Estimator of the ruler length You observe a sample of measurements coming from a fixed length ruler. If the object is shorter than the ruler you observe the actual measurement. Otherwise, you observe the length of the ruler. What would be a good estimator of the ruler length
 A: If the maximum observed value occurs repeatedly, and the probability of two measured things having exactly the same length is $0,$ then that maximum observed number is probably the length of the ruler.
But if all measurements are distinct then the ruler length is probably strictly longer than the observed maximum, although it's possible that the observed maximum is a measurement of an object at least as long as the ruler. If the number of objects measured is large, so that the distinct measurements sorted into increasing order are close together, then the maximum is probably not much longer than the length of the ruler.
Possibly a good answer to this depends on the probability distribution of the lengths of measured objects. Suppose a histogram of the lengths of objects measured so far looks like that of a normal distribution with expected value $8$ and standard deviation $1,$ except that it's truncated at the $70$th percentile of the distribution? Suppose further that the number of measurements is $38.$ Then probably about $11$ or $12$ measurements would be exactly the maximum, and that would be a good estimate.
Suppose numbers were uniformly distributed between $0$ and some unknown number $\theta.$ If $\theta$ were less than the length of the ruler, you'd be estimating the upper bound of a uniform distribution, and an unbiased estimate of that is the maximum observed value multiplied by $(n+1)/n,$ where $n$ is the number of observations. In that case, you might say the ruler's length is probably at least as big as that estimate, but you couldn't rule out its being more than that. On the other hand, suppose $\theta$ is more than the length of the ruler, and you see $10$ observations less than the maximum observation and $5$ that are exactly equal to the maximum. Suppose further that those $10$ are approximately uniformly distributed between $0$ and the maximum observed value. Then it would be reasonable to estimate $\theta$ to be about $15/10$ of the maximum observed value, but the ruler's length then should be estimated to be the maximum observed value that was seen $5$ times.
