The aim of this thesis is to systematically study the statistical guarantee for two representative non-convex optimization problems arsing in the statistics community. The first one is the high-dimensional Gaussian mixture model, which is motivated by the estimation of multiple graphical models arising from heterogeneous observations. The second one is the low-rank tensor estimation model, which is motivated by high-dimensional interaction model. Both optimal statistical rates and numerical comparisons are studied in depth. In the first part of my thesis, we consider joint estimation of multiple graphical models arising from heterogeneous and high-dimensional observations. Unlike most previous approaches which assume that the cluster structure is given in advance, an appealing feature of our method is to learn cluster structure while estimating heterogeneous graphical models. This is achieved via a high dimensional version of Expectation Conditional Maximization (ECM) algorithm. A joint graphical lasso penalty is imposed on the conditional maximization step to extract both homogeneity and heterogeneity components across all clusters. Our algorithm is computationally efficient due to fast sparse learning routines and can be implemented without unsupervised learning knowledge. The superior performance of our method is demonstrated by extensive experiments and its application to a Glioblastoma cancer dataset reveals some new insights in understanding the Glioblastoma cancer. In theory, a non-asymptotic error bound is established for the output directly from our high dimensional ECM algorithm, and it consists of two quantities: statistical error (statistical accuracy) and optimization error (computational complexity). Such a result gives a theoretical guideline in terminating our ECM iterations. In the second part of my thesis, we propose a general framework for sparse and low-rank tensor estimation from cubic sketchings. A two-stage non-convex implementation is developed based on sparse tensor decomposition and thresholded gradient descent, which ensures exact recovery in the noiseless case and stable recovery in the noisy case with high probability. The non-asymptotic analysis sheds light on an interplay between optimization error and statistical error. The proposed procedure is shown to be rate-optimal under certain conditions. As a technical by-product, novel high-order concentration inequalities are derived for studying high-moment sub-Gaussian tensors. An interesting tensor formulation illustrates the potential application to high-order interaction pursuit in high-dimensional linear regression

In this dissertation, we explore the use of three different analytical techniques for approximating the finite-sample properties of estimators and test statistics. These techniques are the saddlepoint approximation, the large-n approximation and the small-disturbance approximation. The first of these enables us to approximate the complete density or distribution function for a statistic of interest, while the other two approximations provide analytical results for the first few moments of the finite-sample distribution. We consider a range of interesting estimation and testing problems that arise in econometrics and empirical economics. Saddlepoint approximations are used to determine the distribution of the half-life estimator that arises in the empirical purchasing power parity literature, and to show that its moments are undefined. They are also applied to the problem of obtaining accurate critical points for the Anderson-Darling goodness-of-fit test. The large-n approximation is used to study the first two moments of the MLE in the binary Logit model. Finally, we use small-disturbance approximations to examine the bias and mean squared error of some commonly used price index numbers, when the latter are viewed as point estimators.

In this dissertation, we explore the use of three different analytical techniques for approximating the finite-sample properties of estimators and test statistics. These techniques are the saddlepoint approximation, the large-n approximation and the small-disturbance approximation. The first of these enables us to approximate the complete density or distribution function for a statistic of interest, while the other two approximations provide analytical results for the first few moments of the finite-sample distribution. We consider a range of interesting estimation and testing problems that arise in econometrics and empirical economics. Saddlepoint approximations are used to determine the distribution of the half-life estimator that arises in the empirical purchasing power parity literature, and to show that its moments are undefined. They are also applied to the problem of obtaining accurate critical points for the Anderson-Darling goodness-of-fit test. The large-n approximation is used to study the first two moments of the MLE in the binary Logit model. Finally, we use small-disturbance approximations to examine the bias and mean squared error of some commonly used price index numbers, when the latter are viewed as point estimators.

Evaluating the statistical properties of Cohen’s κ, this thesis conducts a study of the reliability of patient surveys and medical records for mild cases of Covid-19. Using data collected from Uppsala University Hospital, right-tailed hypothesis testing is performed for the variables Hypertension, Lung disease, Fatigue, Dyspnea and loss of smell and/or taste. Results show only Hypertension rejecting the null hypothesis of low reliability, thus achieving a "Moderate" strength of agreement according to Landis and Koch. Simulating 10000 draws of a multinomial distribution, the thesis finds that the distribution of κ is asymmetric. Simulation of the size-adjusted power of κ shows that the probability of committing Type II errors is significant for the sample sizes of 400 and lower used in the study. In addition, κ appears vulnerable to the effects of sample bias and prevalence and thus risk giving a misleading result. As such,researchers should compare it with the adjusted estimator PABAK if possible.

In dieser Arbeit wird ein neuer Ansatz für die Analyse von Profile Maximierungsschätzern präsentiert. Es werden die Ergebnisse von Spokoiny (2011) verfeinert und angepasst für die Schätzung von Komponenten von endlich dimensionalen Parametern mittels der Maximierung eines Kriteriumfunktionals. Dabei werden Versionen des Wilks Phänomens und der Fisher-Erweiterung für endliche Stichproben hergeleitet und die dafür kritische Relation der Parameterdimension zum Stichprobenumfang gekennzeichnet für den Fall von identisch unabhängig verteilten Beobachtungen und eines hinreichend glatten Funktionals. Die Ergebnisse werden ausgeweitet für die Behandlung von Parametern in unendlich dimensionalen Hilberträumen. Dabei wir die Sieve-Methode von Grenander (1981) verwendet. Der Sieve-Bias wird durch übliche Regularitätsannahmen an den Parameter und das Funktional kontrolliert. Es wird jedoch keine Basis benötigt, die orthogonal in dem vom Model induzierten Skalarprodukt ist. Weitere Hauptresultate sind zwei Konvergenzaussagen für die alternierende Maximisierungsprozedur zur approximation des Profile-Schätzers. Alle Resultate werden anhand der Analyse der Projection Pursuit Prozedur von Friendman (1981) veranschaulicht. Die Verwendung von Daubechies-Wavelets erlaubt es unter natürlichen und üblichen Annahmen alle theoretischen Resultate der Arbeit anzuwenden. / This thesis presents a new approach to analyze profile M-Estimators for finite samples. The results of Spokoiny (2011) are refined and adapted to the estimation of components of a finite dimensional parameter using the maximization of a criterion functional. A finite sample versions of the Wilks phenomenon and Fisher expansion are obtained and the critical ratio of parameter dimension to sample size is derived in the setting of i.i.d. samples and a smooth criterion functional. The results are extended to parameters in infinite dimensional Hilbert spaces using the sieve approach of Grenander (1981). The sieve bias is controlled via common regularity assumptions on the parameter and functional. But our results do not rely on an orthogonal basis in the inner product induced by the model. Furthermore the thesis presents two convergence results for the alternating maximization procedure. All results are exemplified in an application to the Projection Pursuit Procedure of Friendman (1981). Under a set of natural and common assumptions all theoretical results can be applied using Daubechies wavelets.