By Viktor K. Jirsa, A.R. McIntosh
Summarizes the most recent advancements and methods within the box. Highlights components similar to pattern surveys, speculation trying out, time sequence research, Bayesian inference, and extra.
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Extra info for Handbook of Applied Econometrics and Statistical Inference
The estimator 6: with c = 2a: is called the two-stage minimax-linear estimator (2SMMLE). 4. 31) in [l 11, a noncentral x’ distribution may be approximated by a central x’ distribution according to a=- + K 26, K+S, ’ d= (K K + 6,)’ + 26, where, due to the unknown S,, the factor a and the number of degrees of freedom d are also unknown. , approximately (in the case of independence of 3’) we have The desiredconfidence region for 9, at the level 1 - CY is defined by the interior of the ellipsoid (h: - B)’v,-’(b: - B) < ads‘F,I,,-,(l - CY) (40) b .
INTRODUCTION In statistical research of the linear model there have been many attempts to provide estimators of fi which use sample and prior information simultaneously. Examples are the incorporation of prior information in the form of exact or stochastic restrictions (see [l-31) and the use of inequality restrictions which leads to the “minimax” estimation. Minimax estimation is based on the idea that the quadratic risk function for the estimate is not minimized over the entire parameter space R L , but only over an area B(B) that is restricted by a priori knowledge.
2 Non-exponential Families Exponential families of priors play a very prominent role in Bayesian statistical analysis. However, there are interesting cases which fall outside the Table 1. Adjustments in the hyperparameters in the prior family (31), combined with likelihood (37) Parameter Prior value Posterior value Arnold et al. 14 exponential family framework. Each one mustbedealtwith on acase-by-case basis because there will be no analogous theorem in non-exponential family cases that is aparalleltoTheorem I (which allowed us to clearly identify thejoint densities with specified conditional densities).