Question

Suppose X 1 ,X 2 ,…,X n denote a random sample of size n fom Unif orm [β−1,β+1], where the nonnegative parameter β. a) Show that statistics T=(X (1) ,X(2) ) is a minimal sufficient statistics for β b) Find a MVUE for β

Answer 1

The student's question relates to uniform distribution for a parameter β, focusing on the concept of minimal sufficient statistics and seeking a Minimum Variance Unbiased Estimator (MVUE) for β.

Step-by-step explanation:

The student has posed a question related to the statistical properties of a uniform distribution and the concept of a minimal sufficient statistic (MSS) for a nonnegative parameter β. When dealing with uniform distributions, the mean (μ) is given by the average of the lower and upper bounds, and the standard deviation (σ) by the formula σ = (b - a)/√12, where a and b are the bounds of the uniform distribution. In this case, the sample comes from a uniform distribution ranging from β - 1 to β + 1.

To show that the statistics T consisting of the smallest and second smallest observations (X (1), X(2)) is a minimal sufficient statistic (MSS) for β, we could use the Neyman-Fisher factorization theorem. However, given that this is an applied problem, we will not delve into more detail in this context.

For finding a Minimum Variance Unbiased Estimator (MVUE) for β, one could leverage the properties of the uniform distribution and the sample mean, taking into account the fact that the sum of uniform random variables converges to a normal distribution under the Central Limit Theorem.