Coverage Probability of Confidence Intervals for the Normal Mean and Variance with Restricted Parameter Space

Recent articles have addressed the problem to construct the confidence intervals for the mean of a normal distribution where the parameter space is restricted, see for example Wang [Confidence intervals for the mean of a normal distribution with restricted parameter space. Journal of Statistical Computation and Simulation, Vol. 78, No. 9, 2008, 829–841.], we derived, in this paper, analytic expressions of the coverage probability and the expected length of confidence interval for the normal mean when the whole parameter space is bounded. We also construct the confidence interval for the normal variance with restricted parameter for the first time and its coverage probability and expected length are also mathematically derived. As a result, one can use these criteria to assess the confidence interval for the normal mean and variance when the parameter space is restricted without the back up from simulation experiments.

• Sa-aat Niwitpong

• Confidence interval
• coverage probability
• expected length
• restricted parameter space.

[1] BP. Roe, MB. Woodroofe, M.B., 2001, Setting confidence belts, Physical
Review D, 60 (2001) 3009-3015
[2] G. Casella, R. Berger, Statistical Inference (Second Edition), Buxbury,
Los Angeles, 2002.
[3] GJ. Feldman, RD. Cousins, Unified approach to the classical statistical
analysis of small signals, Physical Review D, 57 (1998) 3873-3889.
[4] H. Wang, Confidence intervals for the mean of a normal distribution
with restricted parameter space, Journal of Statistical Computation and
Simulation, 78(9) (2008) 829-841.
[5] Wang, H., Modified P-value of two-sided test for normal distribution
with restricted parameter space, Communication of Statistics Theory and
Method, 35 (2006) 1-14.
[6] M. Mandelkern, Setting confidence intervals for bounded parameters,
Statistical Science, 17 (2002) 149-172.
[7] S. Niwitpong, Sa. Niwitpong, Confidence interval for the difference of
two normal population means with a known ratio of variances, Applied
Mathematical Sciences, 4(8) (2010) 347 - 359.