# Abdollah Safari

### The use of submodels as a basis for ecient estimation of complex models

In this thesis, we consider problems where the true underlying models are complex and obtaining the maximum likelihood estimator (MLE) of the true model is challenging or time-consuming.

In our first paper, we investigate a general class of parameter-driven models for time series of counts. Depending on the distribution of the latent variables, these models can be highly complex. We consider a set of simple models within this class as a basis for estimating the regression coefficients in the more complex models. We conduct a comprehensive simulation study to evaluate the accuracy and efficiency of these estimators. We also derive standard errors (SEs) for the estimators. Our results show that, except in extreme cases, the maximizer of the Poisson generalized linear model (the simplest estimator in our context) is an efficient, consistent, and robust estimator with a well-behaved standard error.

In our second paper, we work in the context of display advertising,where the goal is to estimate the probability of conversion (a pre-defined action such as making a purchase) after a user clicks on an ad. In addition to accuracy, in this context, the speed with which the estimate can be computed is critical. Again, computing the MLEs of the true model for the observed conversion statuses (which depends on the distribution of the delays in observing conversions) is challenging, in this case because of the huge size of the data set. We use a logistic regression model as a basis for estimation, and then adjust this estimate for its bias. We show that our estimation algorithm leads to accurate estimators and requires far less computation time than does the MLE of the true model.

Our third paper also concerns the conversion probability estimation problem in display advertising. We consider a more complicated setting where users may visit an ad multiple times prior to taking the desired action (e.g., making a purchase). We extend the estimator that we developed in our second paper to incorporate information from such visits. We show that this new estimator, the DV-estimator (which accounts for the distributions of both the conversion delay times and the inter-visit times) is more accurate and leads to better confidence intervals than the estimator that accounts only for delay times (the D-estimator). In addition, the time required to compute the DV-estimate for a given data set is only moderately greater than that required to compute the D-estimate – and is substantially less than that required to compute the MLE.

In summary, in a variety of settings, we show that estimators based on simple, misspecified models can lead us to accurate, precise, and computationally efficient estimates of both the key model parameters and their standard deviations.

**Keywords: **Kullback-Leibler information; Bias-adjustment; Parameter-driven models; Time series; Count data; Generalized linear model; Generalized linear mixed model; Hidden Markov model; Display advertising; Conversion probability; Survival times; Censoring.