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## 3.8 Exercises

#### 1. Biased …

Consider the following alternative estimator for \(\mu_Y\), the mean of the \(Y_i\)

\[\widetilde{Y}=\frac{1}{n-1}\sum\limits_{i=1}^n Y_i\]

In this exercise we will illustrate that this estimator is a biased estimator for \(\mu_Y\).

**Instructions:**

Define a function

`Y_tilde`that implements the estimator above.Randomly draw 5 observations from the \(\mathcal{N}(10, 25)\) distribution and compute an estimate using

`Y_tilde()`. Repeat this procedure 10000 times and store the results in`est_biased`.Plot a histogram of

`est_biased`.Add a red vertical line at \(\mu=10\) using the function

`abline()`.

**Hints:**

To compute the sum of a vector you can use

`sum()`, to get the length of a vector you can use`length()`.Use the function

`replicate()`to compute repeatedly estimates of random samples. With the arguments`expr`and`n`you can specify the operation and how often it has to be replicated.A histogram can be plotted with the function

`hist()`.The point on the x-axis as well as the color for the vertical line can be specified via the arguments

`v`and`col`.

#### 2. … but consistent estimator

Consider again the estimator from the previous exercise. It is available in your environment as the function `Y_tilde()`. You are requested to do the same procedure as in the previous exercise. This time, however, increase the number of observations to draw from 5 to 1000.

What do you notice? What can you say about this estimator?

**Instructions:**

Randomly draw 1000 observations from the \(\mathcal{N}(10, 25)\) distribution and compute an estimate of the mean using

`Y_tilde()`. Repeat this procedure 10000 times and store the results in`est_consistent`.Plot a histogram of

`est_consistent`.Add a red vertical line at \(\mu=10\) using the function

`abline()`.

**Hints:**

Use the function

`replicate()`to compute estimates of repeatedly drawn random samples. Using the arguments`expr`and`n`you may specify the operation and how often it will be replicated.A histogram can be plotted with the function

`hist()`.The position on the x-axis as well as the color for the vertical line can be specified via the arguments

`v`and`col`.

#### 3. Efficiency of an Estimator

In this exercise we want to illustrate the result that the sample mean

\[\hat{\mu}_Y=\sum\limits_{i=1}^{n}a_iY_i\] with the equal weighting scheme \(a_i=\frac{1}{n}\) for \(i=1,...,n\) is the best linear unbiased estimator (BLUE) of \(\mu_Y\).

As an alternative, consider the estimator

\[\tilde{\mu}_Y=\sum\limits_{i=1}^{n}b_iY_i\]

where \(b_i\) gives the first \(\frac{n}{2}\) observations a higher weighting than the second \(\frac{n}{2}\) observations (we assume that \(n\) is even for simplicity).

The vector of weights `w` has been defined already and is available in your working environment.

**Instructions:**

Verify that \(\tilde{\mu}\) is an unbiased estimator of \(\mu_Y\), the mean of the \(Y_i\).

Implement the alternative estimator of \(\mu_Y\) as a function

`mu_tilde()`.Randomly draw 100 observations from the \(\mathcal{N}(5, 10)\) distribution and compute estimates with both estimators. Repeat this procedure 10000 times and store the results in

`est_bar`and`est_tilde`.Compute the sample variances of

`est_bar`and`est_tilde`. What can you say about both estimators?

**Hints:**

In order for \(\tilde{\mu}\) to be an unbiased estimator all weights have to sum up to 1.

Use the function

`replicate()`to compute estimates of repeatedly drawn samples. With the arguments`expr`and`n`you can specify the operation and how often it is replicated.You may use

`var()`the compute the sample variance.

#### 4. Hypothesis Test — \(t\)-statistic

Consider the CPS dataset from Chapter 3.6 again. The dataset `cps` is available in your working environment.

We suppose that the average hourly earnings (in prices of 2012) `ahe12` exceed 23.50 \(\$/h\) and wish to test this hypothesis at a significance level of \(\alpha=0.05\). Please do the following:

**Instructions:**

Compute the test statistic by hand and assign it to

`tstat`.Use

`tstat`to accept or reject the null hypothesis. Please do so using the normal approximation.

**Hints:**

We test \(H_0:\mu_{Y_{ahe}}\leq 23.5\) vs. \(H_1:\mu_{Y_{ahe}}>23.5\). That is, we conduct a right-sided test.

The \(t\)-statistic is defined as \(\frac{\bar{Y}-\mu_{Y,0}}{s_{Y}/\sqrt{n}}\) where \(s_Y\) denotes the sample variance.

To decide whether the null hypothesis is accepted or rejected you can compare the \(t\)-statistic with the respective quantile of the standard normal distribution. Use logical operators.

#### 5. Hypothesis Test — \(p\)-value

Reconsider the test situation from the previous exercise. The dataset `cps` as well as the vector `tstat` are available in your working environment.

Instead of using the \(t\)-statistic as decision criterion you may also use the \(p\)-value. Now please do the following:

**Instructions:**

Compute the \(p\)-value by hand and assign it to

`pval`.Use

`pval`to accept or reject the null hypothesis.

**Hints:**

The \(p\)-value for a right-sided test can be computed as \(p=P(t>t^{act}|H_0)\).

We reject the null if \(p<\alpha\). Use logical operators to check for this.

#### 6. Hypothesis Test — One Sample \(t\)-test

In the last two exercises we discussed two ways of conducting a hypothesis test. These approaches are somewhat cumbersome to apply by hand which is why `R` provides the function `t.test()`. It does most of the work automatically. `t.test()` provides \(t\)-statistics, \(p\)-values and even confidence intervals (more on the latter in later exercises). Note that `t.test()` uses the \(t\)-distribution instead of the normal distribution which becomes important when the sample size is small.

The dataset `cps` and the variable `pval` from Exercise 3.4 are available in your working environment.

**Instructions:**

Conduct the hypothesis test from previous exercises using the function

`t.test()`.Extract the \(t\)-statistic and the \(p\)-value from the list created by

`t.test()`. Assign them to the variables`tstat`and`pvalue`.Verify that using the normal approximation here is valid as well by computing the difference between both \(p\)-values.

**Hints:**

The type of the test as well as the null hypothesis can be specified via the arguments

`alternative`and`mu`.The \(t\)-statistic and the \(p\)-value can be obtained via

`$statistic`and`$p.value`, respectively.

#### 7. Hypothesis Test — Two Sample \(t\)-test

Consider the annual maximum sea levels at Port Pirie (Southern Australia) and Fremantle (Western Australia) for the last 30 years.

The observations are made available as vectors `portpirie` and `fremantle` in your working environment.

**Instructions:**

- Test whether there is a significant difference in the annual maximum sea levels at a significance level of \(\alpha=0.05\).

**Hints:**

We test \(H_0:\mu_{P}-\mu_{F}=0\) vs. \(H_1:\mu_{P}-\mu_{F}\ne 0\). That is, we conduct a two sample \(t\)-test.

For a two sample \(t\)-test the function

`t.test()`expects two vectors containing the data.

#### 8. Confidence Interval

Reconsider the test situation concerning the annual maximum sea levels at Port Pirie and Fremantle.

The variables `portpirie` and `fremantle` are again available in your working environment.

**Instructions:**

- Construct a \(95\%\)-confidence interval for the difference in the sea levels using
`t.test()`.

**Hint:**

- The function
`t.test()`computes a \(95\%\) confidence interval by default. This is accessible via`$conf.int`.

#### 9. (Co)variance and Correlation I

Consider a random sample \((X_i, Y_i)\) for \(i=1,...,100\).

The respective vectors `X` and `Y` are available in your working environment.

**Instructions:**

Compute the variance of \(X\) using the function

`cov()`.Compute the covariance of \(X\) and \(Y\).

Compute the correlation between \(X\) and \(Y\).

**Hints:**

The variance is a special case of the covariance.

`cov()`as well as`cor()`expect a vector for each variable.

#### 10. (Co)variance and Correlation II

In this exercise we want to examine the limitations of the correlation as a dependency measure.

Once the session has initialized you will see the plot of 100 realizations from two random variables \(X\) and \(Y\).

The respective observations are available in the vectors `X` and `Y` in your working environment.

**Instructions:**

- Compute the correlation between \(X\) and \(Y\). Interpret your result critically.

**Hint:**

`cor()`expects a vector for each variable.