Let X1,X2,X3, and X4 denote the counts in four cells. Suppose that X1,X2,X3, and X4 follow a multinomial distribution with a total count of n and cell probabilities (2+θ)/4,(1−θ)/4,(1−θ)/4, and θ/4, respectively, where 0 < θ < 1. Given 0 <α<1, find an approximate (1−α)% confidence interval for θ.
This is a past qual question. I tried to find Mle but stuck after finding a quadratic equation in θ. Do we get a nice Mle in this case. Or do I have to find confidence interval without finding MLE and using θ hat instead?