Central limit theorems form the basis for much of the statistical inference used in regression analysis. Thus, it is helpful to provide an explicit statement of one version of the central limit theorem.
Central Limit Theorem. Suppose that $y_1,\ldots,y_n$ are independently distributed with mean $\mu$, finite variance $\sigma ^{2}$ and $\mathrm{E}|y|^{3}$ is finite. Then, $\lim_{n \rightarrow \infty}~ \Pr \left(\frac{\sqrt{n}}{\sigma} (\overline{y} -\mu ) \le x \right) = \Phi \left(x \right)$, for each $x$, where $\Phi \left( \cdot \right)$ is the standard normal distribution function.
Under the assumptions of this theorem, the re-scaled distribution of $\overline{y}$ approaches a standard normal as the sample size, $n$, increases. We interpret this as meaning that, for “large” sample sizes, the distribution of $\overline{y}$ may be approximated by a normal distribution. Empirical investigations have shown that sample sizes of $n=25$ through $50$ provide adequate approximations for most purposes.
When does the central limit theorem not work well? Some insights are provided by another result from mathematical statistics.
$$
\Pr \left( \frac{\sqrt{n}}{\sigma }(\overline{y}-\mu ) \le x \right)
=\Phi \left( x \right) +\frac{1}{6}\frac{1}{\sqrt{2\pi }}e^{-x^{2}/2}\frac{\mathrm{E }(y-\mu )^{3}}{\sigma ^{3}\sqrt{n}}+\frac{h_n}{\sqrt{n}},
$$ for each $x$, where $h_n \rightarrow 0$ as $n \rightarrow \infty$.
This result suggests that the distribution of $\bar{y}$ becomes closer to a normal distribution as the skewness, $\mathrm{E}(\overline{y} -\mu )^{3}$, becomes closer to zero. This is important in insurance applications because many distributions tend to be skewed. Historically, analysts used the second term on the right-hand side of the result to provide a “correction” for the normal curve approximation. See, for example, Beard, Pentikainen and Pesonen (1984) for further discussion of Edgeworth approximations in actuarial science. An alternative (used in this book) that we saw in Section 1.3 is to transform the data, thus achieving approximate symmetry. As suggested by the Edgeworth approximation theorem, if our parent population is close to symmetric, then the distribution of $\overline{y}$ will be approximately normal.