Questions
Single choice
Consider the likelihood of an i.i.d. sample from a Bernoulli population with parameter ๐ ๐ฟ ( ๐ฅ 1 , . . . , ๐ฅ ๐ ) = โ ๐ก = 1 ๐ ๐ ๐ฅ ๐ก ( 1 โ ๐ ) 1 โ ๐ฅ ๐ก . If you estimate the parameter using a Maximum Likelihood estimator, you obtain the point estimate ๐ ฬ = 1 ๐ โ ๐ก = 1 ๐ ๐ฅ ๐ก . The standard error can be computed according to two different approaches as we have seen in class: (1) use the variance-covariance matrix of the score ๐บ 0 ; (2) use the matrix of second derivatives of the standardized log-likelihood ๐ต 0 . What is the formula for the standard error of the estimated parameter, if we follow approach (2)?
Options
A.The standard error of
๐
ฬ
is
๐
๐ผ
(
๐
ฬ
)
=
1
๐
๐
ฬ
(
1
โ
๐
ฬ
)
B.The standard error of
๐
ฬ
is
๐
๐ผ
(
๐
ฬ
)
=
๐
ฬ
(
1
โ
๐
ฬ
)
C.The standard error of
๐
ฬ
is
๐
๐ผ
(
๐
ฬ
)
=
1
๐
๐
ฬ
(
1
โ
๐
ฬ
)
D.The standard error of
๐
ฬ
is
๐
๐ผ
(
๐
ฬ
)
=
2
๐
ฬ
(
1
โ
๐
ฬ
)
E.There is not enough information to compute the standard error of the estimated parameter.
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Step-by-Step Analysis
We begin by restating the problem setup: we have an i.i.d. Bernoulli sample with parameter p, and the MLE is p_hat = (1/T) sum_t x_t. We are asked for the standard error of p_hat when using approach (2), i.e., based on the second derivatives of the standardized log-likelihood B0.
Option 1: The standard error of p_hat is 1/T * p_hat * (1 - p_hat).
This expression misses the essential square root that converts variance to standard error. The Fisher information for Bernoulli trials scales as T * p_hat * (1 - p_hat), and ......Login to view full explanationLog in for full answers
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Which one statement is true?
Letย ย denote a random sample from a distribution with pdfย forย , whereย .ย Consider a dataset and letย the cost function for estimating the model be the negative log-likelihood. Consider the critical point that you found that you found in Question 2 and the formula for the second derivative that you derived in Question 3. What can you conclude? Tick all that apply.ย
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