When sigma is unknown, contaminated normal distributions tend to have (1) the issue of the actual Type 1 error rate being lower than the nominal level and (2) the issue of reduced power. This pattern has been confirmed by data simulations.
The question then is: does knowing the population variance help with (2) keeping the type 1 error at the nominal level and (2) having good power?
Thank you in advance for your help!
To just add some info:
By power, I simply mean correctly rejecting the null hypothesis of mu=mu0 when this h0 is actually false. I thought that contaminated normal distributions tend to cause wider CIs, which can make correct rejections more difficult.
I did some data simulations to check contaminated normal distributions. Say I repeatedly sample from a contaminated normal distribution with mu=0 and sigma=3 for 1000 times. Then I create 90% confidence intervals based on these samples and check how many of these 1000 CIs contain the population mean. As it turned out (I tried this several times), the percentage of CIs containing the population mean is approximately 90% (that is when sigma is known). So in this case, it seems that the the actual type 1 error rate is comparable to the nominal level of 0.1.
I also tried this assuming that sigma is unknown (so using t-statistics, rather than z), I consistently got more than 90% CIs that contained the population mean, so in this case, the actual type 1 error rate seems to be lower than the nominal level of .1.
But I thought the type 1 error rate and power would suffer even when sigma is known (e.g., the actual type 1 error can be lower the nominal level). But my simulations (assuming that I did it correctly) at least did not show this. So I want to check with you guys to see what's going on. Thanks :)