Let $\boldsymbol{X} = (X_1,X_2)^{\rm T}\sim \mathcal{N}_2(\boldsymbol{\mu}, \mathrm{\Sigma})$, where \begin{eqnarray*} \boldsymbol{\mu} = (\mu_1, \mu_2)^{\rm T}& = &(\sqrt{\xi_1\xi_2/(\xi_1+\xi_2)}, 0)^{\rm T}\\ \mathrm{\Sigma} & = &\begin{pmatrix} 1 & -\rho\\ -\rho & 1\end{pmatrix}\\ \rho & = & \sqrt{\xi_1\xi_3/(\xi_1+\xi_2)(\xi_2+\xi_3)}. \end{eqnarray*} It is given that $\xi_1\leq\xi_2\leq\xi_3$, $\xi_i\geq 0$ and $\sum_{i=1}^3\xi_i = 1$. I have a function \begin{equation*} \pi(\boldsymbol{\mu};\boldsymbol{\xi}) = 1-\mathbb{P}(\boldsymbol{X}\leq \boldsymbol{t}) = 1-\mathbb{P}(X_1\leq t \cap X_2\leq t), \end{equation*} where $t>1$. Under the constraints $\xi_1\leq\xi_2\leq\xi_3$, $\xi_i\geq 0$ and $\sum_{i=1}^3\xi_i = 1$, numerically we are getting the maximum of $\pi(\boldsymbol{\mu};\boldsymbol{\xi})$ where the non-zero mean $\mu_1$ is maximized. This occurs when $\xi_1 = \xi_2 = \xi_3 = 1/3$. Can we use some kind of stochastic ordering arguments to prove the result theoretically?


1 Answer 1


$\newcommand{\der}{\mathrm{der}}\newcommand{\tdert}{\mathrm{dert}}\newcommand{\erf}{\operatorname{erf}}\newcommand{\eqs}{\overset{\text{sign}}=}\newcommand{\tder}{\widetilde\der}$The statement about the maximum is true, but the proof has hardly anything to do with stochastic ordering arguments. Rather, the problem is reduced to certain problems of real algebraic geometry -- that is, to solving (rather complicated) systems of algebraic inequalities over $\mathbb R$.

Let $x:=\xi_1$ and $y:=\xi_2$, so that $\xi_3=1-x-y$, $0\le x\le y\le1-x-y$, $$\mu_1=M(x,y):=\sqrt{\frac{x y}{x+y}},$$ $$\rho=-R(x,y),\quad R(x,y):=\sqrt{\frac{x(1-x-y)}{(x+y)(1-x)}}.$$ Let also $$p_t(x,y):=1-P(X_1\le t,X_2\le t)$$ and \begin{equation*} G:=\{(x,y)\colon 0\le x\le y\le1-x-y\}=\{(x,y)\colon 0\le x\le1/3,x\le y\le(1-x)/2\}. \tag{-1} \end{equation*} We want to show that \begin{equation*} p_t(x,y)\le p_t(1/3,1/3) \tag{0} \end{equation*} for all $(x,y)\in G$ and all real $t\ge1$.

Note that \begin{equation*} p_t(x,y):=1-P(X_1\le t,X_2\le t)=P(M(x,y),-R(x,y)), \tag{1} \end{equation*} where \begin{equation*} P(m,r):=1-\int_{-\infty}^t du \int_{-\infty}^t dv\, f_{m,r}(u,v) \end{equation*} and $f_{m,r}$ is the density function of the bivariate normal distribution with means $m,0$, variances $1,1$, and correlation $r$.

The key is Plackett's observation (formula (3)) that \begin{equation*} D_r f_{m,r}(u,v)=D_v D_u f_{m,r}(u,v), \end{equation*} where $D_w$ denote the partial derivative with respect to a variable $w$. It follows that \begin{equation*} \begin{aligned} D_r P(m,r)&:=-\int_{-\infty}^t du \int_{-\infty}^t dv \, D_r f_{m,r}(u,v) \\ &=-\int_{-\infty}^t du \int_{-\infty}^t dv \, D_v D_u f_{m,r}(u,v) \\ &=-f_{m,r}(t,t)<0. \end{aligned} \tag{2} \end{equation*} Next, \begin{equation*} \begin{aligned} P(m,r)&=1-\int_{-\infty}^t du \int_{-\infty}^t dv\, f_{0,r}(u-m,v) \\ &=1-\int_{-\infty}^{t-m} dw \int_{-\infty}^t dv\, f_{0,r}(w,v) \end{aligned} \end{equation*} and hence \begin{equation*} \begin{aligned} D_m P(m,r)=\int_{-\infty}^t dv\, f_{0,r}(t-m,v)=\int_{-\infty}^t dv\, f_{m,r}(t,v)>0. \end{aligned} \tag{3} \end{equation*}

Also, for $(x,y)\in G\setminus\{(0,0)\}$, \begin{equation*} (D_x M(x,y),D_y M(x,y))=\frac{(y^2,x^2)}{(x + y)^2}, \end{equation*} \begin{equation*} (D_x R(x,y),D_y R(x,y))=\frac{((1-2 x-y)y,-(1-x) x)}{(1-x)^2 (x + y)^2}, \end{equation*} so that $D_x M(x,y)\ge0$ and $D_x R(x,y)\ge0$.

So, by (1), a chain rule of differentiation, and the inequalities in (2) and (3), \begin{equation*} D_x p_t(x,y)=D_m P(M(x,y),-R(x,y))D_x M(x,y)-D_r P(M(x,y),-R(x,y))D_x R(x,y)\ge0. \end{equation*}

So, the maximum of $p_t(x,y)$ over $(x,y)\in G$ occurs at one of right boundaries of $G$, that is, where either $y=x$ and $y=(1-x)/2$.

Next, for $x\in[0,1/3]$ (cf. (-1)), \begin{equation*} \frac d{dx}\,M(x,(1-x)/2)=\frac{1-2x-x^2}{(1+x)^2}>0, \end{equation*} \begin{equation*} \frac d{dx}\,R(x,(1-x)/2)=\frac1{(1+x)^2}>0. \end{equation*} So, by (1) and the inequalities in (2) and (3), $p_t(x,(1-x)/2)$ is increasing in $x\in[0,1/3]$.

It remains to show that \begin{equation*} \der(t,x)\overset{\text{(?)}}\ge0 \text{ for $x\in[0,1/3]$ and $t\ge1$,} \tag{4} \end{equation*} where \begin{equation*} \begin{aligned} \der(t,x)&:=8 \sqrt{\pi } \sqrt{x}\, e^{\left(\sqrt{2} \sqrt{x}-2 t\right)^2/8}\, D_x p_t(x,x) \\ &=1+\erf\left(v(t,x)/2\right)-\frac{2 }{\sqrt{\pi } (1-x)}\,\sqrt{\frac{x}{1-2 x}}\,e^{u(t,x)/4}, \\ u(t,x)&:=t^2 \left(-4 \sqrt{2} \sqrt{\frac{1-2x}{1-x}} (1-x)+8 x-6\right) \\ &+2 t \left(-2 \sqrt{2} x^{3/2}+2 (1-x) \sqrt{\frac{1-2x}{1-x}} \sqrt{x}+\sqrt{2} \sqrt{x}\right)+x (2 x-1), \\ v(t,x)&:=\sqrt{1-x} \left(\sqrt{\frac{1-2x}{1-x}} \left(\sqrt{2} t-\sqrt{x}\right)+2 t\right). \end{aligned} \end{equation*}

By a chain rule of differentiation, \begin{equation*} D_x p_t(x,x)=D_m P(M(x),-R(x))M'(x)-D_r P(M(x),-R(x))R'(x), \end{equation*} where \begin{equation*} M(x):=M(x,x),\quad R(x):=R(x,x). \end{equation*} Using (2) and (3), we get \begin{equation*} \der(t,x)=A_1(x)e^{A_2(t,x)}+1+\erf(A_3(t,x));\tag{4.5} \end{equation*} here and in what follows, $A_1,A_2,\dots$ are certain algebraic functions and $\erf$ is the error function. So, \begin{equation*} D_t\der(t,x) =A_4(t,x)e^{A_2(t,x)}+A_5(t,x)e^{-A_3(t,x)^2/2}. \tag{5} \end{equation*} thus getting rid of the function $\erf$. Moreover, \begin{equation*} A_5(t,x)>0 \end{equation*} (for $t$ and $x$ as in (4)).

Letting now \begin{equation*} \tdert(t,x):=\frac{D_t\der(t,x)}{A_5(t,x)e^{-A_3(t,x)^2/2}}, \end{equation*} we get \begin{equation*} D_t\der(t,x)\eqs\tdert(t,x)=A_6(t,x)e^{A_7(t,x)}+1, \tag{6} \end{equation*} where $a\eqs b$ means that $a$ and $b$ are of the same sign; this reduces the two exponential expressions $e^{A_2(t,x)}$ and $e^{-A_3(t,x)^2/2}$ in (5) to one such expression. Now we have \begin{multline*} D_t\tdert(t,x)=e^{A_7(t,x)}(1-x) \sqrt{\frac{(1-2x)(1-x)}{x}} \Big(2+\sqrt2\,\sqrt{\frac{1-2x}{1-x}}\,\Big) \\ \times \big(3-4 x+2 \sqrt{2} \sqrt{(1-2 x) (1-x)}\,\big)>0 \end{multline*} (for $x\in(0,1/3]$).
\begin{equation*} \tdert(0,x) =\frac{(1-2 x) \left(2-2 x+\sqrt{2} \sqrt{(1-2 x) (1-x)}\right)}{\left(2+\sqrt{2} \sqrt{2+\dfrac{1-2 x}{1-x}}\right) (1-x)^2}>0. \end{equation*} So, $\tdert(t,x)>0$ for all $t\ge0$ and all $x\in(0,1/3]$.

So, by (6), it is enough to show that \begin{equation*} \der_1(x):=\der(1,x)\overset{\text{(?)}}\ge0 \tag{7} \end{equation*} for all $x\in(0,1/3]$.

Using (4.5) and some algebra, we get \begin{equation*} \der_1(x)=1-\frac{2/\sqrt\pi}{1-x}\,\sqrt{\frac x{1-2x}}\,e^{p(x)/4}+\erf(q(x)), \end{equation*} where \begin{equation*} p(x):=-6+2 \sqrt{2} \sqrt{x}+7 x-4 \sqrt{2} x^{3/2}+2 x^2+\sqrt{\frac{1-2 x}{1-x}} \left(-4 \sqrt{2}+4 \sqrt{x}+4 \sqrt{2} x-4 x^{3/2}\right), \end{equation*} \begin{equation*} q(x):=\frac{1}{2} \sqrt{1-2 x} \left(\sqrt{2}-\sqrt{x}\right)+\sqrt{1-x}. \end{equation*}

Let \begin{equation*} \der_{11}(x):=\der_1'(x)2\sqrt\pi\, \frac{(1-2 x) (1-x)^2 \sqrt{(1-2x)x} } {p_1(x)e^{p(x)/4}}, \end{equation*} where \begin{equation*} \begin{aligned} p_1(x)&:=(1-2x)(1-x) x p'(x)+2+2x-8 x^2 \\ &=2 + 2 x - 8 x^2 + x (1 - 2 x) ((7 + \sqrt2/\sqrt{x} - 6 \sqrt2 \sqrt{x} + 4 x) (1 - x) +p_2(x)), \end{aligned} \end{equation*} \begin{equation*} p_2(x):=2 (\sqrt2 - \sqrt{x})/ s(x) + (4 \sqrt2 + 2/\sqrt{x} - 6 \sqrt{x}) (1 - x) s(x), \end{equation*} \begin{equation*} s(x):=\sqrt{\frac{1-2 x}{1-x}}. \end{equation*} It is elementary to show that $p_2(x)\ge5$ (for $x\in(0,1/3]$), and hence \begin{equation*} \begin{aligned} p_1(x)&\ge2 + 2 x - 8 x^2 + x (1 - 2 x) ((7 + \sqrt2/\sqrt{x} - 6 \sqrt2 \sqrt{x} + 4 x) (1 - x) +5)>0. \end{aligned} \end{equation*} So, \begin{equation*} \der_{11}(x)\eqs\der_1'(x). \end{equation*} Next, let \begin{equation*} \der_{111}(x):=\der_{11}'(x) \sqrt{(1-2x)x} \, e^{p(x)/4+q(x)^2} \, p_1(x)^2 =\tder_{111}(\sqrt x), \end{equation*} where \begin{equation*} \begin{aligned} \tder_{111}(z)&:= -3 \sqrt{2}+50 z+56 \sqrt{2} z^2-282 z^3-61 \sqrt{2} z^4 \\ &+392 z^5-284 \sqrt{2} z^6+248 z^7+540 \sqrt{2} z^8 \\ &-832 z^9-160 \sqrt{2} z^{10}+384 z^{11}-64 \sqrt{2} z^{12} \\ &-2 \sqrt{\frac{1-2z^2}{1-z^2}} \\ &\times\big(1-6 \sqrt{2} z-32 z^2+48 \sqrt{2} z^3 +33 z^4 \\ & -106 \sqrt{2} z^5 +136 z^6+48\sqrt{2} z^7-230 z^8 \\ &+80 \sqrt{2} z^9+48 z^{10}-64 \sqrt{2} z^{11}+32 z^{12}\big). \end{aligned} \end{equation*} The function $\tder_{111}$ switches its sign only once on the interval $(0,1/\sqrt3]$, from $-$ to $+$, and hence $\der_{11}'$ has the same sign pattern on the interval $(0,1/3]$. So, for some $x_*\in(0,1/3)$, the function $\der_{11}$ is decreasing on $(0,x_*]$ and increasing on $[x_*,1/3]$. Also, $\der_{11}(0+)=-3/2<0$ and $\der_{11}(1/3)=-1.11\ldots<0$. So, $\der_{11}<0$ on $(0,1/3]$. So, $\der_1$ is decreasing on $(0,1/3]$, to $\der_1(0)=1.31\ldots>0$.

Thus, (7) is proved. $\quad\Box$

  • $\begingroup$ @Pinelis Thank you for the detailed solution. Some of things are still not clear to me. 1) You have shown first that $𝐷_{x}p_{t}(x,y)\geq 0$ for all $(x,y)\in G$, why is to prove further for specific value of $y$? 2) What is the function $der(t,x)$ defined in (4)? $\endgroup$ Commented Oct 3, 2021 at 9:18
  • $\begingroup$ Also, how to show that der111 switches its sign only once on the interval $(0, 1/\sqrt{3}]$? $\endgroup$ Commented Oct 3, 2021 at 9:28
  • $\begingroup$ @SatyaPrakash : 1) As I wrote, $D_x p_t(x,y)\ge0$ implies only that the maximum of $p_t(x,y)$ over $(x,y)\in G$ occurs at one of right boundaries of $G$, that is, where either $y=x$ and $y=(1-x)/2$. (Try to draw $G$.) So, to maximize $p_t$ over $G$, we then need to consider $p_t(x,x)$ and $p_t(x,(1-x)/2)$. 2) As you wrote, $\text{der}(t,x)$ was defined by (4). Then you should be able to get an explicit form of $\text{der}(t,x)$, which I now added anyway. $\endgroup$ Commented Oct 3, 2021 at 12:54
  • $\begingroup$ @SatyaPrakash : Since $\widetilde{\text{der}}_{111}(z)$ is an algebraic expression, its sign pattern can be determined purely algorithmically, using simple algebra to reduce the problem to that of the sign patterns of polynomials in $z$. I used Mathematica's command Reduce to do that. $\endgroup$ Commented Oct 3, 2021 at 12:59
  • $\begingroup$ Thank you for expressing $der(t,x)$ further. Still it is not clear to me that how $der(t,x)$ suddenly appears in (4). I mean what is its connection to the problem. I am sure I am missing something. Please clarify. $\endgroup$ Commented Oct 4, 2021 at 18:25

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