Lesson test: What makes an estimator good

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What makes an estimator good

1.[3p]
Why is $\sum (X_{i} - \overline{X})^{2}$ divided by $n - 1$ ?

Its expectation is $(n - 1) σ^{2}$ , because deviations are taken from $\overline{X}$ rather than from $μ$

Correct

Because one observation is discarded when computing $\overline{X}$

The answer is: Its expectation is $(n-1)\sigma^2$, because deviations are taken from $\bar{X}$ rather than from $\mu$

To make the sample standard deviation unbiased for $σ$

The answer is: Its expectation is $(n-1)\sigma^2$, because deviations are taken from $\bar{X}$ rather than from $\mu$

Because the sample is smaller than the population

The answer is: Its expectation is $(n-1)\sigma^2$, because deviations are taken from $\bar{X}$ rather than from $\mu$

2.[2p]
A sample of 20 comes from a population with $σ^{2} = 100$ . Dividing the sum of squared deviations by 20 instead of 19 gives an estimator with what expected value?

CorrectNot quite: 95

3.[2p]
If $s^{2}$ is unbiased for $σ^{2}$ , then $s$ is unbiased for $σ$ .

True

The answer is: False

False

Correct

4.[2p]
Which is the correct decomposition of mean squared error?

Variance plus squared bias

Correct

Bias plus variance

The answer is: Variance plus squared bias

Variance minus squared bias

The answer is: Variance plus squared bias

Squared variance plus bias

The answer is: Variance plus squared bias

5.[3p]
Draws come from a uniform on $[0, θ]$ with $n = 10$ . The variance of $2 \overline{X}$ is $θ^{2} / (3 n)$ and that of the rescaled maximum is $θ^{2} / (n (n + 2))$ . How many times larger is the first?

CorrectNot quite: 4

6.[2p]
For normal data the median's variance is $π / 2$ times the mean's. A median from 1000 observations carries the information of a mean from how many?

CorrectNot quite: 637

7.[3p]
Which statements about consistency are correct?

Select all that apply

An estimator whose bias and variance both tend to zero is consistent

Correct

Consistency says nothing about behaviour at any particular sample size

Correct

A biased estimator can still be consistent

Correct

Consistency implies the estimator is unbiased at every $n$

The answer is: An estimator whose bias and variance both tend to zero is consistent, Consistency says nothing about behaviour at any particular sample size, A biased estimator can still be consistent

8.[3p]
Match each property to its definition.

Bias
Mean squared error
Consistent
Efficient

converges in probability to $θ$
$E [(\hat{θ} - θ)^{2}]$
$E [\hat{θ}] - θ$
smallest variance among unbiased rules

Show the answer

Bias: $E [\hat{θ}] - θ$ Mean squared error: $E [(\hat{θ} - θ)^{2}]$ Consistent: converges in probability to $θ$ Efficient: smallest variance among unbiased rules

9.[2p]
For a normal sample, estimating $σ^{2}$ with divisor $n + 1$ instead of $n - 1$ gives a lower mean squared error. Why is $n - 1$ still standard?

Unbiasedness is preserved through later calculations, which the smaller-MSE choice is not

Correct

The $n + 1$ version is inconsistent

The answer is: Unbiasedness is preserved through later calculations, which the smaller-MSE choice is not

The $n + 1$ version has a larger variance

The answer is: Unbiasedness is preserved through later calculations, which the smaller-MSE choice is not

The difference disappears for every sample size

The answer is: Unbiasedness is preserved through later calculations, which the smaller-MSE choice is not

1.[3p] Why is ∑(Xi-X‾)2 divided by n-1?

2.[2p] A sample of 20 comes from a population with σ2=100. Dividing the sum of squared deviations by 20 instead of 19 gives an estimator with what expected value?

3.[2p] If s2 is unbiased for σ2, then s is unbiased for σ.

4.[2p] Which is the correct decomposition of mean squared error?

5.[3p] Draws come from a uniform on [0,θ] with n=10. The variance of 2X‾ is θ2/(3n) and that of the rescaled maximum is θ2/(n(n+2)). How many times larger is the first?

6.[2p] For normal data the median's variance is π/2 times the mean's. A median from 1000 observations carries the information of a mean from how many?

7.[3p] Which statements about consistency are correct?

8.[3p] Match each property to its definition.

9.[2p] For a normal sample, estimating σ2 with divisor n+1 instead of n-1 gives a lower mean squared error. Why is n-1 still standard?

1.[3p]
Why is $\sum (X_{i} - \overline{X})^{2}$ divided by $n - 1$ ?

2.[2p]
A sample of 20 comes from a population with $σ^{2} = 100$ . Dividing the sum of squared deviations by 20 instead of 19 gives an estimator with what expected value?

3.[2p]
If $s^{2}$ is unbiased for $σ^{2}$ , then $s$ is unbiased for $σ$ .

4.[2p]
Which is the correct decomposition of mean squared error?

5.[3p]
Draws come from a uniform on $[0, θ]$ with $n = 10$ . The variance of $2 \overline{X}$ is $θ^{2} / (3 n)$ and that of the rescaled maximum is $θ^{2} / (n (n + 2))$ . How many times larger is the first?

6.[2p]
For normal data the median's variance is $π / 2$ times the mean's. A median from 1000 observations carries the information of a mean from how many?

7.[3p]
Which statements about consistency are correct?

8.[3p]
Match each property to its definition.

9.[2p]
For a normal sample, estimating $σ^{2}$ with divisor $n + 1$ instead of $n - 1$ gives a lower mean squared error. Why is $n - 1$ still standard?