In this paper, we present a Monte Carlo method for estimating a nonlinear function of the mean of a multivariate normal distribution. Building on this method, we propose a parametric estimation procedure for unimodal regression models, assuming that the response variable follows a gamma distribution while some covariates are contaminated with normal measurement error. Compared to existing approaches, the proposed method accommodates multivariate covariates and features a tractable bias-corrected likelihood function, enabling faster computation and more accurate estimation when the data distribution is correctly specified. To enhance the applicability of the proposed method, we also explore various model adequacy diagnostic tools and evaluate its robustness against distributional misspecifications. Notably, we introduce a goodness-of-fit test based on a unique characterization of the gamma distribution, designed to assess the validity of the distributional assumption for the response variable. Numerical studies and real-world data applications are conducted to evaluate the finite-sample performance of the proposed methods.
He et al. (Mon,) studied this question.
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