The three-level multivariate multilevel model (MVMM) is a multivariate extension of the conventional univariate two-level hierarchical linear model (HLM) and is used for estimating and testing the effects of explanatory variables on a set of correlated continuous outcome measures. Two simulation studies were conducted to investigate the sample size requirements for restricted maximum likelihood (REML) estimation of three-level MVMMs, the effects of sample sizes and other design characteristics on estimation, and the performance of the MVMMs compared to corresponding two-level HLMs. The model for the first study was a random-intercept MVMM, and the model for the second study was a fully-conditional MVMM. Study conditions included number of clusters, cluster size, intraclass correlation coefficient, number of outcomes, and correlations between pairs of outcomes. The accuracy and precision of estimates were assessed with parameter bias, relative parameter bias, relative standard error bias, and 95% confidence interval coverage. Empirical power and type I error rates were also calculated. Implications of the results for applied researchers and suggestions for future methodological studies are discussed. / text

Subsampling is an important method in the analysis of Big Data. Subsample size determination (SSSD) plays a crucial part in extracting information from data and in breaking<br>the challenges resulted from huge data sizes. In this thesis, (1) Sample size determination<br>(SSD) is investigated in multivariate parameters, and sample size formulas are obtained for<br>multivariate normal distribution. (2) Sample size formulas are obtained based on concentration inequalities. (3) Improved bounds for McDiarmid’s inequalities are obtained. (4) The<br>obtained results are applied to nonuniform subsampling in Big Data high dimensional linear<br>regression. (5) Numerical studies are conducted.<br>The sample size formula in univariate normal distribution is a melody in elementary<br>statistics. It appears that its generalization to multivariate normal (or more generally multivariate parameters) hasn’t been caught much attention to the best of our knowledge. In<br>this thesis, we introduce a definition for SSD, and obtain explicit formulas for multivariate<br>normal distribution, in gratifying analogy of the sample size formula in univariate normal.<br>Commonly used concentration inequalities provide exponential rates, and sample sizes<br>based on these inequalities are often loose. Talagrand (1995) provided the missing factor to<br>sharpen these inequalities. We obtained the numeric values of the constants in the missing<br>factor and slightly improved his results. Furthermore, we provided the missing factor in<br>McDiarmid’s inequality. These improved bounds are used to give shrunken sample sizes <br>

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.

In this report we give a brief introduction to the multilevel models, provide a brief summary of the need for using the multilevel model, discuss the assumptions underlying use of multilevel models, and present by means of example the necessary steps involved in model building. This introduction is followed by a discussion of power and sample size determination in multilevel designs. Some formulae are discussed to provide insight into the design aspects that are most influential in terms of power and calculation of standard errors. Finally we conclude by discussing and reviewing the simulation study performed by Maas and Hox (2005) about the influence of different sample sizes at individual as well as group level on the accuracy of the estimates (regression coefficients and variances) and their standard errors. / text

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Em diferentes aplicações de Aprendizado de Máquina podemos estar interessados na minimização do valor esperado de certa função de perda. Para a resolução desse problema, Otimização estocástica e Sample Size Selection têm um papel importante. No presente trabalho se apresentam as análises teóricas de alguns algoritmos destas duas áreas, incluindo algumas variações que consideram redução da variância. Nos exemplos práticos pode-se observar a vantagem do método Stochastic Gradient Descent em relação ao tempo de processamento e memória, mas, considerando precisão da solução obtida juntamente com o custo de minimização, as metodologias de redução da variância obtêm as melhores soluções. Os algoritmos Dynamic Sample Size Gradient e Line Search with variable sample size selection apesar de obter soluções melhores que as de Stochastic Gradient Descent, a desvantagem se encontra no alto custo computacional deles. / In different Machine Learnings applications we can be interest in the minimization of the expected value of some loss function. For the resolution of this problem, Stochastic optimization and Sample size selection has an important role. In the present work, it is shown the theoretical analysis of some algorithms of these two areas, including some variations that considers variance reduction. In the practical examples we can observe the advantage of Stochastic Gradient Descent in relation to the processing time and memory, but considering accuracy of the solution obtained and the cost of minimization, the methodologies of variance reduction has the best solutions. In the algorithms Dynamic Sample Size Gradient and Line Search with variable sample size selection, despite of obtaining better solutions than Stochastic Gradient Descent, the disadvantage lies in their high computational cost.

Em diferentes aplicações de Aprendizado de Máquina podemos estar interessados na minimização do valor esperado de certa função de perda. Para a resolução desse problema, Otimização estocástica e Sample Size Selection têm um papel importante. No presente trabalho se apresentam as análises teóricas de alguns algoritmos destas duas áreas, incluindo algumas variações que consideram redução da variância. Nos exemplos práticos pode-se observar a vantagem do método Stochastic Gradient Descent em relação ao tempo de processamento e memória, mas, considerando precisão da solução obtida juntamente com o custo de minimização, as metodologias de redução da variância obtêm as melhores soluções. Os algoritmos Dynamic Sample Size Gradient e Line Search with variable sample size selection apesar de obter soluções melhores que as de Stochastic Gradient Descent, a desvantagem se encontra no alto custo computacional deles. / In different Machine Learnings applications we can be interest in the minimization of the expected value of some loss function. For the resolution of this problem, Stochastic optimization and Sample size selection has an important role. In the present work, it is shown the theoretical analysis of some algorithms of these two areas, including some variations that considers variance reduction. In the practical examples we can observe the advantage of Stochastic Gradient Descent in relation to the processing time and memory, but considering accuracy of the solution obtained and the cost of minimization, the methodologies of variance reduction has the best solutions. In the algorithms Dynamic Sample Size Gradient and Line Search with variable sample size selection, despite of obtaining better solutions than Stochastic Gradient Descent, the disadvantage lies in their high computational cost.

Over the past decades, bulk metallic glasses (BMGs) have attracted extensive interests because of their unique properties such as good corrosion resistance, large elastic limit, as well as high strength and hardness. However, with the advent of micro-electro-mechanical systems (MEMS) and other microscaled devices, the fundamental properties of micrometer-sized BMGs have become increasingly more important. Thus, in this study, a methodology for performing uniaxial compression tests on BMGs having micron-sized dimensions is presented. Micropillar with diameters of 3.8, 1 and 0.7 £gm are fabricated successfully from the Mg65Cu25Gd10 and Zr63.8Ni16.2Cu15Al5 BMGs using focus ion beam, and then tested in microcompression at room temperature and strain rates from 1 x 10-4 to 1 x 10-2 s-1. Microcompression tests on the Mg- and Zr-based BMG pillar samples have shown an obvious sample size effect, with the yield strength increasing with decreasing sample diameter. The strength increase can be rationalized by the Weibull statistics for brittle materials, and the Weibull moduli of the Mg- and Zr-based BMGs are estimated to be about 35 and 60, respectively. The higher Weibull modulus of the Zr-based BMG is consistent with the more ductile nature of this system. In additions, high temperature microcompression tests are performed to investigate the deformation behavior of micron-sized Au49Ag5.5Pd2.3Cu26.9Si16.3 BMG pillar samples from room to their glass transition temperature (~400 K). For the 1 £gm Au-based BMG pillars, a transition from inhomogeneous flow to homogeneous flow is clearly observed at or near the glass transition temperature. Specifically, the flow transition temperature is about 393 K atthe strain rate of 1 x 10-2 s-1. For the 3.8 £gm Au-based BMG pillars, in order to investigate the homogeneous deformation behavior, microcompression tests are performed at 395.9-401.2 K. The strength is observed to decrease with increasing temperature and decreasing strain rate. Plastic flow behavior can be described by a shear transition zone model. The activation energy and the size of the basic flow unit are deduced and compared favorably with the theory.

This thesis explores the practical use of kernel smoothing in three areas: binary regression, density estimation and Poisson regression sample size calculations. Both nonparametric and semiparametric binary regression estimators are examined in detail, and extended to two bandwidth cases. The asymptotic behaviour of these estimators is presented in a unified way, and the practical performance is assessed using a simulation experiment. It is shown that, when using the ideal bandwidth, the two bandwidth estimators often lead to dramatically improved estimation. These benefits are not reproduced, however, when two general bandwidth selection procedures described briefly in the literature are applied to the estimators in question. Only in certain circumstances does the two bandwidth estimator prove superior to the one bandwidth semiparametric estimator, and a simple rule-of-thumb based on robust scale estimation is suggested. The second part summarises and compares many different approaches to improving upon the standard kernel method for density estimation. These estimators all have asymptotically 'better' behaviour than the standard estimator, but a small-sample simulation experiment is used to examine which, if any, can give important practical benefits. Very simple bandwidth selection rules which rely on robust estimates of scale are then constructed for the most promising estimators. It is shown that a particular multiplicative bias-correcting estimator is in many cases superior to the standard estimator, both asymptotically and in practice using a data-dependent bandwidth. The final part shows how the sample size or power for Poisson regression can be calculated, using knowledge about the distribution of covariates. This knowledge is encapsulated in the moment generating function, and it is demonstrated that, in most circumstances, the use of the empirical moment generating function and related functions is superior to kernel smoothed estimates.