ? α simultaneous probability intervals for augmenting a normal probability plot for a simple random sample are available. The first purpose of this article is to demonstrate that the tests associated with the 1 ? α simultaneous probability intervals for a simple random sample may have a size substantially different from α when applied to the residuals from linear regression. This leads to the second purpose of this article: construction of four normal probability plot-based tests for residuals, which have size α exactly. We then compare the powers of these four graphical tests and a non-graphical test for residuals in order to assess the power performances of the graphical tests and to identify the ones that have better power. Finally, an example is provided to illustrate the methods.     

The Lorenz Curve for Model Assessment in Exponential Order Statistic Models

A goodness-of-fit technique for random samples from the exponential distribution based on the sample Lorenz curve is adapted for use in the exponential order statistic (EOS) model. In the EOS model, only those observations in a random sample from the exponential distribution of unknown size N that are less than some known stopping time T are observable. The model is known as the Jelinski-Moranda model in software reliability, where it is used to estimate the number of bugs in software during development. Distributional results are derived for the distance between the sample Lorenz curve and the population Lorenz curve so that it can be used as a goodness-of-fit test statistic. Simulations show that the test has good power against several alternative distributions. Simulations also indicate that in some cases, model misspecification leads to poor parameter estimation. A plotting procedure provides a means of graphical assessment of fit.     

Change PP plot and continous sample quantile function

Functional inference recommends data analysis of a sample of n observations by functional and graphical representations of its probability models using various functions on 0 < u < 1, including the quantile function. This paper discusses: charge PP plots and a continuous version of the sample quantile function which use the mid-distinct value probability integral transform; comparison density functions; comparison interpretation of probability integral transform; maximum spacings method of one sample parameter estimation.     

A MCMC algorithm to fit a general exchangeable model

Consider the exchangeable Bayesian hierarchical model where observations y_i are independently distributed from sampling densities with unknown means, the means µi, are a random sample from a distribution g, and the parameters of g are assigned a known distribution h. A simple algorithm is presented for summarizing the posterior distribution based on Gibbs sampling and the Metropolis algorithm. The software program Matlab is used to implement the algorithm and provide a graphical output analysis. An binomial example is used to illustrate the flexibility of modeling possible using this algorithm. Methods of model checking and extensions to hierarchical regression modeling are discussed.     

基于状态空间图模型方法的投资组合优化决策

将lasso图理论合并到状态空间模型中,利用条件独立性且通过范数惩罚法对协方差阵进行估计。新方法兼具图模型和动态状态空间模型的优点。最后将该方法应用于欧洲股票市场进行投资组合优化决策,结果表明基于lasso图方法的状态空间模型的投资组合业绩要优于自回归和一般的状态空间模型。     

Checking adequacy of the semiparametric location shift model with censored data

The location shift model is commonly used to quantify the difference between groups in a two-arm study. Nonparametric inference procedures for the location shift parameter with censored observations have recently been extensively studied. However, the validity of these procedures depends heavily on the model assumption. In this article, a class of graphical and numerical methods are proposed for checking the adequacy of the location shift model. Our graphical procedures are much less subjective than the eye-ball method based on the standard Q-Q plot. The proposed methods are illustrated with real-life examples.     

Limits of quantitation for laboratory assays

Summary.  A common problem with laboratory assays is that a measurement of a substance in a test sample becomes relatively imprecise as the concentration decreases. A standard solution is to establish lower limits for reliable measurement. A quantitation limit is a level above which a measurement has sufficient precision to be reliably reported. The paper proposes a new approach to defining the limit of quantitation for the case where a linear calibration curve is used to estimate actual concentrations from measured values. The approach is based on the relative precision of the estimated concentration, using the delta method to approximate the precision. A graphical display is proposed for the assessment of estimated concentrations, as well as the overall reliability of the calibration curve. Our research is motivated by a clinical inhalation experiment. Comparisons are made between the approach proposed and two standard methods, using both real and simulated data.     

On the comparison of the pre-test and shrinkage estimators for the univariate normal mean

The estimation of the mean of an univariate normal population with unknown variance is considered when uncertain non-sample prior information is available. Alternative estimators are defined to incorporate both the sample as well as the non-sample information in the estimation process. Some of the important statistical properties of the restricted, preliminary test, and shrinkage estimators are investigated. The performances of the estimators are compared based on the criteria of unbiasedness and mean square error in order to search for a ‘best’ estimator. Both analytical and graphical methods are explored. There is no superior estimator that uniformly dominates the others. However, if the non-sample information regarding the value of the mean is close to its true value, the shrinkage estimator over performs the rest of the estimators. Received: June 19, 1999; revised version: March 23, 2000     

On the impact of contaminations in graphical Gaussian models

This paper analyzes the impact of some kinds of contaminant on model selection in graphical Gaussian models. We investigate four different kinds of contaminants, in order to consider the effect of gross errors, model deviations, and model misspecification. The aim of the work is to assess against which kinds of contaminant a model selection procedure for graphical Gaussian models has a more robust behavior. The analysis is based on simulated data. The simulation study shows that relatively few contaminated observations in even just one of the variables can have a significant impact on correct model selection, especially when the contaminated variable is a node in a separating set of the graph.     

Elliptical safety region plots for C pk

The process capability index C _pk is widely used when measuring the capability of a manufacturing process. A process is defined to be capable if the capability index exceeds a stated threshold value, e.g. C _pk >4/3. This inequality can be expressed graphically using a process capability plot, which is a plot in the plane defined by the process mean and the process standard deviation, showing the region for a capable process. In the process capability plot, a safety region can be plotted to obtain a simple graphical decision rule to assess process capability at a given significance level. We consider safety regions to be used for the index C _pk . Under the assumption of normality, we derive elliptical safety regions so that, using a random sample, conclusions about the process capability can be drawn at a given significance level. This simple graphical tool is helpful when trying to understand whether it is the variability, the deviation from target, or both that need to be reduced to improve the capability. Furthermore, using safety regions, several characteristics with different specification limits and different sample sizes can be monitored in the same plot. The proposed graphical decision rule is also investigated with respect to power.     

AN EXACT CONDITIONAL TEST FOR COVARIANCE SELECTION MODELS

An exact conditional test is developed for testing the absence of an edge in a graphical covariance selection model and is shown to be equivalent to a test based on the partial correlation coefficient. An example is given.     

MEDIAN ESTIMATES IN SIMPLE LINEAR REGRESSION

Summary
Estimates in simple linear regression which minimize the sum of absolute values of residuals are discussed. These estimates, called median-estimates , are an analogue of the sample median, and their properties reflect this. A simple graphical procedure, related to an early (1923) one of Edgeworth, is outlined for computing median estimates. This method works well in small or moderate samples, and is demonstrated on two examples. The small sample distribution of the median estimates is examined by conditioning on the pattern of signs of the residuals, and the resulting conditional distribution gives qualitative information on the adequacy of the approximation of the exact distribution by the (known) asymptotic distibution.     

The Fate Worse than Death and other Curiosities and Stupidities

The problem of finding confidence intervals for the success parameter of a binomial experiment has a long history, and a myriad of procedures have been developed. Most exploit the duality between hypothesis testing and confidence regions and are typically based on large sample approximations. We instead employ a direct approach that attempts to determine the optimal coverage probability function a binomial confidence procedure can have from the exact underlying binomial distributions, which in turn defines the associated procedure. We show that a graphical perspective provides much insight into the problem. Both procedures whose coverage never falls below the declared confidence level and those that achieve that level only approximately are analyzed. We introduce the Length/Coverage Optimal method, a variant of Sterne's procedure that minimizes average length while maximizing coverage among all length minimizing procedures, and show that it is superior in important ways to existing procedures.