I am reading "Introduction to Mathematical Statistics" to familiarize myself with sufficient statistics. I got stuck by an example in Sec. 7.3 of the book. Below are page 427 and 428 that relate to my question.
My question is the last sentence of Example 7.3.1 underlined in red in the screenshot. The preceding steps of the solution find an unbiased statistic $[(n-1)Y_2]/n$ of $\theta$; let's denote it as $Y_3$. In my understanding, next we need to follow Theorem 7.3.1 in page 427 to construct an unbiased estimator $E(Y_3|Y_1)\buildrel \Delta \over =\varphi(Y_1)$. But:
- The example does not make any attempt to compute $E(Y_3|Y_1)$,
- The theorem requires that the unbiased estimator (here $Y_3$) be "not a function of $Y_1$ alone." But $Y_3$ is a function of sufficient statistic $Y_1$. How to reconcile this violation?
- Even if we compute $E(Y_3|Y_1)$ according to Theorem 7.3.1, we only have that its variance is less than or equal to that of $Y_3$. We don't have, however, that this variance is less than or equal to that of any other statistics, which is the definition of MVUE (minimum-variance unbiased estimator). So, why does the last sentence claim immediately that the unbiased statistic $Y_3$ "is an MVUE of $\theta$"?
I tried my best to read the text to fill the gap but failed, so I ask for help here for a reason for such an immediate conclusion. If you know of this text, you can infer from the fact that I read this book that I am not a major in math and my knowledge of probability and statistics is at naïve level, so please provide an accessible answer to my questions. Thank you.