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I am just interested in $E[L_n], E[U_n], E[L_n U_n], E[L_n^2]$ and $E[U_n^2]$ where $L_n =\min(X_1,\cdots,X_n)$ and $U_n=\max(X_1,\cdots,X_n)$. The $X_k$'s are i.i.d. In fact, I am only interested in $E(R_n)$ and $E(R_n^2)$ where $R_n = U_n - L_n$ is the range. Even more narrowly, I am solely interested in $\mbox{Var}[R_n]$.

I am looking for simple recurrence formulae, for instance $E[L_n] = h(n, E[L_{n-1}])$. I searched Google for recurrence formula for the moments / product moments of order statistics. Clearly, there has been a lot of research on this topic, but I haven't found any answer to my question yet.

What do I try to accomplish?

Answer: to find an asymptotic formula for $\mbox{Var}[R_n]$ that applies to any continuous distribution for $X_1, \cdots, X_n$. In particular, for the Gaussian distribution.

In the case of the uniform distribution, $\mbox{Var}[R_n] = \frac{2(n-1)}{(n+1)^2(n+2)} = O(1/n^2)$.

In the case of the exponential distribution, $\mbox{Var}[R_n] =\frac{1}{\lambda^2}\sum_{k=1}^{n-1}\frac{1}{k^2} \rightarrow \frac{\pi^2}{6\lambda^2} = O(1)$. (see here).

Also, $E[R_n] \sim F^{-1}\Big(\frac{n}{n+1}\Big) - F^{-1}\Big(\frac{1}{n+1}\Big)$ where $F^{-1}$ is in the inverse of the cdf attached to the $X_k$'s. I am not sure if this asymptotic relationship is correct, it probably is (it works both for the uniform and exponential distributions) and it probably is a well-known result. It is based on the fact that the transformed order statistics $F(X_{(k)})$ are uniformly distributed on $[0, 1]$ regardless of the underlying distribution $F$. And for a uniform distribution on $[0, 1]$, the minimum and maximum have expectation $\frac{1}{n+1}$ and $\frac{n}{n+1}$ respectively.

My approach to the problem

I have spent some time on this, and a possible way to solve this (besides finding a solution in the literature) is to find a simple recurrence relation for the moments that I am in interested in. For instance, $E[L_n^2] = \int_0^1\int_0^1\cdots\int_0^1 [F^{-1}(\min(u_1,\cdots u_n))]^2 du_1\cdots du_n$. The $n$-tuple integral can be computed iteratively, leading to a simple relationship between $E[L_n^2]$ and $E[L_{n-1}^2]$. That's where I stand right now. A lot more work needs to be done. Any help is appreciated.

Note that $F^{-1}(\min(u_1,\cdots u_n))=\min(F^{-1}(u_1),\cdots,F^{-1}(u_n))$. This is true regardless of $F$ because $F$ (and thus $F^{-1}$) is always an increasing function, at least for the cases we are interested in.

Vincent Granville
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1 Answers1

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Suppose $X_i$ are iid with continuous cdf $F$, $L_n = \min(X_1,\ldots,X_n)$ and $U_n = \max(X_1,\ldots,X_n)$. Then $$\mathbb P(a \le L_n \le U_n \le b) = \mathbb P( X_1,\ldots,X_n \in [a,b]) = (F(b) - F(a))^n$$ and if $F$ corresponds to a pdf $f$, the joint pdf of $(L_n, U_n)$ is $$f_{L_n, U_n}(x,y) = - \dfrac{\partial^2}{\partial x \partial y} (F(y) - F(x))^n = n (n-1) (F(y)-F(x))^{n-2} f(x) f(y)$$ for $x < y$. Of course $$\mathbb E[U_n - L_n] = \iint_{x < y} dx\; dy\; (y-x) f_{L_n,U_n}(x,y)$$ and $$\mathbb E[(U_n - L_n)^2] = \iint_{x < y} dx\; dy\; (y-x)^2 f_{L_n,U_n}(x,y)$$ I don't know what can be said about the asymptotics in general: Watson's lemma may be helpful.

Robert Israel
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  • This is correct but not useful to explicitly compute the moments of the range for specific distributions. Of course the exact solution is known only in very few cases, but getting an approximation (asymptotic) is what I am looking for. Some simple cases (besides uniform and exponential) could be $F_X(x) = x^2, 0\leq x \leq 1$ or $F_X(x) =(1+a(x+|x|))/(2(ax+1))$ with $a>0$. – Vincent Granville May 24 '19 at 14:58
  • For the Gaussian distribution, the variance of the range is $O(1/n)$. This result is very helpful, it confirms empirically that the variance of the range is a power function of $n$. See https://stats.stackexchange.com/questions/229073/variance-of-maximum-of-gaussian-random-variables – Vincent Granville May 24 '19 at 22:31
  • Also, for the Gaussian distribution, the expectation of the range is $O(\sqrt{\log n})$. See https://math.stackexchange.com/questions/89030/expectation-of-the-maximum-of-gaussian-random-variables. – Vincent Granville May 24 '19 at 22:33
  • I have accepted your answer, but I am posting a new question that focuses on what I am really interested in - the asymptotics. You can check it out at https://math.stackexchange.com/questions/3238891/is-the-following-always-true-mboxvar-mboxrangex-1-cdots-x-n-on?noredirect=1#comment6662191_3238891 – Vincent Granville May 25 '19 at 05:16