Let the population x ~ U (0, θ), x1, X2, ···, xn be a sample taken from the population, and x0 be the average number of samples (1) It is proved that θ 1 = 2x0, θ 2 = (n + 1) / N. x (n) is an unbiased estimate of θ (where x (n) = max {x1, X2, ···, xn}); (2) Which one of theta 1 and theta 2 is more effective (n ≥ 2)?

Let the population x ~ U (0, θ), x1, X2, ···, xn be a sample taken from the population, and x0 be the average number of samples (1) It is proved that θ 1 = 2x0, θ 2 = (n + 1) / N. x (n) is an unbiased estimate of θ (where x (n) = max {x1, X2, ···, xn}); (2) Which one of theta 1 and theta 2 is more effective (n ≥ 2)?

For any I, it is obvious that e (XI) = θ / 2, so e (θ 1) = 2e (x0) = 2 / N ∑ e (XI) = 2 * θ / 2 = θ
Let t = x (n) be the order statistic. According to the density formula of the order statistic, its density is g (T) = NF (T) ^ (n-1) P (T)
Where p () and f () denote the density function and distribution function of uniform distribution respectively, P (T) = 1 / θ, f (T) = t / θ
So g (T) = NT ^ (n-1) / θ ^ n
So e (θ 2) = (n + 1) / Ne (x (n)) = (n + 1) / N * ∫ (NT ^ n / θ ^ n) DT = (n + 1) / N * (θ * n / (n + 1)) = θ
Therefore, both θ 1 and θ 2 are unbiased estimates
Next, compare the variance of θ 1 and θ 2, the effect of small variance is better
VAR(θ1)=4VAR(X0)=4/n^2 ∑VAR(Xi)=4/n*VAR(Xi)
VAR（Xi）=E(Xi^2)-(E(Xi))^2=θ^2/3-θ^2/4=θ^2/12
So var (θ 1) = θ ^ 2 / (3n)
Var (θ 2) = (n + 1) ^ 2 / N ^ 2var (x (n)) x (n) = t var (T) = e (T ^ 2) - (ET) ^ 2 = n / (n + 2) * θ ^ 2 - (θ * n / (n + 1)) ^ 2 = n / ((n + 1) ^ 2 * (n + 2)) θ ^ 2
So var (θ 2) = 1 / (n * (n + 2)) θ ^ 2
While var (θ 1) / var (θ 2) = (n + 2) / 3, when n > = 2, VAR (θ 1) / var (θ 2) > 1, that is, VAR (θ 1) > var (θ 2), so θ 2 is more effective