Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series A, Pt. 1, pp. 86--94

ILL-CONDITIONED DESIGN AND UNIDENTIFIABILITY

By

SOMESH DAS GUPTA, Indian Statistical Institute, Kolkata

and

ABHIJIT DASGUPTA, University of Washington

SUMMARY. Suppose $Y_i \sim N(\alpha + \beta x_i, \sigma^2),\ i=1,\dots,n$ are independent random variables. For testing $\alpha=\alpha_0,\ \beta=\beta_0$, it is shown that the power of the likelihood ratio test (LRT) may be smaller than the power of the corresponding test under ``$x_i$'s are all equal", when $\sum_i (x_i-\bar{x})^2$ is close to 0. A similar phenomenon occurs for the confidence set for $(\alpha,\beta)$. It is shown that the transition to $\sum_i(x_i-\bar{x})^2 =0$ is smooth for Bayesian inference.

AMS (1991) subject classification. Primary 62J05; secondary 62K99, 62F15.

Key words and phrases. Ill-conditioned design, linear model, unidentifiability, power function, confidence interval, Bayes decision, HPD region.

Full paper (PDF)