1. SupposeX1,..., Xniid∼Exponential(λ) is a set of n observations drawn independently from an Exponential distribution. (d) Set the score function equal to zero and solve for the parameterλ. (e) Take the second partial derivative of the score function. (f) Check to make sure this value is negative to ensure that the log-likelihood function is concave down.
1. SupposeX1,..., Xniid∼Exponential(λ) is a set of n observations drawn independently from an Exponential distribution.
(d) Set the score
(e) Take the second partial derivative of the score function.
(f) Check to make sure this value is negative to ensure that the log-likelihood function is concave down.
(g) A scientist needs to estimate the mean time between aftershocks of an earthquake. Observational data from 5 aftershocks gives waiting times of 5, 7.2, 4.3, 6.6, and 4.5 hours. Assuming the waiting time for all of the aftershocks follows an Exponential distribution with the same rate parameterλ, what is the Maximum Likelihood Estimate for the mean length of time between aftershocks? What property of the Maximum Likelihood Estimator do you need to appeal to in order to make this calculation?
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