# Comparing two groups of measurements with missing values

There are two groups of normally distributed measurements, carried out under the same conditions: $$x_i = \theta_{x,i} + \epsilon_x$$, $$y_i = \theta_{y,i} + \epsilon_y$$. I would like to test the hypothesis that $$\theta_{x,i} = \theta_{y,i}$$. I cannot use the two-sided t-test, because the means are not constant (they depend on $$i$$). However I could use a one-sided t-test for $$x_i-y_i$$ with the same effect (testing that the mean of $$x_i - y_i$$ is zero).

The problem: There are missing data in these two sets of measurements. If the means were constant, I could have used t-test for two samples with unequal sizes (Welsh's test). Here I however end up with a one-sided test with the size corresponding to that of the smaller sample. It seems like I am losing lots of useful information about the variance. Any suggestions?