CS代写 F71SM STATISTICAL METHODS – cscodehelp代写
F71SM STATISTICAL METHODS
Tutorial on Section 7 ESTIMATION
1. Let X = (X1,X2,…,Xn) be a random sample of a gamma r.v. X with pdf f(x) = 1 xexp(−x/θ), x > 0.
(a) Show that the MME θ ̃ is X ̄ /2.
(b) Find expressions for the score function U(θ) and Fisher’s information function I(θ)
(c) Find the MLE θˆ for θ. Show that it is unbiased and find the MSE of the estimator. Show that it attains the C-R lower bound and state its asymptotic distribution.
(d) Among estimators of the form θ∗ = aX ̄ for some constant a, the value of a for which MSE [θ∗] is minimised is a = n/(2n + 1), with MSE [θ∗] = θ2/(2n + 1). Compare the MSE of the MLE θˆ with that of the optimum estimator θ∗ for the three sample sizes n = 1, 5, 100, and comment briefly.
parameter λ (mean 1/λ). A random sample of n bulbs is put on test, and testing is stopped after a predetermined time t0, by which time it is noted that x of the n bulbs have failed. Find the MLE of λ.
3. The number of claims made by each policyholder in a certain class of insurance business is thought to follow a Poisson(λ) distribution. A sample of n such policyholders was found to contain exactly x who had made one or more claims. Find the MLE of λ.
Suppose now that you discover that of the x policies on which claims were made, there were m with exactly one claim and x − m with exactly two claims. Find the revised MLE of λ.
4. In each of the following situations use the information given to obtain one or more esti- mates of θ, stating which method(s) of estimation you are using.
(a) A six-sided die has unknown probability θ of landing showing “6” up. The die was thrown repeatedly until a “6” showed up and this process was repeated ten times in all. The numbers of throws required were 4, 7, 1, 5, 2, 4, 6, 8, 1, 3.
[MLE = MME = 0.244]
(b) The lengths of certain mass-produced items are modelled as constituting a random sample from a normal distribution with unknown mean θ. A sample of 100 items had a total length of 429.61cm. [MLE = MME = 4.30cm]
(c) The strength of a signal recorded on a meter is modelled as having a uniform distribu-
tion on the interval (0, θ). The maximum possible strength (θ) is unknown. A series
of ten independent recorded values were as follows: 2.1, 1.7, 1.9, 3.4, 0.9, 2.2, 0.8, 1.3, 2.0, 0.4. [MME = 3.34, MLE = 3.4]
5. Random sample, size n, from X with pdf f(x) = θ/xθ+1, x > 1, for parameter θ > 1.
(a) Find the MME of θ.
(b) Find the MLE of θ, and its asymptotic standard error, and state its asymptotic distribution.
lifetime of a certain type of lightbulb is distributed as an exponential r.v. with
(c) Calculate the two estimates of θ for a sample for which n = 100, xi = 123.42, ln xi = 19.173. [MME = 5.27, MLE = 5.22]
