# inequality in bivariate normal variable

Let $U_1=(X_1,Y_1)^T,\dots,U=(X_n,Y_n)^T$ are i.i.d. copies of $U=(X,Y)^T\sim N_2(0,\Sigma)$ where $$\Sigma= \begin{pmatrix} \sigma^2 & \rho\sigma\tau \\ \rho\sigma\tau & \tau^2 \end{pmatrix}$$ such that $$\sum_{i=1}^n\min\{U_iU_i^T,c\}$$ is invertible where $\min\{A,c\}$ represents the matrix with element $\min\{a_{ij},c\}$. Given $c>0$ and define $$b=\frac{\sum_{i=1}^n\max\{-c,\min\{X_iY_i,c\}\}}{\left(\sum_{i=1}^n\min\{X_i^2,c\}\sum_{i=1}^n\min\{Y_i^2,c\}\right)^{1/2}}.$$

The problem is how to prove that $$-E(\min\{\sigma\tau Z^2,c\})\leq b\leq E(\min\{\sigma\tau Z^2,c\}),$$ where $Z$ is normal standard distribution. I don't know how can I start this problem. Any advice?

• Wouldn't you want $c^2$ to appear in the expressions in the denominator rather than $c$? (And perhaps also in the last inequalities?) What is $Z$? – whuber Feb 17 '14 at 15:08
• @whuber: Thanks for your comments. I have added what $Z$ is. And for $c$, I would write $c$ indeed since I have stated that $c$ is positive – Jlamprong Feb 17 '14 at 15:41

## 1 Answer

Notice that $$\mathbb{E}(\min\{\sigma\tau Z^2, c\}) \to 0$$ as $$\sigma\tau\to 0.$$ However, provided both $$\sigma$$ and $$\tau$$ are nonzero and $$n=2$$ it is possible for the outcomes to be $$\{(X_i,Y_i), i=1,2\} = \{\left(2c,\frac{1}{2}\right), \left(2c,\frac{1}{2}\right)\},$$ in which case if $$c \ge 1/4,$$ then $$\max\{-c,\min\{X_iY_i, c\}\}=c,$$ $$\min\{X_i^2,c\}=c,$$ and $$\min\{Y_i^2,c\}=1/4,$$ entailing $$b = \frac{c + c}{\sqrt{(c+c)(\frac{1}{4}+\frac{1}{4})}} = 2\sqrt{c}.$$ Since there is no assumed relationship among $$c,\sigma,\tau,$$ and $$\rho,$$ choose $$\sigma$$ and $$\tau$$ sufficiently small to ensure that $$\mathbb{E}(\min\{\sigma\tau Z^2, c\}) \lt 2\sqrt{c},$$ contradicting what you are trying to prove.

The basic problem is that you are attempting to prove an inequality about arbitrary real numbers as expressed by the formula for $$b$$; their distribution--provided it is supported on the entire real line--is irrelevant.

### Response to edits

The condition was added in a later edit. Note, though, that in the counterexample

$$\min\{U_iU_i^\prime,c\} = \min\{\pmatrix{4c^2 & c \\ c & \frac{1}{4}}, c\} = \pmatrix{c & c \\ c & \frac{1}{4}}$$

(because $$c\gt \frac{1}{4}$$). Their sum is equal to $$\pmatrix{2c & 2c \\ 2c & \frac{1}{2}}$$ whose determinant $$c - 4c^2$$ is nonzero, showing the sum is invertible, whence the counterexample still applies.

• Thanks @whuber for your counterexample. It turned out that I forgot the other assumptions. Is that still incorrect? – Jlamprong Feb 17 '14 at 16:22
• The edit you made to your question is meaningless: I believe you may have forgotten to include some part of it. Note, too, that the expression $U_iU_i^\prime$ is a two-by-two matrix, calling into question what "min" means. – whuber Feb 17 '14 at 16:23
• OK @Whuber. Thanks for your correction. Now, I have added the missing assumption. – Jlamprong Feb 17 '14 at 16:30
• Your latest edit does not rescue anything, because in the counterexample for $c \gt 1/4$ the matrix is clearly invertible (its determinant is $c-4c^2\ne 0$). I would like to suggest you let this rest, instead of applying a series of quick edits; rethink your question; and pose a new one when you have better determined what it is you need to show. – whuber Feb 17 '14 at 16:35
• OK, Thank you very much for your suggestions. I'll think this problem carefully. – Jlamprong Feb 17 '14 at 16:37