Maximum Likelihood estimate of cell radius from observed cross section radii

Question

Given observations of circular cross sections of a spherical cell ${R_1,R_2.......,R_n}$, with $R_i<R_{i+1}$, find a best estimate of cell(sphere) radius $r$, given that the distances $\sqrt{r^2-R_i^2}$ (say $X$) are uniformly distributed.

I tried to find the distribution of $R_i$ from $X$ as

$F(X)=P(X<=x) = \frac{x}{r}$ so

$P(\sqrt{r^2-X^2}<=t)=P(r^2-X^2<=t^2)=P(X^2>=r^2-t^2)$

$=P(X>=\sqrt{r^2-t^2})=1-P(X<=\sqrt{r^2-t^2})=1-\frac{\sqrt{r^2-t^2}}{r}$

$P(R<=x)= 1-\frac{\sqrt{r^2-x^2}}{r}$

so the density function for $R_i$ is

$\frac{d(P(R<=x))}{dx}=\frac{x}{(r\sqrt{r^2-x^2})}$

so the MLE function is :

$\prod_{i=1}^n\frac{R_i}{r\sqrt{r^2-R_i^2}}$

but this is decreasing with $r$ making $R_1$ the best estimate, which does not seem right as $R_i$ is always less than or equal to $r$, which should make $R_n$ the best estimate, I am not sure where I am going wrong in the process, can someone help me out?

Answer 1

Your math seem right - at least, it seems easy to see both by rigorous proof and intuitive reasoning that the maximum likelihood estimator must be the largest observed radius. Furthermore your problem is quite similar to the ML estimate of uniform distribution parameters, which yields the same result.

However, here the ML estimate is always smaller than the real parameter. That's named bias.

Estimators have a couple (useful) properties that look similar but that are actually different: consistency and unbiasedness. ML estimators are guaranteed to be consistent, but they aren't guaranteed to be unbiased.

Consistency means that as the sample gets larger, the estimate gets closer to the actual parameter. It's easy to see that your ML estimator for radius is consistency (the more sections you observe, the more probable that the biggest one get close to the maximum).

Unbiasedness means that for a given sample size expected value of estimate equals the actual value.

It wouldn't be difficult to build an unbiased estimator based on you ML estimator - you only need to multiply it by a factor which will be function of n - but then it will no longer be a ML estimator, although it will remain consistent and therefore it could be more useful than the ML estimator.