No abstract

Problem statement: In the testing statistic problem for the mean vector of independent and identically distributed multivariate normal random vectors with unknown covariance matrix when the data has sample size less than the dimension n≤p, for example, the data came from DNA microarrays where a large number of gene expression levels are measured on relatively few subjects, the p×p sample covariance matrix S does not have an inverse.. Hence any statistic value involving inversion of S does not exist. Approach: In this study, we showed a version of some modification on S, S+cI and find a real smallest value c≠0 which makes (S+cI)^-1 exist. Results: The result from study provided when the dimension p tends to infinity and smallest change in S' the (S+cI)^-1 do exist when c = 1. Conclusion: In statistical analysis involving with high dimensional data that an inversion of sample covariance matrix do not exist, one way to modify a sample covariance matrix S to have an inverse is to consider a sample covariance matrix, S, as the form S+cI and we recommend to choose c = 1

<p><em>Modern measurement technology has enabled the capture of high-dimensional data by researchers and statisticians and classical statistical inferences, such as </em><em>the renowned Hotelling’s T² test, are no longer valid when the dimension of the data equals or exceeds the sample size. Importantly, when correlations among variables in a dataset exist, taking them into account in the analysis method would provide more accurate conclusions. In this article, we consider the hypothesis testing problem for two mean vectors in high-dimensional data with an underlying normality assumption. A new test is proposed based on the idea of keeping more information from the sample covariances. The asymptotic null distribution of the test statistic is derived. The simulation results show that the proposed test performs well comparing with other competing tests and becomes more powerful when the dimension increases for a given sample size. The proposed test is also illustrated with an analysis of DNA microarray data. </em></p>

In this paper, we proposed a new testing statistic for testing the equality of mean vectors from two multivariate normal populations when the covariance matrices are unknown and unequal in high-dimensional data. A new test is proposed based on the idea of keeping more information from the sample covariance matrices as much as possible. A proposed test is invariant under scalar transformations and location shifts. We showed that the asymptotic distribution of proposed statistic is standard normal distribution when number of random variables approach infinity. We also compared the performance of the proposed test with other three existing tests by the simulation study. The simulation results showed that the attained significance level of proposed test close to setting nominal significance level satisfactorily. The attained power of proposed test outperforms as the other comparative tests under form of covariance matrices considered which can be arranged to block diagonal matrix structure. The attained power becomes more powerful when the dimension increases for a given sample size or vice versa, or relationship level between random variables in each sample increases. Finally, the proposed test is also illustrated with an analysis of DNA microarray data.

Two upper bounds for ruin probability under the discrete time risk model for insurance controlled by two factors: proportional reinsurance and surplus investment are presented. The latter is of interest because of the assumption that insurers invest some or their entire financial surplus on both the stock and bond markets, for which bond interest rates follow a timehomogeneous Markov chain. In addition, the control of reinsurance and stock investment in each time period are assumed to be constant values. The first upper bound for finite time ruin probability and ultimate ruin probability was derived under the condition that the Lundberg coefficient exists. The second upper bound is for finite time ruin probability and was developed from a new worse than used function. Numerical examples are used to illustrate these results, and the upper bound of ruin probability using real-life motor insurance claims data from a broker is also presented.