# fixedpoint.jp

## How to estimate a location parameter when average fails to converge (2018-06-23)

It is sometimes overlooked that increasing the sample size will not always reduce the uncertainty in the estimate of a parameter. [1] illustrates this fact as "Misconception 4" with an example considering a random variable having Cauchy distribution. Its probability density function $$f$$ has two parameters: \begin{align}f(x;x_0, \gamma) = \frac{1}{\pi \gamma \big[1 + (\frac{x - x_0}{\gamma})^2\big]}\end{align} where $$x_0$$ is a real number called the location parameter and $$\gamma (> 0)$$ is its scale parameter.

Since the distribution has no (finite) mean, the law of large numbers cannot ensure that taking its sample average converges to $$x_0$$. Fortunately it is still possible to estimate $$x_0$$ by taking sample median. In fact, the following computational experiment in R code finds sample mean converging to the location parameter quickly:

x <- replicate(16, { s <- rcauchy(10000, 42) h <- head(s, 10) c(median(h), median(s), mean(h), mean(s)) }) boxplot(t(x), names = c("median\n(10 samples)", "median\n(10^4 samples)", "mean\n(10 samples)", "mean\n(10^4 samples)")) 

The resulting figure (obtained with seeding set.seed(1)) shows that the error of sample median from its true value $$42$$ is small, while average of 10 samples is more diverse and averaging 10,000 samples often fails in even bigger errors.