On the convergence rate of sequential fixed-width confidence intervals for normal parameters

We consider the convergence rate of sequential fixed-width confidence intervals for θ=aμ+bσθ=aμ+bσ under a normal N(μ,σ2) model with μμ and σ2σ2 both unknown. We use the fully sequential procedure proposed by [Takada, Y., 1997. Fixed-width confidence intervals for a function of normal parameters. Sequential Anal. 16, 107–117] to construct sequential confidence intervals for θθ and investigate the convergence rate of the coverage probability as the width of the confidence interval approaches zero. We also derive a second-order asymptotic expansion of the average sample size.