1. Let X₁,..., Xn Exponential (Ao) be iid with density given by fo(x) = Ao exp(-Xox) for x>0 (a) Write down an expression for the log-likelihood (b) Compute the score function. (c) By direct calculation show that expected value of the score function eval uated at the true parameter is zero. (d) Compute the Fisher Information for Ao by: i Computing it directly from Exo (u(XoX)²) ii Computing it using Fisher's identity (e) Suppose that you now consider the reparametrisation What is the Fisher Information for o? Ho 2n = (f) A (naughty) statistician makes a claim that they have an unbiased estima tor for the mean of this exponential and that the variance of their estimato is equal to where po is the true mean of the exponential above. Explain with justification why they are lying. h(μ) = 1/
1. Let X₁,..., Xn Exponential (Ao) be iid with density given by fo(x) = Ao exp(-Xox) for x>0 (a) Write down an expression for the log-likelihood (b) Compute the score function. (c) By direct calculation show that expected value of the score function eval uated at the true parameter is zero. (d) Compute the Fisher Information for Ao by: i Computing it directly from Exo (u(XoX)²) ii Computing it using Fisher's identity (e) Suppose that you now consider the reparametrisation What is the Fisher Information for o? Ho 2n = (f) A (naughty) statistician makes a claim that they have an unbiased estima tor for the mean of this exponential and that the variance of their estimato is equal to where po is the true mean of the exponential above. Explain with justification why they are lying. h(μ) = 1/
Chapter6: Exponential And Logarithmic Functions
Section6.8: Fitting Exponential Models To Data
Problem 5SE: What does the y -intercept on the graph of a logistic equation correspond to for a population...
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![1. Let X₁,..., Xn Exponential(X) be iid with density given by
fx (x) = ₁ exp(-λox) for x> 0
(a) Write down an expression for the log-likelihood
(b) Compute the score function.
(c) By direct calculation show that expected value of the score function eval-
uated at the true parameter is zero.
(d) Compute the Fisher Information for λ₁ by:
i Computing it directly from Ex(u(\o]X)²)
ii Computing it using Fisher's identity
=
(e) Suppose that you now consider the reparametrisation A
What is the Fisher Information for o?
4²
2n
h (μ)
where μo is the true mean of the exponential above.
Explain with justification why they are lying.
=
(f) A (naughty) statistician makes a claim that they have an unbiased estima-
tor for the mean of this exponential and that the variance of their estimator
is equal to
1/μ.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53d0abe2-4cb5-4db5-bef8-ebd7977db539%2F5dc92568-31bf-41d3-bd0b-8949682477b6%2F7icpt76e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let X₁,..., Xn Exponential(X) be iid with density given by
fx (x) = ₁ exp(-λox) for x> 0
(a) Write down an expression for the log-likelihood
(b) Compute the score function.
(c) By direct calculation show that expected value of the score function eval-
uated at the true parameter is zero.
(d) Compute the Fisher Information for λ₁ by:
i Computing it directly from Ex(u(\o]X)²)
ii Computing it using Fisher's identity
=
(e) Suppose that you now consider the reparametrisation A
What is the Fisher Information for o?
4²
2n
h (μ)
where μo is the true mean of the exponential above.
Explain with justification why they are lying.
=
(f) A (naughty) statistician makes a claim that they have an unbiased estima-
tor for the mean of this exponential and that the variance of their estimator
is equal to
1/μ.
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Step 1: Write the given information
VIEWStep 2: Write down an expression for the log-likelihood and compute the score function
VIEWStep 3: Show that expected value of the score function evaluated at the true parameter is zero
VIEWStep 4: Compute the Fisher Information for λ0
VIEWStep 5: Find the Fisher Information for μ0 under the re-parametrization λ = h(μ) = 1/μ
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