CONTINUOUSLY UPDATED INDIRECT INFERENCE IN HETEROSKEDASTIC SPATIAL MODELS

Spatial units typically vary over many of their characteristics, introducing potential unobserved heterogeneity which invalidates commonly used homoskedasticity conditions. In the presence of unobserved heteroskedasticity, methods based on the quasi-likelihood function generally produce inconsistent estimates of both the spatial parameter and the coefficients of the exogenous regressors. A robust generalized method of moments estimator as well as a modified likelihood method have been proposed in the literature to address this issue. The present paper constructs an alternative indirect inference (II) approach which relies on a simple ordinary least squares procedure as its starting point. Heteroskedasticity is accommodated by utilizing a new version of continuous updating that is applied within the II procedure to take account of the parameterization of the variance–covariance matrix of the disturbances. Finite-sample performance of the new estimator is assessed in a Monte Carlo study. The approach is implemented in an empirical application to house price data in the Boston area, where it is found that spatial effects in house price determination are much more significant under robustification to heterogeneity in the equation errors.

The tenets of indirect inference in Bayesian models

This paper extends the application of Bayesian inference to probability distributions defined in terms of its quantile function. We describe the method of *indirect likelihood* to be used in the Bayesian models with sampling distributions which lack an explicit cumulative distribution function. We provide examples and demonstrate the equivalence of the "quantile-based" (indirect) likelihood to the conventional "density-defined" (direct) likelihood. We consider practical aspects of the numerical inversion of quantile function by root-finding required by the indirect likelihood method. In particular, we consider a problem of ensuring the validity of an arbitrary quantile function with the help of Chebyshev polynomials and provide useful tips and implementation of these algorithms in Stan and R. We also extend the same method to propose the definition of an *indirect prior* and discuss the situations where it can be useful

Doppler Ultrasound in Prostate Cancer: It’s Utility for the Diagnosis of High-Grade Tumors

Purpose: The aim of the present study was to evaluate the association between prostate vascularization seen in Doppler ultrasound and histopathological grade (Gleason score) in patients with a diagnosis of prostate cancer. Methods: A Gleason score >7 was the dependent variable and Doppler ultrasound findings (vascular analysis, presence of nodule and prostate weight) were the independent variables. Univariate analysis was performed considering advanced tumors (Gleason >7) as the dependent variable and area of hypervascularization, age and PSA as the independent variables. Multivariate analysis was performed using a binary regression model with the occurrence of advanced tumors (Gleason >7) as the dependent variable. Results: In the univariate analysis, samples with Gleason ≤7 had a lower chance of being hypervascularized (OR: 0.44, 95% CI: 0.29-0.69), whereas those with Gleason scores >7 had a fourfold greater chance of being hypervascularized (OR: 4.136, 95% CI: 2.598-6.554, p<0.001). Moreover, hypervascularized tumors had a 7.4-fold greater chance of having a score >7. Conclusion: The present study reveals an association between tumor hypervascularization detected using Doppler ultrasound and higher Gleason scores (more aggressive tumors), enabling an indirect inference of a worse prognosis for hypervascularized prostatic tumors. These findings should be confirmed in longitudinal studies.