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Abstract
Abstract In this paper, we investigate a partially single-index varying-coefficient model, and suggest two empirical log-likelihood ratio statistics for the unknown parameters in the model. The first statistic is asymptotically distributed as a weighted sum of independent chi-square variables under some mild conditions. It is proved that another statistic, with adjustment factor, is asymptotically standard chi-square under some suitable conditions. These useful statistics could be used to construct the confidence regions of the parameters. A simulation study indicates that, with the increase of sample size, the coverage probability of the confidence region constructed by us gradually approaches the theoretical value.
Highlights
Consider a partially single-index varying-coe cient model of the formY = gτ(βτU)X + θτZ + ε, (1.1)where (U, X, Z) ∈ Rp × Rd × Rq is a vector of covariates, Y is the response variable, β is a p × vector of unknown parameters, θ is a q × vector of regression coe cient, g (·) is a d × vector of unknown functions and ε is a random error with E(ε | U, X, Z) = and Var(ε | U, X, Z) = σ
In this paper, we investigate a partially single-index varying-coe cient model, and suggest two empirical log-likelihood ratio statistics for the unknown parameters in the model
Where (U, X, Z) ∈ Rp × Rd × Rq is a vector of covariates, Y is the response variable, β is a p × vector of unknown parameters, θ is a q × vector of regression coe cient, g (·) is a d × vector of unknown functions and ε is a random error with E(ε | U, X, Z) = and Var(ε | U, X, Z) = σ
Summary
Abstract: In this paper, we investigate a partially single-index varying-coe cient model, and suggest two empirical log-likelihood ratio statistics for the unknown parameters in the model. Owen[4] proved the empirical log-likelihood ratio is asymptotically a standard chi-square variable when he applied the empirical likelihood to linear regression model, so that it can be applied to constructing the con dence region of the regression parameter.
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