A New STATISTIC FOR DETECTING INFLUENTIAL OBSERVATIONS IN A SCHEFFE' TYPE CALIBRATION CURVE

A statistic for identifying influential observations in calibration is given. The statistic is easy to interpret, and provides a useful measure of influence for Scheffé type calibration curves.     

A mixture-based approach to robust analysis of generalised linear models

A method for robustness in linear models is to assume that there is a mixture of standard and outlier observations with a different error variance for each class. For generalised linear models (GLMs) the mixture model approach is more difficult as the error variance for many distributions has a fixed relationship to the mean. This model is extended to GLMs by changing the classes to one where the standard class is a standard GLM and the outlier class which is an overdispersed GLM achieved by including a random effect term in the linear predictor. The advantages of this method are it can be extended to any model with a linear predictor, and outlier observations can be easily identified. Using simulation the model is compared to an M-estimator, and found to have improved bias and coverage. The method is demonstrated on three examples.     

Empirical Likelihood for Censored Linear Regression

In this paper we investigate the empirical likelihood method in a linear regression model when the observations are subject to random censoring. An empirical likelihood ratio for the slope parameter vector is defined and it is shown that its limiting distribution is a weighted sum of independent chi-square distributions. This reduces to the empirical likelihood to the linear regression model first studied by Owen (1991) if there is no censoring present. Some simulation studies are presented to compare the empirical likelihood method with the normal approximation based method proposed in Lai et al. (1995). It was found that the empirical likelihood method performs much better than the normal approximation method.     

Bayesian analysis for heteroscedastic normal-Pareto mixture model with application

ABSTRACT

In this article, we propose a new distribution by mixing normal and Pareto distributions, and the new distribution provides an unusual hazard function. We model the mean and the variance with covariates for heterogeneity. Estimation of the parameters is obtained by the Bayesian method using Markov Chain Monte Carlo (MCMC) algorithms. Proposal distribution in MCMC is proposed with a defined working variable related to the observations. Through the simulation, the method shows a dependable performance of the model. We demonstrate through establishing model under a real dataset that the proposed model and method can be more suitable than the previous report.     

Comparisons of the Unbiased Ridge Estimation to the Other Estimations

In this article, we employ the method of empirical likelihood to construct confidence intervals of conditional density for a left-truncation model. It is proved that the empirical likelihood ratio admits a limiting chi-square distribution with one degree of freedom when the lifetime observations with multivariate covariates form a stationary α-mixing sequence.     

Estimation of the linear-plateau segmented regression model in the presence of measurement error

It is well known that when the true values of the independent variable are unobservable due to measurement error, the least squares estimator for a regression model is biased and inconsistent. When repeated observations on each x_i are taken, consistent estimators for the linear-plateau model can be formed. The repeated observations are required to classify each observation to the appropriate line segment. Two cases of repeated observations are treated in detail. First, when a single value of y_i is observed with the repeated observations of x_i the least squares estimator using the mean of the repeated x_i observations is consistent and asymptotically normal. Second, when repeated observations on the pair (x_i, y_i ) are taken the least squares estimator is inconsistent, but two consistent estimators are proposed: one that consistently estimates the bias of the least squares estimator and adjusts accordingly; the second is the least squares estimator using the mean of the repeated observations on each pair.     

Jackknife-After-Bootstrap as Logistic Regression Diagnostic Tool

In this study, we propose using Jackknife-after-Bootstrap (JaB) method to detect influential observations in binary logistic regression model. Performance of the proposed method has been compared with the traditional method for standardized Pearson residuals, Cook's distance, change in the Pearson chi-square and change in the deviance statistics by both real world examples and simulation studies. The results reveal that under the various scenarios considered in this article, JaB performs better than the traditional method and is more robust to masking effect especially for Cook's distance.     

A model with mixed binary responses and censored observations

In this paper, we study the maximum likelihood estimation of a model with mixed binary responses and censored observations. The model is very general and includes the Tobit model and the binary choice model as special cases. We show that, by using additional binary choice observations, our method is more efficient than the traditional Tobit model. Two iterative procedures are proposed to compute the maximum likelihood estimator (MLE) for the model based on the EM algorithm (Dempster et al, 1977) and the Newton-Raphson method. The uniqueness of the MLE is proved. The simulation results show that the inconsistency and inefficiency can be significant when the Tobit method is applied to the present mixed model. The experiment results also suggest that the EM algorithm is much faster than the Newton-Raphson method for the present mixed model. The method also allows one to combine two data sets, the smaller data set with more detailed observations and the larger data set with less detailed binary choice observations in order to improve the efficiency of estimation. This may entail substantial savings when one conducts surveys.     

Regression Methods for Combining Multiple Classifiers

As no single classification method outperforms other classification methods under all circumstances, decision-makers may solve a classification problem using several classification methods and examine their performance for classification purposes in the learning set. Based on this performance, better classification methods might be adopted and poor methods might be avoided. However, which single classification method is the best to predict the classification of new observations is still not clear, especially when some methods offer similar classification performance in the learning set. In this article we present various regression and classical methods, which combine several classification methods to predict the classification of new observations. The quality of the combined classifiers is examined on some real data. Nonparametric regression is the best method of combining classifiers.     

CASE-DELETION DIAGNOSTICS FOR TEST STATISTICS IN MULTIVARIATE REGRESSION

Influence measures in multivariate regression analysis have been widely developed, especially through use of the case-deletion approach. However, there seem to be few accounts of the influence of observations on test statistics in hypothesis testing. This paper examines four common multivariate tests, namely the Wilks' ratio, Lawley-Hotelling trace, Pillai's trace and Roy's greatest root for testing a general linear hypothesis of the regression coefficients in multivariate regression. The influence of observations is measured using the case-deletion approach. The proposed diagnostic measures, except that of Roy's greatest root, can be expressed in terms of statistics without involving the actual deletion of observations. An illustrative example is given with satisfactory results.     

Selection of the Ridge Parameter Using Mathematical Programming

Ridge regression solves multicollinearity problems by introducing a biasing parameter that is called ridge parameter; it shrinks the estimates and their standard errors in order to reach acceptable results. Selection of the ridge parameter was done using several subjective and objective techniques that are concerned with certain criteria. In this study, selection of the ridge parameter depends on other important statistical measures to reach a better value of the ridge parameter. The proposed ridge parameter selection technique depends on a mathematical programming model and the results are evaluated using a simulation study. The performance of the proposed method is good when the error variance is greater than or equal to one; the sample consists of 20 observations, the number of explanatory variables in the model is 2, and there is a very strong correlation between the two explanatory variables.     

Rényi Statistics in Directed Families of Exponential Experiments*

Rényi statistics are considered in a directed family of general exponential models. These statistics are defined as Rényi distances between estimated and hypothetical model. An asymptotically quadratic approximation to the Rényi statistics is established, leading to similar asymptotic distribution results as established in the literature for the likelihood ratio statistics. Some arguments in favour of the Rényi statistics are discussed, and a numerical comparison of the Rényi goodness-of-fit tests with the likelihood ratio test is presented.     

Modified likelihood ratio statistics for inflated beta regressions

The class of inflated beta regression models generalizes that of beta regressions [S.L.P. Ferrari and F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat. 31 (2004), pp. 799–815] by incorporating a discrete component that allows practitioners to model data on rates and proportions with observations that equal an interval limit. For instance, one can model responses that assume values in (0, 1]. The likelihood ratio test tends to be quite oversized (liberal, anticonservative) in inflated beta regressions estimated with a small number of observations. Indeed, our numerical results show that its null rejection rate can be almost twice the nominal level. It is thus important to develop alternative testing strategies. This paper develops small-sample adjustments to the likelihood ratio and signed likelihood ratio test statistics in inflated beta regression models. The adjustments do not require orthogonality between the parameters of interest and the nuisance parameters and are fairly simple since they only require first- and second-order log-likelihood cumulants. Simulation results show that the modified likelihood ratio tests deliver much accurate inference in small samples. An empirical application is presented and discussed.     

Confidence band for percentile line in multiple linear regression

We deal with the problem of estimating constructing a confidence band for the 100γth percentile line in the multiple linear regression model with independent identically normally distributed errors. A method for computing the exact Scheffé type confidence band over a limited space of the particular covariates region is suggested. A confidence band depends on an estimator of the percentile line. The confidence bands based on the different estimators of the percentile line are compared with respect to the average bandwidth.     

Bayesian inference for zero-and-one-inflated geometric distribution regression model using Pólya-Gamma latent variables

Abstract

In the fields of internet financial transactions and reliability engineering, there could be more zero and one observations simultaneously. In this paper, considering that it is beyond the range where the conventional model can fit, zero-and-one-inflated geometric distribution regression model is proposed. Ingeniously introducing Pólya-Gamma latent variables in the Bayesian inference, posterior sampling with high-dimensional parameters is converted to latent variables sampling and posterior sampling with lower-dimensional parameters, respectively. Circumventing the need for Metropolis-Hastings sampling, the sample with higher sampling efficiency is obtained. A simulation study is conducted to assess the performance of the proposed estimation for various sample sizes. Finally, a doctoral dissertation data set is analyzed to illustrate the practicability of the proposed method, research shows that zero-and-one-inflated geometric distribution regression model using Pólya-Gamma latent variables can achieve better fitting results.     

Adjusting and Comparing Survival Curves by Means of an Additive Risk Model

Survival curves may be adjusted for covariates using Aalen's additive risk model. Survival curves may be compared by taking the ratio of two adjusted survival curves; the ratio is denoted the generalized relative survival rate. Adjusting both survival curves for all but one of a common set of covariates gives the partial relative survival rate, which measures the covariate-specific contribution to the generalized relative survival rate. The generalized and partial relative survival rates have interpretations similar to the traditional relative survival rates frequently used in cancer epidemiology. In fact, the traditional relative survival rate can be generalized to a regression context using the additive risk model. This population-adjusted relative survival rate is an alternative and useful method for removing confounding effects of age, cohorts, and sex. The authors use a data set of malignant melanoma patients diagnosed from 1965 to 1974 in Norway. The 25-year survival of 1967 individuals is studied.     

Asymptotic properties of maximum likelihood estimation in the mixed analysis of variance model

Consistency and asymptotic normality of maximum likelihood estimates in the mixed analysis of variance model are presented. These results are direct consequences of the method of Hoadley [2] concerning the case where the observations are independent but not identical. Asymptotic efficiency is given in the sense of the limit of the Cramér-Rao lower bound for the covariance matrix.     

Semiparametric estimation for seasonal long-memory time series using generalized exponential models

We consider a generalized exponential (GEXP) model in the frequency domain for modeling seasonal long-memory time series. This model generalizes the fractional exponential (FEXP) model [Beran, J., 1993. Fitting long-memory models by generalized linear regression. Biometrika 80, 817–822] to allow the singularity in the spectral density occurring at an arbitrary frequency for modeling persistent seasonality and business cycles. Moreover, the short-memory structure of this model is characterized by the Bloomfield [1973. An exponential model for the spectrum of a scalar time series. Biometrika 60, 217–226] model, which has a fairly flexible semiparametric form. The proposed model includes fractionally integrated processes, Bloomfield models, FEXP models as well as GARMA models [Gray, H.L., Zhang, N.-F., Woodward, W.A., 1989. On generalized fractional processes. J. Time Ser. Anal. 10, 233–257] as special cases. We develop a simple regression method for estimating the seasonal long-memory parameter. The asymptotic bias and variance of the corresponding long-memory estimator are derived. Our methodology is applied to a sunspot data set and an Internet traffic data set for illustration.     

Modified likelihood ratio tests for unit gamma regressions

Regression analyses are commonly performed with doubly limited continuous dependent variables; for instance, when modeling the behavior of rates, proportions and income concentration indices. Several models are available in the literature for use with such variables, one of them being the unit gamma regression model. In all such models, parameter estimation is typically performed using the maximum likelihood method and testing inferences on the model''s parameters are usually based on the likelihood ratio test. Such a test can, however, deliver quite imprecise inferences when the sample size is small. In this paper, we propose two modified likelihood ratio test statistics for use with the unit gamma regressions that deliver much more accurate inferences when the number of data points in small. Numerical (i.e. simulation) evidence is presented for both fixed dispersion and varying dispersion models, and also for tests that involve nonnested models. We also present and discuss two empirical applications.     

An Empirical Process View of Inverse Regression

A common approach taken in high‐dimensional regression analysis is sliced inverse regression, which separates the range of the response variable into non‐overlapping regions, called ‘slices’. Asymptotic results are usually shown assuming that the slices are fixed, while in practice, estimators are computed with random slices containing the same number of observations. Based on empirical process theory, we present a unified theoretical framework to study these techniques, and revisit popular inverse regression estimators. Furthermore, we introduce a bootstrap methodology that reproduces the laws of Cramér–von Mises test statistics of interest to model dimension, effects of specified covariates and whether or not a sliced inverse regression estimator is appropriate. Finally, we investigate the accuracy of different bootstrap procedures by means of simulations.