Small Dispersion Asymptotics in Stratified Models

Xuan Mei

doi:https://doi.org/10.21985/N2VH73

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Small Dispersion Asymptotics in Stratified Models

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This dissertation studies the small dispersion asymptotics in highly stratified models. My goal is to show that accurate inferences are possible even if s, the number of strata, is large while m, the number of observations within each stratum, is small, provided that the model ”fit well” in the term that φ, the inverse of dispersion, is large. And second, the use of conditions also depends on the features of the exact model one is facing. To achieve this goal, I will propose the “effective sample size” concept and (φ×m)×s asymptotics which aims to combine both m × s asymptotics and small dispersion(small sigma) asymptotics. On the other hand, if set φ ≡ 1, this dissertation also works as a rigorous derivation of asymptotic properties of highly stratified models. In Chapter 1, I will introduce the importance of stratified and highly stratified models, and give some examples of them. Chapter 2 is a basic literature review section covering exponential dispersion model, two-index asymptotics and small dispersion. Then in Chapter 3 I will explain the “effective sample size” φ×m and summarize the problems need to solve in a formal way. In Chapter 4, for exponential dispersion model with a canonical link function I will derive the consistency and asymptotic normality under the maximum likelihood approach, with rigorous proofs. And I will use normal distribution as a special case to verify my conclusions. I will extend my results to non-canonical link function scenario in Chapter 5 and also point out extra difficulties. In Chapter 6 I will discuss the modified profile likelihood approach which can improve the estimation. Briefly, the requirement on the effective sample size (φ×m) can be lowered to ( . I’ll also derive the modified likelihood function for canonical link function. Multiple Monte Carlo simulations are given in Chapter 7 to verify my conclusions for both maximum likelihood and modified profile likelihood approaches.

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