1. Suppose that Y; takes values 0, 1, 2 with probability f(y;0) = e®y-b(@). (a) Find the function b(0) so that f(y; 0) is a probability mass function (i.e., total probability one). Find the values f(0; 0), f(1;0), and f(2;0) for 0 = 0, 0 = log .5, 0 = log 2 (remember that unless otherwise specified, all logs are natural logs). Find the expected value of Y for each of these three values of 0. (b) If we observe Y1,Y2, ..., Yn, find the log-likelihood function ((0 ;Y1,Y2, ...,Y,). (c) Suppose that we have a sample of size 50 and observe the following summary of values of Y;: Y = 0 Y =1 Y = 2 Total 26 11 13 50 (I.e., 11 of the Y; have value 0, etc.) Find the mean of the Y; and compare to the means from part (a). What does this tell you about the MLE for 0?
1. Suppose that Y; takes values 0, 1, 2 with probability f(y;0) = e®y-b(@). (a) Find the function b(0) so that f(y; 0) is a probability mass function (i.e., total probability one). Find the values f(0; 0), f(1;0), and f(2;0) for 0 = 0, 0 = log .5, 0 = log 2 (remember that unless otherwise specified, all logs are natural logs). Find the expected value of Y for each of these three values of 0. (b) If we observe Y1,Y2, ..., Yn, find the log-likelihood function ((0 ;Y1,Y2, ...,Y,). (c) Suppose that we have a sample of size 50 and observe the following summary of values of Y;: Y = 0 Y =1 Y = 2 Total 26 11 13 50 (I.e., 11 of the Y; have value 0, etc.) Find the mean of the Y; and compare to the means from part (a). What does this tell you about the MLE for 0?
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![1. Suppose that Y; takes values 0, 1, 2 with probability f (y;0) = e®y-b(0).
(a) Find the function b(0) so that f(y; 0) is a probability mass function (i.e., total probability
one). Find the values f(0; 0), f(1;0), and f(2;0) for 0 = 0, 0 = log .5, 0 = log 2
(remember that unless otherwise specified, all logs are natural logs). Find the expected
value of Y for each of these three values of 0.
(b) If we observe Y1,Y2, ..., Yn, find the log-likelihood function l(0 ;Y1,Y2, ..., Y,).
(c) Suppose that we have a sample of size 50 and observe the following summary of values
of Y;:
Y = 0 Y = 1 Y = 2 Total
11
13
26
50
(L.e., 11 of the Y; have value 0, etc.) Find the mean of the Y; and compare to the means
from part (a). What does this tell you about the MLE for 0?
(d) Graph twice the log-likelihood function against 0 given these data for 0 over the interval
(0,1). From the graph estimate the MLE for 0 and an approximate 95% confidence
interval.
(e) Find the score function U(0) (suppressing dependence on Y1,..., Yn) and evaluate at
0 = 0.
(f) Find the Fisher information, I(0) and evaluate at 0 = 0.
(g) Conduct the score test of the null hypothesis Ho: 0 = 0. If µ is the expected value of Y
when 0 = 0 (from part (a)), compare to the one sample t-test of Ho: E[Y] = µ.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8032ea9-534e-4a7d-a600-c74fea44fe9f%2F6b05b326-4b3a-4989-94c7-98881516af4d%2F606nsnh_processed.png&w=3840&q=75)
Transcribed Image Text:1. Suppose that Y; takes values 0, 1, 2 with probability f (y;0) = e®y-b(0).
(a) Find the function b(0) so that f(y; 0) is a probability mass function (i.e., total probability
one). Find the values f(0; 0), f(1;0), and f(2;0) for 0 = 0, 0 = log .5, 0 = log 2
(remember that unless otherwise specified, all logs are natural logs). Find the expected
value of Y for each of these three values of 0.
(b) If we observe Y1,Y2, ..., Yn, find the log-likelihood function l(0 ;Y1,Y2, ..., Y,).
(c) Suppose that we have a sample of size 50 and observe the following summary of values
of Y;:
Y = 0 Y = 1 Y = 2 Total
11
13
26
50
(L.e., 11 of the Y; have value 0, etc.) Find the mean of the Y; and compare to the means
from part (a). What does this tell you about the MLE for 0?
(d) Graph twice the log-likelihood function against 0 given these data for 0 over the interval
(0,1). From the graph estimate the MLE for 0 and an approximate 95% confidence
interval.
(e) Find the score function U(0) (suppressing dependence on Y1,..., Yn) and evaluate at
0 = 0.
(f) Find the Fisher information, I(0) and evaluate at 0 = 0.
(g) Conduct the score test of the null hypothesis Ho: 0 = 0. If µ is the expected value of Y
when 0 = 0 (from part (a)), compare to the one sample t-test of Ho: E[Y] = µ.
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