In lecture (Mon 1/30), we showed that for Bernoulli (p) distribution, the MLE estimator p = x Exercise 1 is sufficient for the parameter p; for Uniform([a, b]), the MLE estimators Geometric (p) are jointly sufficient for the parameters a, b. In this exercise, you will deduce similar results for the following four distributions: • Exp(A) • Poisson(X) • N(μ1,0²) â = y₁ = min(₁, ...,xn), b = y₁ := max(x1, ***, En) (i) For each of these four distributions, write down their likelihood functions. (Hint: the log-likelihood functions for these distributions were computed in previous lecture and homework.) (ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic for each of the first three distributions. (iii) Show that 1/2 is also a sufficient statistic for each of these three distributions. (iv) Use the Fisher-Neyman factorization theorem to show that , v are a pair of jointly sufficient statistics for N(μ, 0²).
In lecture (Mon 1/30), we showed that for Bernoulli (p) distribution, the MLE estimator p = x Exercise 1 is sufficient for the parameter p; for Uniform([a, b]), the MLE estimators Geometric (p) are jointly sufficient for the parameters a, b. In this exercise, you will deduce similar results for the following four distributions: • Exp(A) • Poisson(X) • N(μ1,0²) â = y₁ = min(₁, ...,xn), b = y₁ := max(x1, ***, En) (i) For each of these four distributions, write down their likelihood functions. (Hint: the log-likelihood functions for these distributions were computed in previous lecture and homework.) (ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic for each of the first three distributions. (iii) Show that 1/2 is also a sufficient statistic for each of these three distributions. (iv) Use the Fisher-Neyman factorization theorem to show that , v are a pair of jointly sufficient statistics for N(μ, 0²).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![Exercise 1
In lecture (Mon 1/30), we showed that
for Bernoulli (p) distribution, the MLE estimator
p=i
is sufficient for the parameter p;
• for Uniform([a, b]), the MLE estimators
• Poisson(X)
• N(μ1,0²)
â = y₁ = min(x₁, · · , £n), b = Yn = 1 max(T₁,..., In)
are jointly sufficient for the parameters a, b.
In this exercise, you will deduce similar results for the following four distributions:
• Exp(X)
• Geometric (p)
(i) For each of these four distributions, write down their likelihood functions. (Hint: the
log-likelihood functions for these distributions were computed in previous lecture and
homework.)
(ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic
for each of the first three distributions.
(iii) Show that 1/7 is also a sufficient statistic for each of these three distributions.
(iv) Use the Fisher-Neyman factorization theorem to show that I, v are a pair of jointly
sufficient statistics for N(μ, 0²).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04625ac1-43ff-4999-b93a-55388fc0c5e2%2Ffd03b527-74e0-4672-aff2-101cc116c370%2Ftdne83o_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 1
In lecture (Mon 1/30), we showed that
for Bernoulli (p) distribution, the MLE estimator
p=i
is sufficient for the parameter p;
• for Uniform([a, b]), the MLE estimators
• Poisson(X)
• N(μ1,0²)
â = y₁ = min(x₁, · · , £n), b = Yn = 1 max(T₁,..., In)
are jointly sufficient for the parameters a, b.
In this exercise, you will deduce similar results for the following four distributions:
• Exp(X)
• Geometric (p)
(i) For each of these four distributions, write down their likelihood functions. (Hint: the
log-likelihood functions for these distributions were computed in previous lecture and
homework.)
(ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic
for each of the first three distributions.
(iii) Show that 1/7 is also a sufficient statistic for each of these three distributions.
(iv) Use the Fisher-Neyman factorization theorem to show that I, v are a pair of jointly
sufficient statistics for N(μ, 0²).
Expert Solution
Step 1
The pdf of Exp is for .
The pmf of Geometric is for .
The pmf of Poisson is for .
The pdf of is .
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