**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 X_{1} , X_{2} , . . . , X_{n} from
multivariate normal with mean 0 and covariance matrix Σ = σ^{2}R, where R is the unknown
correlation matrix whose off-diagonal entries are the common correlation ρ between any two
components of X_{1}. We consider two approaches of interval estimation of the common variance σ^{2} 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 σ^{2} 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 σ^{2} is sought
that does not differ from the actual value of σ^{2} by more than a certain percentage of σ^{2} with high
probability. We develop a modified two stage sampling procedure to construct a prescribed
proportional closeness interval of σ^{2} . 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.