Final answer:
A pivotal quantity is a statistic whose distribution doesn't depend on the unknown parameter, making Y/Θ a pivotal quantity for iid Uniform (0,Θ) random variables. The confidence interval for Θ is derived from the CDF of Y and leads to the shortest 1−α interval given by {Θ:y ≤ Θ ≤ y/α1/n}. The combined tails area α corresponds to the confidence level of the interval.
Step-by-step explanation:
In statistics, when dealing with samples and estimations, a pivotal quantity is a function of the observed data and the unknown parameter that has a probability distribution that does not depend on the unknown parameter. For independent and identically distributed (iid) random variables X1,…,Xn from a Uniform (0,Θ) distribution, the largest order statistic, Y, has a particular importance. The pivotal quantity we're investigating is Y/Θ.
The confidence interval for Θ can be derived from the properties of Y. Considering that the cumulative distribution function (CDF) for Y is P(Y ≤ y) = (y/Θ)n, we set the CDF equal to 1-α to find the upper bound of the pivotal quantity, leading to the determination that {Θ:y ≤ Θ ≤ y/α1/n} gives us the shortest 1−α confidence interval for Θ. Notably, the error bound in this context is associated with the range over which Θ can vary within this confidence interval.
The area in both tails combined, denoted by α, represents the total probability of the values of Θ lying outside the confidence interval, which is critical for determining the level of confidence associated with the interval.