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Question: Suppose $X_1, \cdots, X_n$ are $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. Find the UMVUE for $\Phi(\mu)$, where $\Phi(\cdot)$ is the cdf of a standard normal random variable.

I used to guess the desired UMVUE is or similar to $\Phi(\bar X)$ since it is a function of complete and sufficient statistic. However, it is not really unbiased (see herehere). Could anyone shed light on how to find the UMVUE, please? Thank you!

Question: Suppose $X_1, \cdots, X_n$ are $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. Find the UMVUE for $\Phi(\mu)$, where $\Phi(\cdot)$ is the cdf of a standard normal random variable.

I used to guess the desired UMVUE is or similar to $\Phi(\bar X)$ since it is a function of complete and sufficient statistic. However, it is not really unbiased (see here). Could anyone shed light on how to find the UMVUE, please? Thank you!

Question: Suppose $X_1, \cdots, X_n$ are $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. Find the UMVUE for $\Phi(\mu)$, where $\Phi(\cdot)$ is the cdf of a standard normal random variable.

I used to guess the desired UMVUE is or similar to $\Phi(\bar X)$ since it is a function of complete and sufficient statistic. However, it is not really unbiased (see here). Could anyone shed light on how to find the UMVUE, please? Thank you!

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Find the UMVUE for $\Phi(\mu)$

Question: Suppose $X_1, \cdots, X_n$ are $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. Find the UMVUE for $\Phi(\mu)$, where $\Phi(\cdot)$ is the cdf of a standard normal random variable.

I used to guess the desired UMVUE is or similar to $\Phi(\bar X)$ since it is a function of complete and sufficient statistic. However, it is not really unbiased (see here). Could anyone shed light on how to find the UMVUE, please? Thank you!