Expected difference between the largest and second largest observations in a sample of i.i.d. normal variables

Question

Let $X_1,\dots,X_n$ be an i.i.d. sample from the standard normal distribution. Let \begin{align} \mu_n = \mathbb{E}[X_{(n)} - X_{(n-1)}], \end{align} be the expected difference between the largest and second largest observations.

Question: Can we show that $\mu_n$ is decreasing in $n$?

In this answer it is shown that $\mu_n$ is asymptotically decreasing, i.e., it is decreasing for large enough $n$, and in this answer it is argued numerically that $\mu_n$ is also decreasing for finite $n$.

A good first step is to use a known order statistics formula (see e.g. this answer) to write \begin{align} \mu_n = n\int_{-\infty}^{\infty}\Phi^{n-1}(x)(1-\Phi(x))dx, \end{align} where $\Phi$ is the standard normal CDF. However, I am unable to show from this that $\mu_n$ is decreasing.

Addition: The intuition provided in a previously linked answer is that $\mu_n$ is decreasing if the $X_i$'s come from a thin-tailed distributions like the normal distribution, and that $\mu_n$ is constant or even increasing if the $X_i$'s come from a more heavy-tailed distribution (in particular, it is easy to show that $\mu_n$ is constant if the $X_i$'s come from the exponential distribution). Perhaps this intuition can help in showing that $\mu_n$ is decreasing for the normal distribution.

Answer 1

Let’s integrate the expression you provide for $\mu_n$ by parts to reduce the dependence on $n$:

\begin{eqnarray*} \mu_n &=& \int_{-\infty}^\infty n\Phi^{n-1}(1-\Phi)\mathrm dx \\ &=& \int_{-\infty}^\infty n\Phi^{n-1}\Phi'\frac{1-\Phi}{\Phi'}\mathrm dx \\ &=& \left[\Phi^n\frac{1-\Phi}{\Phi'} \right]_{-\infty}^\infty-\int_{-\infty}^\infty\Phi^n\left(\frac{1-\Phi}{\Phi'}\right)'\mathrm dx \\ &=& \int_{-\infty}^\infty\Phi^n\left(1+\frac{\Phi''(1-\Phi)}{\Phi'^2}\right)\mathrm dx\;,\end{eqnarray*}

where the boundary term vanishes for a normal distribution. Since this expression can be extended to real $n$, we can differentiate it with respect to $n$, yielding $$ \frac{\partial\mu_n}{\partial n}=\int_{-\infty}^\infty\Phi^n\log\Phi\left(1+\frac{\Phi''(1-\Phi)}{\Phi'^2}\right)\mathrm dx\;. $$

So far we only assumed a normal distribution in assuming $\Phi'\ne0$ and in concluding that the boundary term vanishes, so this result holds for all distributions that fulfil those conditions (possibly with different limits on the integral). It also holds if the boundary term is non-zero but doesn’t depend on $n$. For instance, for an exponential distribution, with

$$ \Phi=1-\mathrm e^{-\lambda x}\;,\\ \Phi'=\lambda\mathrm e^{-\lambda x}\;,\\ \Phi''=-\lambda^2\mathrm e^{-\lambda x}\;, $$

the boundary term is $1$, independent of $n$, and we have

\begin{eqnarray*} 1+\frac{\Phi''(1-\Phi)}{\Phi'^2} &=& 1+\frac{-\lambda^2\mathrm e^{-\lambda x}\left(1-\left(1-\mathrm e^{-\lambda x}\right)\right)}{\left(\lambda\mathrm e^{-\lambda x}\right)^2} \\ &=& 0\;, \end{eqnarray*}

confirming that $\mu_n$ is constant for this type of distribution.

Since $\Phi^n\log\Phi$ is negative and the remaining factor doesn’t depend on $n$, $\mu_n$ is strictly increasing or strictly decreasing if this factor is everywhere negative or everywhere positive, respectively. If the factor changes sign, then $\frac{\partial\mu_n}{\partial n}$ may change sign as $\Phi^n$ gets increasingly concentrated near $\Phi=1$ for increasing $n$.

For a uniform distribution, the boundary term vanishes, and we have $\Phi''=0$, so $\mu_n$ is strictly decreasing, as expected.

For the standard normal distribution, we have

\begin{eqnarray*} \Phi &=&\frac12\left(1+\operatorname{erf}\left(\frac x{\sqrt2}\right)\right)\;,\\ \Phi' &=& \frac1{\sqrt{2\pi}}\exp\left(-\frac{x^2}2\right)\;,\\ \Phi'' &=& -\frac1{\sqrt{2\pi}}x\exp\left(-\frac{x^2}2\right)\;, \end{eqnarray*}

so

\begin{eqnarray*} 1+\frac{\Phi''(1-\Phi)}{\Phi'^2} &=& 1-\sqrt{\frac\pi2}x\exp\left(\frac{x^2}2\right)\left(1-\operatorname{erf}\left(\frac x{\sqrt2}\right)\right)\;. \end{eqnarray*}

With

\begin{eqnarray*} \sqrt{\frac\pi2}\left(1-\operatorname{erf}\left(\frac x{\sqrt2}\right)\right) &=& \sqrt2\int_{x/\sqrt2}^\infty\exp\left(-t^2\right)\mathrm dt \\ &=& \int_x^\infty\exp\left(-\frac{u^2}2\right)\mathrm du \\ &=& \int_x^\infty\left(-\frac1u\right)\left(-u\exp\left(-\frac{u^2}2\right)\right)\mathrm du \\ &=& \left[-\frac1u\exp\left(-\frac{u^2}2\right)\right]_x^\infty-\int_x^\infty\frac1{u^2}\exp\left(-\frac{u^2}2\right)\mathrm du \\ &=& \frac1x\exp\left(-\frac{x^2}2\right)-\int_x^\infty\frac1{u^2}\exp\left(-\frac{u^2}2\right)\mathrm du\;, \end{eqnarray*}

this becomes

\begin{eqnarray*} 1+\frac{\Phi''(1-\Phi)}{\Phi'^2} &=& 1-x\exp\left(\frac{x^2}2\right)\left(\frac1x\exp\left(-\frac{x^2}2\right)-\int_x^\infty\frac1{u^2}\exp\left(-\frac{u^2}2\right)\mathrm du\right) \\ &=& x\exp\left(\frac{x^2}2\right)\int_x^\infty\frac1{u^2}\exp\left(-\frac{u^2}2\right)\mathrm du\ \\ &\gt&0\;, \end{eqnarray*}

so indeed $\mu_n$ is strictly decreasing for the standard normal distribution.