This thesis develops two variant statistical models for count data based upon compound models for contexts when the counts may be viewed as derived from two generations, which may or may not be independent. Unlike standard compound models, the variants model the sum of both generations. We consider cases where both generations are negative binomial or one is Poisson and the other is negative binomial. The first variant, denoted SVA, follows a zero restriction, where a zero in the first generation will automatically be followed by a zero in the second generation. The second variant, denoted SVB, is a convolution model that does not possess this zero restriction. The main properties of the SVA and SVB models are outlined and compared with standard compound models. The results show that the SVA distributions are similar to standard compound distributions for some fixed parameters. Comparisons of SVA, Poisson hurdle, negative binomial hurdle and their zero-inflated counterpart using simulated SVA data indicate that different models can give similar results, as the generating models are not always selected as the best fitting. This thesis focuses on the use of the variant models to model citation counts. We show that the SVA models are more suitable for modelling citation data than other previously used models such as the negative binomial model. Moreover, the application of SVA and SVB models may be used to describe the citation process. This thesis also explores model selection techniques based on log-likelihood methods, Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The suitability of the models is also assessed using two diagrammatic methods, randomised quantile residual plots and Christmas tree plots. The Christmas tree plots clearly illustrate whether the observed data are within fluctuation bounds under the fitted model, but the randomised quantile residual plots utilise the cumulative distribution, and hence are insensitive to individual data values. Both plots show the presence of citation counts that are larger than expected under the fitted model in the data sets.

Resíduos desempenham um papel importante na verificação do ajuste do modelo e na idenfiticação de observações discrepantes e/ou influentes. Neste trabalho, estudamos duas classes de resíduos para os modelos de regressão gama inflacionados no zero (ZAGA) e gaussiana inversa inflacionados no zero (ZAIG). Essas classes de resíduos são uma função de um resíduo para o componente contínuo do modelo e da estimativa de máxima verossimilhança da probabilidade da observação assumir o valor zero. Estudos de simulação de Monte Carlo foram realizados para examinar as propriedades dessas classes de resíduos em ambos os modelos de regressão (ZAGA e ZAIG). Os resultados mostraram que um resíduo de uma dessas classes tem algumas propriedades semelhantes à da distribuição normal padrão nos modelos estudados. Além desse objetivo principal, descrevemos os modelos de regressão ZAGA e ZAIG, estudamos propriedades de alguns resíduos em modelos lineares generalizados com resposta gama e gaussiana inversa e discutimos outros aspectos de análise de diagnóstico nos modelos ZAGA e ZAIG. Para finalizar, foi feita uma aplicação com dados reais de fundos de investimentos, em que ajustamos o modelo ZAIG, para exemplificar os tópicos discutidos e mostrar a importância desses modelos e as vantagens de um dos resíduos estudados na análise de dados reais. / Residuals play an important role in checking model adequacy and in the identification of outliers and influential observations. In this paper, we studied two class of residuals for the zero adjusted gamma regression model (ZAGA) and the zero adjusted inverse Gaussian regression model (ZAIG). These classes of residuals are function of a residual for the continuous component of the model and the maximum likelihood estimate of the probability of the observation assuming the zero value. Monte Carlo simulation studies are performed to examine the properties of this class of residuals in both models (ZAGA and ZAIG). Results showed that a residual of one of these class has some similar properties to the standard normal distribution in the studied models. We also described ZAGA and ZAIG regression models, studied properties of some residuals in generalized linear models with response gamma and inverse Gaussian and discussed other aspects of diagnostic analysis in ZAGA and ZAIG models. To finsih,we presented a real data set application from invesment funds of Brazil. We fitted the ZAIG model to illustrate the topics discussed and showed the importance of these models and the advantages of one of the studied residuals in the analysis of real dataset.

Submitted by Luciana Sebin (lusebin@ufscar.br) on 2016-09-30T12:41:21Z No. of bitstreams: 1 DissJSR.pdf: 1876095 bytes, checksum: c73d62c08322c1ffad6e30271b52d706 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-13T20:40:15Z (GMT) No. of bitstreams: 1 DissJSR.pdf: 1876095 bytes, checksum: c73d62c08322c1ffad6e30271b52d706 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-13T20:40:24Z (GMT) No. of bitstreams: 1 DissJSR.pdf: 1876095 bytes, checksum: c73d62c08322c1ffad6e30271b52d706 (MD5) / Made available in DSpace on 2016-10-13T20:40:33Z (GMT). No. of bitstreams: 1 DissJSR.pdf: 1876095 bytes, checksum: c73d62c08322c1ffad6e30271b52d706 (MD5) Previous issue date: 2016-03-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Residuals play an important role in checking model adequacy and in the identi cation of outliers and in uential observations. In this paper, we studied two class of residuals for the zero adjusted gamma regression model (ZAGA) and the zero adjusted inverse Gaussian regression model (ZAIG). These classes of residuals are function of a residual for the continuous component of the model and the maximum likelihood estimate of the probability of the observation assuming the zero value. Monte Carlo simulation studies are performed to examine the properties of this class of residuals in both models (ZAGA and ZAIG). Results showed that a residual of one of these class has some similar properties to the standard normal distribution in the studied models. We also described ZAGA and ZAIG regression models, studied properties of some residuals in generalized linear models with response gamma and inverse Gaussian and discussed other aspects of diagnostic analysis in ZAGA and ZAIG models. To nsih, we presented a real dataset application from investment funds of Brazil. We tted the ZAIG model to illustrate the topics discussed and showed the importance of these models and the advantages of one of the studied residuals in the analysis of real dataset. / Resíduos desempenham um papel importante na veri cação do ajuste do modelo e na identi cação de observações discrepantes e/ou in uentes. Neste trabalho, estudamos duas classes de resíduos para os modelos de regressão gama in acionados no zero (ZAGA) e gaussiana inversa in acionados no zero (ZAIG). Essas classes de resíduos são uma função de um resíduo para o componente contínuo do modelo e da estimativa de máxima verossimilhança da probabilidade da observação assumir o valor zero. Estudos de simulação de Monte Carlo foram realizados para examinar as propriedades dessas classes de resíduos em ambos os modelos de regressão (ZAGA e ZAIG). Os resultados mostraram que um resíduo de uma dessas classes tem algumas propriedades semelhantes à da distribuição normal padrão nos modelos estudados. Além desse objetivo principal, descrevemos os modelos de regressão ZAGA e ZAIG, estudamos propriedades de alguns resíduos em modelos lineares generalizados com resposta gama e gaussiana inversa e discutimos outros aspectos de análise de diagnóstico nos modelos ZAGA e ZAIG. Para nalizar, foi feita uma aplicação com dados reais de fundos de investimentos, em que ajustamos o modelo ZAIG, para exempli car os tópicos discutidos e mostrar a importância desses modelos e as vantagens de um dos resíduos estudados na análise de dados reais.