ACCRA TECHNICAL UNIVERSITY
FACULTY OF APPLIED SCIENCES
\begin{tabular}{|l|}
\hline INDEX NUMBER \\
\hline \\
\hline
\end{tabular}
DEPARTMENT OF APPLIED MATHS AND STATS
END OF FIRST SEMESTER EXAMINATION, 2022/2023 ACADEMIC YEAR
COERSE CODE: STA 401
COURSE TITLE: STATISTICAL. ESTIMATION \& INFERENCE
TIME ALLOWED: 2 HOURS
INSTRUCTIONS: ANSWER ANY THREE QUESTIONS
1. a. Explain the following properties of point estimators
i. Relative efficiency,
ii. Conststency,
iil. Unbiasedness
iv. Sufficiency
[4 marks]
b. Suppose that \( X \) is a discrete random variable with the following probabitity miss function
\begin{tabular}{|c|c|c|c|c|}
\hline\( X \) & 0 & 1 & 2 & 3 \\
\hline\( P(X) \) & \( 2 \theta / 3 \) & \( \theta / 3 \) & \( 2(1-\theta) / 3 \) & \( (1-\theta) / 3 \) \\
\hline
\end{tabular}
Where \( 0 \leq \theta \leq 1 \) is a parameter. The following 10 independent observations 3 , \( \mathbf{0 , 2}, \mathbf{1}, \mathbf{3}, \mathbf{2}, \mathbf{1}, \mathbf{0}, \mathbf{2}, \mathbf{1} \) were taken from such a distribution. Find the
i. Method of Moment estimator for \( \theta \).
ii. Maximum Likelihood estimator for \( \theta \)
iii. Compare the two estimators obtained.
[5 marks]
[8 marks]
[3 marks]
2. The Pareto distribution has a probability density function \( f(x)=\theta \alpha^{\theta} x^{-\theta-1} \),
