A discrepancy function is a mathematical function which describes how closely a structural model

Structural equation modeling

Structural equation modeling is a statistical technique for testing and estimating causal relations using a combination of statistical data and qualitative causal assumptions...

 conforms to observed data. Larger values of the discrepancy function indicate a poor fit of the model to data. In general, the parameter estimates for a given model are chosen so as to make the discrepancy function for that model as small as possible.

There are several basic types of discrepancy functions, including maximum likelihood

Maximum likelihood

In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....

 (ML), generalized least squares

Generalized least squares

In statistics, generalized least squares is a technique for estimating the unknown parameters in a linear regression model. The GLS is applied when the variances of the observations are unequal , or when there is a certain degree of correlation between the observations...

 (GLS), and ordinary least squares

Ordinary least squares

In statistics, ordinary least squares or linear least squares is a method for estimating the unknown parameters in a linear regression model. This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear...

 (OLS), which are considered the "classical" discrepancy functions. Discrepancy functions all meet the following basic criteria:

• They are non-negative, i.e., always greater than or equal to zero.
• They are zero only if the fit is perfect, i.e., if the model and parameter estimates perfectly reproduce the observed data.
• The discrepancy function is a continuous function of the elements of S, the sample covariance matrix, and Σ(θ), the "reproduced" estimate of S obtained by using the parameter estimates and the structural model.

In order for "maximum likelihood" to meet the first criterion, it is used in a revised form as the deviance.

See also

• Discrepancy theory
  Discrepancy theory
  In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be. It is also called theory of irregularities of distribution. This refers to the theme of classical discrepancy theory, namely distributing points in some space such that they are evenly...
  
  
• Low-discrepancy sequence
  Low-discrepancy sequence
  In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy....
  
  
• Constructions of low-discrepancy sequences