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I have a sequence of independent, but non-identically distributed Bernoulli random variables $X_i$'s taking on value $1$ with probability $p_{i1}\cdot p_{i2}$, where $i=1,\ldots,n$. Let $X=\sum_{i=1}^n X_i$. Using Lyapunov central limit theorem, I have approximated the probability $\Pr[X \geq k]$ by standard normal distribution. However, for some values of $k$ the approximated value is not close to the exact value. I wanted to ask that why this happens? Thanks in advance for your help.

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    $\begingroup$ It would definitely be helpful to give an explicit example $\endgroup$ – spaceisdarkgreen Sep 5 '18 at 3:58
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    $\begingroup$ To clarify: Is it the case that the approximation doesn't hold for small values of $n$? Or that the approximation doesn't seem to get better as $n$ gets larger? (Thanks) $\endgroup$ – Patrick Hew Sep 5 '18 at 7:43
  • $\begingroup$ The normal approximation may give poor results in the tail, in the sense of relative errors, where tail will depend on the particular question $\endgroup$ – Henry Sep 5 '18 at 8:01
  • $\begingroup$ For instance, let $n=17618$. The approximated value by central limit theorem for small values of $k$ is good but as k increases the error between approximated value and exact value increases. $\endgroup$ – Carol Sep 5 '18 at 18:32
  • $\begingroup$ @PatrickHew The approximation doesn't hold as $k$ increases. $\endgroup$ – Carol Sep 6 '18 at 16:51
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Summary

Notation: Let $F_n$ be the cumulative distribution function for $X = \sum_{I=1}^n X_i$, and let $F$ be the cumulative distribution function $Z$ where $Z$ follows a standard normal distribution.

The result that you have only establishes that for any fixed $k$, $P[Z \geq k]$ will be an increasingly good approximation to $P[X \geq k]$ as $n \rightarrow \infty$.

Details

You have convergence in distribution (by the Lyapunov central limit theorem): for any $x \in \mathbb{R}$, $F_n(x) \rightarrow F(x)$ as $n \rightarrow \infty$. (This holds for any real value $x$ since $F$ is continuous on the real numbers $\mathbb{R}$.)

But (I think that) what you're seeing is that for any given $n$, the closeness of $F_n(x)$ to $F(x)$ can vary with $x$. Hence in particular, the closeness of $F_n(k)$ to $F(k)$ can vary as $k$ increases. Therefore since $P[X \geq k] = 1 - F_n(k)$ and $P[Z \geq k] = 1 - F(k)$, the closeness of $P[Z \geq k]$ to $P[X \geq k]$ can also vary as $k$ increases.

In other words, the convergence in distribution is pointwise convergence and not uniform convergence. That is, in establishing that for any $x \in \mathbb{R}$, $F_n(x) \rightarrow F(x)$ as $n \rightarrow \infty$, the following is true: $$ \text{pointwise convergence:}\ \text{for any}\ x \in \mathbb{R}\ \text{and for any}\ \epsilon > 0, \text{there exists}\ N > 0\ \text{such that if}\ n \geq N \ \text{then}\ |F_n(x) - F(x) |< \epsilon $$ The following is not necessarily true: $$ \text{uniform convergence:}\ \text{for any}\ \epsilon > 0, \text{there exists}\ N > 0\ \text{such that for any}\ x \in \mathbb{R}, \text{if}\ n \geq N \ \text{then}\ |F_n(x) - F(x) |< \epsilon $$

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