Aweight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sumorweighted average. Weight functions occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus"[1] and "meta-calculus".[2]
In the discrete setting, a weight function is a positive function defined on a discreteset, which is typically finiteorcountable. The weight function corresponds to the unweighted situation in which all elements have equal weight. One can then apply this weight to various concepts.
If the function is a real-valued function, then the unweighted sumofon is defined as
but given a weight function, the weighted sumorconical combination is defined as
Weighted means are commonly used in statistics to compensate for the presence of bias. For a quantity measured multiple independent times with variance, the best estimate of the signal is obtained by averaging all the measurements with weight , and the resulting variance is smaller than each of the independent measurements . The maximum likelihood method weights the difference between fit and data using the same weights .
The expected value of a random variable is the weighted average of the possible values it might take on, with the weights being the respective probabilities. More generally, the expected value of a function of a random variable is the probability-weighted average of the values the function takes on for each possible value of the random variable.
Inregressions in which the dependent variable is assumed to be affected by both current and lagged (past) values of the independent variable, a distributed lag function is estimated, this function being a weighted average of the current and various lagged independent variable values. Similarly, a moving average model specifies an evolving variable as a weighted average of current and various lagged values of a random variable.
The terminology weight function arises from mechanics: if one has a collection of objects on a lever, with weights (where weight is now interpreted in the physical sense) and locations , then the lever will be in balance if the fulcrum of the lever is at the center of mass
which is also the weighted average of the positions .