  Let $H_{n}^{+}(\mathbb{R})$ be the cone of all positive semidefinite $n\times
n$ real matrices. Two of the best known partial orders that were mostly studied
on subsets of square complex matrices are the L\"owner and the minus partial
orders. Motivated by applications in statistics we study these partial orders
on $H_{n}^{+}(\mathbb{R})$. We describe the form of all surjective maps on
$H_{n}^{+}(\mathbb{R})$, $n>1$, that preserve the L\"owner partial order in
both directions. We present an equivalent definition of the minus partial order
on $H_{n}^{+}(\mathbb{R})$ and also characterize all surjective, additive maps
on $H_{n}^{+}(\mathbb{R})$, $n\geq3$, that preserve the minus partial order in
both directions.
