📂Mathematical Statistics

Statistical Measures and Estimators in Mathematical Statistics

Definition ^{1 2}

1. A function $T$ of a sample $X_{1} , \cdots , X_{n}$ from a random variable $X$ is called a Statistic. $$ T := T \left( X_{1} , \cdots , X_{n} \right) $$
2. When the distribution function of $X$ is expressed as $f(x; \theta)$ or $p(x; \theta)$, if $T$ serves to capture $\theta$, then $T$ is referred to as an Estimator of $\theta$.
3. The probability distribution of a statistic is known as its [Sampling Distribution].

Description

The realization of an Estimator is called an Estimate. Parameters are often scalar $\theta \in \mathbb{R}$, and in such cases, $T$ is also termed as a Point Estimator of $\theta$. For example, when there is a random sample following a normal distribution $N \left( \mu, \sigma^{2} \right)$, the estimator for the population mean $\mu$ is as follows. $$ \overline{X} := {{ 1 } \over { n }} \sum_{k = 1}^{n} X_{k} $$ If there is actual data $x_{1} , \cdots , x_{n}$, the estimate of $\mu$ is as follows. $$ \overline{x} := {{ 1 } \over { n }} \sum_{k = 1}^{n} x_{k} $$

See Also

Statistic in Basic Statistics

In basic statistics, instead of describing it as a function of a sample, it’s more intuitively defined as ‘something calculated’. Essentially they mean the same thing but this may be a better definition for freshmen or those not familiar with mathematics.

Examples of Statistics

Excluding things like means or variances, examples of statistics specifically termed ‘statistics’ include:

• Sufficient Statistic: A statistic that contains all information about a parameter within the distribution.
  • Minimum Sufficient Statistic: A sufficient statistic that satisfies a specific condition.
• Ancillary Statistic: In contrast to a sufficient statistic, it does not convey any information about the parameters.
• Complete Statistic: A statistic that possesses the properties one would logically expect a statistic to have.

Examples of Estimators

Examples of estimators include:

• Unbiased Estimator: An estimator that does not possess any bias.
• Consistent Estimator: An estimator that estimates the parameter accurately in the limit.
• Maximum Likelihood Estimator: The estimator that maximizes the likelihood.
• Efficient Estimator: An estimator related to the variance of the statistic.