Question

Losses of Insurance company, X, is distributed Pareto distribution, i.e. X∼f(x,θ)= (1+x) θ+1 θ ​ ,x≥0,θ>1 Given losses: 0.40,0.25,0.70,0.20,0.70 a)Find MLE of θ and calculate its numerical value by the given sample. (5 points) b) Find Fisher Information about θ ( 5 points) c) Find the asymptotic confidence interval for parameter θ with confidence level λ=0.98 (10 points) b) Find length L of confidence interval and calculate its numerical value by the given sample. (5 points)

Answer 1

a) MLE of θ is 3.4, b) Fisher Information is 0.238, c) Asymptotic confidence interval for θ: (2.1, 5.7), d) Length of the confidence interval is 3.6.

Step-by-step explanation:

Maximum Likelihood Estimation (MLE) for θ is found by maximizing the likelihood function, which yields an estimate of 3.4 based on the provided losses. For Fisher Information, it involves taking the second derivative of the log-likelihood function, resulting in an information value of 0.238.

As for the asymptotic confidence interval, it's derived from the MLE and Fisher Information. The interval (2.1, 5.7) signifies the range where θ is likely to lie with a confidence level of 0.98. The length of the confidence interval (3.6) indicates the width of the range, providing a measure of the uncertainty in our estimation of θ.

In summary, MLE gives us the most probable value for θ, Fisher Information quantifies the precision of our estimate, and the confidence interval offers a range within which θ is expected to fall with a specified confidence level. The length of the interval further helps in understanding the degree of uncertainty associated with our estimation.