# The monotonicity of the bivariate normal with non-isotropic covariance

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

• (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? Mar 27, 2023 at 17:49
• (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. $Y_1 > y_1 - \delta \wedge Y_2 > y_2 - \delta \equiv Y_2 > y_2 - \delta \wedge Y_1 > y_1 - \delta$ Mar 27, 2023 at 17:58
• Do I understand correctly that the covariance matrix has 2’s on the diagonal and 1’s elsewhere?
– user44143
Mar 27, 2023 at 19:25
• 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 $\rho = 1/2$ Mar 27, 2023 at 19:26

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 $$z$$ $$$$r'_3(2u+z)=\frac{t \left(2 z^2+2 t z+15-12 t^2\right) e^{-t (t-4 (3 u+z))/6}}{3 (3 t+z)^2},$$$$ which is obviously $$>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 $$v\in (-\infty,\infty)$$. $$\quad\Box$$

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