Let $X_1 , X_2 ,...,X_n$ be a random sample of size $n$ from a normal distribution, $X_i \backsim N(\mu,\sigma^2)$, and define $U = \sum_{i=1}^{n}{X_i}$ and $W= \sum_{i=1}^{n}{X_{i}^2}$.
(a) Find a statistic that is a function of $U$ and $W$ and unbiased for the parameter $\theta = 2\mu -5\sigma^2$
(b) Find a statistic that is unbiased for $\sigma^2 + \mu^2$
My question about this problem is the following: Is it necessary to use the transformation theorem to do this, that is, can't this exercise be solved without having a prior concept about the transformation? If so, then what would be the best way to start? or how to apply the theorem in this case in such a way that an unbiased statistic is obtained for $ \theta $ in part (a) and another for part (b)? I appreciate some help to do this.