The Mahalanobis distance is a fundamental statistic in many fields such as Outlier detection, Normality testing and Cluster analysis. However, the standard estimator developed by Mahalanobis (1936) and Wilks (1963) is not well behaved in cases when the dimension (p) of the parent variable increases proportional to the sample size (n). This case is frequently referred to as Increasing Dimension Asymptotics (IDA). Specifically, the sample covariance matrix on which the Mahalanobis distance depends becomes degenerate under IDA settings, which in turn produce stochastically unstable Mahalanobis distances. This research project consists of several parts. It (a) shows that a previously suggested family of “improved” shrinkage estimators of the covariance matrix produce inoperable Mahalanobis distances, both under classical and increasing dimension asymptotics. It (b) develops a risk function specifically designed to assess the Mahalanobis distance and identifies good estimators thereof and (c) develops a family of resolvent-type estimators of the Mahalanobis distance. This family of estimators is shown to remain well behaved even under IDA settings. Suicient conditions for the proposed estimator to outperform the traditional estimator are also supplied. The proposed estimator is argued to be a useful tool for descriptive statistics, such as Assessment of influential values or Cluster analysis, in cases when the dimension of data is proportional to the sample size.

In this paper, we implement the factor model for deriving the covariance matrix that is used for a Mahalanobis distance. The distributional properties of the new Mahalanobis distances are derived. A general case on contamination effects ofoutliers on Mahalanobis distances from separate parts of the factor model are also investigated. An empirical example indicates the difference between the new proposed separated Mahalanobis distances and the original Mahalanobis distance.

A factor model based covariance matrix is used to build a new form of Mahalanobis distance. The distribution and relative properties of the new Mahalanobis distances are derived. A new type of Mahalanobis distance based on the separated part of the factor model is defined. Contamination effects of outliers detected by the new defined Mahalanobis distances are also investigated. An empirical example indicates that the new proposed separated type of Mahalanobis distances predominate the original sample Mahalanobis distance.

In this paper, we study a Kronecker structured model for covariance matrices when data are matrix-valued. Using the modified Cholesky decomposition for Kronecker structured covariance matrix, we propose a regularized covariance estimator by imposing shrinkage and smoothing penalties on the Cholesky factors. A regularized flip-flop (RFF) algorithm is developed to produce a statistically efficient estimator for a large covariance matrix of matrix-valued data. Asymptotic properties are investigated and the performance of the estimator is evaluated by simulations. The results presented are applied to real data example.

Statistical analysis frequently involves methods for reducing high-dimensional data to new variates of lower dimension for the purpose of assessing distributional properties, identification of hidden patterns, for discriminant analysis, etc. In classical multivariate analysis such matters are usually analysed by either using principal components (PC) or the Mahalanobis distance (MD). While the distributional properties of PC’s are fairly well established in high-dimensional cases, no explicit results appear to be available for the MD under such cases. The purpose of this chapter is to bridge that gap by deriving weak limits for the MD in cases where the dimension of the random vector of interest is proportional to the sample size (n, p-asymptotics). The limiting distributions allow for normality-based inference in cases when the traditional low-dimensional approximations do not apply.

This paper derives first-order sampling moments of individual Mahalanobis distances (MD) in cases when the dimension p of the variable is proportional to the sample size n. Asymptotic expected values when n, p → ∞ are derived under the assumption p/n → c, 0 ⩽ c < 1. It is shown that some types of standard estimators remain unbiased in this case, while others are asymptotically biased, a property that appears to be unnoticed in the literature. Second order moments are also supplied to give some additional insight to the matter.

In this paper, we investigate the asymptotic distributions of two types of Mahalanobis distance (MD): leave-one-out MD and classical MD with both Gaussian- and non-Gaussian-distributed complex random vectors, when the sample size n and the dimension of variables p increase under a fixed ratio c=p/n→∞. We investigate the distributional properties of complex MD when the random samples are independent, but not necessarily identically distributed. Some results regarding the F-matrix F=S−12S1—the product of a sample covariance matrix S1 (from the independent variable array (be(Zi)1×n) with the inverse of another covariance matrix S2 (from the independent variable array (Zj≠i)p×n)—are used to develop the asymptotic distributions of MDs. We generalize the F-matrix results so that the independence between the two components S1 and S2 of the F-matrix is not required.

Estimating inverse covariance matrix is an essential part of many statistical methods. This paper proposes a regularized estimator for the inverse covariance matrix. Modified Cholesky decomposition (MCD) is utilized to construct positive definite estimators. Instead of directly regularizing the inverse covariance matrix itself, we impose regularization on the Cholesky factor. The estimated inverse covariance matrix is used to build Mahalanobis distance (MD). The proposed method is evaluated by detecting outliers through simulations and empirical studies.

The thesis consists of three empirical essays on the topics of self-employment, happiness and international trade.

Essay 1 studies how immigrant self-employment entry is affected by the local business cycle in Sweden. Using the unemployment rate at the local labour market level as a proxy for the local business cycle, our study shows that the self-employment entry behaviour for native men and immigrant men is negatively affected by the unemployment rate, except for immigrants from Middle East. However, such a negative effect is quantitatively weaker among the non-European immigrants. Further, the result shows that immigrants from the Middle East are positively affected by the unemployment rate, meaning they are more likely to be pushed into self-employment in recessions. For women, we also find the unemployment rate has a negative impact on the self-employment decision of native women and immigrant women, except for the Middle East group. However, compared with men, the quantitative size of the unemployment rate effect on self-employment is smaller among women, implying the less important role of business cycle in determining females’ entry into self-employment.

Essay 2 investigates the non-pecuniary return of self-employment in China. The results show that the life satisfaction of self-employed men is significantly higher than that of wage-employed men; the life satisfaction of self-employed women is not statistically significant different from that of wage-employed women. Moreover, we show that the life satisfaction of self-employed men in the informal sector is significantly higher than that of wage-employed men in the formal sector. The life satisfaction of wage-employed men in the informal sector is not significantly different from that of wage-employed men in the formal sector. For women, we find that there is no significant life satisfaction disparity between workers in the formal and informal sector. Finally, our job satisfaction data also concludes that self-employment in China is not inferior to wage employment.

Essay 3 evaluates how Swedish manufacturing employment is affected by the increasing import competition from China. The results show that the growth of manufacturing employment is not statistically significant affected by the increasing import competition from China. Moreover, in general, the increasing import exposure from China does not significantly affect the employment growth of non-manufacturing sector either. Regarding the earnings, the analysis shows that the low wage earners in the manufacturing sector is not significantly affected by the increasing import penetration from China while median and high wage earners are positively affected.

Objectives :

In the 2006 yearly report from the Swedish National Register for Lumbar Spine Surgery it was claimed that international studies show obvious differences between private and non-private patients with regard to results from back surgery. Therefore our aim was to reveal such possible differences by comparing the two categories of patients at a private clinic.

Material and Methods :

The material comprises 1184 patients operated on for lumbar disc herniation during the period of 1987 to 2007. Basic pre-operative data were obtained from the medical records and follow-up was performed by a questionnaire around 5 years post-operatively.

Results :

Small but statistically significant differences between private and non-private patients were seen pre-operatively regarding the proportions of a/ men and women in the samples, b/ those with physically demanding jobs, c/ those on sick leave and d/ those with lumbar pain. Over the years the admitted private patients had a decreasing mean duration of symptoms which was not seen in the non-private patients. No apparent differences (n.s.) were seen between the two categories of patients pre-operatively regarding age, presence and level of leg pain or the proportion who smoked. Post-operative improvement in leg and lumbar pain was very similar in private and non-private patients as was satisfaction with the results and the proportion of patients returning to work.

Conclusion :

Despite small pre-operative differences concerning some variables and a significant difference in symptom duration between private and non-private disc herniation patients, the final clinical results were very similar.

Estimation of mean parameters in the growth curve model, when the covariance matrix has a block circular Toeplitz structure, is considered. The purpose of this article is to find the appropriate design matrices so that explicit mean estimators can be obtained.

Under a model having a Kronecker product covariance structure with compound symmetry or circular symmetry, two-sample hypothesis testing for the equality of two correlation parameters is considered. Different tests are proposed by using the ratio of independent F distributions. Several tests are compared with the proposed ones and practical recommendations are made based on their type I error probabilities and powers. Finally, all mentioned tests are applied to a real data example.

Under a model having a Kronecker product covariance structure with compound symmetry or circular symmetry, two-sample hypothesis testing for the equality of two correlation parameters is considered. Different tests are proposed by using the ratio of independent F distributions. Several tests are compared with the proposed ones and practical recommendationsare made based on their type I error probabilities and powers. Finally, all mentioned tests areapplied to a real data example.

The use of contaminated biomass and waste fuels is essential for waste management, waste to energy (WtE) and mitigating carbon emissions. The contamination of heavy metals and metalloids is specially concerned by environmental regulation and waste to energy processes. In this study, comparative characterisation is performed for three typical contaminated biomass and waste fuels. i.e. recycled woods, combustible municipal solid waste, and industrial and commercial wastes. The contamination characteristics are further analysed using statistical methods (e.g. significance, correlation, profile, and principal component analyses) to identify specific contamination features, relations among the contaminants and potential contamination sources. Contamination trend is estimated based on the continuously monitoring fuel qualities, the driving forces for regulating and reduction of the contaminations, and potential changes in major contamination sources. The comparative characterisation combined with statistical analyses provides a better way to understand the contamination mechanisms. The approach can also relate the fuel contamination with the contamination sources and their changes for trend estimation. Generally, the toxic heavy metals and metalloids are expected to be significantly reduced due to stricter regulations, but there is no general trend for the reduction of other metals and metalloids because of the complicated changes in contamination sources and waste recycling streams in the near future. (C) 2019 Elsevier B.V. All rights reserved.