Estimation And Likelihood Ratio Test For The Parameters Of Multiple Normal Populations

Normal distribution is a distribution which is widely applied in statistics. The parameters of normal population about mean and variance are particularly important. It is a wide range of outlook and universal in production and practice. Multiple normal populations are discussed from two aspects.Likelihood ratio tests for parameters of the multiple normal populations are gived;A comparison of method of test by simulating about the mean equality of multiple normal populations is solved.For t normal populations under three assumptions, a likelihood ratio tests about means and variances are discussed in this article. Three kinds of hypothesis are divided into five kinds of cases, testing about the means equality under the unknown or known variances; testing about the variances equality under the unknown or known means; testing about the means and variances equality. Statistics T according to the theorem are gived. Statistics are gradually consistent with the 2x distribution. The simulations by matlab show good performance of the methods in terms of test validity. When the analytical solution of the parameters with the complex equations is not solved, a likelihood ratio test about the unknown equal means and unknown unequal variances is not discussed. A method of the maximum likelihood estimation for the parameters of t normal populations is gived and proved. A likelihood ratio test about the equal means is solved. At the same time other scholars give hypothesis test of the equal means with other methods, comparing with the likelihood ratio test, and the likelihood ratio test is good.Without considering the correlation between the populations, according to the ideas of considering a random variable Y about t populations and n samples, statistics nM , nV , nW are gived under three kinds of hypothesis, using bootstrap p ? value for test, and data simulation. The three kinds of testing method, likelihood ratio test,test of bootstrap p-value and test of bootstrap are summarizes. When sample numbers of t normal populations are different, the problem about test of the mean equality of multiple normal populations is solved by Monte Carlo and comparison.