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统计代写-S203031

时间：2021-01-24

Prof. Elvezio Ronchetti Research Center for Statistics and GSEM

Ass : Edoardo Vignotto University of Geneva

S203031 Probability and Statistics II Fall 2018

EXAM January 22, 2019

— The exam lasts 3h.

— You may answer in English or in French.

— You are not allowed to leave the room during the exam.

— Open book exam.

— Justify your answers and do not forget to write your name on your copy.

Good luck !

Exercise 1 (6 points)

Suppose that the true parameter of a parametric model is θ = 4 and our estimator of θ equals either 3

or 5 with probability 1

2

each. Then our estimator is unbiased and its MSE is 1. Now, suppose we “shrink”

our estimator by a factor r (between 0 and 1), so it equals either 3r or 5r with probability 1

2

each.

1. Compute the bias, the variance, and the MSE of the new estimator as a function of r.

2. What is the value r0 which minimizes the MSE? What is the reduction (in %) in MSE obtained

by shrinking the original estimator by the factor r0 ?

Exercise 2 (9 points)

Let u1, . . . , un be a random sample from a random variable U describing an angle with uniform

density on the interval [−pi

2

, pi

2

]. We make the transformation Z = tg(U).

1. Show that Z follows a Cauchy distribution with density fZ(z) = 1pi

1

1+z2

.

2. What are the expectation and the variance of Z ? Can we apply the Central Limit Theorem to

obtain an approximation for the distribution of the mean Z¯n = 1n

∑n

i=1 Zi ?

3. Compute the exact distribution of Z¯n.

Hint : use the characteristic function.

Exercise 3 (9 points)

Let x1, . . . , xn be a random sample from a random variableX with density f(x|θ) = θ(1 +x)−(θ+1),

where x > 0 and θ > 0 is an unknown parameter.

1. Compute the maximum likelihood estimator γˆ for γ = 1/θ.

2. Compute the density of Y = log(1 +X). Is γˆ unbiased for γ ?

3. Establish if γˆ reaches the Cramer-Rao bound.

Exercise 4 (6 points)

A particular device may be in operating state or under repair. Let X and Y be the random variables

that describe the duration of the operating state and the duration of the repairing operations, respectively.

Assume that X and Y are independent exponential random variables with unknown expectations θ and

λ respectively. Moreover, let ψ = θ/(θ + λ) be a known quantity.

1. Show that ψ = Pr(Y ≤ X) and give an interpretation of ψ.

2. Compute the maximum likelihood estimator for θ and λ based on the random sample

(x1, y1), . . . , (xn, yn) that uses a known value for ψ.

Ass : Edoardo Vignotto University of Geneva

S203031 Probability and Statistics II Fall 2018

EXAM January 22, 2019

— The exam lasts 3h.

— You may answer in English or in French.

— You are not allowed to leave the room during the exam.

— Open book exam.

— Justify your answers and do not forget to write your name on your copy.

Good luck !

Exercise 1 (6 points)

Suppose that the true parameter of a parametric model is θ = 4 and our estimator of θ equals either 3

or 5 with probability 1

2

each. Then our estimator is unbiased and its MSE is 1. Now, suppose we “shrink”

our estimator by a factor r (between 0 and 1), so it equals either 3r or 5r with probability 1

2

each.

1. Compute the bias, the variance, and the MSE of the new estimator as a function of r.

2. What is the value r0 which minimizes the MSE? What is the reduction (in %) in MSE obtained

by shrinking the original estimator by the factor r0 ?

Exercise 2 (9 points)

Let u1, . . . , un be a random sample from a random variable U describing an angle with uniform

density on the interval [−pi

2

, pi

2

]. We make the transformation Z = tg(U).

1. Show that Z follows a Cauchy distribution with density fZ(z) = 1pi

1

1+z2

.

2. What are the expectation and the variance of Z ? Can we apply the Central Limit Theorem to

obtain an approximation for the distribution of the mean Z¯n = 1n

∑n

i=1 Zi ?

3. Compute the exact distribution of Z¯n.

Hint : use the characteristic function.

Exercise 3 (9 points)

Let x1, . . . , xn be a random sample from a random variableX with density f(x|θ) = θ(1 +x)−(θ+1),

where x > 0 and θ > 0 is an unknown parameter.

1. Compute the maximum likelihood estimator γˆ for γ = 1/θ.

2. Compute the density of Y = log(1 +X). Is γˆ unbiased for γ ?

3. Establish if γˆ reaches the Cramer-Rao bound.

Exercise 4 (6 points)

A particular device may be in operating state or under repair. Let X and Y be the random variables

that describe the duration of the operating state and the duration of the repairing operations, respectively.

Assume that X and Y are independent exponential random variables with unknown expectations θ and

λ respectively. Moreover, let ψ = θ/(θ + λ) be a known quantity.

1. Show that ψ = Pr(Y ≤ X) and give an interpretation of ψ.

2. Compute the maximum likelihood estimator for θ and λ based on the random sample

(x1, y1), . . . , (xn, yn) that uses a known value for ψ.