Estimators

Due to obvious reasons, it is not possible to find parameters of a population. In that case, a “good enough” value/range must be guessed/computed for the true values of the parameter to make conclusions (inferences) on the population based on sample.

An estimator is a guess of a population parameter based on a sample data.

Point estimator

A statistics which can be used to get a single number as the result. The estimated value can be considered as the most plausible value of a parameter.

Interval estimator

A statistics which can give an upper limit and a lower limit calculation based on the point estimator. A pair of values calculated as the upper and the lower limits for a population parameter. A confidence is also mentioned.

Confidence Interval

A statistics which gives a range of values for a given population parameter, within which the parameter falls in $(1-\alpha)$ of the time. An interval estimator. $$

To achieve a confidence level of $100(1-\alpha)\%$: $$

$$\bar{x} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$$

If $\sigma$ is unknown, then $s$ is used instead.

Process of Estimation

Estimating a population parameter from a sample data.

Unbiasedness

If a statistic $e$ is used as an estimator for population parameter $\theta$ and $E(e) = θ$, then it is said to be an unbiased estimator of $\theta$.

Accuracy

Mean, variance, standard deviation can be estimated with a sample.

Sample mean is an unbiased estimator for a continuous random variable. Aka. best linear unbiased estimator (BLUE).

Sample variance is an unbiased estimator.

Note

Sample variance can be estimated using:

$$s^2 = \frac{1}{n - 1}\sum_{i=1}^{n} (x_i - \bar{x})^2 = \frac{1}{n-1}\bigg[ \sum_{i=1}^{n} x_i^2 - n\bar{x}^2 \bigg]$$