# How to select optimal observation number n from two marginal distributions?

I have two marginal distributions $(Y_1,Y_2)$ that follow distributions $N(\mu_1,50)$ and $N(\mu_2,100),$ respectively. I can allow for a total of 100 observations to estimate the parameter $\theta = \mu_1 - \mu_2.$ How should I allocate those 100 observations from both the two marginals to model the parameter?

What I think is that I have to generate densities from both normal distributions and optimize with likelihood ratio for all possible combination, but still fuzzy how to conceptualize the problem ...

• Notations for a normal distribution vary. Are 50 and 100 variances or standard deviations? // To answer most usefully, it might help to know your purpose: To estimate $\theta$ or to test $H_0: \mu_1 = \mu_2$ vs. $H_a: \mu_1 \ne \mu_2.$? – BruceET Feb 6 '18 at 18:37
• Could you please explain what it means to "allocate ... from both the two marginals"? Are you saying that you can observe $Y_1$ or $Y_2$ but not both simultaneously? That would be strange, for why would you characterize these as "marginals" unless you were viewing $(Y_1,Y_2)$ as bivariate, which means you do observe them simultaneously? – whuber Feb 6 '18 at 18:49
• Hi, I want to estimate the distance between the two means for two distributions Y1 and Y2. Though it is in the bivariate space (sorry, my mistake). Here, I have 100 observations that I can choose between the observable vector of Y1 and Y2 densities (ie : the marginals from the joint distribution) to find the optimal parameter θ=μ1−μ2. – Gabriel Amiot Feb 6 '18 at 19:09
• You mean "optimal estimator" and not "optimal parameter". But optimal in what sense? If you want an estimator with a minimal variance, and if Y1 and Y2 are independent, than simple calculus shows that you need to allocate n's that are proportional to the standard deviations of Y1 and Y2 (and of course use as estimator the difference between the two sample averages). – Zahava Kor Feb 6 '18 at 19:58
• @ZahavaKor Thanks for your answer. I might arbitrarily fix the criterion to obtain the best variance-bias tradeoff in order to get minimum-variance unbiased estimator. In this case, I want to minimize the standard error of the sample mean for the random variable Z = Y2-Y1. Though I think that all I have to do for a simulation is to select the linear model that maximize the log-likelihood for a vector of allocations.. Is this right according to you? – Gabriel Amiot Feb 6 '18 at 22:17

Without the requested additional information, I can only guess your situation and what you are trying to accomplish. Let me assume that the known population variances are $\sigma_1^2 = 50$ and $\sigma_2^2 = 100.$
Assuming independent samples, the variance of $\hat \theta = \bar Y_1 - \bar Y_2$ is $Var(\hat \theta) = 50/n_1 + 100/n_2,$ where $n_1 + n_2 = 100.$ Then you might want to minimize $Var(\hat \theta)$ by making appropriate choices for sample sizes $n_1 < n_2.$ In round numbers, maybe $n_2 \approx 60.$
n = 40:70