Chapter 2: Confidence intervals for normal quantiles: one- and two-sample problems

1. Introduction

In this chapter, we address the problem of estimating the ratio or difference for the quantiles of two normal populations. In a one-sample problem, [chakraborti2007confidence] have proposed a CI for a normal quantile which is based on a minimum variance unbiased estimate (MVUE) of the population quantile. We show in this chapter that the classical CI and the Chakraborti-Li CI are the same. Furthermore, we propose a simple approximate CI for a normal quantile based on a normal approximation to the NCT distribution. This approximate CI is straightforward to compute and is quite comparable with the classical NCT confidence interval.

An exact method of computing CI for the ratio of normal percentiles is proposed in [huang2006confidence]. However, their CIs are valid only when the variances are equal, not in closed-form and an iterative method is required to find them. Recently, [krishnamoorthy2024conf_sap] have proposed approximate closed-form CIs for the ratio of percentiles for the normal, exponential and Weibull cases. Their CIs for the ratio of percentiles involving these distributions are accurate and easy to compute.

In this chapter, we propose a fiducial approach to find CIs for a ratio of or for the difference between two normal quantiles. Specifically, we propose a fiducial method, based on the normal approximation to the NCT distribution, that can be used to find CIs for a ratio or the difference of two quantiles of normal populations. The rest of this chapter is organized as follows. In the following section, we consider the one-sample problem and show that [chakraborti2007confidence] CI for a normal quantile is the same as the NCT classical CI. Furthermore, we describe the fiducial distribution for a quantile based on the NCT distribution and another one based on the normal approximation to the NCT distribution. Using these fiducial distributions, we develop simple closed-form fiducial CIs for a ratio/difference of quantiles of two normal populations. The proposed approach can be readily extended to find CIs for a ratio of two lognormal quantiles or for a ratio of two gamma quantiles. These are the contents of Section 3. Coverage, precision and comparison studies are carried out in Section 4. Three examples, involving normal, gamma and lognormal distributions, are worked out in Section 5. Some concluding remarks are given in Chapter LABEL:ch5.

2. Confidence Intervals for a Quantile

Let $X_1,\mathrm{\dots },X_n$ be a sample from a normal distribution with mean $\mu$ and variance ${\sigma }^2$, say, $N(\mu ,{\sigma }^2).$ Let $\overline{X}=\frac{1}{n}{\sum }_{i=1}^n{X}_i$ and $S^2=\frac{1}{n-1}{\sum }_{i=1}^n{(X_i-\overline{X})}^2.$ The $p$th quantile of a $N(\mu ,{\sigma }^2)$ distribution is given by ${\xi }_p=\mu +z_p\sigma$, where $z_p$ denotes the $p$th quantile of the standard normal distribution. [chakraborti2007confidence] have proposed a confidence interval (CI) for ${\xi }_p$, referred to as the $T$-interval, based on the minimum variance unbiased estimate (MVUE) of ${\xi }_p$. We show here that the $T$-interval is the same as the simple classical CI based on the noncentral $t$ distribution.

2.1. Noncentral t Confidence Interval

The classical noncentral $t$ (NCT) CI is based on the pivotal quantity $({\xi }_p-\overline{X})/S$ (e.g., see [owen1968survey] and Section 5.3.1.1 of [lawless2011statistical]). Letting $m=n-1$ and using the stochastic representations that

$$\overline{X}\stackrel{d}{=}\mu +Z\frac{\sigma }{\sqrt{n}}\mathrm{and}{S}^2\stackrel{d}{=}{\sigma }^2{U}^2,$$

where $Z$ is the standard normal random variable independently of $U^2\sim {\chi }_m^2/m$ distribution, we see that

(1) ${\displaystyle \frac{{\xi }_p-\overline{X}}{S}}$ $\stackrel{d}{=}$ ${\displaystyle \frac{1}{\sqrt{n}}}{\displaystyle \frac{z_p\sqrt{n}-Z}{U}}$

$\stackrel{d}{=}$ ${\displaystyle \frac{1}{\sqrt{n}}}{t}_m(z_p\sqrt{n}),$

where $t_m(\delta )$ denotes the noncentral $t$ random variable with degrees of freedom (df) $m$ and the noncentrality parameter $\delta$. To get the 2nd step in (1), we used the result that $Z$ and $-Z$ are identically distributed. On the basis of the above distributional result, the $1-2\alpha$ CI for ${\xi }_p$ is given by

(2) $$(\overline{X}+\frac{1}{\sqrt{n}}{t}_{m;\alpha }(z_p\sqrt{n})S,\overline{X}+\frac{1}{\sqrt{n}}{t}_{m;1-\alpha }(z_p\sqrt{n})S),$$

where $t_{m;\alpha }(\delta )$ denotes the 100$\alpha$ percentile of $t_m(\delta )$.

Remark 2.1 [chakraborti2007confidence] noted that the pivotal quantity in (1) is a function of the maximum likelihood estimates (MLEs). These authors have proposed a pivotal quantity based on the minimum variance unbiased estimator of ${\xi }_p$ given by ${\widehat{\xi }}_p=\overline{X}+z_p{c}_{n}S,\mathrm{with}{c}_n=\sqrt{m/2}\mathrm{\Gamma }\left(m/2\right)/\mathrm{\Gamma }(n/2).$ The variance of the estimator ${\widehat{\xi }}_p$ is given by $\mathrm{var}({\widehat{\xi }}_p)=\frac{{\sigma }^2}{n}\left(1+n{z}_p^2(c_n^2-1)\right).$ On the basis of the improved estimate of ${\xi }_p$, [chakraborti2007confidence] have proposed the pivotal quantity

(3) $$T=\frac{{\widehat{\xi }}_p-{\xi }_p}{\sqrt{\widehat{\mathrm{var}}({\widehat{\xi }}_p)}},$$

where the estimate $\widehat{\mathrm{var}}({\widehat{\xi }}_p)=\frac{S^2}{n}\left(1+n{z}_p^2(c_n^2-1)\right)$, to find a CI for ${\xi }_p$. These authors have also provided a method for computing necessary critical values to find CIs based on (3). As shown in Appendix A, the CI based on (3) is the same as the NCT CI in (2). Thus, the percentiles of $T$ are not needed to compute the exact CI for ${\xi }_p$. The methods of computing the percentiles of $T$ given in [chakraborti2007confidence], [zhang2018confidence] and [liu2013simultaneous] are unwarranted.

2.2. Normal Approximation

Let $X$ follow a noncentral $t$ distribution with df = $m$ and the noncentrality parameter $\delta$. Then,

(4) $$\frac{c_{m}X-\delta }{\sqrt{1+\frac{X^2}{2m}}}\sim N(0,1)\text{approximately},$$

where $c_m=1-0.25/m$; see [abramowitz1965handbook]. Let $t_p=\frac{\sqrt{n}({\xi }_p-\overline{X})}{S}$. It follows from (1) and (4) that

(5) $$\frac{c_m{t}_p-z_p}{\sqrt{\frac{1}{n}+\frac{t_p^2}{2m}}}\sim N(0,1),\mathrm{approximately}.$$

Solving the equation

$$\left|\frac{c_m{t}_p-z_p}{\sqrt{\frac{1}{n}+\frac{t_p^2}{2m}}}\right|=q_{\alpha }$$

for $t_p$, we find the two roots as $h(z_{\alpha /2};n,p)$ and $h(z_{1-\alpha /2};n,p)$, where

(6) $$h(z_{\alpha };n,p)=\frac{c_m{z}_p+z_{\alpha }\sqrt{\frac{c_m^2}{n}+\frac{1}{2m}\left(z_p^2-\frac{z_{\alpha }^2}{n}\right)}}{c_m^2-\frac{z_{\alpha }^2}{2m}}.$$

Further, solving the inequality $h(z_{\alpha /2};n,p)\le t_p\le h(z_{1-\alpha /2};n,p)$ for ${\xi }_p$, we find an approximate CI for ${\xi }_p$ as

(7) $$(\overline{X}+h(z_{\alpha /2};n,p)S,\overline{X}+h(z_{1-\alpha /2};n,p)S).$$

On the basis of our extensive simulation study, we found that the above CI is accurate if we choose $c_m=1.0$ for $0.15\le p\le 0.85$ and $c_m=1+.25/m$ otherwise. See the coverage and precision results in Table 1.

2.3. Fiducial Distributions for a Normal Quantile

Let $(\overline{x},s^2)$ be an observed value of $(\overline{X},S^2)$. Using the general approach by [dawid1982functional], a fiducial distribution for ${\xi }_p$ can be obtained from the pivotal quantity (1) .

NCT Fiducial Quantity: Solving the “equation” $\sqrt{n}({\xi }_p-\overline{X})/S\stackrel{d}{=}{t}_m(z_p\sqrt{n}),$ for ${\xi }_p$, and then replacing $(\overline{X},S)$ by its observed value, we find a fiducial quantity for ${\xi }_p$ as

(8) $$Q_{{\xi }_p}=\overline{x}+\frac{1}{\sqrt{n}}{t}_m(z_p\sqrt{n})s.$$

We refer to the above fiducial quantity as the NCT fiducial quantity. For a given $(\overline{x},s)$, the $1-2\alpha$ fiducial CI for ${\xi }_p$ is formed the percentiles of $Q_{{\xi }_p}$, which is the same as exact classical CI (2) for ${\xi }_p$. We shall use the fiducial quantity in (8) to find CIs for the difference between and a ratio of two quantiles in the sequel.

Normal approximate Fiducial Quantity: On the basis of the normal approximation (5), a fiducial quantity for ${\xi }_p$ can be obtained by solving the “equation”

$$\frac{c_m{t}_p-z_p}{\sqrt{\frac{1}{n}+\frac{t_p^2}{2m}}}\stackrel{d}{=}Z,$$

where $t_p=\left(\frac{{\xi }_p-\overline{X}}{S/\sqrt{n}}\right)$, for ${\xi }_p$ and then replacing $(\overline{X},S)$ with $(\overline{x},s)$. It can be readily verified that the fiducial quantity for ${\xi }_p$ is given by

(9) $$\overline{x}+h(Z;n,p)s,$$

where

(10) $$h(Z;n,p)=\frac{c_m{z}_p+Z\sqrt{\frac{c_m^2}{n}+\frac{1}{2m}\left(z_p^2-\frac{Z^2}{n}\right)}}{c_m^2-\frac{Z^2}{2m}}$$

and $Z$ is a standard normal random variable. The percentiles of $h(Z;n,p)$ is given by $h(z_{\alpha };n,p)$ in (6). It is easy to check that $h(z_{\alpha };n,p)$ is a finite number if $n>(z_{\alpha }^2+2)/2$. This means that for most practical values of $\alpha$, $h(z_{\alpha };n,p)$ is defined if $n\ge 5.$

3. Confidence Intervals for the Difference/Ratio of Two Quantiles

Let $({\overline{X}}_i,S_i^2)$ denote the (mean, variance) based on a sample of size $n_i$ from a normal distribution with mean ${\mu }_i$ and variance ${\sigma }_i^2$, $i=1,2.$ Let $({\overline{x}}_i,s_i^2)$ be an observed value of $({\overline{X}}_i,S_i^2)$. Let ${\xi }_{i{p}_i}={\mu }_i+z_{p_i}{\sigma }_i$, the $p$th quantile of the $N({\mu }_i,{\sigma }_i^2)$ distribution.

3.1. Noncentral t Fiducial Confidence Intervals

NCT Fiducial CIs for the Difference of Two Quantiles

Using the fiducial quantity in (8), a fiducial quantity for the difference ${\xi }_{1p}-{\xi }_{2p}$ can be obtained by substitution as

(11) $Q_{{\xi }_{1p_1}-{\xi }_{2p_2}}$ $=$ $Q_{{\xi }_{1p_1}}-Q_{{\xi }_{2p_2}}$

$=$ $({\overline{x}}_1-{\overline{x}}_2)+{\displaystyle \frac{s_1}{\sqrt{n_1}}}{t}_{m_1}(z_{p_1}\sqrt{n_1})-{\displaystyle \frac{s_2}{\sqrt{n_2}}}{t}_{m_2}(z_{p_2}\sqrt{n_2}).$

For a given $({\overline{x}}_1,s_2,{\overline{x}}_2,s_2)$, the lower and upper 100$\alpha$ percentiles of $Q_{{\xi }_{1p_1}-{\xi }_{2p_2}}$ form a $1-2\alpha$ fiducial CI for ${\xi }_{1p_1}-{\xi }_{2p_2}$.

The percentiles of $Q_{{\xi }_{ip}-{\xi }_{jp}}$ can be estimated using Monte Carlo simulation or approximated using the modified normal-based approximation given in [krishnamoorthy2016modified] as follows. For ease of writing, let $t_i=t_{m_i}(z_{p_i}\sqrt{n_i})$ and let $t_{i;\alpha }$ denote the $\alpha$ quantile of $t_i$, $i=1,2.$ Since ${\overline{x}}_1-{\overline{x}}_2$ is fixed, the percentiles of $Q_{{\xi }_{1p_1}-{\xi }_{2p_2}}$ are determined by the percentiles of $D_{12}=\frac{s_1}{\sqrt{n_1}}{t}_1-\frac{s_2}{\sqrt{n_2}}{t}_2$. For ,

(12) $$D_{12;\alpha }\approx \frac{s_1}{\sqrt{n_1}}{t}_{1;0.5}-\frac{s_2}{\sqrt{n_2}}{t}_{2;0.5}-\sqrt{\frac{s_1^2}{n_1}{\left(t_{1;0.5}-t_{1;\alpha }\right)}^2+\frac{s_2^2}{n_2}{\left(t_{2;0.5}-t_{2;1-\alpha }\right)}^2}$$

and

(13) $$D_{12;1-\alpha }\approx \frac{s_1}{\sqrt{n_1}}{t}_{1;0.5}-\frac{s_2}{\sqrt{n_2}}{t}_{2;0.5}+\sqrt{\frac{s_1^2}{n_1}{\left(t_{1;0.5}-t_{1;1-\alpha }\right)}^2+\frac{s_2^2}{n_2}{\left(t_{2;0.5}-t_{2;\alpha }\right)}^2}.$$

Thus, an approximate $1-2\alpha$ CI for ${\xi }_{1p_1}-{\xi }_{2p_2}$ can be expressed as

(14) $$({\overline{x}}_1-{\overline{x}}_2+D_{12;\alpha },{\overline{x}}_1-{\overline{x}}_2+D_{12;1-\alpha }).$$

It should be noted that [krishnamoorthy2024conf_sap] have provided the above CI with the means $E(t_{m_i}(z_{p_i}\sqrt{n_i}))$ instead of the medians of $t_{m_i}(z_{p_i}\sqrt{n_i})$. Our numerical investigation (not reported here) suggested that the approximations with the medians are slightly better than those with means.

NCT Fiducial CIs for the Ratio of Two Quantiles

A fiducial quantity for $R={\xi }_{1p_1}/{\xi }_{2p_2}=({\mu }_1+z_{p_1}{\sigma }_1)/({\mu }_2+z_{p_2}{\sigma }_2)$ can be obtained by substitution as

(15) $$Q_R=\frac{{\overline{x}}_1+t_{m_1}(z_{p_1}\sqrt{n_1})\frac{s_1}{\sqrt{n_1}}}{{\overline{x}}_2+t_{m_2}(z_{p_2}\sqrt{n_2})\frac{s_2}{\sqrt{n_2}}},$$

where the noncentral $t$ random variables $t_{m_1}(z_{p_1}\sqrt{n_1})$ and $t_{m_2}(z_{p_2}\sqrt{n_2})$ are independent. For a given $({\overline{x}}_1,{\overline{x}}_2,s_1,s_2)$ and sample sizes, Monte Carlo simulation can be used to estimate the percentiles of $Q_R$ and appropriate percentiles form a desired CI for the ratio.

Remark 2.2 A ratio of two parameters has a meaningful interpretation only when both parameters are positive. Note that the above fiducial CI is a bona fide positive CI only when both numerator and the denominator in (15) are positive. [krishnamoorthy2024conf_sap] have shown that these quantities are more likely to be positive under the assumption that all population percentiles are positive.

To find an approximation to the percentiles of $Q_R$, let

$$X={\overline{x}}_1+t_{m_1}(z_{p_1}\sqrt{n_1})\frac{s_1}{\sqrt{n_1}}\mathrm{and}Y={\overline{x}}_2+t_{m_2}(z_{p_2}\sqrt{n_2})\frac{s_2}{\sqrt{n_2}}.$$

Let $X_{\alpha }(Y_{\alpha })$ denote the $\alpha$ quantile of $X(Y)$. Note that

$$X_{\alpha }={\overline{x}}_1+t_{m_1;\alpha }(z_{p_1}\sqrt{n_1})\frac{s_1}{\sqrt{n_1}}\mathrm{and}{Y}_{\alpha }={\overline{x}}_2+t_{m_2;\alpha }(z_{p_2}\sqrt{n_2})\frac{s_2}{\sqrt{n_2}}.$$

For convenience, denote the median of $X(Y)$ by $X_{\mathrm{md}}(Y_{\mathrm{md}})$. In terms of these quantities, approximate percentiles of $Q_R$ can be expressed as follows.

(16)

where $r=X_{\mathrm{md}}/Y_{\mathrm{md}}.$ For example, the 95% CI for ${\xi }_{1p_1}/{\xi }_{2p_2}$ is given by $(Q_{R,.025},Q_{R,.975}).$ This CI is essentially the same as the one given in [malekzadeh2020constructing], except that these authors used the means of the noncentral $t$ distributions instead of the medians $t_{m_i;0.5}(z_{p_i}\sqrt{n_i})$, $i=1,2.$ As noted earlier for the case of difference, the approximate percentiles based on the medians are little more accurate than those based on the means.

3.2. Normal Approximate Fiducial Confidence Intervals

Normal Approximate Fiducial CIs for the Difference of Two Quantiles

In a two-sample problem, a fiducial quantity for the difference ${\xi }_{1p_1}-{\xi }_{2p_2}$ is given by

(17) $F_{{\xi }_{1p_1}-{\xi }_{2p_2}}$ $=$ $\left[{\overline{x}}_1+h(Z_1;n_1,p_1)s_1\right]-\left[{\overline{x}}_2+h(Z_2;n_2,p_2)s_1\right]$

$=$ ${\overline{x}}_1-{\overline{x}}_2+s_{1}h(Z_1;n_1,p_1)-s_{2}h(Z_2;n_2,p_2).$

where $Z_1$ and $Z_2$ are independent standard normal random variables. In the above expression for $h(Z_i;n_i,p_i)$, we choose $c_{m_i}=1+0.25/m_i$, $i=1,2$. For a given $({\overline{x}}_1,s_1,{\overline{x}}_2,s_2)$, Monte Carlo simulation can be used to estimate the percentiles of $F_{{\xi }_{1p_1}-{\xi }_{2p_2}}$ in (17). The lower and upper $100\alpha$ percentiles form a $1-2\alpha$ CI for ${\xi }_{1p_1}-{\xi }_{2p_2}$.

The percentiles of $F_{{\xi }_{1p_1}-{\xi }_{2p_2}}$ can also be approximated using the modified normal-based approximation and the approximations like the ones in (12) and (13). Note that $h(z_{\alpha };n_i,p_i)$ is the $100\alpha$ percentile of $h(Z_i;n_i,p_i)$, $i=1,2$. Since the median $z_{.5}=0$, $h(z_{.5};n_i,p_i)$ simplifies to $z_{p_i}/c_{m_i}$, $i=1,2.$ The percentiles of $s_{1}h(Z_1;n_1,p_1)-s_{2}h(Z_2;n_2,p_2)$ in (17) can be approximated as follows. For ,

(18) $$V_{12;\alpha }\simeq s_1\frac{z_{p_1}}{c_{m_1}}-s_2\frac{z_{p_2}}{c_{m_2}}-\sqrt{s_1^2{\left[\frac{z_{p_1}}{c_{m_1}}-h(z_{\alpha };n_1,p_1)\right]}^2+s_2^2{\left[\frac{z_{p_2}}{c_{m_2}}-h(z_{1-\alpha };n_2,p_2)\right]}^2},$$

and

(19) $$V_{12;1-\alpha }=s_1\frac{z_{p_1}}{c_{m_1}}-s_2\frac{z_{p_2}}{c_{m_2}}+\sqrt{s_1^2{\left[\frac{z_{p_1}}{c_{m_1}}-h(z_{1-\alpha };n_1,p_1)\right]}^2+s_2^2{\left[\frac{z_{p_2}}{c_{m_2}}-h(z_{\alpha };n_2,p_2)\right]}^2},$$

where $c_{m_i}=1+0.25/m_i$, $i=1,2$. The 100$(1-2\alpha )$% normal approximate fiducial CI for ${\xi }_{1p_1}-{\xi }_{2p_2}$ is given by

(20) $$({\overline{x}}_1-{\overline{x}}_2+V_{12;\alpha },{\overline{x}}_1-{\overline{x}}_2+V_{12;1-\alpha }).$$

Normal Approximate Fiducial CIs for the Ratio of Two Quantiles

Using the one-sample fiducial quantity in (9), we can write the FQ for the ratio ${\xi }_{1p_1}/{\xi }_{2p_2}$ as

(21) $$F_R=\frac{{\overline{x}}_1+h(Z_1;n_1,p_1)s_1}{{\overline{x}}_2+h(Z_2;n_2,p_2)s_2},$$

where $h(Z_i;n_i,p_i)$ is given in (10) and $Z_1$ and $Z_2$ are independent standard normal random variables. For a given $({\overline{x}}_1,{\overline{x}}_2,s_1,s_2)$ and sample sizes, Monte Carlo simulation can be used to estimate the percentiles of $F_R$. The lower and upper 100$\alpha$ percentiles of $F_R$ form a $1-2\alpha$ CI for the ratio ${\xi }_{1p_1}/{\xi }_{2p_2}$.

As in the preceding section, the percentiles of $F_R$ can be approximated as follows. Let $X={\overline{x}}_1+h(Z_1;n_1,p_1)s_1$ and $Y={\overline{x}}_2+h(Z_2;n_2,p_2)s_2$. Note that $X_{\alpha }={\overline{x}}_1+h(z_{\alpha };n_1,p_1)s_1$ is the $\alpha$ quantile of $X$ and $Y_{\alpha }={\overline{x}}_2+h(z_{\alpha };n_2,p_2)s_2$ is the $\alpha$ quantile of $Y$. Let us denote the median of $X(Y)$ by $X_{\mathrm{md}}(Y_{\mathrm{md}})$. Noting that $z_{0.5}=0$, we see that $h(z_{0.5};n_i,p_i)=z_{p_i}/c_{m_i}$, $i=1,2$. Furthermore,

$$X_{\mathrm{md}}={\overline{x}}_1+\frac{z_{p_1}}{c_{m_1}}{s}_1\mathrm{and}{Y}_{\mathrm{md}}={\overline{x}}_2+\frac{z_{p_2}}{c_{m_2}}{s}_2.$$

Using these quantiles of $X$ and $Y$, and the medians of $X$ and $Y$ in (16) and $r=X_{\mathrm{md}}/Y_{\mathrm{md}}$, we can find the interval $(F_{R;\alpha },F_{R;1-\alpha })$, which is a $1-2\alpha$ CI for ${\xi }_{1p_1}/{\xi }_{2p_2}$. We refer to the CI $(F_{R;\alpha },F_{R;1-\alpha })$ as the normal approximate fiducial CI for the ratio ${\xi }_{1p_1}/{\xi }_{2p_2}$.

Lognormal and Gamma Distributions

A CI for a lognormal percentile or for the ratio of two lognormal percentiles can be obtained in a straightforward manner as follows. Recall that if $X\sim N(\mu ,{\sigma }^2)$, then $Y=\exp (X)$ has a lognormal distribution with parameters $\mu$ and ${\sigma }^2$, say, LN$(\mu ,{\sigma }^2)$. So the $p$th quantile of a lognormal distribution is given by $Q_p=\exp (\mu +z_p\sigma )$, where $z_p$ is the $p$th quantile of the standard normal distribution. Thus, to construct a CI for $Q_p$, we use the normal-based approach to find CI $(L,U)$ for $\mu +z_p\sigma$ using a log-transformed sample, and a CI $Q_p$ is obtained as $(\exp (L),\exp (U))$. Furthermore, finding a CI for the ratio of percentiles

$$\frac{\exp ({\mu }_1+z_{p_1}{\sigma }_1)}{\exp ({\mu }_2+z_{p_2}{\sigma }_2)}=\exp \left(({\mu }_1+z_{p_1}{\sigma }_1)-({\mu }_2+z_{p_2}{\sigma }_2)\right),$$

simplifies to finding a CI for the difference of normal percentiles based on log-transformed samples, which can be regarded as samples from normal distributions. See Example 2.5.3 in the example section.

A CI for a quantile of a gamma distribution can be obtained using the cube root transformation. Let $Y_1,\mathrm{\dots },Y_n$ be a sample from a gamma distribution with the shape parameter $a$ and the scale parameter $b$, say, gamma$(a,b)$ distribution. On the basis of [wilson1931distribution] approximation, we can consider the cube root transformed sample $X_i=Y_i^{1/3}$, $i=1,\mathrm{\dots },n,$ as a sample from a normal distribution. To find an approximate CI for the $p$th quantile of a gamma$(a,b)$ distribution, we first obtain a CI $(L,U)$ for the normal $p$th quantile based on $X_1,\mathrm{\dots },X_n$, and then construct $(L^3,U^3)$ as a CI for the gamma quantile. A CI for the ratio of $p$th gamma quantiles can be found similarly. In particular, we first find CI for the ratio based on cube root transformed samples, and then taking third power of the CI, we can find an approximate CI for the ratio of $p$th quantiles of gamma distributions. See [krishnamoorthy2008normal] for applications of normal-based methods for gamma distributions.

4. Coverage and Precision Studies

To appraise the properties of the CIs for a quantile, we evaluate the coverage probabilities and expected widths of the CIs. Recall that the NCT CIs are exact in the sense that the coverage probability of the CI is always the nominal level for all parameter values. We added the NCT CIs for comparing their expected widths with those of the normal approximate CIs (Norm-CI). We also note that the coverage probability of a CI for $\mu +z_p\sigma$ is the same as that for estimating $\mu +z_{1-p}\sigma$ (see Appendix B), and so we assume $p\ge 0.5$ in our simulation study. Reported estimates in Table 1 are based on 100,000 simulation runs. We first note that coverage probabilities of the exact NCT CIs are 0.950 for all cases as expected. The estimates of the coverage probabilities of Norm-CIs are also very close to the nominal level. In most case the coverage probabilities are very close to the nominal level 0.95, and they are seldom smaller than 0.947. We also note that the precisions of both CIs are practically the same when the coverage probabilities are equal to the nominal level.

Table 1. Coverage probabilities and (average lengths) of 95% CIs for ${\xi }_p$

	$n=5$
	$p=0.5$		$p=0.60$		$p=0.75$		$p=0.90$		$p=0.99$
$\sigma$	NCT-CI	Norm-CI	NCT-CI	Norm-CI	NCT-CI	Norm-CI	NCT-CI	Norm-CI	NCT-CI	Norm-CI
.01	.949(.023)	.947(.023)	.949(.023)	.948(.023)	.949(.021)	.952(.028)	.950(.036)	.947(.033)	.950(.055)	.953(.051)
.05	.950(.117)	.947(.114)	.950(.120)	.948(.119)	.950(.136)	.953(.142)	.950(.178)	.947(.164)	.950(.272)	.953(.255)
.10	.950(.233)	.947(.229)	.950(.239)	.948(.237)	.950(.271)	.953(.284)	.950(.355)	.947(.329)	.950(.545)	.952(.510)
.20	.950(.467)	.947(.457)	.950(.478)	.948(.474)	.950(.543)	.952(.569)	.950(.710)	.947(.658)	.950(1.09)	.952(1.02)
.40	.950(.934)	.947(.914)	.950(.956)	.948(.949)	.950(1.08)	.952(1.13)	.950(1.42)	.947(1.31)	.950(2.18)	.953(2.04)
.60	.950(1.40)	.947(1.37)	.950(1.43)	.948(1.42)	.950(1.62)	.953(1.70)	.950(2.13)	.947(1.97)	.950(3.27)	.952(3.06)
.80	.950(1.86)	.947(1.82)	.951(1.91)	.949(1.89)	.950(2.17)	.953(2.27)	.950(2.84)	.947(2.63)	.950(4.35)	.953(4.08)
.90	.950(2.10)	.947(2.05)	.950(2.15)	.948(2.13)	.950(2.44)	.953(2.55)	.950(3.19)	.947(2.95)	.950(4.90)	.952(4.59)
.95	.950(2.21)	.947(2.17)	.950(2.27)	.948(2.25)	.950(2.57)	.952(2.70)	.950(3.37)	.947(3.12)	.950(5.17)	.952(4.84)
1.0	.950(2.33)	.947(2.28)	.950(2.39)	.948(2.37)	.950(2.71)	.953(2.84)	.950(3.55)	.947(3.28)	.950(5.44)	.953(5.10)
	$n=10$
.01	.949(.013)	.945(.013)	.949(.014)	.946(.013)	.950(.016)	.949(.015)	.949(.019)	.947(.019)	.950(.029)	.952(.028)
.05	.950(.069)	.945(.068)	.950(.070)	.946(.069)	.949(.078)	.949(.078)	.950(.098)	.947(.094)	.950(.146)	.950(.140)
.10	.949(.139)	.945(.136)	.950(.142)	.946(.139)	.950(.157)	.950(.156)	.950(.198)	.947(.190)	.950(.292)	.950(.281)
.20	.950(.278)	.946(.272)	.950(.284)	.946(.278)	.950(.315)	.949(.313)	.950(.395)	.948(.379)	.950(.583)	.950(.561)
.40	.950(.557)	.946(.544)	.950(.567)	.946(.556)	.950(.629)	.950(.625)	.950(.790)	.948(.758)	.950(1.17)	.950(1.12)
.60	.950(.835)	.945(.816)	.950(.851)	.946(.834)	.950(.944)	.950(.937)	.950(1.18)	.947(1.13)	.950(1.75)	.950(1.68)
.80	.950(1.11)	.946(1.08)	.950(1.13)	.947(1.11)	.950(1.25)	.949(1.25)	.950(1.58)	.947(1.51)	.950(2.33)	.950(2.25)
.90	.950(1.25)	.945(1.22)	.950(1.27)	.946(1.25)	.950(1.42)	.950(1.41)	.950(1.77)	.947(1.70)	.950(2.62)	.950(2.53)
.95	.951(1.32)	.946(1.29)	.950(1.34)	.946(1.32)	.950(1.49)	.949(1.48)	.950(1.87)	.948(1.80)	.950(2.77)	.950(2.67)
1.0	.950(1.39)	.945(1.36)	.950(1.41)	.946(1.39)	.950(1.57)	.950(1.56)	.950(1.97)	.947(1.89)	.950(2.91)	.950(2.81)
	$n=15$
.01	.949(.011)	.947(.011)	.949(.011)	.947(.010)	.950(.012)	.949(.012)	.950(.015)	.948(.014)	.950(.022)	.949(.022)
.05	.950(.054)	.947(.054)	.949(.055)	.947(.054)	.950(.061)	.949(.060)	.950(.075)	.948(.073)	.950(.111)	.950(.108)
.10	.950(.109)	.947(.107)	.950(.111)	.947(.109)	.950(.122)	.950(.121)	.950(.152)	.948(.148)	.950(.221)	.950(.216)
.20	.950(.218)	.947(.214)	.950(.222)	.947(.218)	.950(.244)	.949(.242)	.950(.303)	.948(.295)	.950(.442)	.950(.431)
.40	.950(.435)	.947(.428)	.950(.443)	.947(.437)	.950(.488)	.949(.485)	.950(.607)	.949(.590)	.950(.884)	.950(.862)
.60	.950(.653)	.947(.642)	.950(.664)	.947(.655)	.950(.732)	.949(.727)	.950(.910)	.948(.885)	.950(1.33)	.950(1.29)
.80	.950(.870)	.946(.856)	.950(.886)	.947(.873)	.950(.977)	.949(.970)	.950(1.21)	.948(1.18)	.950(1.77)	.950(1.72)
.90	.950(.979)	.947(.963)	.950(.997)	.947(.983)	.950(1.09)	.949(1.09)	.950(1.36)	.948(1.32)	.950(1.99)	.950(1.94)
.95	.950(1.03)	.947(1.01)	.950(1.05)	.947(1.03)	.950(1.16)	.949(1.15)	.950(1.44)	.948(1.40)	.950(2.10)	.950(2.05)
1.0	.950(1.08)	.947(1.07)	.950(1.10)	.947(1.09)	.950(1.22)	.949(1.21)	.950(1.51)	.948(1.47)	.950(2.21)	.950(2.16)
	$n=20$
.01	.950(.009)	.947(.009)	.949(.009)	.947(.009)	.950(.010)	.949(.010)	.950(.012)	.949(.012)	.949(.014)	.949(.014)
.05	.950(.046)	.947(.045)	.949(.047)	.947(.046)	.950(.051)	.949(.051)	.949(.063)	.948(.062)	.950(.073)	.949(.071)
.10	.949(.092)	.946(.091)	.950(.094)	.947(.092)	.950(.103)	.950(.103)	.950(.128)	.949(.125)	.950(.146)	.949(.143)
.20	.950(.185)	.947(.182)	.950(.188)	.948(.186)	.950(.207)	.949(.205)	.950(.255)	.948(.250)	.950(.292)	.949(.287)
.40	.950(.370)	.947(.365)	.950(.376)	.948(.372)	.950(.413)	.949(.411)	.950(.511)	.949(.500)	.950(.584)	.949(.573)
.60	.950(.554)	.947(.547)	.950(.564)	.948(.558)	.950(.620)	.949(.616)	.950(.766)	.948(.750)	.950(.877)	.949(.860)
.80	.950(.739)	.947(.730)	.950(.752)	.947(.744)	.950(.827)	.949(.821)	.950(1.02)	.949(1.00)	.950(1.16)	.949(1.14)
.90	.950(.831)	.947(.821)	.950(.846)	.948(.836)	.950(.930)	.949(.924)	.950(1.14)	.948(1.12)	.950(1.31)	.949(1.29)
.95	.950(.878)	.947(.867)	.950(.893)	.948(.883)	.950(.981)	.949(.975)	.950(1.21)	.948(1.18)	.950(1.38)	.949(1.36)
1.0	.950(.924)	.947(.913)	.950(.940)	.948(.929)	.950(1.03)	.949(1.03)	.950(1.27)	.948(1.25)	.950(1.46)	.949(1.43)

The results of coverage and precision studies for estimating the percentiles of gamma distributions are given in Table 2. For the simulation study, without loss of generality, we take the scale parameter $b=1$. We estimated the coverage probabilities and precisions of the 95% CIs when the shape parameter $a=0.5,1.0,1.5,2.0,3.0$, sample size $n=5,10,15$ and $p=0.05,0.25,0.50,0.75,0.95$. For most cases, the NCT CIs and normal approximate CIs are very similar in terms of coverage probability and precision. For sample sizes of 10 or more, these two CIs are quite similar in terms of coverage probability and precision. For smaller samples, one may notice significant differences in terms of precision. See the estimates for $n=5$ and $p=0.95$.

Table 2. Coverage probabilities and (average lengths) of 95% CIs for quantiles of gamma distributions

$a$	$n=5$		$n=10$		$n=15$
	$p=0.05$
	NCT	Norm-CI	NCT	Norm-CI	NCT	Norm-CI
0.5	.977(2.14)	.979(1.59)	.973(0.21)	.977(0.18)	.968(0.08)	.975(0.07)
1.0	.965(1.32)	.965(1.04)	.966(0.27)	.966(0.27)	.967(0.19)	.967(0.20)
1.5	.959(1.18)	.959(1.04)	.959(0.50)	.959(0.51)	.961(0.41)	.961(0.42)
2.0	.955(1.30)	.955(1.24)	.958(0.77)	.957(0.79)	.958(0.65)	.957(0.65)
3.0	.953(1.83)	.952(1.83)	.954(1.32)	.953(1.33)	.954(1.09)	.954(1.10)
	$p=0.25$
0.5	.964(0.48)	.967(0.51)	.971(0.25)	.970(0.25)	.970(0.19)	.969(0.19)
1.0	.956(0.93)	.959(0.93)	.958(0.62)	.957(0.62)	.961(0.50)	.960(0.50)
1.5	.952(1.40)	.955(1.40)	.955(0.95)	.954(0.95)	.957(0.77)	.956(0.77)
2.0	.951(1.84)	.954(1.84)	.953(1.25)	.952(1.24)	.955(1.00)	.954(1.00)
3.0	.951(2.62)	.954(2.64)	.951(1.74)	.950(1.74)	.953(1.39)	.952(1.39)
	$p=0.50$
0.5	.945(1.26)	.942(1.23)	.947(0.64)	.943(0.62)	.947(0.48)	.944(0.47)
1.0	.947(2.04)	.945(1.99)	.949(1.13)	.945(1.10)	.951(0.86)	.947(0.85)
1.5	.949(2.61)	.946(2.55)	.949(1.49)	.944(1.45)	.949(1.15)	.946(1.13)
2.0	.949(3.10)	.946(3.03)	.950(1.78)	.945(1.74)	.950(1.38)	.947(1.36)
3.0	.949(3.87)	.946(3.78)	.949(2.25)	.945(2.20)	.950(1.75)	.947(1.72)
	$p=0.75$
0.5	.937(3.78)	.941(4.19)	.933(1.54)	.932(1.52)	.930(1.09)	.929(1.08)
1.0	.947(4.96)	.950(5.44)	.945(2.21)	.944(2.19)	.945(1.62)	.944(1.60)
1.5	.948(5.65)	.952(6.15)	.948(2.64)	.947(2.62)	.948(1.95)	.947(1.93)
2.0	.949(6.21)	.952(6.74)	.950(2.98)	.949(2.96)	.947(2.22)	.947(2.20)
3.0	.948(7.09)	.950(7.64)	.948(3.53)	.947(3.50)	.950(2.65)	.950(2.62)
	$p=0.95$
0.5	.940(14.2)	.937(12.0)	.935(4.71)	.929(4.33)	.930(3.18)	.923(3.01)
1.0	.949(15.9)	.948(13.7)	.949(5.82)	.946(5.40)	.948(4.03)	.945(3.84)
1.5	.951(16.8)	.950(14.5)	.950(6.43)	.948(5.99)	.950(4.50)	.948(4.30)
2.0	.950(17.4)	.950(15.1)	.950(6.87)	.949(6.42)	.950(4.86)	.948(4.65)
3.0	.950(18.3)	.950(16.1)	.952(7.55)	.951(7.08)	.951(5.42)	.950(5.20)

Monte Carlo estimates of the coverage probabilities and expected widths of (a) NCT fiducial CIs (14) and (b) normal approximate fiducial CIs (20) are given in Table 3. We have carried out extensive simulation studies, but report here only the results for the case $p_1=p_2=p$ and $p=0.5,0.6,0.75,0.90,0.99$. We observe from the table values that the NCT fiducial CIs could be conservative for small sample sizes. The normal approximate fiducial CIs are also conservative for small samples but less conservative than the NCT fiducial CIs. Also, the normal approximate fiducial CIs have better precision than the NCT fiducial CIs. For example, see the results for $(n_1,n_2)=(5,5)$ and $p\ge 0.50$. Coverage probabilities of both CIs are very close to the nominal level for sample sizes of 10 or more, and the normal approximate CIs offer some improvements over the NCT fiducial CIs in terms of precision.

The coverage probabilities and precisions of the CIs for the ratio ${\xi }_{1p}/{\xi }_{2p}$ are provided in Table 4. Since the estimation problem is scale equivariant, we chose ${\sigma }_2^2=1$ and ${\sigma }_1^2=$ 0.1,0.3,0.6,1.0,2.0,3.0,4.0. Furthermore, we chose ${\mu }_1=10$ and ${\mu }_2=5$, so that both ${\xi }_{1p}$ and ${\xi }_{2p}$ are positive for all $p\in (0,1)$. The estimates in Table 4 indicate that both CIs have coverage probabilities close to the nominal level for the most cases, except that NCT fiducial CIs are little conservative for sample sizes of 10 and 15. Between these two CIs, the normal approximate fiducial CIs have better coverage properties with shorter expected widths for all cases.

We also investigated the properties of the CIs for $({\mu }_1+z_{p_1}{\sigma }_1)-({\mu }_2+z_{p_2}{\sigma }_2)$ and of those for the ratio $({\mu }_1+z_{p_1}{\sigma }_1)/({\mu }_2+z_{p_2}{\sigma }_2)$ with unequal values for $p_1$ and $p_2$. The results are very similar to the ones reported in Tables 3 and 4, and so we did not report them here.

Table 3. Coverage probabilities and (average lengths) of 95% NCT fiducial and normal approximate fiducial (14) CIs for ${\xi }_{1p}-{\xi }_{2p}$

${\sigma }_2^2=1;{\mu }_2=0;{\mu }_1=2.0$
$p=.50$	$(n_1,n_2)=(5,5)$		$(n_1,n_2)=(7,10)$		$(n_1,n_2)=(10,10)$		$(n_1,n_2)=(10,15)$		$(n_1,n_2)=(20,20)$
${\sigma }_1^2$	(a)	(b)	(a)	(b)	(a)	(b)	(a)	(b)	(a)	(b)
0.1	.963(2.47)	.943(2.17)	.959(1.51)	.946(1.42)	.955(1.46)	.944(1.38)	.956(1.18)	.947(1.13)	.953(.971)	.947(.945)
0.3	.971(2.72)	.953(2.38)	.965(1.72)	.953(1.61)	.960(1.60)	.950(1.51)	.960(1.34)	.952(1.28)	.955(1.05)	.950(1.03)
0.6	.975(3.03)	.958(2.66)	.966(1.99)	.954(1.85)	.963(1.78)	.953(1.68)	.961(1.54)	.953(1.47)	.956(1.17)	.951(1.14)
1.0	.976(3.40)	.959(2.98)	.966(2.29)	.953(2.12)	.964(1.99)	.953(1.88)	.961(1.78)	.952(1.69)	.957(1.31)	.951(1.28)
1.5	.976(3.80)	.959(3.33)	.965(2.62)	.952(2.42)	.963(2.23)	.953(2.10)	.960(2.04)	.950(1.94)	.957(1.47)	.951(1.43)
2.0	.974(4.15)	.957(3.64)	.963(2.90)	.950(2.68)	.963(2.44)	.952(2.30)	.959(2.26)	.949(2.15)	.956(1.61)	.950(1.57)
3.0	.972(4.78)	.954(4.19)	.961(3.40)	.947(3.14)	.961(2.81)	.950(2.65)	.957(2.66)	.947(2.52)	.955(1.85)	.950(1.81)
$p=0.60$
0.1	.962(2.54)	.943(2.24)	.959(1.54)	.948(1.46)	.955(1.49)	.944(1.41)	.956(1.20)	.949(1.16)	.953(.988)	.947(.963)
0.3	.971(2.80)	.954(2.47)	.964(1.76)	.954(1.65)	.960(1.64)	.950(1.55)	.960(1.37)	.952(1.31)	.955(1.07)	.949(1.05)
0.6	.974(3.14)	.958(2.77)	.966(2.04)	.954(1.90)	.962(1.82)	.952(1.72)	.961(1.58)	.953(1.51)	.956(1.19)	.951(1.16)
1.0	.975(3.51)	.958(3.10)	.965(2.35)	.954(2.18)	.963(2.04)	.953(1.92)	.961(1.82)	.952(1.74)	.957(1.34)	.951(1.30)
1.5	.974(3.92)	.958(3.46)	.964(2.68)	.952(2.49)	.963(2.28)	.953(2.15)	.960(2.09)	.951(1.98)	.956(1.49)	.951(1.45)
2.0	.973(4.29)	.957(3.78)	.963(2.97)	.951(2.75)	.962(2.49)	.952(2.35)	.959(2.31)	.950(2.20)	.956(1.64)	.951(1.59)
3.0	.971(4.93)	.954(4.35)	.960(3.49)	.947(3.22)	.961(2.87)	.950(2.72)	.957(2.71)	.947(2.57)	.955(1.89)	.950(1.84)
$p=0.75$
0.1	.961(2.94)	.949(2.66)	.958(1.73)	.951(1.65)	.955(1.67)	.947(1.59)	.956(1.33)	.951(1.29)	.952(1.08)	.948(1.06)
0.3	.967(3.28)	.956(2.98)	.962(2.00)	.955(1.90)	.958(1.83)	.951(1.75)	.958(1.53)	.953(1.48)	.954(1.19)	.950(1.16)
0.6	.969(3.69)	.958(3.35)	.962(2.31)	.955(2.19)	.960(2.05)	.953(1.95)	.959(1.77)	.953(1.70)	.955(1.32)	.951(1.29)
1.0	.969(4.14)	.959(3.76)	.962(2.67)	.954(2.51)	.961(2.30)	.953(2.19)	.959(2.04)	.953(1.96)	.955(1.48)	.951(1.45)
1.5	.969(4.62)	.959(4.19)	.961(3.04)	.953(2.86)	.960(2.57)	.953(2.44)	.957(2.33)	.952(2.23)	.955(1.65)	.952(1.62)
2.0	.968(5.04)	.958(4.57)	.960(3.38)	.952(3.17)	.960(2.81)	.953(2.67)	.957(2.59)	.951(2.48)	.955(1.81)	.951(1.77)
3.0	.967(5.77)	.956(5.24)	.959(3.94)	.950(3.70)	.959(3.22)	.952(3.07)	.956(3.03)	.949(2.90)	.954(2.09)	.950(2.04)
$p=0.90$
0.1	.959(3.92)	.956(3.66)	.957(2.22)	.955(2.15)	.954(2.11)	.951(2.03)	.954(1.68)	.953(1.64)	.952(1.35)	.950(1.33)
0.3	.961(4.43)	.958(4.15)	.958(2.58)	.956(2.49)	.956(2.34)	.953(2.26)	.956(1.93)	.954(1.89)	.953(1.48)	.951(1.45)
0.6	.961(5.01)	.959(4.70)	.958(3.00)	.955(2.89)	.956(2.63)	.953(2.53)	.955(2.25)	.954(2.19)	.953(1.65)	.951(1.62)
1.0	.962(5.63)	.959(5.28)	.957(3.46)	.954(3.33)	.957(2.94)	.953(2.84)	.955(2.59)	.953(2.52)	.953(1.85)	.951(1.82)
1.5	.962(6.28)	.959(5.89)	.957(3.95)	.955(3.78)	.957(3.29)	.953(3.17)	.955(2.96)	.953(2.87)	.953(2.06)	.952(2.03)
2.0	.962(6.84)	.959(6.41)	.957(4.37)	.954(4.18)	.956(3.59)	.953(3.46)	.954(3.28)	.952(3.18)	.953(2.26)	.951(2.22)
3.0	.961(7.81)	.958(7.31)	.956(5.10)	.953(4.86)	.956(4.12)	.952(3.97)	.954(3.84)	.951(3.71)	.953(2.60)	.951(2.55)
$p=0.99$
0.1	.957(6.09)	.958(5.78)	.955(3.32)	.956(3.25)	.953(3.14)	.953(3.04)	.953(2.47)	.954(2.44)	.951(1.96)	.951(1.93)
0.3	.957(6.94)	.959(6.62)	.955(3.89)	.957(3.81)	.954(3.50)	.954(3.40)	.954(2.87)	.955(2.83)	.952(2.16)	.951(2.13)
0.6	.957(7.86)	.959(7.53)	.954(4.54)	.956(4.44)	.954(3.93)	.954(3.83)	.953(3.34)	.954(3.29)	.952(2.41)	.952(2.38)
1.0	.956(8.84)	.959(8.47)	.954(5.24)	.955(5.11)	.954(4.41)	.953(4.30)	.953(3.85)	.953(3.79)	.952(2.70)	.951(2.67)
1.5	.957(9.85)	.959(9.43)	.954(5.98)	.955(5.81)	.954(4.92)	.953(4.79)	.953(4.40)	.953(4.31)	.952(3.02)	.952(2.98)
2.0	.957(10.7)	.959(10.2)	.954(6.61)	.955(6.41)	.954(5.37)	.954(5.23)	.953(4.88)	.953(4.77)	.952(3.30)	.952(3.26)
3.0	.958(12.2)	.959(11.6)	.953(7.71)	.955(7.44)	.954(6.15)	.954(5.99)	.953(5.70)	.953(5.56)	.952(3.80)	.952(3.74)

NOTE: (a) NCT fiducial CI; (b) Normal approximate fiducial CI

Table 4. Coverage probabilities of 95% NCT fiducial and normal approximate fiducial CIs for the ratio ${\xi }_{1p}/{\xi }_{2p}$

${\sigma }_2^2=1;{\mu }_2=5;{\mu }_1=10.0$
	$(n_1,n_2)=(10,10)$		$(n_1,n_2)=(10,15)$		$(n_1,n_2)=(15,10)$		$(n_1,n_2)=(15,15)$		$(n_1,n_2)=(20,20)$
$p=.10$
${\sigma }_1^2$	(a)	(b)	(a)	(b)	(a)	(b)	(a)	(b)	(a)	(b)
0.1	.951(1.99)	.948(2.00)	.952(1.35)	.947(1.27)	.952(2.44)	.950(2.00)	.951(1.35)	.949(1.27)	.951(1.04)	.949(1.00)
0.3	.953(2.10)	.950(1.92)	.953(1.35)	.948(1.27)	.951(2.19)	.951(2.00)	.951(1.33)	.949(1.25)	.951(1.03)	.950(0.99)
0.6	.954(2.22)	.951(1.90)	.954(1.37)	.950(1.29)	.952(2.20)	.951(2.21)	.953(1.32)	.950(1.25)	.952(1.02)	.950(0.99)
1.0	.955(2.23)	.952(2.02)	.955(1.39)	.951(1.31)	.953(2.10)	.952(1.90)	.954(1.32)	.951(1.25)	.953(1.02)	.951(0.99)
2.0	.956(2.28)	.953(2.02)	.955(1.47)	.952(1.39)	.954(2.07)	.953(1.90)	.954(1.34)	.952(1.28)	.953(1.05)	.951(1.01)
3.0	.956(2.20)	.953(2.15)	.955(1.55)	.952(1.47)	.955(2.14)	.953(1.94)	.954(1.38)	.952(1.32)	.953(1.08)	.952(1.05)
4.0	.956(2.28)	.953(2.10)	.955(1.63)	.952(1.55)	.955(1.82)	.953(1.83)	.954(1.42)	.952(1.36)	.953(1.12)	.952(1.08)
$p=.25$
0.1	.951(.997)	.944(.922)	.952(.720)	.945(.685)	.951(.990)	.945(.921)	.951(.715)	.945(.682)	.951(.586)	.947(.567)
0.3	.953(1.00)	.946(.928)	.954(.733)	.947(.698)	.952(.980)	.946(.919)	.952(.718)	.947(.687)	.951(.589)	.947(.570)
0.6	.956(1.02)	.948(.943)	.956(.757)	.949(.721)	.953(.990)	.947(.922)	.953(.728)	.948(.697)	.952(.597)	.949(.579)
1.0	.957(1.04)	.950(.965)	.958(.790)	.951(.753)	.955(.999)	.949(.931)	.955(.745)	.950(.714)	.953(.611)	.950(.593)
2.0	.959(1.10)	.952(1.02)	.958(.868)	.952(.828)	.956(1.03)	.951(.961)	.956(.789)	.952(.758)	.955(.648)	.951(.630)
3.0	.960(1.15)	.953(1.08)	.959(.941)	.952(.896)	.958(1.06)	.952(.992)	.957(.834)	.951(.801)	.955(.686)	.951(.666)
4.0	.960(1.21)	.953(1.13)	.958(1.01)	.952(.960)	.958(1.09)	.953(1.03)	.957(.877)	.952(.843)	.955(.722)	.951(.702)
$p=.75$
0.1	.952(.489)	.945(.471)	.953(.390)	.949(.382)	.951(.485)	.942(.465)	.951(.385)	.946(.375)	.951(.328)	.947(.322)
0.3	.954(.515)	.947(.495)	.955(.417)	.951(.408)	.953(.503)	.944(.483)	.953(.404)	.948(.394)	.952(.343)	.948(.337)
0.6	.957(.547)	.949(.526)	.957(.452)	.952(.441)	.954(.525)	.946(.505)	.954(.428)	.949(.417)	.953(.363)	.949(.356)
1.0	.958(.584)	.951(.561)	.958(.492)	.953(.478)	.955(.551)	.948(.530)	.956(.455)	.951(.444)	.954(.386)	.950(.379)
2.0	.960(.663)	.952(.636)	.959(.577)	.953(.558)	.957(.607)	.950(.584)	.956(.515)	.952(.501)	.955(.436)	.951(.427)
3.0	.960(.730)	.953(.700)	.958(.648)	.953(.625)	.958(.655)	.951(.631)	.957(.566)	.952(.551)	.955(.479)	.951(.469)
4.0	.960(.789)	.953(.757)	.958(.710)	.952(.685)	.958(.698)	.952(.674)	.957(.611)	.952(.595)	.955(.517)	.951(.507)
$p=.90$
0.1	.952(.495)	.949(.483)	.953(.398)	.952(.393)	.951(.489)	.947(.475)	.952(.390)	.949(.384)	.951(.332)	.950(.328)
0.3	.954(.534)	.951(.521)	.954(.437)	.953(.432)	.952(.517)	.948(.502)	.952(.418)	.950(.411)	.952(.355)	.951(.351)
0.6	.955(.580)	.952(.565)	.955(.485)	.954(.477)	.953(.550)	.949(.535)	.953(.451)	.951(.444)	.953(.382)	.950(.378)
1.0	.956(.630)	.952(.615)	.956(.537)	.954(.528)	.954(.586)	.950(.571)	.954(.489)	.951(.480)	.953(.413)	.951(.408)
2.0	.956(.734)	.953(.715)	.956(.643)	.954(.630)	.955(.662)	.951(.645)	.954(.565)	.952(.556)	.953(.477)	.952(.470)
3.0	.956(.818)	.953(.797)	.955(.729)	.953(.713)	.955(.724)	.951(.707)	.954(.629)	.952(.618)	.953(.530)	.951(.523)
4.0	.956(.892)	.953(.868)	.955(.805)	.953(.786)	.955(.779)	.951(.761)	.954(.684)	.952(.672)	.953(.576)	.951(.568)

NOTE: (a) NCT fiducial CI; (b) Normal approximate fiducial CI

5. Examples

We shall now illustrate the interval estimation methods by constructing CIs based on different real-life data those can be modelled by normal distributions, gamma distributions and lognormal distributions.

Example 2.5.1 (Normal Distributions) A commonly used measure of overall strength of a wood species is referred to as modulus of rupture (MOR), which is a measure of a material’s ability to resist breaking in bending. The data on the modulus of rupture (MOR) of Douglas fir specimens are presented in Table 2 of [huang2006confidence]. These authors and [krishnamoorthy2024conf_sap] have used the data to construct CIs for a ratio of percentiles. The data can be found in these two paper just cited, and so we report only summary statistics in Table 5. We shall use the data to find confidence intervals for quantiles, the ratio of the 5th percentiles and for the ratio/difference of quartiles. For more details on the need for comparing the 5th percentiles of MOR measurements can be found in [huang2006confidence]. The required summary statistics for computing confidence intervals are given in the following table.

Table 5. Summary statistics of MOR data

Data	sample size	mean, ${\overline{x}}_i$	variance, $s_i^2$
MOR-1	$n_1$ = 107	${\overline{x}}_1=4840.3$	$s_1^2=2354470$
MOR-2	$n_2$ = 100	${\overline{x}}_2=7144.9$	$s_2^2=2414487$

NOTE: MOR-1: MOR (lb/in²) of 2 x 4in Grade 2, Green (30% moisture content)
     MOR-2: MOR (lb/in²) of 2 x 6in select structural, Green (30% moisture content)

Table 6. 95% Confidence intervals for the difference/ratio of the percentiles

Wood Species	method	CI for the 5th	method	CI for the ratio	CI for the difference
		percentile		of 5th percentiles	of 5th percentiles
				(MOR-1)/(MOR-2)	(MOR-1)-(MOR-2)
Grade 2	NCT	(1813.7, 2720.8)	NCT siml. (15)	(.390, .618)	(-2927.9, -1608.9)
	Nor Apprx	(1816.1, 2723.5)	NCT apprx	(.390, .618)	(-2928.2, -1607.9)
structural	NCT	(4060.0, 5011.4)	Normal siml.(21)	(.391, .618)	(-2926.0, -1612.9)
	Nor Apprx	(4062.7, 5014.3)	Normal approx.	(.391, .618)	(-2927.4, -1610.4)

The CIs for the ratio or for the difference of the 5th percentiles are computed using the R code given in Appendix C. The 95% CIs for the ratio or for the difference of the 5th percentiles indicate that the 5th percentiles of breaking strength distributions of MOR-1 and MOR-2 are not significantly different.

Table 6 continued.
Wood Species method CI for the ratio of CI for the difference CI for the ratio of CI for the difference 25th percentiles of 25th percentiles 75th percentiles of 75th percentiles (MOR-1)/(MOR-2) (MOR-1)-(MOR-2) (MOR-1)/(MOR-2) (MOR-1)-(MOR-2) Grade 2 NCT siml. (15) (.560, .689) (-2762.2, -1816.7) (.669, .769) (-2793.2, -1847.4) NCT apprx (.560, .689) (-2763.2, -1815.9) (.669, .769) (-2793.3, -1846.0) structural Normal siml.(21) (.561, .689) (-2760.8, -1818.5) (.669, .769) (-2790.0, -1848.2) Normal approx. (.560, .689) (-2761.7, -1818.3) (.669, .769) (-2790.8, -1847.7)

Example 2.5.2 (Gamma Distributions) In order to analyze the effects of pyrotechnic seeding on rainfall in south Florida, experiments were conducted in 1968 and 1970. The data collected from these experiments consist of $n_1=26$ seeded clouds and $n_2=26$ control clouds, measured in terms of rainfall in acre-feet per cloud. The data are taken from [simpson1972use], and they are given in Table 7. Several articles have used gamma models to analyse and compare the data. The Q-Q plots in [krishnamoorthy2014small] have suggested that both data fit gamma models.

Table 7. Single-cloud data for 1968 and 1970

1	1		2	2
Seeded rain	Seeded rain		Control rain	Control rain
129.6	7.7		26.1	28.6
31.4	1656.0		26.3	830.1
2745.6	978.0		87.0	345.5
489.1	198.6		95.0	1202.6
430.0	703.4		1.0∗	36.6
302.8	1697.8		372.4	4.9
119.0	334.1		17.3	4.9
4.1	118.3		24.4	41.1
92.4	255.0		11.5	29.0
17.5	115.3		321.2	163.0
200.7	242.5		68.5	244.3
274.7	32.7		81.2	147.8
274.7	40.6		47.3	21.7

^∗The actual entry 0 is replaced to fit a gamma model

The statistics based on the cube root transformed data are ${\overline{x}}_1=6.276$, $s_1=3.095$, ${\overline{x}}_2=4.342$ and $s_2=2.359$. The 95% NCT CI for the median of seeded cloud is computed as (127.0, 426.3) and the normal approximate CI is (127.9, 424.3). For the control clouds, the 95% NCT CI for the median is (38.93, 148.4) and the normal approximate CI is (39.24, 147.7). Even though these CIs are in agreement, the normal approximate CIs are shorter than the corresponding NCT CIs.

Table 8. 95% confidence intervals for the rain data

	ratio of the 1st	ratio of the	ratio of the 3rd
	quartiles	medians	quartiles
method	seeded/control	seeded/control	seeded/control
NCT fiducial	(.678, 22.87)	(1.237, 7.572)	(1.356, 5.782)
Normal Approx	(.709, 21.37)	(1.260, 7.426)	(1.369, 5.733)

The 95% CIs for the ratio of the first quartiles of the acre-feet distributions of the seeded and control clouds indicate that the first quartiles are not significantly different. On the other hand, the 95% CIs for the ratios of the medians and the third quartiles indicate that the median and the third quartile for the seeded clouds are greater than the corresponding median and the third quartile for the control clouds.

Example 2.5.3 (Lognormal Distributions) The gate oxide layer in semiconductor devices plays a critical role in ensuring the performance and reliability of the device. This layer, typically made from silicon dioxide, acts as a dielectric between the gate electrode and the substrate. Its integrity is vital because any breakdown can lead to significant device failure, impacting overall functionality and lifespan. To estimate oxide life, a time-dependent dielectric breakdown (TDDB) test is used on special test structures during device development and manufacturing process. The results of the test are useful to assess the time it takes for the oxide to break which in turn leading to the failure of the device.

The TDDB data from samples of devices built with two nominally identical oxidation furnaces A and B are given in [doganaksoy2021simplified]. The data were analyzed on the basis of lognormal models. We also checked that Q-Q plots for log-transformed data fit normal distributions. The summary statistics based on the data are given in Table 9.

Table 9. Summary statistics of TDDB data (in min) from furnaces A and B

	Furnace A	Furnace B
sample size	$n_1=32$	$n_2=32$
sample mean	${\overline{x}}_1=3.986$	${\overline{x}}_2=4.171$
sample variance	$s_1^2=0.04796$	$s_2^2=0.05093$

Suppose an engineer wanted to compare the oxide TDDB distributions of devices from furnaces A and B, by comparing the 95th percentiles of the two life distributions.

Table 10. 95% Confidence intervals for the percentiles based on TDDB data

Furnace		CI for the 95th		CI for the ratio of
	method	percentile	method	95th percentiles (A/B)
A	NCT	(69.94, 89.52)	NCT siml. (15)	(.687, .982)
	Nor Apprx	(69.73, 88.98)	NCT apprx	(.686, .983)
B	NCT	(84.83, 109.39)	Normal siml.(21)	(.685, .985)
	Nor Apprx	(84.56, 108.72)	Normal approx.	(.687, .981)

We first computed the CIs for the 95th percentiles of TDDB distribution based on the sample mean and variance reported in Table 9, and then by taking exponentiation we found CIs for the lognormal percentiles. CIs for the 95th percentiles of TDDB distributions of both furnaces are given in Table 10. Both NCT and normal approximation methods produced similar CIs. CIs for the ratio of the 95th percentiles are also reported in the same table. The NCT methods with simulation and closed-form approximation produced practically the same CI for the ratio. Normal approximate method based on simulation and the closed-form normal approximate CI are also quite close to each others. All the CIs indicate that 95th percentiles of TDDB distribution from the furnace A is significantly less than that of the furnace B.