The generalized linear model (GLM) is a popular model in many research areas. In the GLM, each outcome of the dependent variable is assumed to be generated from a particular distribution function in the exponential family. The mean of the distribution depends on the independent variables. The link function provides the relationship between the linear predictor and the mean of the distribution function. In this dissertation, two semiparametric extensions of the GLM will be developed. In the first part of this dissertation, we have proposed a new model, called a semiparametric generalized linear model with a log-concave random component (SGLM-L). In this model, the estimate of the distribution of the random component has a nonparametric form while the estimate of the systematic part has a parametric form. In the second part of this dissertation, we have proposed a model, called a generalized semiparametric single-index mixed model (GSSIMM). A nonparametric component with a single index is incorporated into the mean function in the generalized linear mixed model (GLMM) assuming that the random component is following a parametric distribution. In the first part of this dissertation, since most of the literature on the GLM deals with the parametric random component, we relax the parametric distribution assumption for the random component of the GLM and impose a log-concave constraint on the distribution. An iterative numerical algorithm for computing the estimators in the SGLM-L is developed. We construct a log-likelihood ratio test for inference. In the second part of this dissertation, we use a single index model to generalize the GLMM to have a linear combination of covariates enter the model via a nonparametric mean function, because the linear model in the GLMM is not complex enough to capture the underlying relationship between the response and its associated covariates. The marginal likelihood is approximated using the Laplace method. A penalized quasi-likelihood approach is proposed to estimate the nonparametric function and parameters including single-index coe±cients in the GSSIMM. We estimate variance components using marginal quasi-likelihood. Asymptotic properties of the estimators are developed using a similar idea by Yu (2008). A simulation example is carried out to compare the performance of the GSSIMM with that of the GLMM. We demonstrate the advantage of my approach using a study of the association between daily air pollutants and daily mortality adjusted for temperature and wind speed in various counties of North Carolina. / Ph. D.

Bakgrund: Aktieindex används världen över som placeringsalternativ, jämförelsemått inom portföljförvaltning och som underlag för portföljoptimering. Forskare har under senare tid ifrågasatt index viktade efter börsvärdet och alternativa viktningsmetoder för index har framtagits som substitut till det kapitalviktade indexet och prestationsjämförelser har gjorts. Studier har främst gjorts i USA och denna studie ämnar göra en liknande undersökning på den svenska marknaden.  Syfte: Syftet med vår studie är att undersöka alternativa viktningsmetoder på det svenska aktieindexet OMXS30 och dess historiska prestation under åren 1995-2011 i förhållande till det traditionellt kapitalviktade OMXS30. Syftet är vidare att analysera de alternativt viktade indexen som grund för portföljoptimering enligt Single-Index Model.  Genomförande: Fem alternativt viktade index konstrueras i studien där viktningen grundas på fundamentala värden, Sharpekvoter, standardavvikelse, likaviktning och handelsvolym och jämförs prestationsmässigt mot OMXS30. Indexen används sedan vid portföljoptimering enligt SIM där aktiers och portföljers karakteristika analyseras. Indexens prognostisering av betavärden utvärderas i studien för att urskilja om något index är mer träffsäkert gällande aktiens beta för nästkommande period.  Slutsats: Ett flertal av de konstruerade alternativa indexen genererar signifikant högre avkastning till lägre risk i den nedgångsperiod som analyserats varför dessa kan ses som en mer lönsam investering. Tendenser till högre avkastning för den totala perioden finns även om signifikanta skillnader inte föreligger. De alternativa indexen har föranlett skilda allokeringsbeslut vid portföljoptimeringen vilket har gett stora utslag i portföljernas förväntade prestation såväl som faktisk prestation efter optimeringen genomförts. / Background: Stock market indexes are widely used as investment strategies and as a benchmark when portfolios are being constructed and evaluated. Researchers have recently questioned the capital weighted index in favor of other available weighting schemes. By comparing alternative weighted indexes to the traditionally capital weighted index one has been made aware of the significantly lower risk adjusted performance for the capital weighted index.  Aim: Our aim is to investigate in alternative weighting schemes used on the Swedish index OMXS30 and evaluate the historical performance of these alternative indexes in comparison to the traditionally capital weighted index during the period 1995-2011. Furthermore, our objective is to analyze the effects alternative weighting schemes have on portfolio optimization through Single-Index Model.  Completion: To fulfill the purpose of this thesis, five alternative weighting schemes have been applied on the Swedish index OMXS30. The weights have been calculated on fundamental measures, Sharpe ratios, standard deviation, equally weighted and trade volume and they have been compared to the traditionally cap-weighted index. Furthermore, the constructed indexes will be used to optimize portfolios with Single Index Model to compare the portfolios characteristics when different indexes have been used.  Results/Findings: The majority of the alternative weighted indexes generate significantly higher returns in one of our analyzed periods and this was during a market recession. For the total analyzed period no statistical differences among the indexes could be determined even though differences in total return are made clear. The indexes had a big effect on the portfolio optimization in terms of different share allocation.

This thesis examines the impact of company announcements on the daily stock returns of lowly capitalised companies. A total of 105 companies comprise the sample and 1464 events are examined over the period 21110/97 to 03/0412000. The methodology employed is primarily, empirical in nature. Event studies are conducted to gauge the impact of company announcements on stock returns using the single index market model (SIMM) as the chosen equilibrium market model for modelling abnormal returns. The study professes three mam contributions to knowledge. The empirical evidence suggests that financial announcement have a more timely impact on stock returns than non-financial announcements. Secondly, there appears to be significant over-reaction and mean-reversion exhibited by lowly capitalised firms. Thirdly, the speed of adjustment of stock prices to new information is increased in cases where shareholder concentration is high while over-reactions appear inversely proportionate to shareholder concentration. This may be a consequence of smaller firms experiencing leakage of boardroom level information prior to public announcement days.

We provide a Frequentist-Bayesian hybrid test statistic in this dissertation for two testing problems. The first one is to design a test for the significant differences between non-parametric functions and the second one is to design a test allowing any departure of predictors of high dimensional X from constant. The implementation is also given in construction of the proposal test statistics for both problems. For the first testing problem, we consider the statistical difference among massive outcomes or signals to be of interest in many diverse fields including neurophysiology, imaging, engineering, and other related fields. However, such data often have nonlinear system, including to row/column patterns, having non-normal distribution, and other hard-to-identifying internal relationship, which lead to difficulties in testing the significance in difference between them for both unknown relationship and high-dimensionality. In this dissertation, we propose an Adaptive Bayes Sum Test capable of testing the significance between two nonlinear system basing on universal non-parametric mathematical decomposition/smoothing components. Our approach is developed from adapting the Bayes sum test statistic by Hart (2009). Any internal pattern is treated through Fourier transformation. Resampling techniques are applied to construct the empirical distribution of test statistic to reduce the effect of non-normal distribution. A simulation study suggests our approach performs better than the alternative method, the Adaptive Neyman Test by Fan and Lin (1998). The usefulness of our approach is demonstrated with an application in the identification of electronic chips as well as an application to test the change of pattern of precipitations. For the second testing problem, currently numerous statistical methods have been developed for analyzing high-dimensional data. These methods mainly focus on variable selection approach, but are limited for purpose of testing with high-dimensional data, and often are required to have explicit derivative likelihood functions. In this dissertation, we propose ``Hybrid Omnibus Test'' for high-dimensional data testing purpose with much less requirements. Our Hybrid Omnibus Test is developed under semi-parametric framework where likelihood function is no longer necessary. Our Hybrid Omnibus Test is a version of Freqentist-Bayesian hybrid score-type test for a functional generalized partial linear single index model, which has link being functional of predictors through a generalized partially linear single index. We propose an efficient score based on estimating equation to the mathematical difficulty in likelihood derivation and construct our Hybrid Omnibus Test. We compare our approach with a empirical likelihood ratio test and Bayesian inference based on Bayes factor using simulation study in terms of false positive rate and true positive rate. Our simulation results suggest that our approach outperforms in terms of false positive rate, true positive rate, and computation cost in high-dimensional case and low-dimensional case. The advantage of our approach is also demonstrated by published biological results with application to a genetic pathway data of type II diabetes. / Ph. D.

Dans cette thèse, nous nous proposons d'étudier quelques paramètres fonctionnels lorsque les données sont générées à partir d'un modèle de régression à indice simple. Nous étudions deux paramètres fonctionnels. Dans un premier temps nous supposons que la variable explicative est à valeurs dans un espace de Hilbert (dimension infinie) et nous considérons l'estimation de la densité conditionnelle par la méthode de noyau. Nous traitons les propriétés asymptotiques de cet estimateur dans les deux cas indépendant et dépendant. Pour le cas où les observations sont indépendantes identiquement distribuées (i.i.d.), nous obtenons la convergence ponctuelle et uniforme presque complète avec vitesse de l'estimateur construit. Comme application nous discutons l'impact de ce résultat en prévision non paramétrique fonctionnelle à partir de l'estimation de mode conditionnelle. La dépendance est modélisée via la corrélation quasi-associée. Dans ce contexte nous établissons la convergence presque complète ainsi que la normalité asymptotique de l'estimateur à noyau de la densité condtionnelle convenablement normalisée. Nous donnons de manière explicite la variance asymptotique. Notons que toutes ces propriétés asymptotiques ont été obtenues sous des conditions standard et elles mettent en évidence le phénomène de concentration de la mesure de probabilité de la variable fonctionnelle sur des petites boules. Dans un second temps, nous supposons que la variable explicative est vectorielle et nous nous intéressons à un modèle de prévision assez général qui est la régression robuste. A partir d'observations quasi-associées, on construit un estimateur à noyau pour ce paramètre fonctionnel. Comme résultat asymptotique on établit la vitesse de convergence presque complète uniforme de l'estimateur construit. Nous insistons sur le fait que les deux modèles étudiés dans cette thèse pourraient être utilisés pour l'estimation de l'indice simple lorsque ce dernier est inconnu, en utilisant la méthode d'M-estimation ou la méthode de pseudo-maximum de vraisemblance, qui est un cas particulier de la première méthode. / In this thesis, we propose to study some functional parameters when the data are generated from a model of regression to a single index. We study two functional parameters. Firstly, we suppose that the explanatory variable take its values in Hilbert space (infinite dimensional space) and we consider the estimate of the conditional density by the kernel method. We establish some asymptotic properties of this estimator in both independent and dependent cases. For the case where the observations are independent identically distributed (i.i.d.), we obtain the pointwise and uniform almost complete convergence with rateof the estimator. As an application we discuss the impact of this result in fuctional nonparametric prevision for the estimation of the conditional mode. In the dependent case we modelize the later via the quasi-associated correlation. Note that all these asymptotic properties are obtained under standard conditions and they highlight the phenomenon of concentration properties on small balls probability measure of the functional variable. Secondly we suppose that the explanatory variable takes values in the _nite dimensional space and we interest in a rather general prevision model whichis the robust regression. From the quasi-associated data, we build a kernel estimator for this functional parameter. As an asymptotic result we establish the uniform almost complete convergence rate of the estimator. We point out by the fact that these two models studied in this thesis could be used for the estimation of the single index of the model when the latter is unknown, by using the method of M-estimation or the pseudo-maximum likelihood method which is a particular case of the first method.

本研究主要是探討台灣股票市場 79 至 81 年各分類指數的報酬率，以比 較各產業間報酬的多寡與風險的大小。並介紹投資組合的觀念以降低非系 統風險，而系統風險的部分可由單一指數模式估計。再以主成分分析法， 找出影響八個分類指數報酬率的主要成分，得到了第一個主成分做單一指 數模式的迴歸估計，以比較與加權指數為因變數的估計結果。最後探討這 三年來分類指數的報酬率，是否具有隨機性以及分類指數的行為。

In an experiment to understand colon carcinogenesis, all animals were exposed to a carcinogen while half the animals were also exposed to radiation. Spatially, we measured the existence of aberrant crypt foci (ACF), namely morphologically changed colonic crypts that are known to be precursors of colon cancer development. The biological question of interest is whether the locations of these ACFs are spatially correlated: if so, this indicates that damage to the colon due to carcinogens and radiation is localized. Statistically, the data take the form of binary outcomes (corresponding to the existence of an ACF) on a regular grid. We develop score??type methods based upon the Matern and conditionally autoregression (CAR) correlation models to test for the spatial correlation in such data, while allowing for nonstationarity. Because of a technical peculiarity of the score??type test, we also develop robust versions of the method. The methods are compared to a generalization of Moran??s test for continuous outcomes, and are shown via simulation to have the potential for increased power. When applied to our data, the methods indicate the existence of spatial correlation, and hence indicate localization of damage. Assuming that there are correlations in the locations of the ACF, the questions are how great are these correlations, and whether the correlation structures di?er when an animal is exposed to radiation. To understand the extent of the correlation, we cast the problem as a spatial binary regression, where binary responses arise from an underlying Gaussian latent process. We model these marginal probabilities of ACF semiparametrically, using ?xed-knot penalized regression splines and single-index models. We ?t the models using pairwise pseudolikelihood methods. Assuming that the underlying latent process is strongly mixing, known to be the case for many Gaussian processes, we prove asymptotic normality of the methods. The penalized regression splines have penalty parameters that must converge to zero asymptotically: we derive rates for these parameters that do and do not lead to an asymptotic bias, and we derive the optimal rate of convergence for them. Finally, we apply the methods to the data from our experiment.

In an experiment to understand colon carcinogenesis, all animals were exposed to a carcinogen while half the animals were also exposed to radiation. Spatially, we measured the existence of aberrant crypt foci (ACF), namely morphologically changed colonic crypts that are known to be precursors of colon cancer development. The biological question of interest is whether the locations of these ACFs are spatially correlated: if so, this indicates that damage to the colon due to carcinogens and radiation is localized. Statistically, the data take the form of binary outcomes (corresponding to the existence of an ACF) on a regular grid. We develop score??type methods based upon the Matern and conditionally autoregression (CAR) correlation models to test for the spatial correlation in such data, while allowing for nonstationarity. Because of a technical peculiarity of the score??type test, we also develop robust versions of the method. The methods are compared to a generalization of Moran??s test for continuous outcomes, and are shown via simulation to have the potential for increased power. When applied to our data, the methods indicate the existence of spatial correlation, and hence indicate localization of damage. Assuming that there are correlations in the locations of the ACF, the questions are how great are these correlations, and whether the correlation structures di?er when an animal is exposed to radiation. To understand the extent of the correlation, we cast the problem as a spatial binary regression, where binary responses arise from an underlying Gaussian latent process. We model these marginal probabilities of ACF semiparametrically, using ?xed-knot penalized regression splines and single-index models. We ?t the models using pairwise pseudolikelihood methods. Assuming that the underlying latent process is strongly mixing, known to be the case for many Gaussian processes, we prove asymptotic normality of the methods. The penalized regression splines have penalty parameters that must converge to zero asymptotically: we derive rates for these parameters that do and do not lead to an asymptotic bias, and we derive the optimal rate of convergence for them. Finally, we apply the methods to the data from our experiment.

Semiparametric modeling is a hybrid of the parametric and nonparametric modelings where some function forms are known and others are unknown. In this dissertation, we have made several contributions to semiparametric modeling based on the single index model related to the following three topics: the first is to propose a model for detecting change points simultaneously with estimating the unknown function; the second is to develop two models for spatially correlated data; and the third is to further develop two models for spatially-temporally correlated data. To address the first topic, we propose a unified approach in its ability to simultaneously estimate the nonlinear relationship and change points. We propose a single index change point model as our unified approach by adjusting for several other covariates. We nonparametrically estimate the unknown function using kernel smoothing and also provide a permutation based testing procedure to detect multiple change points. We show the asymptotic properties of the permutation testing based procedure. The advantage of our approach is demonstrated using the mortality data of Seoul, Korea from January, 2000 to December, 2007. On the second topic, we propose two semiparametric single index models for spatially correlated data. One additively separates the nonparametric function and spatially correlated random effects, while the other does not separate the nonparametric function and spatially correlated random effects. We estimate these two models using two algorithms based on Markov Chain Expectation Maximization algorithm. Our approaches are compared using simulations, suggesting that the semiparametric single index nonadditive model provides more accurate estimates of spatial correlation. The advantage of our approach is demonstrated using the mortality data of six cities, Korea from January, 2000 to December, 2007. The third topic involves proposing two semiparametric single index models for spatially and temporally correlated data. Our first model has the nonparametric function which can separate from spatially and temporally correlated random effects. We refer it to "semiparametric spatio-temporal separable single index model (SSTS-SIM)", while the second model does not separate the nonparametric function from spatially correlated random effects but separates the time random effects. We refer our second model to "semiparametric nonseparable single index model (SSTN-SIM)". Two algorithms based on Markov Chain Expectation Maximization algorithm are introduced to simultaneously estimate parameters, spatial effects, and times effects. The proposed models are then applied to the mortality data of six major cities in Korea. Our results suggest that SSTN-SIM is more flexible than SSTS-SIM because it can estimate various nonparametric functions while SSTS-SIM enforces the similar nonparametric curves. SSTN-SIM also provides better estimation and prediction. / Ph. D.

影響股票價格之因素，基本上可分為三大類，分別是1.市場因素、2.行業因素、3.公 司因素。本研究主要在了解台灣地區股票投資報酬率與行業因素之關系。 理論基礎部份，介紹單一指數模式(single-indel model)與多重指數模式(multi-in- dex model)，以了解兩模式之主要內容以及有關研究行業因素過程中應注意的問題。 文獻探討部份在說明以往國內外學者之研究大部份證實了行業因素之存在，並敘述將 行業因素引入多重指數模式中，以了解是否能降低殘差相關，提高對股價變動之解釋 能力的有關研究之研究方法與結果。以往國內有關行業因素之研究皆按現行證券市場 之分類標準予以分類，本研究則以陳發輝依六變數( 包括流動比率、負債比率、 EPS 、資產週轉率、資本額、稅後盈餘成長率、交易週轉率) 所區分出之五種類型( 包括 成長績優型、高度投機型、穩健成長型、保守停滯型、穩定獲利型 )來將研究資料中 上市公司予以分類。 研究設計在說明研究的設計過程，包括證券種類的選擇、研究期間、選樣標準、變數 之操作性定義、資料來源、研究步驟等。 實證結果與分析部份對實證研究的統計結果加以解釋並對本研究實證部分的研究限制 加以說明。