Confidence Intervals for the Reciprocal of a Normal Mean with a Known Coefficient of Variation and a Restricted Parameter Space

The natural parameter space is known to be restricted in many real applications such as engineering, sciences and social sciences. The confidence interval derived from the classical Neyman procedure is unsatisfactory in the case of a restricted parameter space. New confidence intervals for the reciprocal of a normal mean with a known coefficient of variation and a restricted parameter space are proposed in this paper. A simulation study has been conducted to compare the performance of the proposed confidence intervals.


Introduction
The reciprocal of a normal mean, defined by 1 ,    where  is the population mean, is widely used in many areas, such as experimental nuclear physics, biological sciences, agriculture and econometrics. For example, Lamanna et al. (1981) studied charged particle momentum, 1 p    where  is the track curvature of a particle. The inverse of the common mean of structural econometric models was estimated by Zellner (1978). Furthermore, several researchers have studied the reciprocal of a normal mean. For example, Zaman (1981) proposed the estimators without moments in the case of the reciprocal of a normal mean. A class of estimators with a finite moment for the reciprocal of the mean was developed by Srivastava and Bhatnager (1981). Zaman (1985) also proposed the maximum likelihood estimate of the reciprocal of a normal mean with a class of zero-one loss functions. Voinov (1985) suggested the unbiased estimators of power for the reciprocal of the mean and related problems. Recently, Withers and Nadarajah (2013) discussed a theorem to construct the point estimators for the inverse powers of a normal mean. Two confidence intervals for the reciprocal of a normal mean with a known coefficient of variation were proposed by Wongkhao et al. (2013). Their confidence intervals can be applied when the coefficient of variation of a control group is known. The exact method was used to construct one of their confidence intervals from the pivotal statistics , Z where Z follows the standard normal distribution. The other confidence interval is developed based on the generalized confidence interval (Weerahandi, 1993). However, the lower and upper limits of the confidence interval based on the exact method are difficult to compute since they depend on an infinite summation. Panichkitkosolkul (2017) proposed the approximate confidence interval for the reciprocal of a normal population mean with a known coefficient of variation. The approximate confidence interval performs as well as the exact confidence interval in terms of coverage probability. However, the approximate confidence interval is very easy to calculate compared with the exact confidence interval.
Although statistical inference is studied in a natural parameter space, the parameter space is restricted in several real applications, such as engineering, sciences and social sciences. For example, the blood pressure of patients or the weight of a human body are restricted or bounded. Furthermore, Mandelkern (2002) indicated the importance of statistical inference where the parameter space is known to be restricted. Additionally, he gave the example that the classical Neyman procedure is unsatisfactory in the case of a restricted parameter space. The main reason is that the information regarding the restriction is simply ignored. The other related works are Feldman and Cousins (1998) and Roe and Woodroofe (2001). Although research has been done on confidence intervals for the reciprocal of a normal mean, the confidence intervals for the reciprocal of a normal mean with a restricted parameter space have not been the subject of much study. Therefore, it would be of significant interest to develop confidence intervals for the reciprocal of a normal mean that include additional information on the population mean being restricted in order to improve the accuracy of the confidence interval. Motivated by the recent work of Panichkitkosolkul (2017), we propose confidence intervals for the reciprocal of a normal mean with a known coefficient of variation and a restricted parameter space in this paper.
The structure of this paper is as follows: Section 2 reviews two confidence intervals for the reciprocal of a normal mean with a known coefficient of variation. Confidence intervals for the reciprocal of a normal mean with a known coefficient of variation and a restricted parameter space are proposed in Section 3. The performance of all confidence intervals is investigated through a Monte Carlo simulation study in Section 4. We then conclude this paper in Section 5.

Confidence Intervals for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
In this section, we review the confidence intervals for the reciprocal of a normal mean with a known coefficient of variation proposed recently by Wongkhao et al. (2013) and Panichkitkosolkul (2017). The exact and the approximate confidence intervals for the reciprocal of a normal mean with a known coefficient of variation are discussed.
The theorem and corollary proposed by Wongkhao Next, the following theorem suggested by Panichkitkosolkul (2017)

Confidence Intervals for the Reciprocal of a Normal Mean with a Known Coefficient of Variation and a Restricted Parameter Space
Confidence intervals for the mean of a normal distribution with restricted parameter space were derived by Wang (2008). Following the method proposed by Wang (2008), we present confidence intervals for the reciprocal of a normal mean with a known coefficient of variation when the population mean is restricted.
The true value of a parameter of interest is usually unknown. However, parameter space is often known to be restricted and the bounds of parameter space are known. We denote  where L  and U  are the lower and upper limits of the confidence intervals for ,  respectively.
In addition, the exact and approximate confidence intervals for  reviewed in the previous section are used in order to obtain confidence intervals for  when the population mean is restricted.

Simulation Results
The performances of the confidence intervals for the reciprocal of a normal mean with a known coefficient of variation and a restricted parameter space derived in the previous section were investigated through simulation studies in this section. A simulation was conducted using the R statistical software (Ihaka and Gentleman, 1996)   The number of simulation runs was 10,000 and the nominal confidence level 1   was fixed at 0.95.
In the simulation study, the estimated coverage probabilities of the confidence intervals with a restricted population mean are the same as those of the confidence intervals with an unrestricted population mean. Additionally, all confidence intervals have estimated coverage probabilities close to the nominal confidence level in most situations. The estimated coverage probabilities of all the confidence intervals do not increase or decrease according to the values of  and . n The confidence intervals with a restricted population mean have shorter expected lengths than those of the confidence intervals with an unrestricted population mean when the sample sizes are not too large and the values of  are not too small.