In the practice of auditing, a sample of accounts is chosen to verify if the accounts are materially misstated, as opposed to auditing all accounts; it would be too expensive to audit all acounts. This paper seeks to find a method for choosing a sample size of accounts that will give a more accurate estimate than the current methods for sample size determination that are currently being used. A review of methods to determine sample size will be investigated under both the frequentist and Bayesian settings, and then our method using the Zero-Inflated Poisson (ZIP) model will be introduced which explicitly considers zero versus non-zero errors. This model is favorable due to the excess zeros that are present in auditing data which the standard Poisson model does not account for, and this could easily be extended to data similar to accounting populations.

Zero inflation occurs when the proportion of zeros of a model is greater than the proportion of zeros of the corresponding Poisson model. This situation is very common in count data. In order to model zero inflated count time series data, we propose the zero inflated autoregressive conditional Poisson (ZIACP) model by the extending the autoregressive conditional poisson (ACP) model of Ghahramani and Thavaneswaran (2009). The stationarity conditions and the autocorrelation functions of the ZIACP model are provided. Based on the expectation maximization (EM) algorithm an estimation method is developed. A simulation study shows that the estimation method is accurate and reliable as long as the sample size is reasonably high. Three real data examples, syphilis data Yang (2012), arson data Zhu (2012) and polio data Kitromilidou and Fokianos (2015) are studied to compare the performance of the proposed model with other competitive models in the literature. / February 2016

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<p>Syftet med uppsatsen är att försöka förklara vad det är som påverkar tiden som en mamma ammar. För att undersöka vad det är som påverkar tiden som en mamma ammar, har en Zero inflated negative binomial-modell (ZINB-modell) tagits fram. Resultaten visar att det som avgör hur länge en mamma kommer att amma är: Graviditetens längd, mammans ålder, mammans rökvanor under graviditetens sista månader, mammans rökvanor samt mammans nationella ursprung.</p>

Syftet med uppsatsen är att försöka förklara vad det är som påverkar tiden som en mamma ammar. För att undersöka vad det är som påverkar tiden som en mamma ammar, har en Zero inflated negative binomial-modell (ZINB-modell) tagits fram. Resultaten visar att det som avgör hur länge en mamma kommer att amma är: Graviditetens längd, mammans ålder, mammans rökvanor under graviditetens sista månader, mammans rökvanor samt mammans nationella ursprung.

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Doctor of Philosophy / Department of Statistics / Wei-Wen Hsu / Evaluating heterogeneity in the class of zero-inflated models has attracted considerable attention in the literature, where the heterogeneity refers to the instances of zero counts generated from two different sources. The mixture probability or the so-called mixing weight in the zero-inflated model is used to measure the extent of such heterogeneity in the population. Typically, the homogeneity tests are employed to examine the mixing weight at zero. Various testing procedures for homogeneity in zero-inflated models, such as score test and Wald test, have been well discussed and established in the literature. However, it is well known that these classical tests require the correct model specification in order to provide valid statistical inferences. In practice, the testing procedure could be performed under model misspecification, which could result in biased and invalid inferences. There are two common misspecifications in zero-inflated models, which are the incorrect specification of the baseline distribution and the misspecified mean function of the baseline distribution. As an empirical evidence, intensive simulation studies revealed that the empirical sizes of the homogeneity tests for zero-inflated models might behave extremely liberal and unstable under these misspecifications for both cross-sectional and correlated count data. We propose a robust score statistic to evaluate heterogeneity in cross-sectional zero-inflated data. Technically, the test is developed based on the Poisson-Gamma mixture model which provides a more general framework to incorporate various baseline distributions without specifying their associated mean function. The testing procedure is further extended to correlated count data. We develop a robust Wald test statistic for correlated count data with the use of working independence model assumption coupled with a sandwich estimator to adjust for any misspecification of the covariance structure in the data. The empirical performances of the proposed robust score test and Wald test are evaluated in simulation studies. It is worth to mention that the proposed Wald test can be implemented easily with minimal programming efforts in a routine statistical software such as SAS. Dental caries data from the Detroit Dental Health Project (DDHP) and Girl Scout data from Scouting Nutrition and Activity Program (SNAP) are used to illustrate the proposed methodologies.

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Semicontinuous data consist of a combination of a point-mass at zero and a positive skewed distribution. This type of non-negative data distribution is found in data from many fields, but presents unique challenges for analysis. Specifically, these data cannot be analyzed using positive distributions, but distributions that are unbounded are also likely a poor fit. Two-part models incorporate both the zero values from semicontinuous data and the positive continuous values. In this dissertation, we compare zero-inflated gamma (ZIG) and zero-inflated log-normal (ZILN) two-part models. For both of these models, the probability that an outcome is non-zero is modeled via logistic regression. Then the distribution of the non-zero outcomes is modeled via gamma regression with a log-link for ZIG regression and via log-normal regression for ZILN. In this dissertation we propose tests which combine the two parts of the ZIG and ZILN models in meaningful ways for performing a two group comparison. Then we compare these tests in terms of observed Type 1 error rates and power levels under both correctly specified and misspecified ZIG and ZILN models. Tests falling under two main hypotheses are examined. First, we look at two-part tests which come from a two-part hypothesis of no difference between the two groups in terms of the probability of non-zero values and in terms of the the mean of the non-zero values. The second type of tests are mean-based tests. These combine the two parts of the model in ways related to the overall group means of the semicontinuous variable. When not adjusting for covariates, two tests are developed based on a difference of means (DM) and a ratio of means (RM). When adjusting for covariates, tests using mean-based hypotheses are developed which marginalize over the values of the adjusting covariates. Under the adjusting framework, two ratio of means statistics are proposed and examined, an average of the subject specific ratio of means (RMSS) and a ratio of the marginal group means (RMMAR). Simulations are used to compare Type 1 error and power for these tests and standard two group comparison tests. Simulation results show that when ZIG and ZILN models are misspecified and the coefficient of variation (CoV) and/or sample size is large, there are differences in Type 1 error and power results between the misspecified and correctly specified models. Specifically, when ZILN data with high CoV or sample size are analyzed as ZIG, Type 1 error rates are prohibitively high. On the other hand, when ZIG data are analyzed as ZILN under these scenarios, power levels are much lower for ZILN analyses than for ZIG analyses. Examination of Q-Q plots show, however, that in these settings, distinguishing between ZIG and ZILN data can be relatively straightforward. When the coefficient of variation is small it is harder to distinguish between ZIG and ZILN models, but the differences between Type 1 error rates and power levels for misspecified or correctly specified models is also slight. Finally, we use the proposed methods to analyze a data set involving Parkinson's disease (PD) and driving. A number of these methods show that PD subjects exhibit poorer lane keeping ability than control subjects.