Lesson test: Maximum likelihood

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Maximum likelihood

1.[2p]
What does the likelihood $L (θ)$ measure?

How probable the observed data is, read as a function of $θ$

Correct

The probability that $θ$ takes a particular value

The answer is: How probable the observed data is, read as a function of $\theta$

The density of $θ$ given the data

The answer is: How probable the observed data is, read as a function of $\theta$

The proportion of samples in which $θ$ is correct

The answer is: How probable the observed data is, read as a function of $\theta$

2.[1p]
A coin gives 60 heads in 100 tosses. What is the maximum likelihood estimate of $p$ ?

CorrectNot quite: 0.6

3.[2p]
In 400 voters, 240 support a proposal. What is the standard error of $\hat{p}$ , to four decimal places?

CorrectNot quite: 0.0245

4.[2p]
200 corps-years of Prussian cavalry records contain 122 deaths from horse kicks. What is the maximum likelihood Poisson rate per corps-year?

CorrectNot quite: 0.61

5.[3p]
Match each model to its maximum likelihood estimator.

Bernoulli $p$
Poisson $λ$
Exponential rate $λ$
Uniform upper limit $θ$

the sample maximum
the sample mean
one over the sample mean
the sample proportion $k / n$

Show the answer

Bernoulli $p$ : the sample proportion $k / n$ Poisson $λ$ : the sample mean Exponential rate $λ$ : one over the sample mean Uniform upper limit $θ$ : the sample maximum

6.[2p]
The maximum likelihood estimator of $σ^{2}$ for a normal sample divides the sum of squared deviations by $n$ , and is therefore biased low.

True

Correct

False

The answer is: True

7.[3p]
Why is the curvature of the log likelihood at its maximum related to the standard error?

A sharp peak means nearby parameter values fit much worse, so the data pins $θ$ down tightly

Correct

Curvature measures how far the estimate is from the true value

The answer is: A sharp peak means nearby parameter values fit much worse, so the data pins $\theta$ down tightly

The second derivative equals the bias of the estimator

The answer is: A sharp peak means nearby parameter values fit much worse, so the data pins $\theta$ down tightly

A flat likelihood means the estimate is unbiased

The answer is: A sharp peak means nearby parameter values fit much worse, so the data pins $\theta$ down tightly

8.[2p]
From $\hat{p} = 0.6$ , what is the maximum likelihood estimate of the odds $p / (1 - p)$ ?

CorrectNot quite: 1.5

9.[3p]
Where does the standard large-sample theory of maximum likelihood break down?

Select all that apply

When the range of possible data values depends on the parameter, as for a uniform upper limit

Correct

When the sample is small, so that asymptotic results have not taken effect

Correct

When one group in a logistic regression has no failures, giving an infinite maximum

Correct

Whenever the model has more than one parameter

The answer is: When the range of possible data values depends on the parameter, as for a uniform upper limit, When the sample is small, so that asymptotic results have not taken effect, When one group in a logistic regression has no failures, giving an infinite maximum

1.[2p] What does the likelihood L(θ) measure?

2.[1p] A coin gives 60 heads in 100 tosses. What is the maximum likelihood estimate of p?

3.[2p] In 400 voters, 240 support a proposal. What is the standard error of pˆ, to four decimal places?

4.[2p] 200 corps-years of Prussian cavalry records contain 122 deaths from horse kicks. What is the maximum likelihood Poisson rate per corps-year?

5.[3p] Match each model to its maximum likelihood estimator.

6.[2p] The maximum likelihood estimator of σ2 for a normal sample divides the sum of squared deviations by n, and is therefore biased low.

7.[3p] Why is the curvature of the log likelihood at its maximum related to the standard error?

8.[2p] From pˆ=0.6, what is the maximum likelihood estimate of the odds p/(1-p)?

9.[3p] Where does the standard large-sample theory of maximum likelihood break down?

1.[2p]
What does the likelihood $L (θ)$ measure?

2.[1p]
A coin gives 60 heads in 100 tosses. What is the maximum likelihood estimate of $p$ ?

3.[2p]
In 400 voters, 240 support a proposal. What is the standard error of $\hat{p}$ , to four decimal places?

4.[2p]
200 corps-years of Prussian cavalry records contain 122 deaths from horse kicks. What is the maximum likelihood Poisson rate per corps-year?

5.[3p]
Match each model to its maximum likelihood estimator.

6.[2p]
The maximum likelihood estimator of $σ^{2}$ for a normal sample divides the sum of squared deviations by $n$ , and is therefore biased low.

7.[3p]
Why is the curvature of the log likelihood at its maximum related to the standard error?

8.[2p]
From $\hat{p} = 0.6$ , what is the maximum likelihood estimate of the odds $p / (1 - p)$ ?

9.[3p]
Where does the standard large-sample theory of maximum likelihood break down?