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Despite a well-designed and controlled study, missing values are consistently present inresearch. It is well established that when disregarding missingness by analyzing completecases only, statistical power is reduced and parameter estimates are biased. The existing traditional methods of imputing missing data are incapable of accounting for misleading representation of data. Research shows that these traditional methods like single imputation, often underestimate the variance. This problem can be bypassed by imputing a missing value multiple times and taking the uncertainty of imputing correctly into consideration. In this thesis a simulation study is conducted to compare two different multiple imputation models. A comparison between a defined linear stochastic regression model and a non defined flexible neural network model, where the validation MSE loss is used to account for variance in the imputed values, is done. In total there are three simulated data sets sampled from a multiple bivariate linear regression model where som of the values in Y2 are MAR given the Y1 variable. When applying a neural network on the datasets with 25, 50 and 75 percent missing values a total of 30 times and the result from the regression analysis on the complete data is pooled, the results show that almost all confidence intervals of the intercept are covering the expected value. The only exception was in the case of 75 percent missingness. When applying Multiple imputation by chained equations on the data sets, the true intercept is covered by all confidence intervals. When 25 percent of the data is missing, both models yield unbiased results.

Random matrix theory has found many applications in various fields such as physics, statistics, number theory and so on. One important approach of studying random matrices is based on their spectral properties. In this thesis, we investigate the limiting behaviors of condition numbers of suitable random matrices in terms of large deviations. The thesis is divided into two parts. Part I, provides to the readers an short introduction on the theory of large deviations, some spectral properties of random matrices, and a summary of the results we derived, and in Part II, two papers are appended. In the first paper, we study the limiting behaviors of the 2-norm condition number of p x n random matrix in terms of large deviations for large n and p being fixed or p = p(n) → ∞ with p(n) = o(n). The entries of the random matrix are assumed to be i.i.d. whose distribution is quite general (namely sub- Gaussian distribution). When the entries are i.i.d. normal random variables, we even obtain an application in statistical inference. The second paper deals with the β-Laguerre (or Wishart) ensembles with a general parameter β > 0. There are three special cases β = 1, β = 2 and β = 4 which are called, separately, as real, complex and quaternion Wishart matrices. In the paper, large deviations of the condition number are achieved as n → ∞, while p is either fixed or p = p(n) → ∞ with p(n) = o(n/ln(n)).

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The birth of the first child is a special event for a mother whose life can change dramatically. In Ethiopia women’s timing to enter motherhood vary between the regions. This paper is therefore focusing on how birth cohort, education and residence affect the rate of entering motherhood for Ethiopian women in the different regions and the entire country. The dataset is extracted from the 2016 Ethiopia Demographic and Health Survey (EDHS) and contains 15,019 women from 487 different households. For more accurate estimations and results, the correlation within households is taken into consideration with multilevel survival analysis. The methods used are the Cox proportional hazard model and two frailty models. The results of the paper show that women residing in rural areas have an increased rate of entering motherhood compared to those residing in urban areas, every age group older than those born 1997 to 2001 have a higher intensity to enter parenthood and those with education have a decreased intensity ratio compared to the women with no education. It also shows that there is a regional difference in the effect of the estimated ratios of the covariates. Performing the multilevel analysis only changes the estimated effects of the covariates in the cities and one region. It is concluded that the estimated intensity ratio of multilevel survival analysis only varies from the standard Cox regression when the region is heterogeneous.

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