Robustness of Bayesian and classical inference under distribution bands and shape-restricted families

Sanjib Basu, Purdue University

Abstract

Robustness of classical and Bayesian inference is considered when exact model assumptions are violated. Two families of problems are addressed: (i) Sensitivity of Bayesian posterior measures to the choice of the prior; (ii) Robustness of the coverage probability of standard confidence intervals for location parameters under departure from standard assumptions such as an i.i.d. sample from a normal distribution. Under (i), we consider a dimensional probability distribution with a scalar parameter $\theta$ and compute the ranges of various posterior quantities such as posterior moments and posterior probabilities of an interval when the CDF of the prior of $\theta$ lies between two fixed CDF's. We also consider the further shape restricted family of prior CDF's which are in addition symmetric and unimodal about a fixed point. This includes as special cases well known metric neighborhoods of a fixed CDF such as Kolmogorov and Levy neighborhoods. The derivation usually involves a low dimensional numerical optimization in spite of the obvious infinite dimensional nature of the problem. It is found that despite the richness of the family of priors, robustness obtains when the band bounding the prior CDF's is tight. Several explicit examples are worked out, in particular the normal and the Binomial. Under (ii), the following problems are addressed: (a) Let $X\sb i$ = $\theta$ + $Z\sb i$, where $Z\sb i$ are i.i.d. with CDF F and a known variance $\sigma\sp2$. If F is N(0,1), a standard confidence interval for $\theta$ is $\bar X\pm z\sb{\alpha/2}{\sigma\over\sqrt{n}}.$ The problem of evaluating the infimum of its coverage probability is considered when F belongs to appropriate families of CDF's, such as $\rm{\cal F}\sb1 = \{F : F$ is a scale mixture of Normal distributions with mean zero$\},$ $\rm{\cal F}\sb2 = \{F : F$ is symmetric and unimodal about zero$\},$ $\rm{\cal F}\sb3 = \{F : F$ is unimodal (not necessarily at zero) and has mean zero$\}.$ Our findings suggest that the infimum coverage of the z-confidence interval over these classes is, indeed, quite close to the nominal coverage, thus indicating a good degree of robustness. (b) Let $X\sb i$ = $\theta$ + $\sigma Z\sb i$, where $\theta$, $\sigma$ are both unknown. The problems described in (a) above are considered when the interval under consideration is the well known t-confidence interval $\bar X\pm t\sb{\alpha/2}{s\over\sqrt{n}}$. For example, when the family ${\cal F}\sb2$ is considered, it is found that for sufficiently small $\alpha$, the minimum of the coverage probability is attained at the Uniform ($-$1,1) distribution, and the t interval is, indeed, quite robust. But for sufficiently large $\alpha$ (when $t\sb{\alpha/2}< 1)$, the infimum coverage is found to be zero, thus indicating drastic sensitivity to nonnormality. The effect of skewness in the parent distribution is also examined by introducing a contamination class. It is found that the t interval is also very sensitive to skewness.

Degree

Ph.D.

Advisors

DasGupta, Purdue University.

Subject Area

Statistics

COinS