Approximation Results Of Gamma Function And Its Application In Statistical Calculation

As a special function,Gamma function is widely used in many fields.It originated from the study of the factorial function,which is obtained by the interpolation method by extending the factorial function from the integer set to the real number set.Gamma function is used in the approximation of Hypergeometric function and Mellin-Barnes integral,and it is also an important tool for constructing some density functions of distributions.In the Sequential Probability Ratio Test,the approximation of Gamma function is applied to Confluent Hypergeometric function to invert the expression,and the effective termination time and number of samples are obtained.Obviously,the approximation of Gamma function has important theoretical and practical significance.In this thesis,we obtain different principal approximation formulas of gamma function with fast convergence effect from different angles,then we use different approximation tools to improve the approximation accuracy.The statistical problems related to the approximation of Gamma function are also studied.We first make a brief background on Gamma function,as well as some theoretical knowledge that will be used during the research of this thesis.In chapter 3,we get the approximation formula of Gamma function from three aspects of Completely monotonic function,Burnside formula and Digamma function,and give the numerical calculation to show that our results are better.In chapter 4,we use the chi-square distribution median calculation from the perspective of statistical calculation to illustrate the necessity of improving the accuracy of the approximate function.At the same time,we propose an approximate formula with parameters and obtain a recursive algorithm for Gamma function,which can improve the accuracy of Gamma approximation function with small variables.In the last chapter,we consider the hypothesis testing problem.By using the classical asymptotic expansion of Gamma function,we obtain the conclusion that the log-likelihood ratio test statistic has an asymptotic normal distribution under certain assumptions,and prove that our results are valid by numerical simulation.