Point estimation

Builds on Estimator Statistic Statistical model Maximum likelihood estimation

Definition

Let $(\mathcal{X}, \mathcal{A})$ be a sample space and let $\set{P_\theta : \theta \in \Theta}$ be a statistical model on it. Let $g \colon \Theta \to D$ be a function, the estimand, with values in a measurable space $D$ . A point estimator of $g(\theta)$ is a statistic $T \colon \mathcal{X} \to D$ , that is, a measurable map that does not depend on $\theta$ . For observed data $x \in \mathcal{X}$ , the value $T(x)$ is the point estimate of $g(\theta)$ . Point estimation is the problem of choosing such a $T$ so that $T(X)$ is close to $g(\theta)$ when $X \sim P_\theta$ , for every $\theta \in \Theta$ .

Remarks

The general notion of estimator covers the same object; the word "point" stresses that the output is a single element of $D$ , not a set.

Example: in the model of $n$ independent Bernoulli( $p$ ) trials with $\Theta = [0, 1]$ and $g(p) = p$ , the map $T(x) = \frac{1}{n}\sum_i x_i$ is a point estimator; it is the maximum likelihood estimator.

Non-example: the map $T(x) = \frac{1}{n}\sum_i x_i - p$ is not a point estimator of $p$ , because it depends on the unknown parameter $p$ . Also, a confidence interval $[T_1(x), T_2(x)]$ is an interval estimator, not a point estimator, since its value is a set, not an element of $D$ .

Closeness is measured by a loss function, most often the squared error $(T - g(\theta))^2$ , whose expectation is the mean squared error.

Reading path

To understand this page from the ground up, read these 75 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Statistical model concept
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Order on the reals definition
Power set definition
Complement definition
Function definition
Interval definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Conditional probability definition
Random variable definition
Cumulative distribution function definition
Discrete random variable definition
Law of a random variable definition
Independence of two events definition
Convergence in probability definition
Random variable with a density definition
Joint probability distribution definition
Probability mass function definition
Mutual independence of events definition
Expected value definition
Identically distributed random variables definition
Independence of two random variables definition
Probability density function definition
Marginal distribution definition
Independent sigma-algebras definition
Variance definition
Linearity of expectation statement
Independence of random variables definition
Independence means the joint law is the product of the marginals statement
Covariance definition
Independent and identically distributed random variables definition
Variance of a sum statement
Expectation of a product of independent random variables statement
Statistic definition
Likelihood function definition
Independent random variables are uncorrelated statement
Estimator definition
Variance of a sum of independent random variables statement
Sample mean definition
Consistent estimator definition
Maximum likelihood estimation concept
Point estimation definition

Tag	`ZRMWNF10FA`
URI	`m2://ZRMWNF10FA`
Kind	definition
Writing	W0 stub
MSC	62F10
Defines	Point estimation
Uses	Estimator, Statistic, Statistical model, Maximum likelihood estimation

Point estimation

Definition

Remarks

Reading path

Further reading

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