First please forgive me of using this non-mathematical terminology in the title.
Assume we have two series of real numbers $X_1, X_2, \ldots, X_M$ and $Y_1, Y_2, \ldots, Y_N$ where it is possible for $X_i=X_j$ and $i \neq j$. Now if we combine series $X$ and $Y$, the resulting set of real numbers $X_1, X_2, \ldots, X_M, Y_1, Y_2, \ldots, Y_N$ is denoted as $X+Y$. Finally let us assume a statistics $T$ can be defined for $X$, $Y$, and $X+Y$ similarly.
My question is: what statistics is guaranteed to have the property: $T(X+Y) >= T(X)$?
For example, suppose we define $T(X)=max(X_1, X_2, \ldots, X_M)$. It follows that $T(X+Y)=max(X_1, X_2, \ldots, X_M, Y_1, Y_2, \ldots, Y_N ) \geq max(X_1, X_2, \ldots, X_M) = T(X)$.
So the maximum statistics satisfy this property. Another example would be $T(X)=card(X)$. In both cases, as data grows (from $X$ to $X+Y$), the statistics is monotonically non-decreasing.
On the other hand, the median statistics does not necessarily satisfy the property.
My question is, in general, what types of statistics would hold the property of $T(X+Y) >= T(X)$? I am particular in range statistics.
