We investigate the problem of minimizing a nonconvex function with respect to convex constraints, and we study different techniques to compute a lower bound on the optimal value: The method of using convex envelope functions on one hand, and the method of exploiting nonconvex duality on the other hand. We investigate which technique gives the better bound and develop conditions under which the dual bound is strictly better than the convex envelope bound. As a byproduct, we derive some interesting results on nonconvex duality. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Recent studies have found that large numbers of consumers innovate. In our study, we provide a re-estimation of the figures provided in the extant literature. We do so by conducting a study in which we apply two different methods of data collection: (1) telephone interviews, the method considered most valid in previous research, and (2) personal interviews, which involve much higher effort but induce better individual recollection. Using telephone interviews, we measured a user-innovator frequency of 10.8% in our sample. In stark contrast, personal follow-up interviews resulted in a frequency of 39.7%, indicating a considerable underestimation in extant research. We then used the correction factor generated to re-estimate findings on user innovation frequency in Finland, Japan, Korea, Sweden, the UK, and the USA. It appears that user innovation is indeed a mass phenomenon that should not be overlooked by policymakers or firms. (authors' abstract)

This thesis investigates a technique to estimate the risk of the mean-variance (MV) portfolio optimization problem. We call this technique the Scaling technique. It provides a better estimator of the risk of the MV optimal portfolio. We obtain this result for a general estimator of the covariance matrix of the returns which includes the correlated sampling case as well as the independent sampling case and the exponentially weighted moving average case. This gave rise to the paper, [CMcS]. Our result concerning the Scaling technique relies on the moments of the inverse of compound Wishart matrices. This is an open problem in the theory of random matrices. We actually tackle a much more general setup, where we consider any random matrix provided that its distribution has an appropriate invariance property (orthogonal or unitary) under an appropriate action (by conjugation, or by a left-right action). Our approach is based on Weingarten calculus. As an interesting byproduct of our study - and as a preliminary to the solution of our problem of computing the moments of the inverse of a compound Wishart random matrix, we obtain explicit moment formulas for the pseudo-inverse of Ginibre random matrices. These results are also given in the paper, [CMS]. Using the moments of the inverse of compound Wishart matrices, we obtain asymptotically unbiased estimators of the risk and the weights of the MV portfolio. Finally, we have some numerical results which are part of our future work.

This thesis investigates a technique to estimate the risk of the mean-variance (MV) portfolio optimization problem. We call this technique the Scaling technique. It provides a better estimator of the risk of the MV optimal portfolio. We obtain this result for a general estimator of the covariance matrix of the returns which includes the correlated sampling case as well as the independent sampling case and the exponentially weighted moving average case. This gave rise to the paper, [CMcS]. Our result concerning the Scaling technique relies on the moments of the inverse of compound Wishart matrices. This is an open problem in the theory of random matrices. We actually tackle a much more general setup, where we consider any random matrix provided that its distribution has an appropriate invariance property (orthogonal or unitary) under an appropriate action (by conjugation, or by a left-right action). Our approach is based on Weingarten calculus. As an interesting byproduct of our study - and as a preliminary to the solution of our problem of computing the moments of the inverse of a compound Wishart random matrix, we obtain explicit moment formulas for the pseudo-inverse of Ginibre random matrices. These results are also given in the paper, [CMS]. Using the moments of the inverse of compound Wishart matrices, we obtain asymptotically unbiased estimators of the risk and the weights of the MV portfolio. Finally, we have some numerical results which are part of our future work.

Objective: The purpose of this study was to determinate the rate of underestimation of core biopsy (CB) of nonpalpable breast lesions, under image guidance, with validation at surgical excision histologic examination at Instituto Nacional de Câncer (INCA). Materials e methods: We retrospectively reviewed 352 CB that were submmited to surgery, from February 2000 to December 2005, and which histopathologic findings were at INCA database system. CB results were compared to surgical findings and underestimation rate was determined by dividing the number of lesions that proved to be carcinoma at surgical excision by the total number of high risk lesions and intracuctal carcinoma evaluated with excisional biopsy. Clinical, imaging, core biopsy and pathologic features were analyzed to identify factors that affect the rate of underestimation. Results: All patients were female, which mean age was 56,1 years old (26-86). Mass lesion was the most frequent finding (71,3%), being 69,9% less than 20mm, as well as BI-RADS® 4 (71,0%). The main guidance was stereotactic (57,1%), all using 14-gauge needles. The mean number of core samples was eight (4-22), being 99,7% at least five samples. The patients tolerated CB in 99,1% of cases, but bleeding occurred in 6,6%. The inconclusive CB findings occurred in 15,6%, (55/352). It was benign in 26,4%, high risk lesion in 12,8% and malignant in 45,2%. The segmentectomia was the more frequent surgery (70,2%), being benign in 26,7%, high risk in 18,2% and malign in 55,1%. There was agreement between CB and surgery in 82,1% (Kappa = 0,75). False-negative rate was 5,4% and the lesion was completely removed in 3,4%. Underestimation rate was 9,1%, and was associated with BI-RADS® 5 (p = 0,049), microcalcifications (p < 0,001) and stereotactic guidance (p = 0,002). All underestimated cases were less than 20 mm of diameter and there were at least five fragments. Underestimation rate of high risk lesions was 31,1% and there was no significant associations. Atypical ductal hyperplasia underestimation rate was 41,2% and there was not associations. Papillary lesions underestimation was 31,2% and was associated with stereotactic guidance (p = 0,036). Phyllodes tumor underestimation was 16,7% but it was not possible to make associations. There was one lobular neoplasia case that was concordant to surgery. Ductal carcinoma in situ underestimation was 41,9% and there was not significant associations. Conclusions: The core breast biopsy under image guidance is a reliable procedure but it remains the recommendation of surgical excision for high risk lesions detected at CB as well as it was not possible to assess clinical, imaging, core biopsy and pathologic features that could predict underestimation and avoid excision. Representative samples are much more important than number of fragments. / Objetivo: Determinar o grau de subestimação de core biopsy (CB), guiada por imagem, de lesões impalpáveis da mama e que foram subsequentemente submetidas à exérese cirúrgica no Hospital do Câncer III/ Instituto Nacional de Câncer (INCA). Materiais e métodos: Foram revisadas retrospectivamente 352 CB que foram submetidas à cirurgia entre fevereiro de 2000 e dezembro de 2005, e cujo laudo histopatológico estava registrado no sistema interno de informação do INCA. Os resultados da CB foram comparados com os da cirurgia e a taxa de subestimação foi calculada dividindo o número de carcinoma in situ e/ou invasivo à cirurgia pelo número de lesões de alto risco ou carcinoma in situ que foram submetidas à cirurgia. Foram analisadas características clínicas, imaginológicas, da CB e patológicas que poderiam influir na subestimação. Resultados: Todas as pacientes eram do sexo feminino, com média de idade de 56,1 anos (26-86). O nódulo foi o tipo de lesão mais frequente (71,3%) com 69,9% menor do que 20 mm, bem como BI-RADS® categoria 4 (71,0%). O tipo de guia mais utilizado foi a estereotaxia (57,1%), todos utilizando agulhas 14-gauge. O número médio de fragmentos foi de oito (4-22), com 99,7% apresentando pelo menos cinco fragmentos. O procedimento foi bem tolerado em 99,1% dos casos, mas ocorreu hematoma em 6,6%. A CB foi inconclusiva em 15,6%. O laudo histopatológico da CB foi benigno em 26,4%, lesão de alto risco em 12,8% e maligno em 45,2%. A segmentectomia foi a cirurgia mais frequente (70,2%), com laudo benigno em 26,7%, lesão de alto risco em 18,2% e maligno em 55,1%. A concordância entre a CB e a cirurgia foi de 82,1% (Kappa = 0,75). Falso-negativo foi de 5,4% e a lesão foi completamente removida em 3,4%. A taxa de subestimação foi de 9,1% e esteve associada com BI-RADS® categorias 5 (p = 0,049), microcalcificações (p < 0,001) e estereotaxia (p = 0,002). Todos os casos subestimados possuíam menos de 20 mm e em todos foram retirados pelo menos cinco fragmentos. A taxa de subestimação para lesões de alto risco foi de 31,1% e não apresentou variáveis associadas à subestimação. Já a taxa de subestimação de hiperplasia ductal atípica foi de 41,2% e também não houve associações. A subestimação de lesões papilíferas foi 31,2% e apresentou associação com estereotaxia (p = 0,036). Tumor filóides foi subestimado em 16,7%, mas não foi possível estabelecer associações. Houve apenas um caso de neoplasia lobular à CB que foi concordante com a cirurgia. A subestimação de carcinoma ductal in situ foi 41,9% e também não apresentou associações relevantes. Conclusões: CB guiada por imagem é um procedimento confiável, contudo permanece a recomendação de ressecção cirúrgica de lesões de alto risco detectadas à CB. Além disso, não foi possível estabelecer características clínicas, imaginológicas, da CB e patológicas que pudessem predizer subestimação e evitar a cirurgia. Amostras representativas da lesão são mais importantes que o número de fragmentos.

This thesis investigates a technique to estimate the risk of the mean-variance (MV) portfolio optimization problem. We call this technique the Scaling technique. It provides a better estimator of the risk of the MV optimal portfolio. We obtain this result for a general estimator of the covariance matrix of the returns which includes the correlated sampling case as well as the independent sampling case and the exponentially weighted moving average case. This gave rise to the paper, [CMcS]. Our result concerning the Scaling technique relies on the moments of the inverse of compound Wishart matrices. This is an open problem in the theory of random matrices. We actually tackle a much more general setup, where we consider any random matrix provided that its distribution has an appropriate invariance property (orthogonal or unitary) under an appropriate action (by conjugation, or by a left-right action). Our approach is based on Weingarten calculus. As an interesting byproduct of our study - and as a preliminary to the solution of our problem of computing the moments of the inverse of a compound Wishart random matrix, we obtain explicit moment formulas for the pseudo-inverse of Ginibre random matrices. These results are also given in the paper, [CMS]. Using the moments of the inverse of compound Wishart matrices, we obtain asymptotically unbiased estimators of the risk and the weights of the MV portfolio. Finally, we have some numerical results which are part of our future work.