The Discussion And Numerical Simulation About The Best Confidence Region Of Pivot Quantity Which Follows Single Peak Distribution

Pivot quantity method is a common method of seeking the confidence interval of unknown parameters.The usual distribution of pivot including normal distribution, chi-square distribution, t distribution, F distribution, gamma distribution, lognormal distribution and so on is single peak distribution.This paper discusses estimation problems about the best confidence region of the pivot with common single peak distribution in the case of one-dimensional and two-dimensional. The "best" confidence region here means the shortest interval lengths(one-dimensional) or the smallest areas of confidence region(two-dimensional) under the same confidence degree, and then the relationship of best confidence region between the pivot quantity and the corresponding parameters is discussed.The paper also shows numerical simulation in Excel 2010.The main contents are as follows:1.This paper proves that the shortest confidence interval of the pivot with single peak density function is to keep the confidence interval endpoint density function values “equal”,rather than the confidence interval by traditional way which makes the areas on both sides equal.2.The paper conducts a numerical calculations in Excel and under a certain confidence degree compares the confidence interval lengths in two different situations for the pivot quantity that follows normal distribution, t distribution, chi-square distribution, F distribution, gamma distribution and lognormal distribution. The results confirm the above analysis conclusions.3.This paper also discusses the relationship of best confidence region between the pivot and the corresponding parameters contained in the pivot.It proves that if there is a linear relationship between pivot quantity and the corresponding parameters, then we can get the shortest confidence interval of corresponding parameters by calculating the shortest confidence interval of the pivot quantity.4.The conclusion about shortest confidence interval of pivot which follows single peak distribution in the one-dimensional case is extended to the confidence region of pivot quantity which follows single peak surface in the two-dimensional case.Make an example of a binary standard normal distribution, the numerical simulation and dynamic presentation are done in Excel.It is not hard to get a conclusion that the smallest confidence area of pivot quantity is to keep the surface function values equal in the same confidence degree.