A confidence interval for the population mean of a one-parameter exponential distribution based on the Wilson-Hilferty transformation

Abstract

In this paper, an approximate confidence interval (CI) is proposed for the population mean of a one-parameter exponential distribution. The Wilson-Hilferty approximation is used to transform the exponential random variable to a normal random variable. The efficiency of this proposed confidence interval is evaluated using an extensive Monte-Carlo simulation study. Through this method, the coverage probabilities and average widths of the proposed CI are compared with those of the other two commonly existing CIs, namely, the exact and asymptotic confidence intervals. The simulation results show that the proposed confidence interval performs well in terms of coverage probability and average width. Additionally, the average width of the proposed confidence interval is lower than that of the asymptotic confidence interval for a small sample size and all levels of the parameter ( $\theta$ ). Furthermore, the three confidence interval estimations get systematically closer to the nominal level for all levels of the sample size and parameter. In addition, the efficiencies of the three confidence interval estimations seem to have no difference for a large sample size and all levels of the parameter. Real-life data was used for illustration and performing a comparison that support the findings obtained from the simulation study.

Keywords

Confidence interval exponential distribution Wilson-Hilferty transformation normal distribution coverage probability average width

1. Introduction

One-parameter exponential distribution is a continuous distribution approach that widely used in statistical applications in various fields. It has several important statistical properties, and exhibits great mathematical tractability (Balakrishnan & Basu, 1995). Therefore, we often perform one-parameter exponential distribution for various applications such as survival analysis, reliability analysis, extreme values analysis, and others (Lupas, 2017). Furthermore, this distribution is also involved in the queuing theory, which is applied to many situations, including bank teller queues, airline check-in lines, and supermarket checkout progression (Bhat, 2008; Ibe, 2009; Forbes et al., 2011; Haviv, 2013). The use of one-parameter exponential distribution in such situations can yield precise and accurate results based on the parameter-estimation method used. In this study, the confidence interval estimations for an unknown parameter $\theta$ that provides a range of values with a known probability of capturing the true value of the parameter $\theta$ are investigated. Therefore, a confidence interval (CI) can be defined as a range of values that provides the user with an understanding of how precise the estimates of a parameter are (Abu-Shawiesh et al., 2019).

Neyman (1937) developed the estimation theory for constructing the confidence interval (CI) of a parameter using the pivotal quantities approach, which is well known as an exact confidence interval as the mentions of Hogg and Tanis (2001), and Casella and Berger (2002). The estimation approach using pivotal quantities provides valid results for any sample size $n$ (Balakrishnan et al., 2014; Cho et al., 2015. In the case of a large sample size $n$ , the asymptotic confidence interval is extensively used to construct a sequence of estimators $\widehat{\theta}_{n}$ for an unknown parameter $\theta$ with a probability density function $f(.^{\prime}\theta)$ that is asymptotically normally distributed with mean $\theta$ and variance $\sigma^{2}_{n}(\theta)$ (Mood et al., 1974; Mukhopadhyay, 2000; Abu-Shawiesh, 2010).

In this study, an approximate confidence interval (CI) is proposed for the population mean ( $\theta$ ) of the one-parameter exponential distribution. This confidence interval is derived based on the Wilson and Hilferty (WH) approximation (1931) for transforming an exponential random variable to a normal random variable. The efficiency of this proposed confidence interval is evaluated using an extensive Monte-Carlo simulation study. Through this method, the coverage probabilities and average widths of the proposed CI are compared with those of the other two commonly existing CIs, namely, the exact and asymptotic confidence intervals. Furthermore, the three confidence interval estimation methods are illustrated using real-life data to support the findings obtained from the simulation study.

This paper is organized as follows: In Section 2, the Wilson-Helferty (WH) transformation for normal approximation is presented. In Section 3, the materials and methods used in the study are discussed. In Section 4, two important and useful confidence intervals for the population mean ( $\theta$ ) of the one-parameter exponential distribution are reviewed, and the proposed confidence interval is presented. To compare the performance of the three confidence interval methods, a Monte-Carlo simulation study is conducted in Section 5. Two real-life data are analyzed to illustrate the implementation of several methods in Section 6. Finally, some concluding remarks are presented in Section 7.

2. The Wilson-Hilferty transformation

The Wilson-Hilferty (WH) transformation is a simple normal approximation for a chi-square distribution, which is a special form of the gamma distribution with the shape parameter $\alpha={v}/{2}$ and scale parameter $\theta=$ 2, where $v$ is the degrees of freedom. By considering $(\chi^{2}/v)^{\frac{1}{3}}$ to be the approximate normal distribution with a mean of 1–2/(9 $v$ ) and a standard deviation of $\sqrt{2/(9v)}$ , Wilson and Hilfrty (1931) derived their normal approximation value (Zar, 1978). For this approximation, the cube root of a gamma random variable was considered to be approximately normally distributed (Krishnamoorthy et al., 2008). Specifically, for this approximation, the Wilson-Hilferty (WH) transformation states that if $X$ is a random variable follows a two-parameter gamma distribution with shape parameter $\alpha$ and scale parameter $\theta$ , i.e., $X\sim G(\alpha,\theta)$ , then the distribution of the random variable $X^{1/3}$ can be approximated using a normal distribution with mean ( $\mu_{1/3}$ ) and variance ( $\sigma^{2}_{1/3}$ ), i.e., $X^{1/3}\sim N(\mu_{1/3},\sigma^{2}_{1/3}$ ) where $\mu=\mu_{1/3}=E(X^{1/3})=\frac{\theta^{1/3}\Gamma(\alpha+\frac{1}{3})}{\Gamma(% \alpha)}$ and $\sigma^{2}=\sigma^{2}_{1/3}=\textit{Var}(X^{1/3})=\frac{\theta^{2/3}\Gamma(% \alpha+\frac{2}{3})}{\Gamma(\alpha)}-{\mu}^{2}_{1/3}$ . For a justification for the choice of 1/3, see Hernandez and Johnson (1980), Section 3.2.

Therefore, if $X_{1},X_{2},\ldots,X_{n}$ be a random sample of size $n$ in the $G(\alpha,\theta)$ distribution, then the transformed sample $X^{*}_{1}=X^{1/3}_{1},X^{*}_{2}=X^{1/3}_{2},\ldots,X^{*}_{n}=X^{1/3}_{n}$ is a random sample from a normal distribution. In particular, the upper and lower confidence limits derived by using the Wilson-Hilferty (WH) transformation provide approximate confidence limits and are expected to provide satisfactory coverage probabilities. In addition to the simplicity advantage of this approximation, i.e., it does not depend on any difficult sample statistics that would makes the calculation of the confidence interval limits difficult, one can also compute the confidence interval limits by using tabulated values given in this paper. In his work, Zar (1978) compared several normal approximations at different values of significance level and concluded that the Wilson-Hilferty (WH) transformation for normal approximation is fairly reliable for most of the considered significance levels.

3. Materials and methods

In this section, the criteria for efficiency of the considered confidence intervals and the essential conditions for conducting study are discussed. Also, the pivotal quantity that will be used in this study to construct the ( $1-\alpha$ ) 100% proposed confidence interval (CI) for the population mean ( $\theta$ ) of the one-parameter exponential distribution is derived.

3.1 Criteria for the efficiency comparison

The efficiency comparison criteria for the three estimation methods-exact, asymptotic and proposed confidence intervals–are the coverage probability (CP) and the average width or expected length (AW) of the confidence intervals. It is acknowledged that the CP and AW are useful factors for evaluating confidence intervals. Let $\text{CI}=(L(\bm{X}),U(\bm{X}))$ be a confidence interval of a parameter $\theta$ based on data $\bm{X}$ , where $L(\bm{X})$ and $U(\bm{X})$ , respectively, are the lower and upper limits of this confidence interval. The following are the definitions for the efficiency comparison used in this study:

Definition 1. The coverage probability (CP) associated with a confidence interval $\text{CI}=(L(\bm{X}),U(\bm{X}))$ for an unknown parameter $\theta$ of a probability density function $f(x;\theta)$ is measured by $P_{\theta}\{\theta\in L(\bm{X}),U(\bm{X}))\}$ (see Mukhopadhyay, 2000).

Definition 2. The length of a confidence interval, $W=U(\bm{X})-L(\bm{X})$ , is the difference between the upper $U(\bm{X})$ and lower $L(\bm{X})$ limits of a confidence interval $\text{CI}=(L(\bm{X}),U(\bm{X}))$ . The expected length of a confidence interval $\text{CI}=(L(\bm{X}),U(\bm{X}))$ is given by $E_{\theta}(W)$ (see Barker, 2002; Swift, 2009; Patil & Kulkarni, 2012).

3.2 Essential conditions for the study

Throughout the following discussion, some essential conditions for this study are denoted by (C1)–(C3), which are as follows:

(C1)
Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample of size $n$ from a population of the one-parameter exponential distribution with scale parameter $\theta\in\Omega$ , where $\Omega=\{\theta:0<\theta<\infty\}$ . The probability density function ( $p d f$ ) of a one-parameter exponential random variable $X$ , $f(x;\theta)$ , is given by Eq. (1).

$\displaystyle f(x;\theta)=\left\{\begin{array}[]{ll}\frac{1}{\theta}e^{-\frac{% x}{\theta}},&x>0,\theta>0\\ 0,&\textit{otherwise}\end{array}\right.$ (1)

For $X\sim\textit{Exp}(\theta)$ , we have $\mu=E(X)=\theta,\sigma^{2}=\textit{Var}(X)={\theta}^{2}$ and $\sigma=\textit{SD}(X)=\theta$ . The theoretical coefficient of variation $\gamma=\textit{CV(X)}=\frac{\sigma}{\mu}=\frac{\theta}{\theta}=$ 1. This is a useful indicator, especially if the observation data has a coefficient of variation that close to 1, sampled from the one-parameter exponential distribution.
(C2)
Let $\chi^{2}_{(\frac{\alpha}{2},2n)}$ and $\chi^{2}_{(1-\frac{\alpha}{2},2n)}$ , respectively, be the ${\left(\frac{\alpha}{2}\right)}^{th}$ and ${\left(1-\frac{\alpha}{2}\right)}^{th}$ percentiles points (quantiles) of the chi-square distribution with $2n$ degrees of freedom, $n>$ 0.
(C3)
Let $Z_{\frac{\alpha}{2}}$ , and $Z_{1-\frac{\alpha}{2}}$ , respectively, be the ${\left(\frac{\alpha}{2}\right)}^{th}$ and ${\left(1-\frac{\alpha}{2}\right)}^{th}$ percentiles points (quantiles) of the standard normal distribution, $Z\sim N(0,1)$ , satisfying the following relation: $P\left(\left|Z\right|<Z_{1-\frac{\alpha}{2}}\right)=P(-Z_{1-\frac{\alpha}{2}}<% Z<Z_{1-\frac{\alpha}{2}})$ $=P\left(Z_{\frac{\alpha}{2}}<Z<Z_{1-\frac{\alpha}{2}}\right)=1-\alpha$ .

3.3 Pivotal quantity for the proposed confidence interval

In this section, the pivotal quantity is derived, based on the Wilson and Hilferty (WH) approximation (1931) for the transformation of an exponential random variable to a normal random variable. The result is then used to construct the ( $1-\alpha$ ) 100% proposed confidence interval (CI) for the population mean ( $\theta$ ) of the one-parameter exponential distribution in this study.

Let $X$ be a random variable from a gamma distribution with shape and scale parameters given by $\alpha$ and $\theta$ , respectively. When a shape parameter $\alpha=$ 1, the gamma distribution is $X\sim\textit{Exp}(\theta)$ , i.e., a one-parameter exponential distribution with a scale parameter $\theta$ , as per the probability density function given in Eq. (1). The cumulative distribution function (cdf) of the one-parameter exponential distribution is given by Eq. (2).

$\displaystyle F\left(x;\ \theta\right)=P(X\leqslant x)=\left\{\begin{array}[]{% ll}1-e^{-\ \frac{x}{\theta}},&x>0,\theta>0\\ 0,&\textit{otherwise}\end{array}\right.$ (2)

Wilson and Hilferty (1931) suggested that the transformation of $X$ to $X*=X^{1/3}$ is normally distributed, and mean, variance and standard deviation, respectively, can be derived from the Eqs (3) to (5) (Khan et al., 2017).

$\displaystyle\mu_{X^{*}}=E\left(X^{*}\right)=\theta^{1/3}\Gamma\left(1+\frac{1% }{3}\right)={\theta}^{1/3}\Gamma\left(\frac{4}{3}\right)=0.8930\theta^{1/3}$ (3) $\displaystyle{\sigma}^{2}_{X^{*}}=\textit{Var}(X^{*})={\theta}^{2/3}\Gamma% \left(1+\frac{2}{3}\right)-{\mu}^{2}_{X^{*}}={\theta}^{2/3}\Gamma\left(\frac{5% }{3}\right)-{\left({\theta}^{{1}/{3}}\Gamma\left(\frac{4}{3}\right)\right)}^{2% }=\theta^{{2}/{3}}\ \left(\Gamma\left(\frac{5}{3}\right)-{\left(\Gamma\left(% \frac{4}{3}\right)\right)}^{2}\right)={0.105251\ \theta}^{{2}/{3}}$ (4) $\displaystyle\sigma_{X^{*}}=\textit{SD}\left(X^{*}\right)=\sqrt{{\sigma}^{2}_{% X^{*}}}=\sqrt{0.105251\ \theta^{2/3}}=0.32442\ \theta^{1/3}$ (5)

That is, the transformation $X^{*}=X^{1/3}\sim N\left(0.8930\theta^{1/3},0.105251\theta^{2/3}\right)$ is normally distributed, as suggested by Wilson and Hilferty (1931). Therefore, the sample mean ( ${\overline{X}}^{*}$ ) for the transformed data can be given as follows:

$\displaystyle{\overline{X}}^{*}=\frac{\sum^{n}_{i=1}{X^{*}_{i}}}{n}=\frac{{(X}% ^{*}_{1}=X^{{1}/{3}}_{1})\ +\ \ \left(X^{*}_{2}=X^{{1}/{3}}_{2}\right)\ +\ % \ldots\ +\ (\ X^{*}_{n}=X^{{1}/{3}}_{n})}{n}$ (6)

This can be normally distributed using the mean, variance, and standard deviation, are given in the Eqs (7) to (9).

$\displaystyle{\mu}_{{\overline{X}}^{*}}=E\left({\overline{X}}^{*}\right)={\mu}% _{X^{*}}={\theta}^{1/3}\Gamma\left(1+\frac{1}{3}\right)={\theta}^{{1}/{3}}% \Gamma\left(\frac{4}{3}\right)={0.8930}{\ \theta}^{1/3}$ (7) $\displaystyle{\sigma}^{2}_{{\overline{X}}^{*}}=\text{Var}({\overline{X}}^{*})=% \frac{{\sigma}^{2}_{X^{*}}}{n}=\frac{{\theta}^{{2}/{3}}\left(\Gamma\left(\frac% {5}{3}\right)-{\left(\Gamma\left(\frac{4}{3}\right)\right)}^{2}\right)}{n}=% \frac{0.105251\ {\theta}^{{2}/{3}}}{n}$ (8) $\displaystyle{\sigma}_{{\overline{X}}^{*}}=\text{SD}\left({\overline{X}}^{*}% \right)=\sqrt{{\sigma}^{2}_{{\overline{X}}^{*}}}=\frac{{\sigma}_{X^{*}}}{\sqrt% {n}}=\frac{\sqrt{0.105251\ {\theta}^{{2}/{3}}}}{\sqrt{n}}=\ \frac{0.32442\ {% \theta}^{{1}/{3}}}{\sqrt{n}}$ (9)

As suggested by Wilson and Hilferty (1931), the exponential data becomes the approximate normal data when the transformation of $X^{*}=X^{1/3}$ is used. Accordingly, the central limit theorem (CLT) can be applied to derive the pivotal quantity for the proposed distribution of the sample mean. Hence, the modified $Z^{*}$ using the transformation $X^{*}=X^{1/3}$ is given in Eq. (10), which is then used as the pivotal quantity measure for this study.

$\displaystyle Z^{*}=\frac{{\overline{X}}^{*}-{\mu}_{{\overline{X}}^{*}}}{{% \sigma}_{{\overline{X}}^{*}}}=\frac{\overline{X}^{*}-0.8930\theta^{1/3}}{\frac% {0.32442\theta^{1/3}}{\sqrt{n}}}=\frac{\sqrt{n}\ \left({\overline{X}}^{*}-0.89% 30{\ \theta}^{1/3}\right)}{0.32442\ {\theta}^{1/3}}\ \sim N\left(0,\ 1\right)$ (10)

The pivotal quantity ( $Z^{*}$ ) will be used to construct the ( $1-\alpha$ ) 100% proposed confidence interval (CI) for the population mean ( $\theta$ ) of the one-parameter exponential distribution.

4. Confidence intervals for the one-parameter exponential distribution

For $0<\alpha<1$ , the following methods with a ( $1-\alpha$ ) 100% confidence interval have been studied for the efficiency comparison in this study. They are the two widely used confidence interval methods for the population mean ( $\theta$ ) of the one-parameter exponential distribution, namely, the exact and asymptotic methods. The proposed confidence interval can be derived using the pivotal quantity ( $Z^{*}$ ) given in Eq. (10).

4.1 The exact confidence interval for the one-parameter exponential distribution

Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample of size ( $n$ ) from the one-parameter exponential distribution with a parameter $\theta$ , that is $\textit{Exp}(\theta)$ ). The pivotal quantity for the exact confidence interval is denoted by $Q=\frac{2}{\theta}\sum^{n}_{i=1}{X_{i}}\sim{\chi}^{2}_{(2n)}$ . For $0<\alpha<1$ , after carrying out the proof, the ( $1-\alpha$ ) 100% exact confidence interval for the population mean ( $\theta$ ) of the one-parameter exponential distribution can be obtained as in Eq. (11) (Trivedi, 2001).

$\displaystyle\textit{CI}_{\textit{Exact}}=\left(\begin{array}[]{cc}\frac{2\sum% ^{n}_{i=1}{X_{i}}}{\chi^{2}_{\left(\frac{\alpha}{2},2n\right)}},&\frac{2\sum^{% n}_{i=1}{X_{i}}}{\chi^{2}_{\left(1-\ \frac{\alpha}{2},2n\right)}}\end{array}\right)$ (11)

where $\chi^{2}_{(\frac{\alpha}{2},2n)}$ and $\chi^{2}_{(1-\frac{\alpha}{2},2n)}$ are based on the condition (C2).

4.2 The asymptotic confidence interval for the one-parameter exponential distribution

The asymptotic confidence interval is valid only for a sufficiently large sample size ( $n$ ). Again, let $X_{1},X_{2},\ldots,X_{n}$ be a random sample of size ( $n$ ) from the one-parameter exponential distribution with a parameter $\theta$ , that is $\textit{Exp}(\theta)$ ). Then, using the maximum likelihood estimation (MLE) method to estimate the parameter $\theta$ , the unbiased estimator $\widehat{\theta}=\frac{\sum^{n}_{i=1}{X_{i}}}{n}=\overline{X}$ is obtained. Using the central limit theorem (CLT), we know that the random variable $\overline{X}$ has an approximate normal distribution with mean $\theta$ and variance $\frac{{\theta}^{2}}{n}$ , that is $\overline{X}\sim N(\theta,\frac{{\theta}^{2}}{n})$ . Using the asymptotic distribution of the estimator $\widehat{\theta}=\overline{X}$ , we can determine the confidence interval (CI) for the scale parameter $\theta$ . To do, the standard normal random variable $Z=\frac{\sqrt{n}\left(\widehat{\theta}-\theta\right)\ }{\theta}$ can be considered as the pivotal quantity, and the significance level $\alpha$ based on condition (C3) can be used to get Eq. (12).

$\displaystyle P\left(Z_{\frac{\alpha}{2}}<Z<Z_{1-\frac{\alpha}{2}}\right)=P% \left(Z_{\frac{\alpha}{2}}<\frac{\sqrt{n\ }\left(\widehat{\theta}-\theta\right% )\ }{\theta}<Z_{1-\frac{\alpha}{2}}\right)=1-\alpha$ (12)

After carrying out the proof, we obtain the ( $1-\alpha$ ) 100% asymptotic confidence interval for the population mean ( $\theta$ ) of the one-parameter exponential distribution based on the maximum likelihood estimator $\widehat{\theta}=\overline{X}$ as follows (Mood et al., 1974; Lupas, 2017):

$\displaystyle\textit{CI}_{\textit{Asymptotic}}=\left(\begin{array}[]{cc}\frac{% \widehat{\theta}}{1-\ \frac{Z_{\frac{\alpha}{2}}}{\sqrt{n}}},&\frac{\widehat{% \theta}}{1-\ \frac{Z_{1-\ \frac{\alpha}{2}}}{\sqrt{n}}}\end{array}\right)=% \left(\begin{array}[]{cc}\frac{\overline{X}}{1-\ \frac{Z_{\frac{\alpha}{2}}}{% \sqrt{n}}},&\frac{\overline{X}}{1-\ \frac{Z_{1-\ \frac{\alpha}{2}}}{\sqrt{n}}}% \end{array}\right)$ (13)

where $Z_{\frac{\alpha}{2}}$ , and $Z_{1-\frac{\alpha}{2}}$ , are based on the condition (C3).

4.3 The proposed confidence interval for the One-Parameter exponential distribution

In this section, the proposed confidence interval for the population mean ( $\theta$ ) of the one-parameter exponential distribution based on the pivotal quantity ( $Z^{*}$ ) given in Eq. (10) is constructed. Since this confidence interval proposed by Abu-Shawiesh and Juthaphorn based on the Wilson and Hilferty (WH) transformation (1931), we will refer to it by using the symbol $\text{CI}_{\textit{WHTAJ}}$ . The procedure to construct the ( $1-\alpha$ )100% proposed confidence interval for the population mean ( $\theta$ ) of the one-parameter exponential distribution is as follows:

Step 1:
Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample of size $n$ under the condition (C1).
Step 2:
Calculate $X^{}=X^{{1}/{3}}$ for the random sample $X_{1},X_{2},\ldots,X_{n}$ to get the new random sample $X^{}_{1},X^{}_{2},\ldots,X^{}_{n}$ where $X^{}_{1}=X^{1/3}_{1},X^{}_{2}=X^{1/3}_{2},\ldots,X^{}_{n}=X^{1/3}_{n}$ .
Step 3:
Calculate the sample mean ( ${\overline{X}}^{}$ ) for the transformed data in Step 2 as follows:

$\displaystyle{\overline{X}}^{}=\frac{\sum^{n}_{i=1}{X^{}_{i}}}{n}=\frac{{(X}% ^{}_{1}=X^{{1}/{3}}_{1})\ +\left(X^{}_{2}=X^{{1}/{3}}_{2}\right)+\ldots+(X^{% }_{n}=X^{{1}/{3}}_{n})}{n}.$
Step 4:
Consider $Z_{\frac{\alpha}{2}}$ , and $Z_{1-\frac{\alpha}{2}}$ , values based on the condition (C3).
Step 5:
Consider the pivotal quantity $Z^{}=\frac{\sqrt{n}\left({\overline{X}}^{}-0.8930{\theta}^{{1}/{3}}\right)\ % }{0.32442\ {\theta}^{{1}/{3}}}$ derived in Eq. (10) and the significance level $\alpha$ , then based on the relation given in condition (C3), the ( $1-\alpha$ ) 100% proposed confidence interval for the population mean ( $\theta$ ) of the one-parameter exponential distribution ( $\textit{CI}_{\textit{WHTAJ}}$ ) can be derived as follows:

$\displaystyle P\left(Z_{\frac{\alpha}{2}}<Z^{}<Z_{1-\frac{\alpha}{2}}\right)=% 1-\alpha$ $\displaystyle P\left(Z_{\frac{\alpha}{2}}<\frac{\sqrt{n}\left({\overline{X}}^{% }-0.8930{\theta}^{{1}/{3}}\right)}{0.32442\ {\theta}^{{1}/{3}}}<Z_{1-\frac{% \alpha}{2}}\right)=1-\alpha$ $\displaystyle P\left(Z_{\frac{\alpha}{2}}<\sqrt{n\ }\left[\frac{{\overline{X}}% ^{}}{0.32442\ {\theta}^{{1}/{3}}}-2.75260\right]<Z_{1-\frac{\alpha}{2}}\right% )=1-\alpha$ $\displaystyle P\left(\frac{Z_{\frac{\alpha}{2}}}{\sqrt{n}}<\left[\frac{{% \overline{X}}^{}}{0.32442\ {\theta}^{{1}/{3}}}-2.75260\right]<\frac{Z_{1-% \frac{\alpha}{2}}}{\sqrt{n}}\right)=1-\alpha$ $\displaystyle P\left(\frac{Z_{\frac{\alpha}{2}}}{\sqrt{n}}+2.75260<\frac{{% \overline{X}}^{}\ }{0.32442\ {\theta}^{{1}/{3}}}<\frac{Z_{1-\frac{\alpha}{2}}% }{\sqrt{n}}+2.75260\right)=1-\alpha$ $\displaystyle P\left(\frac{{\overline{X}}^{}}{\left[\left(\frac{Z_{1-\frac{% \alpha}{2}}}{\sqrt{n}}+2.75260\right)\left(0.32442\right)\right]}\ <\ {\theta}% ^{{1}/{3}}\ <\ \frac{{\overline{X}}^{}}{\left[\left(\frac{Z_{\frac{\alpha}{2}% }}{\sqrt{n}}+2.75260\right)\left(0.32442\right)\right]}\right)=1-\alpha$ $\displaystyle P\left(\frac{{\overline{X}}^{}}{\left(\ \frac{{0.32442\ Z}_{1-% \frac{\alpha}{2}}}{\sqrt{n}}+0.89300\right)}\ <\ {\theta}^{{1}/{3}}\ <\ \frac{% {\overline{X}}^{}}{\left(\ \frac{{0.32442\ Z}_{\frac{\alpha}{2}}}{\sqrt{n}}+0% .89300\right)}\right)=1-\alpha$ $\displaystyle P\left(\ \ {\left[\frac{{\overline{X}}^{}}{\left(\ \frac{{0.324% 42\ Z}_{1-\frac{\alpha}{2}}}{\sqrt{n}}+0.89300\right)}\right]}^{3}\ <\ \theta% \ <\ {\left[\frac{{\overline{X}}^{}}{\left(\ \frac{{0.32442\ Z}_{\ \frac{% \alpha}{2}}}{\sqrt{n}}+0.89300\right)}\right]}^{3}\ \ \right)=1-\alpha$

Hence, the ( $1-\alpha$ )100% proposed confidence interval for the population mean ( $\theta$ ) of the one-parameter exponential distribution can be obtained from the Eqs (14) and (16).

$\displaystyle\textit{CI}_{\textit{WHTAJ}}=\left(\ \ {\left[\frac{{\overline{X}% }^{}}{\left(\ \frac{{0.32442\ Z}_{1-\frac{\alpha}{2}}}{\sqrt{n}}\ +\ 0.89300% \right)}\right]}^{3}\ ,\ \ \ {\left[\frac{{\overline{X}}^{}}{\left(\ \frac{{0% .32442\ Z}_{\ \frac{\alpha}{2}}}{\sqrt{n}}\ +\ 0.89300\right)}\right]}^{3}\ \ \right)$ (14)

Let $k_{1}$ and $k_{2}$ be the two constants in Eq. (15):

$\displaystyle k_{1}=\left(\ \frac{{0.32442\ Z}_{1-\frac{\alpha}{2}}}{\sqrt{n}}% \ +\ 0.89300\right)\ \ \text{and}\ k_{2}=\left(\ \frac{{0.32442\ Z}_{\frac{% \alpha}{2}}}{\sqrt{n}}\ +\ 0.89300\right)$ (15)

Then, Eq. (14) can be simplified as follows:

$\displaystyle\textit{CI}_{\textit{WHTAJ}}=\left({\left[\frac{{\overline{X}}^{% }}{k_{1}}\right]}^{3},{\left[\frac{{\overline{X}}^{}}{k_{2}}\right]}^{3}\right)$ (16)

The constants $k_{1}$ and $k_{2}$ are required for the most common confidence interval used in real applications, i.e., the confidence level of 95% ( $\alpha=$ 0.05). Hence, the constants $k_{1}$ and $k_{2}$ for sample sizes not greater than 100 are provided in Table 1.

Table 1
The Values of $k_{1}$ and $k_{2}$ for ( $1-\alpha$ )100% $=$ 95%

n $k_{1}$ $k_{2}$ n $k_{1}$ $k_{2}$ n $k_{1}$ $k_{2}$

2 1.34262 0.443377 35 1.00048 0.785520 68 0.970110 0.815890

3 1.26012 0.525884 36 0.99898 0.787023 69 0.969549 0.816451

4 1.21093 0.575068 37 0.99754 0.788465 70 0.969000 0.817000

5 1.17737 0.608633 38 0.99615 0.789849 71 0.968463 0.817537

6 1.15259 0.633410 39 0.99482 0.791180 72 0.967937 0.818063

7 1.13333 0.652666 40 0.99354 0.792461 73 0.967422 0.818578

8 1.11781 0.668188 41 0.99231 0.793695 74 0.966918 0.819082

9 1.10495 0.681046 42 0.99112 0.794884 75 0.966423 0.819577

10 1.09408 0.691922 43 0.98997 0.796032 76 0.965939 0.820061

11 1.08472 0.701280 44 0.98886 0.797140 77 0.965463 0.820537

12 1.07656 0.709442 45 0.98779 0.798211 78 0.964997 0.821003

13 1.06936 0.716643 46 0.98675 0.799247 79 0.964540 0.821460

14 1.06294 0.723058 47 0.98575 0.800250 80 0.964092 0.821908

15 1.05718 0.728821 48 0.98478 0.801221 81 0.963651 0.822349

16 1.05197 0.734034 49 0.98384 0.802162 82 0.963219 0.822781

17 1.04722 0.738781 50 0.98292 0.803075 83 0.962795 0.823205

18 1.04287 0.743126 51 0.98204 0.803961 84 0.962378 0.823622

19 1.03888 0.747123 52 0.98118 0.804822 85 0.961969 0.824031

20 1.03518 0.750817 53 0.98034 0.805657 86 0.961567 0.824433

21 1.03176 0.754243 54 0.97953 0.806470 87 0.961172 0.824828

22 1.02857 0.757434 55 0.97874 0.807260 88 0.960783 0.825217

23 1.02559 0.760413 56 0.97797 0.808029 89 0.960401 0.825599

24 1.02280 0.763205 57 0.97722 0.808778 90 0.960026 0.825974

25 1.02017 0.765827 58 0.97649 0.809507 91 0.959657 0.826343

26 1.01770 0.768297 59 0.97578 0.810218 92 0.959293 0.826707

27 1.01537 0.770628 60 0.97509 0.810910 93 0.958936 0.827064

28 1.01317 0.772833 61 0.97441 0.811586 94 0.958584 0.827416

29 1.01108 0.774923 62 0.97375 0.812245 95 0.958238 0.827762

30 1.00909 0.776908 63 0.97311 0.812889 96 0.957898 0.828102

31 1.00720 0.778796 64 0.97248 0.813517 97 0.957562 0.828438

32 1.00541 0.780594 65 0.97187 0.814131 98 0.957232 0.828768

33 1.00369 0.782310 66 0.97127 0.814731 99 0.956907 0.829093

34 1.00205 0.783950 67 0.97068 0.815317 100 0.956586 0.829414

Table 2
The coverage probabilities and average widths of the 95% CIs for the parameter $\theta$ when $\theta=$ 0.5

$n$ Confidence interval method

Exact Asymptotic Proposed

CP AW CP AW CP AW

5 0.9520 1.30 0.9571 3.79 0.9504 1.48

10 0.9520 0.75 0.9557 1.01 0.9505 0.84

30 0.9527 0.38 0.9541 0.41 0.9516 0.42

50 0.9520 0.29 0.9519 0.30 0.9515 0.32

100 0.9515 0.20 0.9513 0.20 0.9503 0.22

1000 0.9489 0.06 0.9485 0.06 0.9465 0.07

5. Simulation study and results

n	$k_{1}$	$k_{2}$	n	$k_{1}$	$k_{2}$	n	$k_{1}$	$k_{2}$
2	1.34262	0.443377	35	1.00048	0.785520	68	0.970110	0.815890
3	1.26012	0.525884	36	0.99898	0.787023	69	0.969549	0.816451
4	1.21093	0.575068	37	0.99754	0.788465	70	0.969000	0.817000
5	1.17737	0.608633	38	0.99615	0.789849	71	0.968463	0.817537
6	1.15259	0.633410	39	0.99482	0.791180	72	0.967937	0.818063
7	1.13333	0.652666	40	0.99354	0.792461	73	0.967422	0.818578
8	1.11781	0.668188	41	0.99231	0.793695	74	0.966918	0.819082
9	1.10495	0.681046	42	0.99112	0.794884	75	0.966423	0.819577
10	1.09408	0.691922	43	0.98997	0.796032	76	0.965939	0.820061
11	1.08472	0.701280	44	0.98886	0.797140	77	0.965463	0.820537
12	1.07656	0.709442	45	0.98779	0.798211	78	0.964997	0.821003
13	1.06936	0.716643	46	0.98675	0.799247	79	0.964540	0.821460
14	1.06294	0.723058	47	0.98575	0.800250	80	0.964092	0.821908
15	1.05718	0.728821	48	0.98478	0.801221	81	0.963651	0.822349
16	1.05197	0.734034	49	0.98384	0.802162	82	0.963219	0.822781
17	1.04722	0.738781	50	0.98292	0.803075	83	0.962795	0.823205
18	1.04287	0.743126	51	0.98204	0.803961	84	0.962378	0.823622
19	1.03888	0.747123	52	0.98118	0.804822	85	0.961969	0.824031
20	1.03518	0.750817	53	0.98034	0.805657	86	0.961567	0.824433
21	1.03176	0.754243	54	0.97953	0.806470	87	0.961172	0.824828
22	1.02857	0.757434	55	0.97874	0.807260	88	0.960783	0.825217
23	1.02559	0.760413	56	0.97797	0.808029	89	0.960401	0.825599
24	1.02280	0.763205	57	0.97722	0.808778	90	0.960026	0.825974
25	1.02017	0.765827	58	0.97649	0.809507	91	0.959657	0.826343
26	1.01770	0.768297	59	0.97578	0.810218	92	0.959293	0.826707
27	1.01537	0.770628	60	0.97509	0.810910	93	0.958936	0.827064
28	1.01317	0.772833	61	0.97441	0.811586	94	0.958584	0.827416
29	1.01108	0.774923	62	0.97375	0.812245	95	0.958238	0.827762
30	1.00909	0.776908	63	0.97311	0.812889	96	0.957898	0.828102
31	1.00720	0.778796	64	0.97248	0.813517	97	0.957562	0.828438
32	1.00541	0.780594	65	0.97187	0.814131	98	0.957232	0.828768
33	1.00369	0.782310	66	0.97127	0.814731	99	0.956907	0.829093
34	1.00205	0.783950	67	0.97068	0.815317	100	0.956586	0.829414

$n$	Confidence interval method
	Exact	Asymptotic	Proposed
	CP	AW	CP	AW	CP	AW
5	0.9520	1.30	0.9571	3.79	0.9504	1.48
10	0.9520	0.75	0.9557	1.01	0.9505	0.84
30	0.9527	0.38	0.9541	0.41	0.9516	0.42
50	0.9520	0.29	0.9519	0.30	0.9515	0.32
100	0.9515	0.20	0.9513	0.20	0.9503	0.22
1000	0.9489	0.06	0.9485	0.06	0.9465	0.07

To evaluate the efficiencies of the exact, asymptotic, and proposed confidence intervals, an extensive Monte-Carlo simulation study was conducted using the SAS version 9.4 program to examine the coverage probabilities and average widths of the three confidence intervals. The most common confidence level of 95% used. The six populations of a size $N=$ 100,000 were each generated for a one-parameter exponential distribution with the parameter $\theta=$ 0.5, 1, 5, 10, 50, and 100. Each population was randomly generated 50,000 times for sample sizes $\theta=$ 5, 10, 30, 50, 100, and 1,000. For each of the samples, the 95% confidence intervals of the parameter $\theta$ were constructed for the three methods. The coverage probability (CP) and the average width (AW) were obtained by using the following two formulas:

$\displaystyle\textit{CP}=\frac{\#(L\leqslant\theta\leqslant U)}{50000}\text{% and}\ \textit{AW}=\frac{\sum^{50000}_{i=1}{(U_{i}-L_{i})}}{50000}$ (17)

The simulation results are shown in Tables 2–7. The coverage probabilities of the proposed, exact, and asymptotic confidence intervals were found to get systematically closer to the nominal level (0.95) for all levels of the sample size and parameter $\theta$ . When considering the average widths of the three confidence intervals, the proposed and exact confidence intervals were found to have similar average widths for all the sample sizes and all levels of the parameter $\theta$ . Further, the average widths of the proposed and exact confidence intervals were lower than that of the asymptotic confidence interval for a small sample size $\left(n\leqslant 10\right)$ and all the levels of the parameter $\theta$ . For a large sample size, the efficiencies of the three confidence intervals showed no differences for all the levels of the parameter $\theta$ . In addition, the average widths of all methods tended to decrease when the sample size increase for all the levels of the parameter $\theta$ . It can be concluded that both the exact and proposed confidence intervals can be captured using a common parameter that provides the highest coverage probability and shortest width. Based on the results from this simulation study, we concluded that the exact and proposed confidence intervals are possible candidates for the confidence interval of the mean ( $\theta$ ) for a one-parameter exponential distribution.

Table 3

The coverage probabilities and average widths of the 95% CIs for the parameter $\theta$ when $\theta=$ 1

$n$	Confidence interval method
	Exact		Asymptotic		Proposed
	CP	AW	CP	AW	CP	AW
5	0.9518	2.60	0.9570	7.60	0.9511	2.96
10	0.9492	1.51	0.9530	2.02	0.9492	1.68
30	0.9511	0.76	0.9529	0.82	0.9505	0.84
50	0.9503	0.58	0.9506	0.60	0.9485	0.63
100	0.9496	0.40	0.9498	0.41	0.9479	0.44
1000	0.9501	0.12	0.9496	0.12	0.9473	0.14

Table 4

The coverage probabilities and average widths of the 95% CIs for the parameter $\theta$ when $\theta=$ 5

$n$	Confidence interval method
	Exact		Asymptotic		Proposed
	CP	AW	CP	AW	CP	AW
5	0.9507	12.98	0.9568	37.89	0.9497	14.77
10	0.9512	7.53	0.9561	10.10	0.9502	8.40
30	0.9497	3.83	0.9507	4.12	0.9488	4.20
50	0.9513	2.89	0.9522	3.01	0.9500	3.17
100	0.9521	2.01	0.9517	2.05	0.9514	2.19
1000	0.9506	0.62	0.9497	0.62	0.9489	0.68

Table 5

The coverage probabilities and average widths of the 95% CIs for the parameter $\theta$ when $\theta=$ 10

$n$	Confidence interval method
	Exact		Asymptotic		Proposed
	CP	AW	CP	AW	CP	AW
5	0.9506	26.01	0.9571	75.93	0.9497	29.54
10	0.9495	15.05	0.9540	20.20	0.9499	16.79
30	0.9494	7.66	0.9503	8.25	0.9495	8.42
50	0.9498	5.77	0.9499	6.02	0.9493	6.33
100	0.9511	4.01	0.9507	4.09	0.9505	4.39
1000	0.9517	1.25	0.9510	1.25	0.9487	1.36

6. Applications using real data

In this section, two real-life data examples taken from the civil aviation and healthcare sectors are used to illustrate the applications and performances of the exact, asymptotic, and proposed ( $\textit{CI}_{\textit{WHTAJ}}$ ) confidence intervals.

6.1 Example 1: AC failure time data

This example is demonstrated using the data given by Proschan (1963) and introduced by Shi and Kibria (2007). The data represents the time (in days) between the successive failures of the air conditioning (AC) equipment in the Boeing 720 airplane. According to Shi and Kibria (2007), the data has been well fitted to an exponential distribution with mean $\theta=$ 122 days. The statistical summary of the AC failure time data is as follows: $n=15,\sum^{n=15}_{i=1}{X_{i}}=1819,\overline{X}=121.267,{\overline{X}}^{*}=4.2% 87,k_{1}=1.05718,k_{2}=0.728821$ . The 95% confidence intervals for the population mean ( $\theta$ ) and the corresponding confidence widths for the three methods considered in this study are given below in Table 8.

Table 6
The coverage probabilities and average widths of the 95% CIs for the parameter $\theta$ when $\theta=$ 50

$n$	Confidence interval method
	Exact		Asymptotic		Proposed
	CP	AW	CP	AW	CP	AW
5	0.9510	129.91	0.9564	379.25	0.9512	147.39
10	0.9512	75.43	0.9551	101.21	0.9504	84.10
30	0.9516	38.28	0.9525	41.24	0.9497	42.08
50	0.9517	28.90	0.9514	30.16	0.9500	31.68
100	0.9502	20.05	0.9504	20.46	0.9491	21.93
1000	0.9506	6.23	0.9494	6.25	0.9485	6.81

Table 7

The coverage probabilities and average widths of the 95% CIs for the parameter $\theta$ when $\theta=$ 100

$n$	Confidence interval method
	Exact		Asymptotic		Proposed
	CP	AW	CP	AW	CP	AW
5	0.9505	260.65	0.9554	760.93	0.9512	296.02
10	0.9498	150.51	0.9542	201.97	0.9495	167.70
30	0.9529	76.54	0.9533	82.46	0.9521	84.15
50	0.9514	57.83	0.9516	60.34	0.9484	63.40
100	0.9504	40.09	0.9508	40.92	0.9490	43.83
1000	0.9502	12.47	0.9490	12.49	0.9478	13.61

Table 8

The 95% CIs for the population mean ( $\theta$ ) of the AC failure time data

Confidence interval method	Confidence interval limits
	Lower limit	Upper limit	Width
Exact	77.439	216.664	139.225
Asymptotic	80.519	245.514	164.995
Proposed	66.699	203.567	136.868

From Table 8, we can see that all the three confidence interval types encompass the true population mean ( $\theta=$ 122). The widths of the exact and proposed confidence intervals show no difference, and the proposed confidence interval has the shortest interval width. Furthermore, the widths of the exact and proposed confidence intervals are shorter than that of the asymptotic confidence interval for a sample size $n=$ 15. Both the exact and proposed confidence intervals performed well compared to the asymptotic confidence interval. Hence, these results support the results obtained from the simulation study.

6.2 Example 2: Urinary ract infection (UTI) data

This example demonstrate the application of the exact, asymptotic, and proposed confidence intervals in healthcare regarding the duration of male patient urinary tract infections (UTIs) in days (see Santiago & Smith, 2013), obtained from a large hospital system, are considered. The same data set used by Aslam et al. (2014) and Azam et al. (2017). The data represents the duration between the admission and discharge of male patients with UTI. According to Santiago and Smith (2013), the data are well fitted to an exponential distribution with a mean time of $\theta=$ 0.21 days. The statistical summary of the urinary tract infection (UTI) data is as follows: $n=$ 54, $\sum^{n=54}_{i=1}{X_{i}}=$ 11.3542 , $\overline{X}=$ 0.210262, ${\overline{X}}^{*}=$ 0.534447, $k_{1}=$ 0.97953, $k_{2}=$ 0.806470. The 95% confidence intervals for the population mean ( $\theta$ ) and the corresponding confidence widths for the three methods considered in this study are given below in Table 9.

Table 9
The 95% CIs for the population mean ( $\theta$ ) of the Urinary tract infection data

Confidence interval method	Confidence interval limits
	Lower limit	Upper limit	Width
Exact	0.16378	0.27989	0.11611
Asymptotic	0.16599	0.28674	0.12075
Proposed	0.16243	0.29104	0.12861

As shown in Table 9, all the three confidence interval types encompass the true population mean ( $\theta=$ 0.21). The widths of the three confidence intervals show no difference for a large sample size of $n=$ 54, and the exact confidence interval has the shortest interval width. All the three confidence intervals performed well. Hence, these results support the results obtained from the simulation study.

7. Concluding remarks

A new confidence interval estimation method has been proposed for the mean of the one-parameter exponential distribution. The proposed confidence interval can be used when the time between events is exponentially distributed. The Wilson-Hilferty (WH) transformation was used to establish the limits of the proposed confidence interval. Considering $n\ \leqslant$ 100, Table 1 provides the values of $k_{1}$ and $k_{2}$ for the proposed confidence interval for the confidence level of 95%, the most common confidence level used in real applications. A Monte-Carlo simulation study was performed to compare the efficiency of the proposed CI with the efficiencies of the other two commonly existing CIs, namely the exact and asymptotic confidence intervals, in terms of their coverage probabilities and average widths. This simulation study show that the coverage probabilities of the three confidence intervals were close to the nominal level for all the levels of the sample size. Furthermore, the comparison results show that for all levels of $\theta$ , the average width of the proposed confidence interval is shorter than that of the asymptotic confidence interval for a small sample size $n$ . For a large sample size $n$ , there seemed to be no difference between the efficiency of the three methods for all the levels of $\theta$ and methods, which they perform equivalently well. To illustrate the findings, two real-life applications of the exact, asymptotic, and proposed ( $\textit{CI}_{\textit{WHTAJ}}$ ) confidence intervals considered, and the results support the simulation study. Moreover, both exact and proposed confidence intervals are easy to compute and can be recommended for practitioners. For $n>$ 100, the values $k_{1}$ and $k_{2}$ for the proposed confidence interval can be easily computed using the following two formulas:

$\displaystyle k_{1}=\left(\ \frac{{0.32442\ Z}_{1-\frac{\alpha}{2}}}{\sqrt{n}}% \ +\ 0.89300\right)$

and

$\displaystyle k_{2}=\left(\ \frac{{0.32442\ Z}_{\frac{\alpha}{2}}}{\sqrt{n}}\ % +\ 0.89300\right).$

The ${\left(\frac{\alpha}{2}\right)}^{th}$ and ${\left(1-\frac{\alpha}{2}\right)}^{th}$ percentile points of a chi-squared distribution with $2n$ degrees of freedom are denoted by the symbols $\chi^{2}_{(\frac{\alpha}{2},2n)}$ and $\chi^{2}_{(1-\frac{\alpha}{2},2n)}$ , respectively, for the exact confidence interval is not easy to compute without computer programming. Even though the calculation for the proposed confidence interval seems to be more complicated than the exact confidence interval calculate without computer programming. Therefore, the proposed confidence interval method may be a reliable alternative for both small and large sample sizes, whereas the exact confidence interval cannot practically calculated without computer programming for a large sample size. For practitioners convenience use of the proposed confidence interval, R code for this proposed confidence interval were provide in the appendix.

Footnotes

Acknowledgments

The authors are deeply thankful to the editor and three anonymous referees for their invaluable constructive comments and suggestions, which helped clarify several ideas and improved the quality and presentation of this paper.

Appendix

An example of the programming for the proposed confidence interval; the R code depicts how a population can be generated

(A.1) The 95% proposed confidence interval of the AC failure time data can be calculated without difficulty using R programming as follows:

The following is the output of this R programming for the 95% proposed confidence interval of the AC failure time data.

(A.2) The following R code is an example of how a population of size np $=$ 10,000 can be generated, considering nitr $=$ 50,000 samples each of size ns $=$ 5. Then, using $k_{1}$ $<$ $-$ 1.17737 and $k_{2}<$ $-$ 0.608633 values from Table 1, an almost nominal coverage of 95% for theta1 $=$ 5 is obtained.

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A confidence interval for the population mean of a one-parameter exponential distribution based on the Wilson-Hilferty transformation

Abstract

Keywords

1. Introduction

2. The Wilson-Hilferty transformation

3. Materials and methods

3.1 Criteria for the efficiency comparison

3.2 Essential conditions for the study

4.1 The exact confidence interval for the one-parameter exponential distribution

6.1 Example 1: AC failure time data

Table 6 The coverage probabilities and average widths of the 95% CIs for the parameter θ when θ = 50

Table 9 The 95% CIs for the population mean ( θ ) of the Urinary tract infection data

Footnotes

Acknowledgments

Appendix

References

Table 6
The coverage probabilities and average widths of the 95% CIs for the parameter $\theta$ when $\theta=$ 50

Table 9
The 95% CIs for the population mean ( $\theta$ ) of the Urinary tract infection data