Estimating the area of a circle somwhere within a square

Question

Consider the following simulation experiment:

In order to estimate the area ($\theta$) of a circle drawn somewhere on a square of known side=10, a collection of random vectors $(X_i,Y_i), i=1,...,10$ are generated. Each $X_i$ and $Y_i$ are iid $\text{Uniform}[0,10]$.

From this, iid random variables $Z_i$ are obtained. Each $Z_i$ is $1$ if $(X_i,Y_i)$ is within the circle and $0$ otherwise.

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I asked a version of this questions earlier and the comments below have been very helpful in getting me to an answer. However, while reviewing it again today, I encountered an issue that I would like to ask about.

From comments below, I got that the $Z_i \sim \text{Bern}(\theta/100)$ where $\theta$ is the area of the circle. Using this, I was able to get the estimator $100\frac{\sum_{i=1}^{10} z_i}{10}$ and prove that this is unbiased for $\theta$.

I was also able to show that $\sum_{i=1}^{10} Z_i$ is a sufficient complete statistic for $\theta$ and so $100\frac{\sum_{i=1}^{10} z_i}{10}$ is UMVUE for $\theta$.

However, if I use this estimator I can end up with an estimate of the area that is not possible (one larger than the largest possible circle). That is if all or almost all my points happen to fall in the circle (8, 9 or 10), I can get an estimate that's $(>25\pi)$.

My question is this: why isn't $25\pi\frac{\sum_{i=1}^{10} z_i}{10}$ a better estimator of $\theta$? This never gives an impossible value.

If I let my $Z_i's$ be $Bern(\frac{\theta}{25\pi})$ I think the rest of the solution for all of the requirements of the problem will follow, but I can't seem to justify that when the instruction of the problem is to have $Z_i$ be the indicator function that the point $(X_i,Y_i)$ is in the circle.

Answer 1

While it never gives an impossible value for the area, it's certainly no longer unbiased (its expectation will be $\frac{\pi}{4}\, \theta$, about 22% too small on average). Bias isn't necessarily a bad idea, but in this case that's a lot of bias in cases where the estimate was not an issue.

For example you might easily have six points in the circle (a reasonably likely number to come up if the circle isn't small - say about half the area or more).

The unbiased estimator you started with would estimate the area to be 60, but your scaled down estimator would say it was about 47.

Note further that you're much more likely (about 1.4 times as likely) to get 6 points in there when the area is near 60 than when it's near 47.

For fewer than 8 points it doesn't seem to make a lot of sense to scale it down like that - certainly not that much, though you may be able to reduce MSE if you shrink it a little (I haven't checked this though). For 8+ points falling within the circle, there's a clear argument for not saying the circle is larger than $25\pi$ (you know for sure it isn't) -- and your estimate cannot be worse (cannot be further from $\theta$) for doing so. You would lose unbiasedness by never saying more than $25\pi$ but you'd reduce the variance substantially.

(You might like to consider what the MLE of the area would be.)

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Of course in large samples -- ones large enough for this to be a practical tool -- impossible values will become so rare as to make it essentially a nonissue. I do a lot of simulations, and for me $n=10^4$ when estimating a proportion like this is usually something I see as "too small", except as a rough feasibility check. I often do $10^6$ or $10^7$ simulations for proportions if they're not really close to a boundary - sometimes more.