Research For The Shortest Length Of C.I. About Parameters From Some Distributions

The confidence interval (C.I.) of unknown parameter represents the range of the value and the reliability about estimation of parameter. Under given confidence level, the length of C.I. represents the precision of the estimation. It is very necessary to research the problem how to construct good random variables to make the length of C.I. as short as possible. In practical applications, people generally get C.I. by probability symmetry. But the length of this kind of C.I. is not the shortest.Firstly, the author gets the general C.I. of parameters by approximate Likelihood Ratiotests. Then, the author proves the condition that the shortest C.I. about 2 and a of NormalDistribution should satisfy by derivation, and gives the shortest C.I. about 2 under theconfidence level 0.90 , 0.95 , 0.99. The difference between the two kinds of length of the C.I., computed by the general method and the method in this paper, is compared and analyzed. After that , the author changes the question of C.I. for the parameter of four kinds of distribution into the question of nonlinear programming , and proves the condition that the C.I. about parameters of the four distributions should satisfy and gives the shortest C.I.about cr2 under the confidence level 0.90, 0.95, 0.99. Finally, the author gives theconditions that the shortest C.I. about and of two Normal Distribution and ofGamma distribution. When sample size is not large, conclusion shows that the precision of the interval estimation of parameter is remarkably increasing if the data from Tables in this paper are used.