Point estimation

We want to estimate an unknown parameter \(\theta\) of the distribution of a characteristic/variable in a population
- To do this: draw random sample of size 𝒏𝒏 from the population
- We assume that \(X_1, \dots, X_n\) are independent replications of the following random variable: X = „Value of the characteristic in a randomly selected statistical entity of the population.“


Now we want to infer the unknown parameter \(\theta\) of the distribution of X
This is done by estimators or estimation statistics:
- estimation function or estimation statistic for the parameter \(\theta\) is a function \(T = g(X_1, \dots, X_n)\)
- the estimator is the numerical value \(t = g(x_1, \dots, x_n)\) resulting from the realizations \(x_1, \dots, x_n\)
- the value of \(T = g(X_1, \dots, X_n)\) is again a random variable

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Estimators for the expected value \(\mu\) = E(X)

• \(\overline{X} = g(X_1, \dots, X_n) = \frac{1}{n}\sum_{i = 1}^{n} Xi\)
• the estimator \(\overline{x} = g(x_1, \dots, x_n) = \frac{1}{n}\sum_{i = 1}^{n} xi\) corresponds to the arithmetic mean of the sample (sample mean)

This estimator is unbiased and consistent.

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Estimators for the variance \(o^2 = Var(X)\)

• both \(\tilde{S^2} = g(X_1, \dots, X_n) = \frac{1}{n}\sum_{i = 1}^{n}(X_i - \overline{X})^2\) and \(S^2 = g(X_1, \dots, X_n) = \frac{1}{n - 1}\sum_{i = 1}^{n}(X_i - \overline{X})^2\) are estimation functions of the variance.
  For the first estimator the bias is \(-\frac{1}{n}o^2\). The second estimator is unbiased and consistent.

• the estimators \(\tilde{s^2} = g(x_1, \dots, x_n) = \frac{1}{n}\sum_{i = 1}^{n}(x_i - \overline{x})^2\) and \(s^2 = g(x_1, \dots, x_n) = \frac{1}{n - 1}\sum_{i = 1}^{n}(x_i - \overline{x})^2\) correspond to the empirical variance and the sample variance

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Estimators for the parameter \(\pi\) (probability of occurrence) of a Bernoulli distribution

• if X is a Bernoulli variable then \(X_i \in {0,1}\) for all i = 1, …, and \(\overline{X} = g(X_1, \dots, X_n) = \frac{1}{n}\sum_{i = 1}^{n}X_i\) is an estimation function for the probability of occurrence
• the estimator \(\overline{x} = g(x_1, \dots, x_n) = \frac{1}{n}\sum_{i = 1}^{n}x_i\) corresponds to the relative frequency of \(x_i = 1\) in the sample

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