Question

#4. Let $f(x|\theta) =\frac{1}{2\theta}$ , $-\theta<x<\theta$,  $\theta>0$ and $X_1, X_2, ..., X_n$ be a random sample from f. Find the UMVUE for $\theta$

Answer 1

Solution

Back-up Theory

If a continuous random variable, X, is uniformly distributed over the interval (a, b), then the pdf (probability density function) of X is given by f(x) = 1/(b – a) ………………..……….................................................……(1)

Mean or Expected Value, E(X) = (a + b)/2 ………………………………………………….(2)

If X₁, X₂, …….., X_n is a random sample from a population with mean µ, then the sample mean Xbar = (1/n)Σ(i = 1 to n)X_i is a UMVUE for µ……………………………………...............................................…………. (3)

Now, to work out the solution,

Vide (1), the given function, f(x/θ) = 1/(2θ), - θ < x < θ, is the probability density function of Uniform(-θ, θ) distribution………………………………………………………....................……………. (4)

(2) and (4) => θ is the population mean. This, in conjunction with (3) =>

Sample mean Xbar = (1/n)Σ(i = 1 to n)X_i is a UMVUE for θ. Answer

DONE