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Consider the likelihood of an i.i.d. sample from a Bernoulli population with parameter ๐‘ ๐ฟ ( ๐‘ฅ 1 , . . . , ๐‘ฅ ๐‘‡ ) = โˆ ๐‘ก = 1 ๐‘‡ ๐‘ ๐‘ฅ ๐‘ก ( 1 โˆ’ ๐‘ ) 1 โˆ’ ๐‘ฅ ๐‘ก . If you estimate the parameter using a Maximum Likelihood estimator, you obtain the point estimate ๐‘ ฬ‚ = 1 ๐‘‡ โˆ‘ ๐‘ก = 1 ๐‘‡ ๐‘ฅ ๐‘ก . 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 ๐›บ 0 ; (2) use the matrix of second derivatives of the standardized log-likelihood ๐ต 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•Š ๐”ผ ( ๐‘ ฬ‚ ) = 1 ๐‘‡ ๐‘ ฬ‚ ( 1 โˆ’ ๐‘ ฬ‚ )
B.The standard error of ๐‘ ฬ‚ is ๐•Š ๐”ผ ( ๐‘ ฬ‚ ) = ๐‘ ฬ‚ ( 1 โˆ’ ๐‘ ฬ‚ )
C.The standard error of ๐‘ ฬ‚ is ๐•Š ๐”ผ ( ๐‘ ฬ‚ ) = 1 ๐‘‡ ๐‘ ฬ‚ ( 1 โˆ’ ๐‘ ฬ‚ )
D.The standard error of ๐‘ ฬ‚ is ๐•Š ๐”ผ ( ๐‘ ฬ‚ ) = 2 ๐‘ ฬ‚ ( 1 โˆ’ 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 explanation

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