Editing Log-normal distribution (section)

====Confidence interval for {{math|E(''X'')}}====

The literature discusses several options for calculating the [[confidence interval]] for <math>\mu</math> (the mean of the log-normal distribution). These include [[Bootstrapping (statistics)|bootstrap]] as well as various other methods.<ref name = "Olsson2005">Olsson, Ulf. "Confidence intervals for the mean of a log-normal distribution." ''Journal of Statistics Education'' 13.1 (2005).[https://www.tandfonline.com/doi/pdf/10.1080/10691898.2005.11910638 pdf] [https://jse.amstat.org/v13n1/olsson.html html]</ref><ref>user10525, How do I calculate a confidence interval for the mean of a log-normal data set?, URL (version: 2022-12-18): https://stats.stackexchange.com/q/33395</ref>

The Cox Method{{efn|The Cox Method was quoted as "personal communication" in Land, 1971,<ref>Land, C. E. (1971), "Confidence intervals for linear functions of the normal mean and variance," Annals of Mathematical Statistics, 42, 1187–1205.</ref> and was also given in CitationZhou and Gao (1997)<ref>Zhou, X-H., and Gao, S. (1997), "Confidence intervals for the log-normal mean," ''Statistics in Medicine'', 16, 783–790.</ref> and Olsson 2005<ref name = "Olsson2005" />{{rp|Section 3.3}}}} proposes to plug-in the estimators <math display="block">\widehat \mu = \frac {\sum_i \ln x_i}{n}, \qquad S^2 = \frac {\sum_i \left( \ln x_i - \widehat \mu \right)^2} {n-1}</math>

and use them to construct [[Confidence_interval#Approximate_confidence_intervals|approximate confidence intervals]] in the following way:
<math>\mathrm{CI}(\operatorname{E}(X)) : \exp\left(\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)}} \right)</math>

{{hidden begin|style=width:100%|ta1=center|border=1px #aaa solid|title=[Proof]}}
We know that {{nowrap|<math>\operatorname{E}(X) = e^{\mu + \frac{\sigma^2}{2}}</math>.}} Also, <math>\widehat \mu</math> is a normal distribution with parameters: <math>\widehat \mu \sim N\left(\mu, \frac{\sigma^2}{n}\right)</math>

<math>S^2</math> has a [[chi-squared distribution]], which is [[Chi-squared_distribution#Related_distributions|approximately]] normally distributed (via [[Central limit theorem|CLT]]), with [[Variance#Distribution of the sample variance|parameters]]: {{nowrap|<math>S^2 \dot \sim N\left(\sigma^2, \frac{2\sigma^4}{n-1}\right)</math>.}} Hence, {{nowrap|<math>\frac{S^2}{2} \dot \sim N\left(\frac{\sigma^2}{2}, \frac{\sigma^4}{2(n-1)}\right)</math>.}}

Since the sample mean and variance are independent, and the sum of normally distributed variables is [[Normal distribution#Operations on two independent normal variables|also normal]], we get that:
<math>\widehat \mu + \frac{S^2}{2} \dot \sim N\left(\mu + \frac{\sigma^2}{2}, \frac{\sigma^2}{n} + \frac{\sigma^4}{2(n-1)}\right)</math>
Based on the above, standard [[Normal distribution#Confidence intervals|confidence intervals]] for <math>\mu + \frac{\sigma^2}{2}</math> can be constructed (using a [[Pivotal quantity]]) as: <math>\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)} } </math>
And since confidence intervals are preserved for monotonic transformations, we get that:
<math>\mathrm{CI}\left(\operatorname{E}[X] = e^{\mu + \frac{\sigma^2}{2}}\right): \exp\left(\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)}} \right)</math>

As desired.

{{hidden end}}

Olsson 2005, proposed a "modified Cox method" by replacing <math>z_{1-\frac{\alpha}{2}}</math> with <math>t_{n-1, 1-\frac{\alpha}{2}}</math>, which seemed to provide better coverage results for small sample sizes.<ref name = "Olsson2005" />{{rp|Section 3.4}}