Question

In a pasta factory, due to a miscalibration of the machinery, all spaghetti sticks produced had different lengths on a given day, then randomly packed & shipped to different locations and consumers. We wonder the length of the longest spaghetti stick produced that day. Offer a good statistical estimator to address this problem and discuss/evaluate its properties.

Answer 1

Suppose the length of the ith stick be Xi. We can assume Xi follows Rectangular(0, theta) independently of the other Xs.

We take a random sample of size n from all the sticks produced that day and measure the longest stick. Let that value be X(n), n the order statistic. By WLLN X(n) converges in probability to theta as n tends to infinity. Hence we take a large sample say of size N from all the sticks produced that day. Then the estimate of the longest stick on that day is X(N), the largest stick present in that sample