Wald Test - Mathematical Details

Mathematical Details

Under the Wald statistical test, the maximum likelihood estimate of the parameter(s) of interest is compared with the proposed value, with the assumption that the difference between the two will be approximately normally distributed. Typically the square of the difference is compared to a chi-squared distribution. In the univariate case, the Wald statistic is


\frac{ ( \widehat{ \theta}-\theta_0 )^2 }{\operatorname{var}(\hat \theta )}

which is compared against a chi-squared distribution.

Alternatively, the difference can be compared to a normal distribution. In this case the test statistic is

where is the standard error of the maximum likelihood estimate. A reasonable estimate of the standard error for the MLE can be given by, where is the Fisher information of the parameter.

In the multivariate case, a test about several parameters at once is carried out using a variance matrix. A common use for this is to carry out a Wald test on a categorical variable by recoding it as several dichotomous variables.

Read more about this topic:  Wald Test

Famous quotes containing the words mathematical and/or details:

    An accurate charting of the American woman’s progress through history might look more like a corkscrew tilted slightly to one side, its loops inching closer to the line of freedom with the passage of time—but like a mathematical curve approaching infinity, never touching its goal. . . . Each time, the spiral turns her back just short of the finish line.
    Susan Faludi (20th century)

    If my sons are to become the kind of men our daughters would be pleased to live among, attention to domestic details is critical. The hostilities that arise over housework...are crushing the daughters of my generation....Change takes time, but men’s continued obliviousness to home responsibilities is causing women everywhere to expire of trivialities.
    Mary Kay Blakely (20th century)