Let X1, X2, ..., Xn be a random sample from an absolutely continuous distribution with the probability density function
\(f(x \mid \theta)=\left\{\begin{array}{cl} e^{\theta-x}, & \text { if } x \geq \theta \\ 0, & \text { if } x<\theta \end{array},\right.\)
where θ ∈ ℝ is unknown. Define \(\bar{X}=\frac{1}{n} \sum_{i=1}^n X_i\) and X(1) = min{X1, ..., Xn}. Then
which of the following statements are true?
1
X̅ is the method of moments estimator of θ
2
X(1) is the maximum likelihood estimator of θ
3
X(1) = \(\frac{1}{n}\)is the uniformly minimum variance unbiased estimator of θ
4
X(1) is a sufficient statistic for θ