# A Computationally Efficient Wald Test in M-Estimation

#### Abstract

Under the maximum likelihood framework, three asymptotic overall tests have been well developed in generalized linear models (GLM) for testing the single null hypothesis H0 : θ = θ0, namely, the Wald test, Likelihood Ratio Test (LRT) and Score test also known as the Lagrange Multiplier test (LM). Modified versions of Wald, LR and LM tests can also be found for testing the significance of a portion of the parameter θ, i.e., if θ = (θT1, θT2 )T it is of interest to test H0 : θ2 = 0. However, with the constant increase of dimensionality in data, the three tests becomes unfeasible to compute. The computational cost one has to pay seems to be unrealistic and difficult or even untractable. The approach taken in this document to deal with this issue follows the profile likelihood framework which consists of partitioning the p-dimensional parameter vector θ into two parameter vectors θ1 and θ2 of dimension q and p − q, respectively, estimate θ1 under H0, say θ1, and use θ1 to estimate θ2. With this approach, one could reduce considerably the execution time when estimating a big number of parameters in the model without losing the asymptotic properties and the power of the traditional tests. Also, one could test the null hypothesis even if the dimension of θ is moderately bigger than the sample size n as long as both q and p − q are smaller than n.

#### Subject Area

Statistics

#### Recommended Citation

Urenda Castañeda, Denisse, "A Computationally Efficient Wald Test in M-Estimation" (2022). *ETD Collection for University of Texas, El Paso*. AAI29324236.

https://scholarworks.utep.edu/dissertations/AAI29324236