scholarly journals Instability and change detection in exponential families and generalized linear models, with a study of Atlantic tropical storms

2014 ◽  
Vol 21 (6) ◽  
pp. 1133-1143
Author(s):  
Y. Lu ◽  
S. Chatterjee
Keyword(s):  
Change Detection ◽  
Generalized Linear Models ◽  
Exponential Family ◽  
Linear Models ◽  
Exponential Families ◽  
Tropical Storms ◽  
Ratio Test ◽  
Test Statistic ◽  
Distributional Assumption ◽  
Data Collection Process

Abstract. Exponential family statistical distributions, including the well-known normal, binomial, Poisson, and exponential distributions, are overwhelmingly used in data analysis. In the presence of covariates, an exponential family distributional assumption for the response random variables results in a generalized linear model. However, it is rarely ensured that the parameters of the assumed distributions are stable through the entire duration of the data collection process. A failure of stability leads to nonsmoothness and nonlinearity in the physical processes that result in the data. In this paper, we propose testing for stability of parameters of exponential family distributions and generalized linear models. A rejection of the hypothesis of stable parameters leads to change detection. We derive the related likelihood ratio test statistic. We compare the performance of this test statistic to the popular normal distributional assumption dependent cumulative sum (Gaussian CUSUM) statistic in change detection problems. We study Atlantic tropical storms using the techniques developed here, so to understand whether the nature of these tropical storms has remained stable over the last few decades.

scholarly journals Instability and change detection in exponential families and generalized linear models, with a study of Atlantic tropical storms

2014 ◽  
Vol 1 (1) ◽  
pp. 371-401
Author(s):  
Y. Lu ◽  
S. Chatterjee
Keyword(s):  
Change Detection ◽  
Generalized Linear Models ◽  
Exponential Family ◽  
Linear Models ◽  
Exponential Families ◽  
Tropical Storms ◽  
Ratio Test ◽  
Test Statistic ◽  
Distributional Assumption ◽  
Data Collection Process

Abstract. Exponential family statistical distributions, including the well-known Normal, Binomial, Poisson, and exponential distributions, are overwhelmingly used in data analysis. In the presence of covariates, an exponential family distributional assumption for the response random variables results in a generalized linear model. However, it is rarely ensured that the parameters of the assumed distributions are stable through the entire duration of data collection process. A failure of stability leads to nonsmoothness and nonlinearity in the physical processes that drive the data under. In this paper, we propose testing for stability of parameters of exponential family distributions and generalized linear models. A rejection of the hypothesis of stable parameters leads to change detection. We derive the related likelihood ratio test statistic. We compare the performance of this test statistic to the popular Normal distributional assumption dependent cumulative sum (Gaussian-CUSUM) statistic in change detection problems. We study Atlantic tropical storms using the techniques developed here, to understand whether the nature of these tropical storms has remained stable over the last few decades.

scholarly journals Testing for treatment effect in covariate-adaptive randomized trials with generalized linear models and omitted covariates

2021 ◽  
pp. 096228022110082
Author(s):  
Yang Li ◽  
Wei Ma ◽  
Yichen Qin ◽  
Feifang Hu
Keyword(s):  
Generalized Linear Models ◽  
Treatment Effect ◽  
Linear Models ◽  
Type I Error ◽  
Type I ◽  
Test Statistic ◽  
Asymptotic Results ◽  
Adaptive Randomization ◽  
Adjustment Method ◽  
Inflated Type

Concerns have been expressed over the validity of statistical inference under covariate-adaptive randomization despite the extensive use in clinical trials. In the literature, the inferential properties under covariate-adaptive randomization have been mainly studied for continuous responses; in particular, it is well known that the usual two-sample t-test for treatment effect is typically conservative. This phenomenon of invalid tests has also been found for generalized linear models without adjusting for the covariates and are sometimes more worrisome due to inflated Type I error. The purpose of this study is to examine the unadjusted test for treatment effect under generalized linear models and covariate-adaptive randomization. For a large class of covariate-adaptive randomization methods, we obtain the asymptotic distribution of the test statistic under the null hypothesis and derive the conditions under which the test is conservative, valid, or anti-conservative. Several commonly used generalized linear models, such as logistic regression and Poisson regression, are discussed in detail. An adjustment method is also proposed to achieve a valid size based on the asymptotic results. Numerical studies confirm the theoretical findings and demonstrate the effectiveness of the proposed adjustment method.

scholarly journals The Empirical Cressie-Read Test Statistics for Longitudinal Generalized Linear Models

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Junhua Zhang ◽  
Ruiqin Tian ◽  
Suigen Yang ◽  
Sanying Feng
Keyword(s):  
Generalized Linear Models ◽  
Correlation Matrix ◽  
Linear Models ◽  
Real Data ◽  
Regularity Conditions ◽  
Finite Sample ◽  
Test Statistic ◽  
Quadratic Inference Function ◽  
Working Correlation ◽  
Statistic Approach

For the marginal longitudinal generalized linear models (GLMs), we develop the empirical Cressie-Read (ECR) test statistic approach which has been proposed for the independent identically distributed (i.i.d.) case. The ECR test statistic includes empirical likelihood as a special case. By adopting this ECR test statistic approach and taking into account the within-subject correlation, the efficiency theory results of estimation and testing based on ECR are established under some regularity conditions. Although a working correlation matrix is assumed, there is no need to estimate the nuisance parameters in the working correlation matrix based on the quadratic inference function (QIF). Therefore, the proposed ECR test statistic is asymptotically a standardχ2limit under the null hypothesis. It is shown that the proposed method is more efficient even when the working correlation matrix is misspecified. We also evaluate the finite sample performance of the proposed methods via simulation studies and a real data analysis.

scholarly journals Fitting Tweedie's Compound Poisson Model to Insurance Claims Data: Dispersion Modelling

2002 ◽  
Vol 32 (1) ◽  
pp. 143-157 ◽  
Author(s):  
Gordon K. Smyth ◽  
Bent Jørgensen
Keyword(s):  
Generalized Linear Models ◽  
Exponential Family ◽  
Linear Models ◽  
Likelihood Function ◽  
Poisson Model ◽  
Insurance Claims ◽  
Poisson Arrival ◽  
Insurance Claims Data ◽  
The Cost ◽  
Insurance Data

AbstractWe reconsider the problem of producing fair and accurate tariffs based on aggregated insurance data giving numbers of claims and total costs for the claims. Jørgensen and de Souza (Scand Actuarial J., 1994) assumed Poisson arrival of claims and gamma distributed costs for individual claims. Jørgensen and de Souza (1994) directly modelled the risk or expected cost of claims per insured unit, μ say. They observed that the dependence of the likelihood function on μ is as for a linear exponential family, so that modelling similar to that of generalized linear models is possible. In this paper we observe that, when modelling the cost of insurance claims, it is generally necessary to model the dispersion of the costs as well as their mean. In order to model the dispersion we use the framework of double generalized linear models. Modelling the dispersion increases the precision of the estimated tariffs. The use of double generalized linear models also allows us to handle the case where only the total cost of claims and not the number of claims has been recorded.

A Global Test for the Goodness of Fit of Generalized Linear Models : An Estimating Equation Approach

2005 ◽  
Vol 56 (1-4) ◽  
pp. 251-282
Author(s):  
R. Prabhakar Rao ◽  
B.C. Sutradhar
Keyword(s):  
Generalized Linear Models ◽  
Goodness Of Fit ◽  
Exponential Family ◽  
Linear Models ◽  
Estimating Equation ◽  
Goodness Of Fit Test ◽  
Global Test ◽  
Goodness Of Fit Tests ◽  
Exponential Family Of Distributions ◽  
Family Of Distributions

Summary Generalized linear models are used to analyze a wide variety of discrete and continuous data with possible overdispersion under the assumption that the data follow an exponential family of distributions. The violation of this assumption may have adverse effects on the statistical inferences. The existing goodness of fit tests for checking this assumption are valid only for a standard exponential family of distributions with no overdispersion. In this paper, we develop a global goodness of fit test for the general exponential family of distributions which may or may not contain overdispersion. The proposed statistic has asymptotically standard Gaussian distribution which should be easy to implement.

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