Abstract:The strength of statistical evidence is measured by the likelihood ratio. Two key performance properties of this measure are the probability of observing strong misleading evidence and the probability of observing weak evidence. For the likelihood function associated with a parametric statistical model, these probabilities have a simple large sample structure when the model is correct. Here we examine how that structure changes when the model fails. This leads to criteria for determining whether a given likeli… Show more
“…Royall and Tsou [5] referred to the preserved consistency property as the first condition of robustness. They showed that likelihood functions on the basis of such working models could be adjusted to become robust if the condition is fulfilled.…”
This paper introduces an exploratory way to determine how variance relates to the mean in generalized linear models. This novel method employs the robust likelihood technique introduced by Royall and Tsou.A urinary data set collected by Ginsberg et al. and the fabric data set analysed by Lee and Nelder are considered to demonstrate the applicability and simplicity of the proposed technique. Application of the proposed method could easily reveal a mean-variance relationship that would generally be left unnoticed, or that would require more complex modelling to detect.
“…Royall and Tsou [5] referred to the preserved consistency property as the first condition of robustness. They showed that likelihood functions on the basis of such working models could be adjusted to become robust if the condition is fulfilled.…”
This paper introduces an exploratory way to determine how variance relates to the mean in generalized linear models. This novel method employs the robust likelihood technique introduced by Royall and Tsou.A urinary data set collected by Ginsberg et al. and the fabric data set analysed by Lee and Nelder are considered to demonstrate the applicability and simplicity of the proposed technique. Application of the proposed method could easily reveal a mean-variance relationship that would generally be left unnoticed, or that would require more complex modelling to detect.
“…In order to make inference about θ in the presence of φ, we can consider the profile likelihood function L(θ,φ(θ)), whereφ(θ) is the ML estimate of φ given θ (Kalbfleisch and Sprott 1970). Royall (1997) has elaborated that the profile likelihood is a proper representation of the evidence about θ . Now, let θ be the population mean and use the inverse Gaussian with the density function…”
Section: Representing Evidence About the Parameter: The Profile Likelmentioning
confidence: 99%
“…If one applies the adjustment introduced by Royall and Tsou (2003) to L P (θ ), the resulting rescaled adjusted likelihood, denoted by L AP (θ ), is…”
“…situations one can modify the two likelihood functions for counts to become asymptotically identical to the profile likelihood for the mean parameter for general nonnegative continuous data. The idea of robust likelihood introduced by Royall and Tsou (2003) suffices to accomplish this task.…”
In independent and identically distributed situations, we show that one can properly correct the Poisson and the negative binomial likelihood functions to become asymptotically identical to the profile likelihood function for the mean parameter of nonnegative continuous distributions under mild conditions. We present theoretical justifications and use data analyses to demonstrate the merit of our new robust likelihood method.
“…Robust likelihood functions can be made insensitive to model misspecification (Royall and Tsou, 2003) in a regression setting (Blume et al, 2007). This is accomplished by adjusting the likelihood function, so that the adjusted likelihood behaves as if the model were correctly specified.…”
Section: Wrong Likelihood; Right Answer?mentioning
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