Sufficient statistics for $\mu_1 – \mu_2$

If $ X_1, …, X_n$ is a random sample from $ X \sim N(\mu_1, \sigma^2)$ and $Y_1,…, Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $ \sigma^2$ is known, can we say that $\bar{X}-\bar{Y}$ is sufficient for $\mu_1 – \mu_2$ ?

My guess is that it is true. I thought we could define

$$W_i=X_i-Y_i \sim N(\mu_1 – \mu_2, 2 \sigma^2)$$

n independent random variables and use it to show that $\bar{W} = \bar{X} – \bar{Y} $ is sufficient to the mean. Is that correct?



Source : Link , Question Author : Giiovanna , Answer Author : Community

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