Distributed Random Variables - Statistics Assignment Help

Assignment Task :

In this assignment you may use standard properties of common probability distributions, as described in the document Common Probability Distributions available on iLearn. 

Question 1. Consider a sample of n > 2 independent and identically distributed random variables Y1,...,Yn from an exp(λ) distribution with probability density function 

Answer the questions on the following page.  

3. Derive the relative efficiency of ˆθ1 compared to ˆθ2. Which estimator do you prefer?

4. Determine whether ˆθ1 and ˆθ2 are consistent estimators of θ. 

5. Write down the likelihood function and the log-likelihood function, as functions of λ. 

6. Find a sufficient statistic for λ. 

7. Derive the maximum likelihood estimator (MLE) of λ. Comment on your result in relation to part (6). 

8. Find the mean and variance of the MLE of λ and hence show that it is biased but consistent. 

9. Obtain the expected information and hence write down an approximate large sample  confidence interval for λ. 

10. Based on the MLE, construct an unbiased estimator of λ. 

11. What is the minimum variance bound for unbiased estimators of λ? 

12. Does the estimator you constructed in part (10) attain the minimum variance bound? 

13. What is the MLE of θ? 

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