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Let $X\in\{0,1\}^2$ have mean $\mu=\left[\begin{smallmatrix}p_1\\p_2\end{smallmatrix}\right]$ and $\Pr[X_1 = X_2 = 1] = p\le \min\{p_1,p_2\}$. (Note we must have $1-p_1-p_2+p\ge 0$ for the distribution to be well defined.) We can then compute the covariance matrix $\Sigma = E[(X-\mu)(X-\mu)^T] = \left[\begin{smallmatrix}p_1(1-p_1)&p-p_1p_2\\p-p_1p_2&p_2(1-p_2)\end{smallmatrix}\right]$.

I would like to use the Berry Essen bound, and for that we need to upper bound the quantity $\gamma=E[\|\Sigma^{-1/2}(X-\mu)\|_2^3]$.

I believe one should be able to show $$\gamma \le C\left(\tfrac1{\sqrt{p_1(1-p_1)}}+\tfrac1{\sqrt{p_2(1-p_2)}}+\tfrac1{\sqrt{\min\{p_1,p_2\}-p}}\right) , $$ for some universal constant $C>0$.

The symbolic computation of $\Sigma^{-1/2}$ is a bit unwieldy though, and so I wonder if there is some tricks I may use to arrive at this result more neatly?

Or if not, any proof would be appreciated.

Update: Using Pinelli's observation, we can compute $$\begin{align}\gamma |\Sigma|^{3/2} &= p[(1-p_1)(1-p_2)v]^{3/2} \\&+ (p_1-p)[(1-p_1)p_2(1-v)]^{3/2} \\&+ (p_2-p)[p_1(1-p_2)(1-v)]^{3/2} \\&+ (1-p_1-p_2+p)[p_1 p_2 v]^{3/2} , \end{align}$$ where $v = p_1+p_2-2p$ and $|\Sigma|=p_1p_2(1-v)-p^2$.

It seems like I forgot a factor $1/\sqrt{p}$. In fact it seems that bounds of $\gamma \le \frac{2} {\sqrt{p\,(\min\{p_1,p_2\}-p)}} $ or $\gamma \le \frac{1} {\sqrt{p}}+ \frac{2} {\sqrt{\min\{p_1,p_2\}-p}} $ are sufficient and correct.

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Your inequality does not hold in general:

You don't need to compute to compute $\Sigma^{-1/2}$, because $\|\Sigma^{-1/2} x\|_2^2=x^T\Sigma^{-1}x$ for all $x\in\mathbb R^2$. Using this simple observation with e.g. $p=0$, $p_1=1/2$, $p_2=1/2-\epsilon$, and $\epsilon\downarrow0$, we find $$\gamma =\frac{\left(1-\epsilon -\epsilon ^2\right)^{3/2}}{\sqrt{\epsilon }}+\frac{1}{2} (1+\epsilon )^{3/2}+\frac{\left(1-\epsilon +2 \epsilon ^2\right)^{3/2}}{2 \sqrt{1-2 \epsilon }}\to\infty,$$ whereas $$\frac1{\sqrt{p_1(1-p_1)}}+\frac1{\sqrt{p_2(1-p_2)}}+\frac1{\sqrt{\min(p_1,p_2)-p}}\to4+\sqrt2.$$

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  • $\begingroup$ Good point, and good approach. It looks like the term I need is $\frac{1}{\sqrt{(p_1+p_2-2p)(1-p_1-p_2+2p)}}$ rather than $\frac{1}{\sqrt{\min\{p_1,p_2\}-p}}$. What do you think about that version? $\endgroup$ – Thomas Dybdahl Ahle Apr 19 at 10:34
  • $\begingroup$ @ThomasDybdahlAhle : I think the new version requires a rather different approach. Therefore and because your current question has been answered, the new version should be posted as a separate question. $\endgroup$ – Iosif Pinelis Apr 19 at 13:40
  • $\begingroup$ My conjecture was disproved, and you did point in a good direction, but I still don't have a useful upper bound, which was the main point of the question. Do you have a suggestion for how to state the new question without too much overlap with this one? $\endgroup$ – Thomas Dybdahl Ahle Apr 19 at 19:07
  • $\begingroup$ @ThomasDybdahlAhle : Your original question was fully answered. Therefore, I think any additional questions you may have should be posted separately. $\endgroup$ – Iosif Pinelis Apr 21 at 16:50
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We have the simplex $0\le p, 0\le p_1-p, 0\le p_2-p, 0\le 1-p_1-p_2+p$. From this, we can deduce $v\in[0,1]$ and so we may bound

$$\begin{align} \gamma |\Sigma|^{3/2} &= p[(1-p_1)(1-p_2)v]^{3/2} \\&\quad+ (p_1-p)[(1-p_1)p_2(1-v)]^{3/2} \\&\quad+ (p_2-p)[p_1(1-p_2)(1-v)]^{3/2} \\&\quad+ (1-p_1-p_2+p)[p_1 p_2 v]^{3/2} \\&\le p[(1-p_1)(1-p_2)v] \\&\quad+ (p_1-p)[(1-p_1)p_2(1-v)] \\&\quad+ (p_2-p)[p_1(1-p_2)(1-v)] \\&\quad+ (1-p_1-p_2+p)[p_1 p_2 v] \\&=2|\Sigma| . \end{align}$$ In other words $\gamma\le2|\Sigma|^{-1/2}$.

This bound is tight except at the four 0-faces/corners of the simplex.

In particular, the determinant is 0 exactly in the 1-faces of the simplex (see the picture), and $|\Sigma|^{-1/2}$ approaches those as $1/\sqrt\epsilon$, which $\gamma$ does too.

The bound still isn't quite tight though, as it approaches the 0-faces/corners as $1/\epsilon$, while $\gamma$ approaches them as $1/\sqrt\epsilon$. (E.g. if we let $p=\epsilon, p_1=2\epsilon, p_2=3\epsilon$.)

enter image description here

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