Question

Let X1, · · · ,Xn be iid from Uniform(−θ,θ), where θ > 0. Let X(1) < X(2) < ... < X(n) denotes the order statistics.

(a) Find a minimal sufficient statistics for θ

(d) Find the UMVUE for θ.

(e) Find the UMVUE for τ(θ) = P(X1 > k).

Answer 1

Answer 2

To find the minimal sufficient statistics for θ in the given problem, we'll start with the definition of sufficient statistics. A statistic T(X) is said to be sufficient for a parameter θ if the conditional distribution of the sample X, given T(X), does not depend on θ.

(a) Minimal Sufficient Statistics for θ: In this case, we have a sample X1, X2, ..., Xn from a Uniform(-θ, θ) distribution. The joint probability density function (PDF) for the sample is given by:

f(x1, x2, ..., xn | θ) = (1 / (2θ)^n) for -θ ≤ xi ≤ θ (for all i)

To find the minimal sufficient statistics, we can use the Factorization Theorem. According to the Factorization Theorem, a statistic T(X) is a minimal sufficient statistic if and only if the joint PDF can be factored into two functions, one that depends on the data only through T(X) and the other that does not depend on the parameter θ.

Let's find the factorization for the sample:

f(x1, x2, ..., xn | θ) = (1 / (2θ)^n) for -θ ≤ xi ≤ θ (for all i)

Now, we can rewrite the joint PDF as:

f(x1, x2, ..., xn | θ) = [1 / (2θ)^n] * [1 for -θ ≤ xi ≤ θ (for all i)]

Now, it is evident that the above expression can be factorized into:

f(x1, x2, ..., xn | θ) = g(T(X) | θ) * h(X)

where: T(X) = (X1, X2, ..., Xn) (the sample itself) g(T(X) | θ) = 1 (since it does not depend on θ) h(X) = [1 for -θ ≤ xi ≤ θ (for all i)]

Since we can find a factorization where the factor g(T(X) | θ) does not depend on θ, the sample T(X) = (X1, X2, ..., Xn) is a minimal sufficient statistic for θ.

(d) UMVUE for θ: The UMVUE (Uniformly Minimum Variance Unbiased Estimator) for θ can be obtained using the Rao-Blackwell Theorem, which states that taking the conditional expectation of an unbiased estimator with respect to a minimal sufficient statistic produces a new estimator that is still unbiased and has a smaller variance (or at least the same).

We know that the sample mean (X̄) is an unbiased estimator for the mean of the Uniform(-θ, θ) distribution, which is 0. Now, we need to find the conditional expectation of X̄ given the minimal sufficient statistic T(X) = (X1, X2, ..., Xn).

E(X̄ | T(X) = t) = E(X̄ | X1 = t1, X2 = t2, ..., Xn = tn)

Since the X's are independent and identically distributed (iid), the conditional expectation for each Xi given Ti is simply the same as the overall expectation, which is 0.

Therefore, the UMVUE for θ is simply the sample mean X̄.

(e) UMVUE for τ(θ) = P(X1 > k): To find the UMVUE for τ(θ) = P(X1 > k), where k is a constant, we'll use the invariance property of UMVUE. If g(T) is an unbiased estimator for θ, then g(h(T)) is an unbiased estimator for g(θ) for any function h.

The indicator function I(X1 > k) takes the value 1 when X1 > k and 0 otherwise. Therefore, E[I(X1 > k)] gives us P(X1 > k).

E[I(X1 > k)] = P(X1 > k)

Now, since X̄ is the UMVUE for θ (as shown in part (d)), we can apply the invariance property:

g(T) = X̄ g(θ) = θ

So, the UMVUE for τ(θ) = P(X1 > k) is simply the sample mean X̄.