I'm very confused by what these are. My definition reads:

Let $Z_1,\dots,Z_n$ take the values of random variables $X_1,\dots, X_n,$ arranged in increasing order so that $Z_1\leq \cdots\leq Z_n$. We call $Z_1,\dots ,Z_n$ order statistics.

What is bothering me is arranging random variables in increasing order: I thought random variables were functions, so I don't understand how we can do this. Can someone explain what these are, more intuitively?

  • $\begingroup$ Random variables are functions that take different "values" depending on outcomes of the random experiment. So, $X_1$ is a function but when the random experiment is done, it takes on a "value" that we denote by $Z_1$. $\endgroup$ – Hoda Apr 9 '14 at 0:24
  • $\begingroup$ When you do your experience, the random variables $X_i$ take some values. Once these values are known, arrange them in increasing order. Note that $Z_1 = \min(X_i)$ and $Z_n = \max(X_i)$. Similarly, for example, $Z_2$ takes the value of the second lowest value attained by the random variables $X_i$. For different realizations of the experience, the second lowest value will differ and it will not always be the same $X_j$ that will take it. $\endgroup$ – Amateur Apr 9 '14 at 0:25

Think of an order statistic as a function defined on a vector of random variables. Therefore, an order statistic is itself a random variable, whose value is dependent not on any single, particular random variable, but on the entire set of random variables.

We know that there is a natural way to think of making new random variables from existing ones; e.g., suppose $X_1, X_2$ are random variables (not necessarily IID) and we can define $Y = g(X_1, X_2)$ for some function $g$. For instance, $g(a,b) = a+b$ and so $Y = X_1 + X_2$.

When we think of the $k^{\rm th}$ order statistic, the function $g$ is simply a particular type of function, namely $g(x_1, x_2, \ldots, x_n) = x_{\pi(k)}$, where $\pi(k)$ is the index of the $k^{\rm th}$ smallest value in the set $\{x_1, \ldots, x_n\}$.

It is important to emphasize here that $g$ is necessarily a function of all of the random variables being ordered, even though the value of $g(x_1, \ldots, x_n)$ is simply one of those $x_i$'s. The randomness of an order statistic is inherited from the entire set of random variables being ordered, in as much as $Y = X_1 + X_2$ has randomness from both $X_1$ and $X_2$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.