What is the symbol ≙ most commonly used for in a mathematical or math-related context?

Question

What is the symbol ≙ most commonly used for in a mathematical or math-related context?

• LaTeX produces the symbol with \hateq.
• The symbol has Unicode codepoint U+2259.
• The respective Unicode code chart shows it with the name ESTIMATES and the annotation = corresponds to. Note that the names and annotations there are often not helpful, incomplete, or even misleading.
• Entering it into the English Wikipedia currently redirects to the article Correspondence (mathematics).

Where and how is ≙ used in math or related fields? (By "related" I am thinking of statistics and physics.)

In German secondary-school math and science classes, it is used to denote "corresponds to" (German: "entspricht") when doing calculations using the rule of three. For example: 3 apples ≙ €15; 7 apples ≙ €x.

Answer 1

The hat above the equals sign is an estimator. Originaly estimators are used on terms. For example:

$$\widehat{\theta}(X)$$

But it has appeared (sometimes) more convenient to use it with an equals sign, especially in statistics. It has transformed a substantive (θ is an estimator for . . .) into a verb.

x ≙ . . . then reads as x estimates . . .

You use it instead of a plain equals sign when you do not have a strict equality between the left term and the right expression. Statistically speaking, it symbolizes an estimation of an unobservable quantity.

Answer 2

In statistics, it is to define the sample minimum $x_{(1)}$ or sample maximum $x_{(n)}$.

For example,

$$x_{(1)} ≙ min\{x_1, x_2, ..., x_n\}$$

$x_{(1)}$ is also called the smallest observation.