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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 $y \in \mathbb{R}^2$.

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  • $\begingroup$ (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

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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 $v\in (-\infty,\infty)$. $\quad\Box$

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  • $\begingroup$ 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

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