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## 3.1 Estimation of the Population Mean

### Key Concept 3.1

### Estimators and Estimates

*Estimators* are functions of sample data drawn from an unknown population. *Estimates* are numeric values computed by estimators based on the sample data. Estimators are random variables because they are functions of *random* data. Estimates are nonrandom numbers.

Think of some economic variable, for example hourly earnings of college graduates, denoted by \(Y\). Suppose we are interested in \(\mu_Y\) the mean of \(Y\). In order to exactly calculate \(\mu_Y\) we would have to interview every working graduate in the economy. We simply cannot do this due to time and cost constraints. However, we can draw a random sample of \(n\) i.i.d. observations \(Y_1, \dots, Y_n\) and estimate \(\mu_Y\) using one of the simplest estimators in the sense of Key Concept 3.1 one can think of, that is,

\[ \overline{Y} = \frac{1}{n} \sum_{i=1}^n Y_i, \]

the sample mean of \(Y\). Then again, we could use an even simpler estimator for \(\mu_Y\): the very first observation in the sample, \(Y_1\). Is \(Y_1\) a good estimator? For now, assume that

\[ Y \sim \chi_{12}^2 \]

which is not too unreasonable as hourly income is non-negative and we expect many hourly earnings to be in a range of \(5€\,\) to \(15€\). Moreover, it is common for income distributions to be skewed to the right — a property of the \(\chi^2_{12}\) distribution.

```
# plot the chi_12^2 distribution
curve(dchisq(x, df=12),
from = 0,
to = 40,
ylab = "density",
xlab = "hourly earnings in Euro")
```

We now draw a sample of \(n=100\) observations and take the first observation \(Y_1\) as an estimate for \(\mu_Y\)

```
# set seed for reproducibility
set.seed(1)
# sample from the chi_12^2 distribution, use only the first observation
<- rchisq(n = 100, df = 12)
rsamp 1]
rsamp[#> [1] 8.257893
```

The estimate \(8.26\) is not too far away from \(\mu_Y = 12\) but it is somewhat intuitive that we could do better: the estimator \(Y_1\) discards a lot of information and its variance is the population variance:

\[ \text{Var}(Y_1) = \text{Var}(Y) = 2 \cdot 12 = 24 \]

This brings us to the following question: What is a *good* estimator of an unknown parameter in the first place? This question is tackled in Key Concepts 3.2 and 3.3.

### Key Concept 3.2

### Bias, Consistency and Efficiency

Desirable characteristics of an estimator include unbiasedness, consistency and efficiency.

**Unbiasedness:**

If the mean of the sampling distribution of some estimator \(\hat\mu_Y\) for the population mean \(\mu_Y\) equals \(\mu_Y\),
\[ E(\hat\mu_Y) = \mu_Y, \]
the estimator is unbiased for \(\mu_Y\). The *bias* of \(\hat\mu_Y\) then is \(0\):

\[ E(\hat\mu_Y) - \mu_Y = 0\]

**Consistency:**

We want the uncertainty of the estimator \(\mu_Y\) to decrease as the number of observations in the sample grows. More precisely, we want the probability that the estimate \(\hat\mu_Y\) falls within a small interval around the true value \(\mu_Y\) to get increasingly closer to \(1\) as \(n\) grows. We write this as

\[ \hat\mu_Y \xrightarrow{p} \mu_Y. \]

**Variance and efficiency:**

We want the estimator to be efficient. Suppose we have two estimators, \(\hat\mu_Y\) and \(\overset{\sim}{\mu}_Y\) and for some given sample size \(n\) it holds that

\[ E(\hat\mu_Y) = E(\overset{\sim}{\mu}_Y) = \mu_Y \] but \[\text{Var}(\hat\mu_Y) < \text{Var}(\overset{\sim}{\mu}_Y).\]

We then prefer to use \(\hat\mu_Y\) as it has a lower variance than \(\overset{\sim}{\mu}_Y\), meaning that \(\hat\mu_Y\) is more *efficient* in using the information provided by the observations in the sample.