t-Statistics for Weighted Means with Application to Risk Factor Models
Lisa R. Goldberg and Alec N. Kercheval
The standard t-statistic test for equality of the mean for two normal iid samples assumes the two samples have a common (but unknown) variance. If we allow each sample value to be drawn from a distribution with a different variance, the best linear unbiased estimate of the common mean becomes a weighted average, with larger weights for sample values with lower variances.
Given two such samples, we provide a formula for a generalized t-statistic for testing whether the means of the two samples are equal. The need for this generalization arises in comparing average spreads for like-rated bonds in different markets, where spread variances are assumed to depend on bond duration.