Suppose $X_1,\ldots, X_m$ is a random sample of size $m$ from the normal distribution $N(\mu_1,\sigma^2)$ with mean $\mu_1$ and standard deviation $\sigma$, and that $Y_1,\ldots, Y_n$ is a random sample of size $n$ from the normal distribution $N(\mu_2,\sigma^2)$ with mean $\mu_2$ and standard deviation $\sigma$. Also, suppose that the samples $X$ and $Y$ are independent. What are the probability density function and the minimal sufficient statistic for $(\mu_1,\mu_2,\sigma)$?.
[Math] Probability density function and the minimal sufficient statistics for two samples from normal distribution
normal distributionprobabilitystatistical-inferencestatistics
Best Answer
The probability density function for the data is
\begin{align} f_\theta\left(\{X\}_i,\{Y\}_j\right)&=\prod_i\exp\left(\frac12\left(\frac{X_i-\mu_1}\sigma\right)^2\right)\prod_j\exp\left(\frac12\left(\frac{Y_j-\mu_2}\sigma\right)^2\right)\\ &=\exp\left(\frac1{2\sigma^2}\left(\sum_iX_i^2+\sum_jY_j^2-2\mu_1\sum_iX_i-2\mu_2\sum_jY_j+m\mu_1^2+n\mu_2^2\right)\right)\;, \end{align}
so $\left(\sum_iX_i^2+\sum_jY_j^2,\sum_iX_i,\sum_jY_j\right)$ is a sufficient statistic. Since
$$ \frac{f_\theta\left(\{X\}_i,\{Y\}_j\right)}{f_\theta\left(\{X'\}_i,\{Y'\}_j\right)} $$
is independent of $\theta$ if and only if this statistic is the same for the two sets of data, this is also a minimal sufficient statistic. (There's no such thing as "the" minimal sufficient statistic, since you can apply any bijective function to a minimal sufficient statistic to obtain another one.)