Abstract
This paper considers sequentially two main problems. First, we estimate both the mean and the variance of the normal distribution under a unified one decision framework using Hall’s three-stage procedure. We consider a minimum risk point estimation problem for the variance considering a squared-error loss function with linear sampling cost. Then we construct a confidence interval for the mean with a preassigned width and coverage probability. Second, as an application, we develop Fortran codes that tackle both the point estimation and confidence interval problems for the inverse coefficient of variation using a Monte Carlo simulation. The simulation results show negative regret in the estimation of the inverse coefficient of variation, which indicates that the three-stage procedure provides better estimation than the optimal.
Highlights
Given pre-defined α, 0 < α < 1 and d(> 0), where (1 − α) is the confidence coefficient and 2d is the fixed-width of the interval, we want to construct a fixed-width (= 2d) confidence interval for the mean μ whose confidence coefficient is at least the nominal value 100(1 − α)%, where at the same time, we will be able to use the same available data to estimate the population variance σ2 under squared-error loss function with linear sampling cost
A series of 50,000 replications were generated from a normal distribution with different values of μ and σ2
Regarding the estimated asymptotic regret, ωwe obtain negative regret, which agrees with the result of Theorem 5
Summary
Let {Xi , i ≥ 1} be a sequence of independent and identically distributed IID random variables from a normal distribution with mean μ ∈ < and variance σ2 ∈ 0), where (1 − α) is the confidence coefficient and 2d is the fixed-width of the interval, we want to construct a fixed-width (= 2d) confidence interval for the mean μ whose confidence coefficient is at least the nominal value 100(1 − α)%, where at the same time, we will be able to use the same available data to estimate the population variance σ2 under squared-error loss function with linear sampling cost We combine both optimal sample sizes in one decision rule to propose the three-stage sampling decision framework. For more details about Equation (1), see Mukhopadhyay and de Silva ([1]; chapter 6, p. 97)
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