Date of Award
Master of Science
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 = (θ T 1 , θ T 2 ) 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 θ˜ 1, and use θ˜ 1 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.
This document is organized as follows. Chapter 2 gives an extensive background where topics as linear regression, generalized linear models, profile likelihood, and stochastic convergence are covered. Chapter 3 describes the two proposed methods and shows the derivation of the asymptotic distribution. Several applications are also discussed at the end of this chapter. In chapter 4, simulations to study the empirical distribution, power, and size of the proposed tests will be performed as well as the execution time. Comparison of the 1 proposed methods and ordinary counterparts will be done. In chapter 5, it will be explored the practical use of the proposed method with the use of a real data, one for each of the three models considered in the simulation. Comparison of the performance among the ordinary and proposed tests is made. Finally, in Chapter 6, summary of the procedure followed to derive the proposed tests is made. Advantages and disadvantages of the proposed tests are stated. Conclusions and future work will be discussed.
Received from ProQuest
Denisse Urenda CastaÃ±eda
Urenda Castañeda, Denisse, "A Computationally Efficient Wald Test in M-Estimation" (2022). Open Access Theses & Dissertations. 3632.