Asymptotic Confidence Intervals for the Mean with Increased Finite-Sample Coverage Probabilities

Abstract

We consider a Student process based on independent copies of a random variable X. If X is in the domain of attraction of the normal law (DAN), a weighted version of the Student process is known to follow a functional Central Limit Theorem (FCLT). Accordingly, appropriate functionals of such a process converge in distribution to the same functionals of the similarly weighted standard Wiener process. We use such a convergence for an integral functional and derive asymptotic confidence intervals (CIs) for the mean of X. For right-skewed distributions of X in DAN, we show that the obtained CIs have higher finite-sample coverage probabilities than, and may be preferred over, a CI $I_1$ of the same asymptotic confidence level $1-\alpha$ that is based on the CLT for the Student t-statistic, since the finite-sample coverage probabilities of the latter CI may be lower than $1-\alpha$. Moreover, for such distributions, the finite-sample coverage probabilities of our best two CIs are also higher than those of their respective equal-expected-length $I_1$ counterparts.

1. Introduction

Let $\{X,X_i,i\ge 1\}$ be a sequence of independent and identically distributed (i.i.d.) random variables (r.v.) with an unknown population mean $EX=\mu$ throughout this paper. Consider the Student t-statistic

$$T_n(X_1,\dots ,X_n):={\displaystyle \frac{{\sum }_{i=1}^n{X}_i}{s_n\sqrt{n}}},$$

(1)

where $s_n:=\sqrt{{\sum }_{i=1}^n{(X_i-{\overline{X}}_n)}^2/(n-1)}$ and ${\overline{X}}_n:={\sum }_{i=1}^n{X}_i/n$.

When $0<VarX<\infty$, it is well-known that

$$\begin{array}{c}\hfill T_n(X_1-\mu ,\dots ,X_n-\mu )\stackrel{\mathcal{D}}{\to }N(0,1),n\to \infty ,\end{array}$$

(2)

which leads to the classical $100(1-\alpha )\%$ asymptotic confidence interval (CI) for the mean $\mu$:

$$I_1:=\left[{\overline{X}}_n-{\displaystyle \frac{z_{\alpha /2}{s}_n}{\sqrt{n}}},{\overline{X}}_n+{\displaystyle \frac{z_{\alpha /2}{s}_n}{\sqrt{n}}}\right],$$

(3)

where $\alpha \in (0,1)$ and $z_{\alpha /2}$ are the $1-\alpha /2$ quantile of the standard normal distribution $N(0,1)$. In fact, ref. [1] extended (2) and hence validated (3) for the distributions in the domain of attraction of the normal law (DAN), where $X\in$ DAN means that there exist constants $a_n$ and $b_n$> 0 such that

$${\displaystyle \frac{{\sum }_{i=1}^n{X}_i-a_n}{b_n}}\stackrel{\mathcal{D}}{\to }N(0,1),n\to \infty .$$

(4)

Remark 1.

The distributions in DAN have finite moments of order $\nu \in (0,2)$, while their variances are positive, but may be infinite. Clearly, if $0<VarX<\infty$, then (4) is satisfied on account of the Central Limit Theorem (CLT), with $a_n$ = $nEX$ and $b_n$ = $\sqrt{nVarX}$. If $VarX=\infty$, then $a_n$ can also be taken as $nEX$, while $b_n=\sqrt{n}{l}_x\left(n\right)$, where $l_x\left(n\right)$ is a slowly varying function at infinity, meaning that $l_x\left(az\right)/l_x\left(z\right)\to 1$, as $z\to \infty$, for any $a>0$. Moreover, in this case $l_x\left(n\right)$ is also nondecreasing and converges to ∞, as $n\to \infty$. Consider a Pareto distribution of the first kind with the tail index α and the scale parameter β, denoted by $Pareto(\alpha ,\beta )$ hereafter. $Pareto(2,1)$ with the probability density function $2x^{-3}{11}_{\{x\ge 1\}}$ is a well-known example of a distribution in DAN with an infinite variance. Moreover, $b_n$ in (4) can be taken as $\sqrt{nlogn}$ in this case. Clearly, if $\alpha >$ 2, then $Pareto(\alpha ,1)$ has a finite variance and hence is also in DAN.

A Student process in $D[0,1]$, the space of real-valued functions on $[0,1]$ that are right-continuous and have left-hand limits, is defined as follows:

$$T_n^t(X_1,\dots ,X_n):={\displaystyle \frac{{\sum }_{i=1}^{\left[nt\right]}{X}_i}{s_n\sqrt{n}}},0\le t\le 1,$$

(5)

where ${\sum }_{i=1}^0{X}_i:=$ 0. Thus, $T_n^t(X_1,\dots .,X_n)$ is a random step function on $[0,1]$:

$$T_n^t(X_1,\dots ,X_n)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if} 0\le t<{\displaystyle \frac{1}{n}},\hfill \\ {\displaystyle \frac{X_1}{s_n\sqrt{n}},}\hfill & \mathrm{if} {\displaystyle \frac{1}{n}}\le t<{\displaystyle \frac{2}{n}},\hfill \\ ⋮\hfill & ⋮\hfill \\ {\displaystyle \frac{X_1+X_2+\cdots +X_{n-1}}{s_n\sqrt{n}},}\hfill & \mathrm{if} {\displaystyle \frac{n-1}{n}}\le t<1,\hfill \\ {\displaystyle \frac{X_1+X_2+\cdots +X_n}{s_n\sqrt{n}},}\hfill & \mathrm{if} t=1.\hfill \end{array}\right.$$

(6)

When $t=1$, the Student process in (6) becomes the Student t-statistic of (1).

Let $\left\{W\right(t),0\le t\le 1\}$ denote a standard Wiener process. Theorem 1 of [2] extended (2) for $X\in$ DAN as follows:

$$\begin{array}{cc}& X\in \mathrm{DAN}\mathrm{and}EX=\mu ⟹\\ & h\left(T_n^t(X_1-\mu ,\dots ,X_n-\mu )\right)\stackrel{\mathcal{D}}{\to }h\left(W\left(t\right)\right),n\to \infty ,\\ & \mathrm{for} \mathrm{all} \mathrm{functionals} h:D[0,1]\to \mathrm{I}\mathrm{R}\mathrm{that}\mathrm{are}\mathcal{D}\text{-}\mathrm{measurable}\mathrm{and}\rho \text{-}\mathrm{continuous},\end{array}$$

(7)

where $\mathcal{D}$ is the $\sigma$-field of subsets of $D[0,1]$ that is generated by the finite-dimensional subsets of $D[0,1]$ and $\rho$ is the sup-norm metric in $D[0,1]$. The convergence in distribution in (7) is known as a functional Central Limit Theorem (FCLT). We also note that the functionals to be considered in this paper are $\mathcal{D}$-measurable and $\rho \text{-}\mathrm{continuous}$.

In order to see that the CLT of (2) for the Student t-statistic is a special case of the FCLT of (7) for the Student process, one reads (7) with the projection functional $h_{1,t_0}(·)$ with $t_0=1$, where

$$h_{1,t_0}\left(f\left(t\right)\right)=f\left(t_0\right),\mathrm{for} \mathrm{any} f\left(t\right)\in D[0,1].$$

(8)

Works [3,4] considered several concrete functionals as in (7) and derived respective asymptotic CIs, abbreviated as FACIs, for the mean of X based on convergence in distribution of these functionals. The focus of these works was to explore finite-sample properties of the obtained asymptotic CIs and compare them to those of $I_1$ of (3). Accordingly, from the comparison of the expected lengths and finite-sample coverage probabilities of these CIs, which was mostly performed numerically, it was concluded that some of the FACIs were shorter than $I_1$ on average, or had higher coverage probabilities than those of $I_1$, and thus could potentially serve as alternatives to $I_1$. Situations in which these FACIs could be preferred over $I_1$ were not discussed in these works.

Now, we present an FCLT for a weighted version of the Student process of (5) needed for an introduction of the current work.

Let Q be the class of positive functions $q\left(t\right):(0,1)\to (0,\infty )$, i.e., ${inf}_{\delta \le t\le 1-\delta }q\left(t\right)>0$ for all $\delta \in (0,1/2)$, which are non-decreasing near 0 and non-increasing near 1. For $q\left(t\right)\in Q$, let

$$\mathcal{I}(q,c):={\int }_0^1{\displaystyle \frac{e^{-c{q}^2\left(t\right)/\left(t(1-t)\right)}}{t(1-t)}}dt,c>0.$$

(9)

Assuming that

$$\mathcal{I}(q,c)<\infty \mathrm{for} \mathrm{all} c>0,$$

(10)

it follows from Corollary 5 of [5] that (7) holds true for the weighted Student process $T_n^t(X_1-\mu ,\dots ,X_n-\mu )/q(\left[nt\right]/n)$:

$$\begin{array}{cc}& X\in DAN\mathrm{and}EX=\mu \Rightarrow \\ & h\left({\displaystyle \frac{T_n^t(X_1-\mu ,\dots ,X_n-\mu )}{q\left(\frac{\left[nt\right]}{n}\right)}}\right)\stackrel{\mathcal{D}}{\to }h\left({\displaystyle \frac{W\left(t\right)}{q\left(t\right)}}\right),n\to \infty ,\\ & \mathrm{for} \mathrm{all} \mathrm{functionals} h:D[0,1]\to \mathrm{I}\mathrm{R}\mathrm{that}\mathrm{are}\mathcal{D}\text{-}\mathrm{measurable}\mathrm{and}\rho \text{-}\mathrm{continuous}.\end{array}$$

(11)

Remark 2.

For $q\left(t\right)\in Q$,

$$\left(\mathsf{10}\right) holds true\mathit{if}\mathit{and}\mathit{only}\mathit{if}\underset{t\downarrow 0}{lim}{\displaystyle \frac{\left|W\right(t\left)\right|}{q\left(t\right)}}=\underset{t\uparrow 1}{lim}{\displaystyle \frac{\left|W\right(t\left)\right|}{q\left(t\right)}}=0\mathit{a}.\mathit{s}.$$

(12)

(see Theorem 3.4 of [6]). Using (12) and Khinchine’s local law of the iterated logarithm for the standard Wiener process, one can obtain examples of weight functions in Q satisfying (10), also known as Chibisov–O’Reilly functions, by simply multiplying the function

$$q_1\left(t\right):=\sqrt{t(1-t)loglog{\left(t(1-t)\right)}^{-1}},t\in (0,1),$$

(13)

by an appropriate function on $(0,1)$ that converges to infinity as $t\downarrow 0$ and $t\uparrow 1$.

Among other functionals, the convergence in distribution of (11) can be read with the integral functional $h(·)=h_2(·)$, where

$$h_2\left(f\left(t\right)\right)={\int }_0^1{f}^2\left(t\right)dt,\mathrm{for} \mathrm{any} f\left(t\right)\in D[0,1].$$

(14)

Moreover, such a convergence can also be concluded by using an $L_2$-approximation in probability (see Theorem 3 of [5]), bypassing the FCLT and using a subclass of weight functions $q\left(t\right)$ from Q that satisfy the condition

$${\int }_0^1{\displaystyle \frac{t(1-t)}{q^2\left(t\right)}}dt<\infty .$$

(15)

Thus, in particular, (11) with $h_2(·)$ holds true with the weight function

$$q_{2,a}\left(t\right):={\left(t(1-t)\right)}^a,\mathrm{where} 0\le a<0.5.$$

(16)

The FCLT in (11) also holds true with a linear combination of the projection functional $h_{1,t_0}$ of (8) and the integral functional $h_2$ of (14), in particular, with the following functional:

$$h_3\left(f\left(t\right)\right)=d_1{h}_{1,0.99}\left(f\left(t\right)\right)+d_2{h}_2\left(f\left(t\right)\right),\mathrm{for} \mathrm{any} f\left(t\right)\in D[0,1],$$

(17)

where $d_1,d_2\in \mathrm{I}\mathrm{R}$.

In the present work, we derive FACIs for the mean $\mu$ by using convergence in distribution of the integral functional $h_2$ of (14) and its modification $h_3$ of (17) applied to the Student process $T_n^t(X_1-\mu ,\dots ,X_n-\mu )/q(\left[nt\right]/n)$ weighted with some appropriate functions $q\left(t\right)$.

We study, numerically, the finite-sample performance of the obtained FACIs and compare it to that of the CLT-based CI $I_1$ of (3). For asymmetric distributions in DAN, the finite-sample coverage probabilities of $I_1$ may be lower than its asymptotic confidence level $1-\alpha$. Moreover, $I_1$ may suffer from significant undercoverage for smaller sample sizes and more skewed distributions, especially the ones with fewer moments, while the rates of convergence of the coverage probabilities of $I_1$ to $1-\alpha$ may be slow. We show that our $1-\alpha$ FACIs have higher finite-sample coverage probabilities than, and may be reasonable alternatives to, the CI $I_1$ for right-skewed distributions in DAN, although the FACIs are longer than $I_1$ on average.

2. Main Results

2.1. Preliminaries

In order to derive our FACIs for $\mu$ from (11), the distribution function of the limiting r.v. $h\left(W\left(t\right)/q\left(t\right)\right)$ must either have a closed form expression or be tabulated. This is only the case for $h_2\left(W\left(t\right)\right)$. The rest of the limiting distributions have to be tabulated by us, similarly to the way of [3,4] for their functionals, which is based on the invariance principle approach of [7]. Accordingly, since the limiting distribution in (11) does not depend on the underlying distribution structure of X, we estimate its quantiles by those of the respective empirical distribution of the r.v. $h(T_n^t(X_1-\mu ,X_2-\mu ,\dots ,X_n-\mu )/q(\left[nt\right]/n))$, where the latter distribution is based on 100,000 simulated values that are each computed from an $N(0,1)$ random sample of size 100,000.

After deriving our FACIs, we conduct a detailed numerical study of their finite-sample coverage probabilities and expected lengths in comparison to those of the CLT-based CI $I_1$ of (3). The exact finite-sample coverage probabilities of the FACIs and $I_1$ are approximated by their empirical counterparts, which are based on 10,000 random samples of size n from a distribution of X.

In order to compare the expected length of a FACI (which is also not feasible to obtain in general in a closed form expression) to the expected length of $I_1$, we use the ratios of the empirical expected lengths of a FACI to that of $I_1$ from 10,000 random samples of size n.

In our simulation study, we use seven right-skewed and two symmetric distributions from DAN with different numbers of moments: $Pareto(2,1)$, $Weibull(0.4,1)$, $InverseNormal(7,1)$, Log-logistic(2.1,1), Log-normal$(0,1.3)$, $Frechet\left(2.5\right)$, Half-t(3), $SymPareto(2,1)$, and $t\left(3\right)$. The probability density functions for some of these distributions are provided in Table 1.

Table 1. Probability density functions for some distributions.

Distribution	Probability Density Function
Weibull$(0.4,1)$	$0.4x^{-0.6}{e}^{-x^{0.4}},x>0$
InverseNormal$(7,1)$	${\left(2\pi x^3\right)}^{-0.5}{e}^{-{(x-7)}^2/\left(98x\right)},x>0$
Log-logistic$(2.1,1)$	$2.1x^{1.1}{(1+x^{2.1})}^{-2},x>0$
Log-normal$(0,1.3)$	${\left(1.3x\sqrt{2\pi }\right)}^{-1}{e}^{-{ln}^2\left(x\right)/\left(2{\left(1.3\right)}^2\right)},x>0$
Frechet$\left(2.5\right)$	$2.5x^{-3.5}{e}^{-x^{-2.5}},x>0$
Half-t$\left(3\right)$	$[4/\left(\sqrt{3}\pi \right)]{(1+x^2/3)}^{-2},x>0$
SymPareto$(2,1)$	${\left|x\right|}^{-3},\left|x\right|\ge 1$

2.2. FACIs Based on $h_2$ and $h_3$

Substituting $h(·)=h_2(·)$ of (14) into (11), we have the following, as $n\to \infty$:

$${\int }_0^1{\displaystyle \frac{{\left(T_n^t(X_1-\mu ,\dots ,X_n-\mu )\right)}^2}{q^2\left(\frac{\left[nt\right]}{n}\right)}}dt\stackrel{\mathcal{D}}{\to }{\int }_0^1{\displaystyle \frac{W^2\left(t\right)}{q^2\left(t\right)}}dt.$$

(18)

Since $T_n^t(X_1,\dots ,X_n)$ is a random step function on $[0,1]$ as in (6), the convergence in (18) amounts to

$${\displaystyle \frac{1}{s_n^2{n}^2}}\sum _{k=1}^{n-1}{\displaystyle \frac{{\left({\sum }_{i=1}^k(X_i-\mu )\right)}^2}{q^2\left(\frac{k}{n}\right)}}\stackrel{\mathcal{D}}{\to }{\int }_0^1{\displaystyle \frac{W^2\left(t\right)}{q^2\left(t\right)}}dt.$$

(19)

We consider (19) with two weight functions $q\left(t\right)=q_{2,a}\left(t\right)={\left(t(1-t)\right)}^a$ from Q with $a=0$ and 0.4, where both $q_{2,0}\left(t\right)$ and $q_{2,0.4}\left(t\right)$ are Chibisov–O’Reilly functions satisfying (10) as well as (15). The cumulative distribution function of ${\int }_0^1(W^2\left(t\right)/q^2\left(t\right))dt$ has only been available for $q\left(t\right)\equiv 1$ (see [8]). We retabulated the quantiles of ${\int }_0^1{W}^2\left(t\right)dt$ and they closely match the ones in the latter paper. We also obtained the required quantiles of ${\int }_0^1(W^2\left(t\right)/q_{2,0.4}^2\left(t\right))dt$. All these quantiles are listed in Table 2.

Table 2. The $1-\alpha$ quantile of ${\displaystyle {\int }_0^1\frac{W^2\left(t\right)}{{\left(t(1-t)\right)}^{2a}}dt}$.

	$\mathit{\alpha }=0.10$	$\mathit{\alpha }=0.05$	$\mathit{\alpha }=0.02$
$a=0$	1.198	1.668	2.292
$a=0.4$	10.734	14.888	20.690

Let $\alpha \in (0,1)$ and define $b>0$ such that $P\left({\int }_0^1(W^2\left(t\right)/q^2\left(t\right))dt\le b\right)$ = $1-\alpha$. It follows from (19) that

$$P\left({\displaystyle \frac{1}{s_n^2{n}^2}}\sum _{k=1}^{n-1}{\displaystyle \frac{{\left({\sum }_{i=1}^k(X_i-\mu )\right)}^2}{q^2\left(\frac{k}{n}\right)}}\le b\right)\stackrel{\mathcal{D}}{\to }1-\alpha ,n\to \infty .$$

(20)

The quadratic inequality in $\mu$ under the probability sign in (20) reads

$${\mu }^2\sum _{k=1}^{n-1}{\displaystyle \frac{k^2}{q^2\left(\frac{k}{n}\right)}}-2\mu \sum _{k=1}^{n-1}{\displaystyle \frac{k{\sum }_{i=1}^k{X}_i}{q^2\left(\frac{k}{n}\right)}}+\sum _{k=1}^{n-1}{\displaystyle \frac{{\left({\sum }_{i=1}^k{X}_i\right)}^2}{q^2\left(\frac{k}{n}\right)}}-b{n}^2{s}_n^2\le 0.$$

(21)

This leads to the following $1-\alpha$ FACI for $\mu$:

$$I_{2,q}:=\left[{\displaystyle \frac{2{\sum }_{k=1}^{n-1}\frac{k{\sum }_{i=1}^k{X}_i}{q^2\left(\frac{k}{n}\right)}-\sqrt{D_n}}{2{\sum }_{k=1}^{n-1}\frac{k^2}{q^2\left(\frac{k}{n}\right)}}},{\displaystyle \frac{2{\sum }_{k=1}^{n-1}\frac{k{\sum }_{i=1}^k{X}_i}{q^2\left(\frac{k}{n}\right)}+\sqrt{D_n}}{2{\sum }_{k=1}^{n-1}\frac{k^2}{q^2\left(\frac{k}{n}\right)}}}\right],$$

(22)

where $D_n$ is the discriminant of the quadratic function of (21):

$$D_n:=4{\left(\sum _{k=1}^{n-1}{\displaystyle \frac{k{\sum }_{i=1}^k{X}_i}{q^2\left(\frac{k}{n}\right)}}\right)}^2-4\sum _{k=1}^{n-1}{\displaystyle \frac{k^2}{q^2\left(\frac{k}{n}\right)}}\left(\sum _{k=1}^{n-1}{\displaystyle \frac{{\left({\sum }_{i=1}^k{X}_i\right)}^2}{q^2\left(\frac{k}{n}\right)}}-b{n}^2{s}_n^2\right),$$

(23)

where $D_n>0$ in all the Monte Carlo simulations of this paper. The FACI $I_{2,q}$ with $q\left(t\right)\equiv 1$ reduces to one of the FACIs obtained in [3,4] from (7).

Next, we similarly derive the FACIs based on the convergence in distribution in (11) with the Chibisov–O’Reilly weight function

$$q_3\left(t\right)={\left[t(1-t)\right]}^{0.25}{[loglog{\left(t(1-t)\right)}^{-1}]}^{-0.5}$$

(24)

satisfying (10) and with $h(·)=h_3(·)$ of (17), with two pairs of values for $(d_1,d_2)$,

$$(0.3·q_3\left(0.99\right),0.7)=(0.07652005,0.7)\mathrm{and}(0.5·q_3\left(0.99\right),0.5)=(0.1275334,0.5).$$

(25)

Accordingly, for $\alpha \in (0,1)$ and e defined as $P\left(h_3(W\left(t\right)/q_3\left(t\right))\le e\right)$ = $1-\alpha$, we have

$$P\left(d_1{\displaystyle \frac{{\sum }_{i=1}^{\left[0.99n\right]}(X_i-\mu )}{s_n\sqrt{n}{q}_3\left(\frac{\left[0.99n\right]}{n}\right)}}+{\displaystyle \frac{d_2}{s_n^2{n}^2}}\sum _{k=1}^{n-1}{\displaystyle \frac{{\left({\sum }_{i=1}^k(X_i-\mu )\right)}^2}{q_3^2\left(\frac{k}{n}\right)}}\le e\right)\stackrel{\mathcal{D}}{\to }1-\alpha ,n\to \infty .$$

(26)

Solving the quadratic inequality in $\mu$ under the probability sign in (26) yields the following $1-\alpha$ FACI for $\mu$:

$$I_{3,q}:=\left[{\displaystyle \frac{\frac{d_1{s}_{n}n\sqrt{n}\left[0.99n\right]}{q_3\left(\frac{\left[0.99n\right]}{n}\right)}+2d_2{\sum }_{k=1}^{n-1}\frac{k{\sum }_{i=1}^k{X}_i}{q_3^2\left(\frac{k}{n}\right)}-\sqrt{{\Delta }_n}}{2d_2{\sum }_{k=1}^{n-1}\frac{k^2}{q_3^2\left(\frac{k}{n}\right)}}},{\displaystyle \frac{\frac{d_1{s}_{n}n\sqrt{n}\left[0.99n\right]}{q_3\left(\frac{\left[0.99n\right]}{n}\right)}+2d_2{\sum }_{k=1}^{n-1}\frac{k{\sum }_{i=1}^k{X}_i}{q_3^2\left(\frac{k}{n}\right)}+\sqrt{{\Delta }_n}}{2d_2{\sum }_{k=1}^{n-1}\frac{k^2}{q_3^2\left(\frac{k}{n}\right)}}}\right],$$

(27)

where

$$\begin{array}{ccc}\hfill {\Delta }_n:=& & {\left({\displaystyle \frac{d_1{s}_{n}n\sqrt{n}\left[0.99n\right]}{q_3\left(\frac{\left[0.99n\right]}{n}\right)}}+2d_2\sum _{k=1}^{n-1}{\displaystyle \frac{k{\sum }_{i=1}^k{X}_i}{q_3^2\left(\frac{k}{n}\right)}}\right)}^2\hfill \\ & & -4d_2\sum _{k=1}^{n-1}{\displaystyle \frac{k^2}{q_3^2\left(\frac{k}{n}\right)}}\left({\displaystyle \frac{d_1{s}_{n}n\sqrt{n}{\sum }_{i=1}^{\left[0.99n\right]}{X}_i}{q_3\left(\frac{\left[0.99n\right]}{n}\right)}}+d_2\sum _{k=1}^{n-1}{\displaystyle \frac{{\left({\sum }_{i=1}^k{X}_i\right)}^2}{q_3^2\left(\frac{k}{n}\right)}}-e{s}_n^2{n}^2\right),\hfill \end{array}$$

(28)

where, for the discriminant ${\Delta }_n$, we have ${\Delta }_n>0$ in all the Monte Carlo simulations in the upcoming tables, for both pairs of values for $(d_1,d_2)$ of (25). Our simulated values of $1-\alpha$ quantiles e of $h_3(W\left(t\right)/q_3\left(t\right))$ can be found in Table 3.

Table 3. The $1-\alpha$ quantile of ${\displaystyle h_3\left(\frac{W\left(t\right)}{q_3\left(t\right)}\right)=d_1\frac{W\left(0.99\right)}{q_3\left(0.99\right)}+d_2{\int }_0^1\frac{W^2\left(t\right)}{q_3^2\left(t\right)}dt}$.

	$\mathit{\alpha }=0.10$	$\mathit{\alpha }=0.05$	$\mathit{\alpha }=0.02$
$(d_1,d_2)=(0.07652005,0.7)$	2.414540	3.360469	4.661709
$(d_1,d_2)=(0.12753340,0.5)$	1.875837	2.652915	3.697833

Hereafter, we denote the FACIs $I_{3,q}$ of (27) with $(d_1,d_2)=(0.07652005,0.7)$ and $(0.1275334,0.5)$ by $I_3$ and $I_4$, respectively.

In the following Table 4, we numerically compare the finite-sample properties of $I_1$, $I_{2,q}$ with the weight functions $q\left(t\right)\equiv 1$ and $q\left(t\right)={\left(t(1-t)\right)}^{0.4}$, $I_3$, and $I_4$ at the $1-\alpha =0.95$ asymptotic confidence level. Table 4 lists the values of the empirical coverage probabilities ${\widehat{CP}}_i$ of $I_i$, $i=1$, 3, and 4, and ${\widehat{CP}}_{2,q}$ of $I_{2,q}$, together with the corresponding Monte Carlo standard errors in parentheses, all multiplied by 100%, as well as the values of ${\widehat{r}}_{2,q}$ and ${\widehat{r}}_i$, the ratios of the empirical expected lengths of $I_{2,q}$ and $I_1$, and $I_i$ and $I_1$, $i=3$ and 4. The Monte Carlo standard errors for ${\widehat{r}}_{2,q}$ (with both $q\left(t\right)\equiv 1$ and $q\left(t\right)={\left(t(1-t)\right)}^{0.4}$), ${\widehat{r}}_3$, and ${\widehat{r}}_4$ are in the range of $[0.000154,0.000504]$ across all nine distributions from DAN and the sample sizes $n=50$, 100, 300, 500, and 1000.

Table 4. Comparison of $I_1$, $I_{2,q}$ with $q\left(t\right)={\left(t(1-t)\right)}^a$, $I_3$, and $I_4$ at $95\%$ confidence level.

			FACI Coverages				Ratios of Average Lengths
Distribution	$\mathit{n}$	${\widehat{\mathit{CP}}}_{\mathbf{1}}·\mathbf{100}\%$	$\begin{array}{cc}{\widehat{\mathit{CP}}}_{\mathbf{2},\mathit{q}}·\mathbf{100}\%& \\ \mathit{a}=\mathbf{0}\end{array}$	$\begin{array}{cc}{\widehat{\mathit{CP}}}_{\mathbf{2},\mathit{q}}·\mathbf{100}\%& \\ \mathit{a}=\mathbf{0}.\mathbf{4}\end{array}$	${\widehat{\mathit{CP}}}_{\mathbf{3}}·\mathbf{100}\%$	${\widehat{\mathit{CP}}}_{\mathbf{4}}·\mathbf{100}\%$	$\begin{array}{cc}{\widehat{\mathit{r}}}_{\mathbf{2},\mathit{q}}& \\ \mathit{a}=\mathbf{0}\end{array}$	$\begin{array}{cc}{\widehat{\mathit{r}}}_{\mathbf{2},\mathit{q}}& \\ \mathit{a}=\mathbf{0}.\mathbf{4}\end{array}$	${\widehat{\mathit{r}}}_{\mathbf{3}}$	${\widehat{\mathit{r}}}_{\mathbf{4}}$
$Pareto(2,1)$	50	81.41 (0.39)	83.40 (0.37)	88.49 (0.32)	88.65 (0.32)	92.05 (0.27)	1.124	1.354	1.253	1.339
	100	83.74 (0.37)	85.66 (0.35)	89.51 (0.31)	89.94 (0.30)	93.16 (0.25)	1.115	1.274	1.183	1.272
	300	87.05 (0.34)	88.75 (0.32)	91.10 (0.28)	91.83 (0.27)	94.64 (0.23)	1.109	1.187	1.115	1.196
	500	88.06 (0.32)	89.18 (0.31)	91.14 (0.28)	91.81 (0.27)	94.77 (0.22)	1.108	1.157	1.094	1.174
	1000	89.82 (0.30)	90.81 (0.29)	92.12 (0.27)	92.74 (0.26)	95.34 (0.21)	1.107	1.124	1.074	1.151
$Weibull(0.4,1)$	50	81.51 (0.39)	83.04 (0.38)	87.92 (0.33)	87.99 (0.33)	90.89 (0.29)	1.124	1.354	1.252	1.338
	100	86.04 (0.35)	87.77 (0.33)	90.98 (0.29)	91.27 (0.28)	93.83 (0.24)	1.115	1.274	1.183	1.273
	300	90.19 (0.30)	90.90 (0.29)	93.21 (0.25)	93.37 (0.25)	95.48 (0.21)	1.109	1.187	1.115	1.196
	500	91.17 (0.28)	92.03 (0.27)	93.84 (0.24)	93.82 (0.24)	95.90 (0.20)	1.108	1.157	1.095	1.174
	1000	92.96 (0.26)	93.31 (0.25)	94.90 (0.22)	94.70 (0.22)	95.92 (0.20)	1.107	1.124	1.074	1.151
$InverseNormal(7,1)$	50	82.52 (0.38)	83.97 (0.37)	88.21 (0.32)	88.38 (0.32)	90.84 (0.29)	1.124	1.354	1.253	1.338
	100	87.13 (0.33)	88.26 (0.32)	91.52 (0.28)	91.88 (0.27)	94.20 (0.23)	1.115	1.274	1.183	1.271
	300	91.80 (0.27)	92.10 (0.27)	94.58 (0.23)	94.62 (0.23)	96.02 (0.20)	1.109	1.187	1.114	1.195
	500	92.93 (0.26)	93.52 (0.25)	95.19 (0.21)	95.04 (0.22)	96.25 (0.19)	1.108	1.157	1.094	1.174
	1000	93.78 (0.24)	94.12 (0.24)	95.60 (0.21)	95.21 (0.21)	96.15 (0.19)	1.107	1.124	1.074	1.151
Log-logistic(2.1,1)	50	84.44 (0.36)	86.48 (0.34)	91.34 (0.28)	91.51 (0.28)	94.40 (0.23)	1.124	1.354	1.253	1.339
	100	87.46 (0.33)	88.93 (0.31)	92.76 (0.26)	93.11 (0.25)	95.79 (0.20)	1.115	1.274	1.183	1.271
	300	89.87 (0.30)	90.96 (0.29)	93.25 (0.25)	93.67 (0.24)	96.11 (0.19)	1.109	1.187	1.115	1.196
	500	90.51 (0.29)	91.44 (0.28)	93.56 (0.25)	94.01 (0.24)	96.03 (0.20)	1.108	1.157	1.094	1.173
	1000	91.07 (0.29)	92.13 (0.27)	93.50 (0.25)	93.63 (0.24)	95.73 (0.20)	1.107	1.124	1.074	1.151
Log-normal(0,1.3)	50	85.31 (0.35)	86.86 (0.34)	91.08 (0.29)	91.30 (0.28)	93.71 (0.24)	1.124	1.354	1.253	1.339
	100	87.54 (0.33)	88.64 (0.32)	92.31 (0.27)	92.41 (0.26)	94.78 (0.22)	1.115	1.274	1.183	1.272
	300	91.70 (0.28)	92.39 (0.27)	94.64 (0.23)	94.70 (0.22)	96.19 (0.19)	1.109	1.187	1.115	1.196
	500	92.17 (0.27)	92.90 (0.26)	94.89 (0.22)	94.73 (0.22)	96.24 (0.19)	1.108	1.157	1.094	1.173
	1000	93.25 (0.25)	93.82 (0.24)	95.35 (0.21)	95.17 (0.21)	96.43 (0.19)	1.107	1.124	1.074	1.151
$Frechet\left(2.5\right)$	50	87.93 (0.33)	89.24 (0.31)	93.45 (0.25)	93.52 (0.25)	95.64 (0.20)	1.124	1.354	1.253	1.339
	100	89.61 (0.31)	90.57 (0.29)	94.10 (0.24)	94.27 (0.23)	96.36 (0.19)	1.115	1.274	1.183	1.272
	300	91.90 (0.27)	92.13 (0.27)	94.75 (0.22)	94.82 (0.22)	96.45 (0.19)	1.109	1.187	1.114	1.196
	500	92.22 (0.27)	92.91 (0.26)	94.85 (0.22)	94.94 (0.22)	96.76 (0.18)	1.108	1.157	1.094	1.173
	1000	93.54 (0.25)	93.85 (0.24)	95.41 (0.21)	95.36 (0.21)	96.80 (0.18)	1.107	1.124	1.074	1.151
Half-t(3)	50	90.98 (0.29)	91.77 (0.27)	95.75 (0.20)	95.62 (0.20)	97.30 (0.16)	1.124	1.354	1.253	1.339
	100	91.99 (0.27)	92.86 (0.26)	96.11 (0.19)	96.03 (0.20)	97.57 (0.15)	1.115	1.274	1.183	1.272
	300	93.66 (0.24)	93.92 (0.24)	96.33 (0.19)	96.09 (0.19)	97.06 (0.17)	1.109	1.187	1.115	1.196
	500	93.89 (0.24)	94.13 (0.24)	96.08 (0.19)	95.77 (0.20)	96.78 (0.18)	1.108	1.157	1.095	1.174
	1000	94.11 (0.24)	94.51 (0.23)	96.06 (0.19)	95.49 (0.21)	96.32 (0.19)	1.107	1.124	1.074	1.151
SymPareto(2,1)	50	95.12 (0.22)	95.45 (0.21)	99.10 (0.09)	98.17 (0.13)	97.86 (0.14)	1.124	1.354	1.253	1.339
	100	95.51 (0.21)	95.34 (0.21)	98.56 (0.12)	97.41 (0.16)	97.10 (0.17)	1.115	1.274	1.183	1.271
	300	95.33 (0.21)	95.26 (0.21)	97.96 (0.14)	96.69 (0.18)	96.33 (0.19)	1.109	1.187	1.115	1.196
	500	95.62 (0.20)	95.67 (0.20)	97.89 (0.14)	96.62 (0.18)	96.37 (0.19)	1.108	1.157	1.094	1.173
	1000	95.35 (0.21)	95.20 (0.21)	97.12 (0.17)	96.04 (0.20)	95.96 (0.20)	1.106	1.123	1.074	1.151
$t\left(3\right)$	50	94.80 (0.22)	95.29 (0.21)	98.99 (0.10)	98.09 (0.14)	97.84 (0.15)	1.123	1.354	1.253	1.339
	100	95.02 (0.22)	95.02 (0.22)	98.40 (0.13)	97.37 (0.16)	97.03 (0.17)	1.115	1.274	1.183	1.272
	300	95.11 (0.22)	95.22 (0.21)	97.77 (0.15)	96.49 (0.18)	96.14 (0.19)	1.109	1.187	1.115	1.196
	500	94.85 (0.22)	95.17 (0.21)	97.25 (0.16)	95.98 (0.20)	95.96 (0.20)	1.108	1.157	1.094	1.174
	1000	94.87 (0.22)	94.86 (0.22)	96.60 (0.18)	95.36 (0.21)	95.40 (0.21)	1.107	1.124	1.074	1.151

Based on Table 4, we make the following observations in Remarks 3–6.

Remark 3.

All four FACIs exhibit trade-offs between their respective empirical coverage probabilities and expected lengths. Indeed, they have mostly higher coverage probabilities than those of $I_1$, while being longer than $I_1$ on average. More precisely, the ratios ${\widehat{r}}_{2,q}$, ${\widehat{r}}_3$, and ${\widehat{r}}_4$ are greater than one, being at most $1.354$ for $n=50$ and only at most $1.151$ for $n=1000$, and it appears these ratios do not depend on a distribution and decrease as n increases. As for the empirical finite-sample coverage probabilities, ${\widehat{CP}}_{2,q}$, ${\widehat{CP}}_3$, and ${\widehat{CP}}_4$ are higher than ${\widehat{CP}}_1$, except for a small number of larger samples for ${\widehat{CP}}_{2,q}$ when $q\left(t\right)\equiv 1$. For all the asymmetric distributions, ${\widehat{CP}}_1·100\%$ is lower than $(1-\alpha )100\%=95\%$ (more so for smaller n) and converges to 95% slowly. For the $Pareto(2,1)$ samples, $I_1$ has the most severe undercoverage, where the rate of convergence of ${\widehat{CP}}_1·100\%$ is so slow that for $n=1000$, ${\widehat{CP}}_1·100\%$ is still below 95%, at 89.82%. Moreover, as seen from Table 4 (for the 95% confidence level) and Table A1 and Table A2 from Appendix A with similar comparisons of $I_1$, $I_{2,q}$, $I_3$, and $I_4$ at the 90% and 98% confidence levels, the four FACIs provide coverage improvements of various degree, with the most significant one being the case of $I_4$, and thus may be reasonable alternatives to $I_1$ for right-skewed distributions. On the other hand, for the symmetric $SymPareto(2,1)$ and $t\left(3\right)$ distributions, the coverage ${\widehat{CP}}_1·100\%$ is very close to the asymptotic confidence level for all our sample sizes and does not require an improvement. We note in passing that the ratios of the empirical expected lengths ${\widehat{r}}_{2,q}$, ${\widehat{r}}_3$, and ${\widehat{r}}_4$ obtained in Table A1 and Table A2 are very similar to the ones in Table 4, and that the corresponding Monte Carlo standard errors are in the range of $[0.000219,0.000715]$ in Table A1 and $[0.000109,0.000359]$ in Table A2.

Remark 4.

Various Berry–Esseen upper bounds for

$$E_n:=\underset{x}{sup}\left|P\left(\sum _{i=1}^n{X}_i{\left(\sum _{i=1}^n{X}_i^2\right)}^{-1/2}\le x\right)-P(N(0,1)\le x)\right|$$

(29)

were established by a number of authors under different conditions on the sequence $\{X_i,i\ge 1\}$ (see section 3 of [9]). For example, assuming that $X_1,\dots ,X_n$ are independent but not necessarily identically distributed r.v.s with zero means and finite variances, ref. [10] proved that

$$E_n\le {\delta }_n,$$

(30)

where

$${\delta }_n:={\displaystyle \frac{10.2}{B_n^2}}\sum _{i=1}^{n}E\left(X_i^2{11}_{\left\{\right|X_i|\ge B_n/2\}}\right)+{\displaystyle \frac{25}{B_n^3}}\sum _{i=1}^{n}E\left(\right|X_i{|}^3{11}_{\left\{\right|X_i|\le B_n/2\}}),with B_n^2:=\sum _{i=1}^{n}E{X}_i^2.$$

(31)

When $\{X_i,i\ge 1\}$ are i.i.d. r.v.s, the expression ${\delta }_n$ of (31) converges to zero and becomes a uniform upper bound of the rate of convergence in the self-normalized CLT.

Assuming that $\{X,X_i,i\ge 1\}$ are i.i.d. r.v.s with $X\in \mathit{DAN}$ and $EX=0$ and defining

$${\tilde{b}}_n:=sup\{x:n{x}^{-2}E\left(X^2{11}_{\left\{\right|X|\le x\}}\right)\ge 1\}$$

(32)

and

$${\tilde{\delta }}_n:=nP\left(\right|X|>{\tilde{b}}_n)+n{\tilde{b}}_n^{-1}\left|E\left(X{11}_{\left\{\right|X|\le {\tilde{b}}_n\}}\right)\right|+n{\tilde{b}}_n^{-3}{E\left(\right|X|}^3{11}_{\left\{\right|X|\le {\tilde{b}}_n\}}),$$

(33)

ref. [11] showed that

$$E_n\le A{\tilde{\delta }}_n,$$

(34)

where A is an absolute constant and ${\tilde{\delta }}_n\to 0$, as $n\to \infty$.

The bounds ${\delta }_n$ and $A{\tilde{\delta }}_n$ can also be used to estimate the rate of convergence in the Studentized CLT (2) and how fast the finite-sample coverage probability ${\widehat{CP}}_1$ of the corresponding CI $I_1$ of (3) converges to the asymptotic level $1-\alpha$.

The rate of convergence in the self-normalized and Studentized CLTs and hence that of ${\widehat{CP}}_1$ to $1-\alpha$ are known to be in general faster for symmetric distributions and slower for asymmetric distributions, more so when the latter ones have fewer moments, larger skewness, or both (see, for example, [11] and references therein). This has properly been seen for ${\widehat{CP}}_1$ for the symmetric $SymPareto(2,1)$ and $t\left(3\right)$ distributions from Table 4, Table A1, and Table A2. On the other hand, the asymmetric version of $SymPareto(2,1)$, the ordinary $Pareto(2,1)$ distribution with an infinite variance, has the lowest ${\widehat{CP}}_1$ in these tables and converges slowly. Similarly, for the asymmetric Half-$t\left(3\right)$, its ${\widehat{CP}}_1$ is lower compared to that of $t\left(3\right)$. For the asymmetric $Log$-$logistic(2.1,1)$, $Frechet\left(2.5\right)$, and Half-$t\left(3\right)$ distributions, we can see from Table 4, Table A1, and Table A2 that their respective ${\widehat{CP}}_1$ values decrease, as so do their numbers of finite moments.

Interestingly enough, the behavior of the finite-sample coverage probabilities of our FACIs closely resembles the just described one of ${\widehat{CP}}_1$, as evident from Table 4, Table A1, and Table A2. Moreover, in Table 5, not only ${\widehat{CP}}_1$ but also the four FACI coverages show nearly the same ordering as those of the values of ${\delta }_n$ and ${\tilde{\delta }}_n$ computed for our distributions for $n=100$. The orderings of the values of ${\delta }_n$, ${\tilde{\delta }}_n$, ${\widehat{CP}}_1$, ${\widehat{CP}}_{2,q}$, ${\widehat{CP}}_3$, and ${\widehat{CP}}_4$ vary insignificantly for other sample sizes, while still being in close agreement with each other. Thus, for the right-skewed distributions with larger values of ${\delta }_n$ (if defined) or ${\tilde{\delta }}_n$, one may like to rely on one of the FACIs to estimate the mean μ, as $I_1$ may have undesirably low coverage probabilities.

Table 5. ${\delta }_{100}$, ${\tilde{\delta }}_{100}$, and coverages ${\widehat{CP}}_1$, ${\widehat{CP}}_{2,q}$, ${\widehat{CP}}_3$, ${\widehat{CP}}_4$ ($·100\%$) for $n=100$ at 95% confidence level.

				${\widehat{\mathit{CP}}}_{2,\mathit{q}}·100\%$
Distribution	${\mathit{\delta }}_{\mathbf{100}}$	${\tilde{\mathit{\delta }}}_{\mathbf{100}}$	${\widehat{\mathit{CP}}}_{\mathbf{1}}·\mathbf{100}\%$	$\mathit{q}\left(\mathit{t}\right)\equiv \mathbf{1}$	$\mathit{q}\left(\mathit{t}\right)={\left(\mathit{t}(\mathbf{1}-\mathit{t})\right)}^{\mathbf{0}.\mathbf{4}}$	${\widehat{\mathit{CP}}}_{\mathbf{3}}·\mathbf{100}\%$	${\widehat{\mathit{CP}}}_{\mathbf{4}}·\mathbf{100}\%$
Pareto(2,1)	NA	1.590	83.74	85.66	89.51	89.94	93.16
Weibull(0.4,1)	8.700	1.429	86.04	87.77	90.98	91.27	93.83
InverseNormal(7,1)	8.574	1.247	87.13	88.26	91.52	91.88	94.20
Log-logistic(2.1,1)	9.365	1.232	87.46	88.93	92.76	93.11	95.79
Log-normal(0,1.3)	8.310	1.138	87.54	88.64	92.31	92.41	94.78
Frechet(2.5)	8.215	0.974	89.61	90.57	94.10	94.27	96.36
Half-t(3)	7.922	0.603	91.99	92.86	96.11	96.03	97.57
SymPareto(2,1)	NA	0.451	95.51	95.34	98.56	97.41	97.10
t(3)	6.186	0.328	95.02	95.02	98.40	97.37	97.03

Remark 5.

Upon further inspection of Table 4, Table A1, and Table A2, for the asymmetric distributions, we conclude that the $I_3$ FACI has the finite-sample coverages that are similar to those of $I_{2,q}$ with $q\left(t\right)={\left(t(1-t)\right)}^{0.4}$, being shorter than the latter FACI on average. On the other hand, $I_4$ has a similar average length to that of $I_{2,q}$ with $q\left(t\right)={\left(t(1-t)\right)}^{0.4}$, but has higher finite-sample coverages than the latter FACI. Thus, $I_3$ and $I_4$ improve on $I_{2,q}$ with $q\left(t\right)={\left(t(1-t)\right)}^{0.4}$ in terms of the expected length and finite-sample coverage probability, respectively. In addition to Table 4, for our right-skewed distributions, we compare the 95% confidence level $I_3$ and $I_4$ FACIs to their respective $I_1$ counterparts of the same average lengths (and thus of higher nominal levels, since the ratio of the average lengths ${\widehat{r}}_3$ and ${\widehat{r}}_4$ reported in Table 4 are greater than one). Thus, in Table 6, for each sample size used in Table 4, we find the respective $I_1$ matches for $I_3$ and $I_4$ (so that the absolute difference between the average lengths of the FACI and $I_1$ divided by the average length of the FACI is less than or equal to 0.00001) and report the finite-sample coverages ${\widehat{CP}}_{1,\mathrm{adj}}\times 100\%$ of the thus adjusted $I_1$ together with their nominal asymptotic coverages and the difference in the finite-sample coverage between the 95% confidence level FACI and the adjusted $I_1$ CI. Both $I_3$ and $I_4$ have higher coverages than their adjusted CLT-based $I_1$ counterparts, especially for smaller sample sizes and Pareto(2,1), with a few exceptions for large sample sizes where $I_3$ and $I_4$ do not provide more overcoverage than the respective adjusted $I_1$ CIs. Similar comparisons with analogous conclusions are performed for the 90% and 98% confidence level $I_3$ and $I_4$ FACIs with their respective equal-average-length $I_1$ CIs in Table A3 and Table A4 from Appendix A.

Table 6. Comparison of coverages ${\widehat{CP}}_3·100\%$ of $I_3$ and ${\widehat{CP}}_4·100\%$ of $I_4$ (both at 95% confidence level) with coverages ${\widehat{CP}}_{1,\mathrm{adj}}$ of equal-average-length adjusted $I_1$ counterparts (at higher nominal levels) for right-skewed distributions.

		${\mathit{I}}_3$				${\mathit{I}}_4$
Distribution	$\mathit{n}$	$\begin{array}{cc}{\mathit{I}}_{\mathbf{3}}& \\ \mathbf{avg}.\mathbf{len}.& \end{array}$	$\begin{array}{cc}{\widehat{\mathit{CP}}}_{\mathbf{3}}& \\ \times \mathbf{100}\%& \end{array}$	$\begin{array}{cc}{\widehat{\mathit{CP}}}_{\mathbf{1},\mathbf{adj}}\left(\mathbf{Nominal}\right)& \\ \times \mathbf{100}\%& \end{array}$	$\begin{array}{cc}({\widehat{\mathit{CP}}}_{\mathbf{3}}-{\widehat{\mathit{CP}}}_{\mathbf{1},\mathbf{adj}})& \\ \times \mathbf{100}\%& \end{array}$	$\begin{array}{cc}{\mathit{I}}_{\mathbf{4}}& \\ \mathbf{avg}.\mathbf{len}.& \end{array}$	$\begin{array}{cc}{\widehat{\mathit{CP}}}_{\mathbf{4}}& \\ \times \mathbf{100}\%& \end{array}$	$\begin{array}{cc}{\widehat{\mathit{CP}}}_{\mathbf{1},\mathbf{adj}}\left(\mathbf{Nominal}\right)& \\ \times \mathbf{100}\%& \end{array}$	$\begin{array}{cc}({\widehat{\mathit{CP}}}_{\mathbf{4}}-{\widehat{\mathit{CP}}}_{\mathbf{1},\mathbf{adj}})& \\ \times \mathbf{100}\%& \end{array}$
Pareto(2,1)	50	1.334	88.65	87.08 (98.595)	1.57	1.426	92.05	88.59 (99.133)	3.46
	100	0.966	89.94	87.82 (97.959)	2.12	1.038	93.16	89.71 (98.734)	3.45
	300	0.614	91.83	90.20 (97.108)	1.63	0.658	94.64	91.74 (98.092)	2.90
	500	0.481	91.81	90.13 (96.803)	1.68	0.516	94.77	91.84 (97.857)	2.93
	1000	0.357	92.74	91.54 (96.472)	1.20	0.382	95.34	92.93 (97.591)	2.41
Weibull(0.4,1)	50	6.001	87.99	86.62 (98.590)	1.37	6.412	90.89	87.84 (99.128)	3.05
	100	4.249	91.27	89.88 (97.962)	1.39	4.570	93.83	91.14 (98.738)	2.69
	300	2.479	93.37	92.50 (97.109)	0.87	2.660	95.48	93.66 (98.092)	1.82
	500	1.917	93.82	93.10 (96.806)	0.72	2.056	95.90	94.42 (97.856)	1.48
	1000	1.355	94.70	94.42 (96.476)	0.28	1.452	95.92	95.77 (97.596)	0.15
InverseNormal(7,1)	50	11.086	88.38	87.00 (98.591)	1.38	11.845	90.84	88.02 (99.129)	2.82
	100	7.857	91.88	90.55 (97.955)	1.33	8.446	94.20	91.68 (98.730)	2.52
	300	4.534	94.62	93.78 (97.102)	0.84	4.864	96.02	94.89 (98.086)	1.13
	500	3.469	95.04	94.52 (96.804)	0.52	3.721	96.25	95.53 (97.856)	0.72
	1000	2.434	95.21	95.10 (96.472)	0.11	2.609	96.15	96.19 (97.593)	−0.04
Log-logistic(2.1,1)	50	1.252	91.51	90.09 (98.594)	1.42	1.338	94.40	91.41 (99.132)	2.99
	100	0.918	93.11	91.53 (97.955)	1.58	0.987	95.79	92.96 (98.727)	2.83
	300	0.548	93.67	92.38 (97.109)	1.29	0.588	96.11	93.82 (98.093)	2.29
	500	0.439	94.01	92.63 (96.804)	1.38	0.471	96.03	94.02 (97.855)	2.01
	1000	0.319	93.63	92.84 (96.475)	0.79	0.342	95.73	94.34 (97.596)	1.39
Log-normal(0,1.3)	50	2.808	91.30	89.88 (98.596)	1.42	3.001	93.71	91.06 (99.134)	2.65
	100	1.962	92.41	91.31 (97.959)	1.10	2.109	94.78	92.69 (98.735)	2.09
	300	1.153	94.70	93.92 (97.107)	0.78	1.237	96.19	95.02 (98.090)	1.17
	500	0.880	94.73	94.03 (96.803)	0.70	0.944	96.24	95.21 (97.854)	1.03
	1000	0.626	95.17	94.74 (96.476)	0.43	0.671	96.43	96.03 (97.596)	0.40
Frechet(2.5)	50	0.813	93.52	92.37 (98.593)	1.15	0.869	95.64	93.50 (99.132)	2.14
	100	0.564	94.27	93.29 (97.961)	0.98	0.607	96.36	94.45 (98.736)	1.91
	300	0.330	94.82	94.09 (97.105)	0.73	0.354	96.45	95.27 (98.090)	1.18
	500	0.255	94.94	94.22 (96.801)	0.72	0.274	96.76	95.35 (97.850)	1.41
	1000	0.182	95.36	94.87 (96.472)	0.49	0.195	96.80	95.98 (97.592)	0.82
Half-t(3)	50	0.834	95.62	94.98 (98.596)	0.64	0.891	97.30	95.80 (99.134)	1.50
	100	0.571	96.03	95.42 (97.959)	0.61	0.614	97.57	96.52 (98.735)	1.05
	300	0.321	96.09	95.80 (97.109)	0.29	0.344	97.06	96.85 (98.093)	0.21
	500	0.246	95.77	95.66 (96.807)	0.11	0.264	96.78	96.80 (97.858)	−0.02
	1000	0.172	95.49	95.65 (96.474)	−0.16	0.185	96.32	96.91 (97.596)	−0.59

Remark 6.

As per Table 4, Table A1, and Table A2, the FACIs $I_{2,q}$ with $q\left(t\right)={\left(t(1-t)\right)}^{0.4}$, $I_3$, and $I_4$ exhibit a more expensive, yet, in case of the right-skewed distributions with higher ${\delta }_{100}$ (if defined) or ${\tilde{\delta }}_{100}$, potentially desirable trade-off between their finite-sample coverage probabilities and expected lengths as compared to the FACI $I_{2,q}$ with $q\left(t\right)\equiv 1$ that was obtained from the FCLT in (7) for the unweighted Student process in [3,4]. Moreover, the finite-sample coverage probabilities of other FACIs considered in these works can be seen to be significantly exceeded by those of $I_{2,q}$ with $q\left(t\right)={\left(t(1-t)\right)}^{0.4}$, $I_3$, and $I_4$ for our right-skewed distributions (we did not include our computations in this regard here).

3. Conclusions

In this paper, we derived FACIs $I_{2,q}$, $I_3$, and $I_4$ for a population mean $\mu$ of a distribution in DAN by using convergence in distribution of the integral functional $h_2$ of (14) and its modification $h_3$ of (17) applied to the Student process $T_n^t(X_1-\mu ,\dots ,X_n-\mu )/q(\left[nt\right]/n)$ weighted, respectively, with the function $q\left(t\right)={\left(t(1-t)\right)}^a$ ($a=0$ and 0.4) and the function $q\left(t\right)={\left[t(1-t)\right]}^{0.25}{[loglog{\left(t(1-t)\right)}^{-1}]}^{-0.5}$. We showed, numerically, that our $1-\alpha$ FACIs have higher finite-sample coverage probabilities than, and may be reasonable alternatives to, the CLT-based $1-\alpha$ CI $I_1$ for right-skewed distributions in DAN, although the FACIs are longer than $I_1$ on average by at most $1.361$ for $n=50$ and only at most $1.156$ for $n=1000$. The obtained FACIs may be especially appealing for right-skewed distributions in DAN with larger values of the Berry–Esseen bounds ${\delta }_n$ of (31) (if defined) or ${\tilde{\delta }}_n$ of (33), since for such distributions, the finite-sample coverage probabilities of $I_1$ may be lower, sometimes significantly, than the asymptotic confidence level $1-\alpha$, especially for smaller samples, and may have slow convergence rates. Moreover, $I_{2,q}$ with $q\left(t\right)={\left(t(1-t)\right)}^{0.4}$, $I_3$, and $I_4$ can be seen to be better potential alternatives to the CLT-based CI $I_1$ than the FACIs from the FCLT in (7) for the unweighted Student process, in particular, the $I_{2,q}$ FACI with $q\left(t\right)\equiv 1$ that were obtained in [3,4]. Furthermore, our $h_3$-based FACIs, namely, $I_3$ and $I_4$, improve on the $h_2$-based $I_{2,q}$ FACI with $q\left(t\right)={\left(t(1-t)\right)}^{0.4}$ in terms of the expected length and finite-sample coverage probability, respectively.