Let $(X,Y)$ and $(S,T)$ be two continuous random vectors. It is shown that if $S$, $[Y|X=x]$ and $[T|S=x]$, for all $x$ are $DFR$, $Y$ is stochastically increasing in $X$ and $(X,Y) \le_{sst} (S,T)$, then $H (X,Y) \le H(S,T)$, where $H(Z)$ is Shannon entropy of a random variable $Z$. Let $(X_i,Y_i)$, $i=1,\ldots,max\{m,n\}$ be a set of independent copies of $(X,Y)$. It is also shown that if $X$ and $[Y|X=x]$, for all $x$ have $DFR$ distributions and $Y$ is stochastically increasing in $X$, then for $i \le j$ and $n-i \ge m-j$, $H(X_{i:n},Y_{[i:n]}) \le H(X_{j:m},Y_{[j:m]})$. Let $(S_i,T_i)$, $i=1,\ldots,max\{m,n\}$ be a set of independent copies of $(S,T)$. It is observed that under certain set of mild conditions on $F_{X,Y}$ and $F_{S,T}$, for $i \le j$ and $n-i \ge m-j$, $H(X_{i:n},Y_{[i:n]}) \le H(S_{j:m},T_{[j:m]})$. Finally, we discuss some conjectures about entropy properties of vector of order statistics corresponding to a random sample of size $n$ from a symmetric distribution which admits density.
