Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance.

Define $y = (y_1, y_2)$ and let \begin{align} F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, Y_2 > y_2 - \delta]. \end{align}

Consider \begin{align} G_{\delta}(y) = \frac{F_{\delta}(y) }{ F_0(y) }, \end{align}

We wish to show that $G_{\delta}(y)$ is monotonically increasing in $y$ for $1 > \delta > 0$ and $y \in \mathbb{R}^2$.

  • $\begingroup$ (i) "Show that $G_{\delta}(y)$ is monotonically increasing in $y$ for $1 > \delta > 0$ and $y \in \mathbb{R}^2$." Is this a command? (ii) What is the order on $\mathbb{R}^2$ with respect to which the monotonicity is to be considered? $\endgroup$ Mar 27, 2023 at 17:49
  • $\begingroup$ (i) For example, given $a < b$, I want to show that $\max y \in [a, b]^2 G_{\delta}(y) = G_{\delta}(b, b)$ (ii) there is no specific order, e.g. $Y_1 > y_1 - \delta \wedge Y_2 > y_2 - \delta \equiv Y_2 > y_2 - \delta \wedge Y_1 > y_1 - \delta $ $\endgroup$ Mar 27, 2023 at 17:58
  • $\begingroup$ Do I understand correctly that the covariance matrix has 2’s on the diagonal and 1’s elsewhere? $\endgroup$
    – user44143
    Mar 27, 2023 at 19:25
  • $\begingroup$ Yes that's correct: $\Sigma \triangleq = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$. Alternatively, you could consider $\sigma_1 = \sigma_2 = \sqrt{2}$ and correlation coefficient $\rho = 1/2$ $\endgroup$ Mar 27, 2023 at 19:26

1 Answer 1


For real $u,v$, let \begin{equation*} Q(u,v):=P(Y_1>u\sqrt2,Y_2>v\sqrt2)=\int_u^\infty dz\,\varphi(z)\Big(1-\Phi\Big(\frac{2v-z}{\sqrt3}\Big)\Big), \end{equation*} where $\varphi$ and $\Phi$ are the standard normal pdf and cdf, respectively.

The question in the OP can be restated as follows: show that \begin{equation*} r(u,v):=r_t(u,v):=\frac{Q(u-t,v-t)}{Q(u,v)} \tag{10}\label{10} \end{equation*} is increasing in real $u$ and in real $v$ for each $t\in(0,1/\sqrt2)$. Since $u$ and $v$ are interchangeable, it is enough to shown that \begin{equation*} r(v):=r(u,v) \end{equation*} is increasing in real $v$ for each $t\in(0,1)$.

This will be done by using so-called l'Hospital-type rules for monotonicity. Indeed, consider the "derivative ratio" for the ratio in \eqref{10} with the extra constant factor $e^{t^2/2}$: \begin{equation*} r_1(v):=e^{t^2/2}\frac{Q_v(u-t,v-t)}{Q_v(u,v)}, \end{equation*} where the subscript $_v$ denotes the partial derivative in $v$. We have \begin{equation*} Q_v(u,v)=-\int_u^\infty dz\,\varphi(z)\,\frac2{\sqrt3}\,\varphi\Big(\frac{2v-z}{\sqrt3}\Big) =-\frac{\varphi(v)}2\, \left(\text{erf}\left(\frac{v-2 u}{\sqrt{6}}\right)+1\right). \end{equation*} So, \begin{equation*} r_1(v)=\frac{f_1(v)}{g_1(v)}, \tag{20}\label{20} \end{equation*} where \begin{equation*} f_1(v):=e^{t v} \left(\text{erf}\left(\frac{t-2 u+v}{\sqrt{6}}\right)+1\right),\quad g_1(v):=\text{erf}\left(\frac{v-2 u}{\sqrt{6}}\right)+1. \end{equation*} Next, consider the "derivative ratio" for the ratio in \eqref{20}: \begin{equation*} r_2(v):=\frac{f'_1(v)}{g'_1(v)}=\frac{f_2(v)}{g_2(v)}, \tag{30}\label{30} \end{equation*} where \begin{equation*} f_2(v):=\sqrt{6 \pi } t \left(\text{erf}\left(\frac{t-2 u+v}{\sqrt{6}}\right)+1\right)+2 e^{-(t-2 u+v)^2/6},\quad g_2(v):=2 e^{-t v-(v-2 u)^2/6}. \end{equation*} Further, consider the third (and final) "derivative ratio" for the ratio in \eqref{30}: \begin{equation*} r_3(v):=\frac{f'_2(v)}{g'_2(v)}=\frac{e^{-t (t-4 (u+v))/6}\, (-2 t-2 u+v)}{3 t-2 u+v}. \tag{40}\label{40} \end{equation*} Note that for all real $z$ \begin{equation} r'_3(2u+z)=\frac{t \left(2 z^2+2 t z+15-12 t^2\right) e^{-t (t-4 (3 u+z))/6}}{3 (3 t+z)^2}, \end{equation} which is obviously $>0$ for all $t\in(0,1)$ if $z\ne-3t$.

So, $r_3$ is increasing on $(-\infty,2u-3t)$ and on $(2u-3t,\infty)$. Also, $g_2>0$. Also, $g'_2>0$ on $(-\infty,2u-3t)$ and $g'_2<0$ on $(2u-3t,\infty)$.

So, by lines 1 and 3 of Table 1.1, (the continuous function) $r_2$ is down-up on $(-\infty,2u-3t]$ and up-down on $[2u-3t,\infty)$ -- that is, (i) there is some $a\in[-\infty,2u-3t]$ such that $r_2$ is decreasing on $(-\infty,a]$ and increasing on $[a,2u-3t]$ and (ii) there is some $b\in[2u-3t,\infty]$ such that $r_2$ is increasing on $[2u-3t,b]$ and decreasing on $[b,\infty)$.

So, $r_2$ is down-up-down on $(-\infty,\infty)$. But $r_2$ is a positive function such that $r_2(v)\to0$ as $v\to-\infty$ and $r_2(v)\to\infty$ as $v\to\infty$. So, $r_2$ is increasing on $(-\infty,\infty)$.

But $f_1(v)\to0$ and $g_1(v)\to0$ as $v\to-\infty$. So, by Proposition 4.1 of the cited paper, $r_1$ is increasing on $(-\infty,\infty)$.

Also, clearly $Q(u,v)\to0$ as $v\to\infty$. So, again by Proposition 4.1 of the cited paper, $r(u,v)$ is increasing in $v\in (-\infty,\infty)$. $\quad\Box$

  • $\begingroup$ Thank you so much for the response! I hadn't seen these results regarding l'hôpital's rule for derivative ratios. $\endgroup$ Mar 28, 2023 at 15:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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