Suppose X follows a distribution with two parameters, and we want to estimate both via MLE.  Once we have the log-likelihood function ℓ ( 𝜃 1 , 𝜃 2 ) , what should we do next?单项选择题

A

Set ℓ ( 𝜃 1 , 𝜃 2 ) equal to 0 and solve for 𝜃 1 and 𝜃 2 .

B

Set 𝛿 2 𝛿 𝜃 1 𝛿 𝜃 2 ℓ ( 𝜃 1 , 𝜃 2 )  equal to 0 and solve for 𝜃 1 and 𝜃 2 .

C

Set 𝛿 𝛿 𝜃 1 ℓ ( 𝜃 1 , 𝜃 2 ) equal to 0, and set 𝛿 𝛿 𝜃 2 ℓ ( 𝜃 1 , 𝜃 2 ) equal to 0.  Solve that system of equations for 𝜃 1 and 𝜃 2 .

D

Set 𝛿 𝛿 𝜃 1 ℓ ( 𝜃 1 , 𝜃 2 )  equal to 0 and solve for 𝜃 1 .  Then plug 𝜃 ^ 1 back into the log-likelihood and solve for  𝜃 2 .

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类似问题

Suppose  𝑋 ∼ 𝐵 𝑖 𝑛 𝑜 𝑚 ( 𝑛 = 10 , 𝑝 ) and we observe a random sample of size 3:  𝑥 1 = 0 , 𝑥 2 = 3 , and  𝑥 3 = 1 .  Once we have the log-likelihood function ℓ ( 𝑝 ) , what value will be 𝑝 ^ , the MLE for 𝑝 ?

Consider a Normal population with mean 𝜇 and variance 𝜎 2 < ∞ . We collect a sample of 𝑇 = 3 independent and identically distributed observations 𝑥 = ( 4 , 10 , 7 ) In class we have shown that the standardized log-likelihood of the sample for this case is 1 𝑇 log ( 𝐿 ( 𝑥 , 𝜇 , 𝜎 2 ) ) = − 1 2 log ( 2 𝜋 ) − 1 2 log ( 𝜎 2 ) − 1 𝑇 ∑ 𝑡 = 1 𝑇 ( 𝑥 𝑡 − 𝜇 ) 2 2 𝜎 2 . Then the Maximum Likelihood estimate of the variance 𝜎 2 is:

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