We introduce a new class of measures for testing independence between two random vectors, which uses expected difference of conditional and marginal characteristic functions. By choosing a particular weight function in the class, we propose a new index for measuring independence and study its property. Two empirical versions are developed, their properties, asymptotics, connection with existing measures and applications are discussed. Implementation and Monte Carlo results are also presented. We propose a two-stage sufficient variable selections method based on the new index to deal with large p small n data. The method does not require model specification and especially focuses on categorical response. Our approach always improves other typical screening approaches which only use marginal relation. Numerical studies are provided to demonstrate the advantages of the method. We introduce a novel approach to sufficient dimension reduction problems using the new measure. The proposed method requires very mild conditions on the predictors, estimates the central subspace effectively and is especially useful when response is categorical. It keeps the model-free advantage without estimating link function. Under regularity conditions, root-n consistency and asymptotic normality are established. The proposed method is very competitive and robust comparing to existing dimension reduction methods through simulations results.

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This thesis aims to familiarize the reader with the topic of life insurance, especially insurance taken out for more people. Its focus is empirical rather as combining the use of several actuarial and statistical methods through three statistical software and applications created in MS Excel using VBA. First, thesis is focused to multiple life insurance with intention to show reader the diversity of this interesting but rarely offered product in business world, especially through the application created for the purpose of this work. Sense of mentioned application lies in the fact it is able to calculate amount of the net premium for a user-selected type of insurance. Based on this outcome insurer can determine the net price corresponding to the chosen risk coverage (regardless of the amount of the costs associated with this type of insurance and required profitability). Age of insured person isn't usually part of calculation of premium for accident insurance. One of the aims of this study is to assess whether this parameter actually has or doesn't have any effect on the price of insurance. With help of SPSS and RStudio software relationship between age of insured person and amount of indemnification from daily compensation insurance has been researched. Based on the test of variable independence were detected dependencies of these variables where all of which used measure of association suggests a weak dependence amount of indemnification on the age of the insured. It surely worth insurer should consider different insurance rates for various ages of insured person. Following by examination whether the rate used to calculate premiums in case of hospitalization extra insurance corresponds to the actual risk coverage. It has been calculated the rate could be lower than currently used due to the loss experience of insurance for the past 9 years and the assumption of certain expenses and desired profit. It would be cheaper for insurance group contracts than it is today, when the insurer has done so. It could also attract new clientele. The last stated goal in this thesis is to create prognosis of the amount of indemnity paid for injuries covered by additional daily compensation insurance. Using the software EViews is analyzed this trend in the past 32 months first. Based on its outcome there is a future level forecast created for the next four years. These resulting values indicate the insurance indemnity for group contract will grow. Therefore, based on this analysis insurer should to conclude in the contract it would be appropriate to increase the premium of the insurance coverage. With these practical tasks the reader can get idea not only of problems solved in the scope of multiple life insurance and group insurance but also to become familiar with some statistical software used for analyzes carried out not only in the insurance industry. The conclusions of this thesis can also be beneficial for insurer who may adjust premium prices of analyzed insurances.