Date & Time: Monday, April 21, 2014, 15:00-15:30
Venue: IRCC Conference Room II, SOM-IRCC Building

Title: Interval Estimation of The Common Variance of Equi-correlated Normal Distributions: Two Sampling Approaches

Speaker: Shyamal K. De, Binghamton University (SUNY), USA

Abstract: In an ANOVA design, where the m × 1 measurement vector from a unit is considered as the response variable, one can model the error part by an m−variate normal distribution with mean 0 and covariance matrix Σ. This work investigates the estimation of Σ with a special structure. Formally, consider n i.i.d. m−dimensional vectors X1 , X2 , . . . , Xn from multivariate normal with mean 0 and covariance matrix Σ = σ2R, where R is the unknown correlation matrix whose off-diagonal entries are the common correlation ρ between any two components of X1. We consider two approaches of interval estimation of the common variance σ2 in the presence of nuisance parameter ρ.

In the first approach, we develop a modified three-stage sampling procedure to construct a fixed-width (1 − α)-level confidence interval of σ2 for a predetermined α ∈ (0, 1). The modified three-stage sampling substantially reduces the expected sample size compared to that of the two- stage sampling scheme of Haner and Zacks (2013). The coverage probabilities of the proposed interval estimators are computed exactly and compared with the coverage probabilities obtained by the two-stage sampling. The exact distributions of the stopping variables and the estimators of the common variance at stopping are derived.

In the second approach, we consider a different criterion for interval estimation known as prescribed proportional closeness criterion. According to this criterion, an estimator of σ2 is sought that does not differ from the actual value of σ2 by more than a certain percentage of σ2 with high probability. We develop a modified two stage sampling procedure to construct a prescribed proportional closeness interval of σ2 . We establish asymptotic first-order efficiency and consistency properties of this procedure. Moreover, the exact distribution of the stopping variable and the coverage probabilities are obtained and compared with that of the first approach.