Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Distribution
Let X1....Xn (n ≥ 3) be a random sample from a distribution having probability density function \(\rm f(|x|θ)=\left\{\begin{matrix}θ e^{-θ x}, &if\ x >0\\\ 0, &otherwise\end{matrix}\right.\)
where θ > 0 is an unknown parameter. Let \(\rm T_n=\frac{1}{n}\Sigma_{i=1}^nX_i\) Which of the following statements are true?
1
Uniformly minimum variance unbiased estimator of θ is \(\rm \frac{n-1}{nT_n}\)
2
Cramer-Rao lower bound for the variance of any unbiased estimator of θ is \(\rm \frac{\theta^2}{n}\)
3
Uniformly minimum variance unbiased estimator of θ attains the Cramer-Rao lower bound
4
\(\rm (1-e^{-\frac{1}{T_n}})\) is a consistent estimator of pθ (X1 ≤ 1)