Simultaneous confidence bands for the distribution function of a finite population in stratified sampling

Abstract

Stratified sampling is one of the most important survey sampling approaches and is widely used in practice. In this paper, we consider the estimation of the distribution function of a finite population in stratified sampling by the empirical distribution function (EDF) and kernel distribution estimator (KDE), respectively. Under general conditions, the rescaled estimation error processes are shown to converge to a weighted sum of transformed Brownian bridges. Moreover, simultaneous confidence bands (SCBs) are constructed for the population distribution function based on EDF and KDE. Simulation experiments and illustrative data example show that the coverage frequencies of the proposed SCBs under the optimal and proportional allocations are close to the nominal confidence levels.

Appendix

In this Appendix, we use $a_{n}=o\left( b_{n}\right)$ to denote that $\lim _{n\rightarrow \infty }a_{n}/b_{n}=0$, and $a_{n}=O\left( b_{n}\right)$ to denote that $\limsup _{n\rightarrow \infty }a_{n}/b_{n}=c$, where c is a constant. In addition, we denote by $o_{p}$$\left( O_{p}\right)$ and $o_{a.s.}$ a sequence of random variables of order o$\left( O\right)$ in probability and almost surely, respectively, while $u_{a.s.}$ means $o_{a.s.}$ uniformly in the domain.

In the following we will prove Lemma 1 and Theorems 2–4.

1.1 A.1 Proof of Lemma 1

Our framework given in Sect. 2 and Condition (C2) ensure that, for any $s\in \left\{ 1,\ldots ,S\right\}$,

$$\begin{aligned} \lim _{k\rightarrow \infty }\left( N_{sk}/N_{k}\right) =W_{s},\ \ \lim _{k\rightarrow \infty }\left( n_{sk}/n_{k}\right) =w_{s},\ \ \lim _{k\rightarrow \infty }\left( n_{k}/N_{k}\right) =C. \end{aligned}$$

Hence,

$$\begin{aligned} CW_{s}^{-1}w_{s}={\normalsize \lim _{k\rightarrow \infty }}\left( n_{k}/N_{k}\right) \left( N_{sk}/N_{k}\right) ^{-1}\left( n_{sk}/n_{k}\right) ={\normalsize \lim _{k\rightarrow \infty }}\left( n_{sk}/N_{sk}\right) \le 1. \end{aligned}$$

Making use of the simple inequality

$$\begin{aligned} n_{k}/N_{k}=\frac{\sum \nolimits _{s=1}^{S}n_{sk}}{\sum \nolimits _{s=1}^{S}N_{sk}}\ge \min _{1\le s\le S}\left( n_{sk}/N_{sk}\right) \end{aligned}$$

and letting $k\rightarrow \infty$, one obtains that

$$\begin{aligned} C= & {} \lim _{k\rightarrow \infty }n_{k}/N_{k}\ge \lim _{k\rightarrow \infty }\min _{1\le s\le S}\left( n_{sk}/N_{sk}\right) =\min _{1\le s\le S} \lim _{k\rightarrow \infty }\left( n_{sk}/N_{sk}\right) \\= & {} \min _{1\le s\le S}CW_{s}^{-1}w_{s}, \end{aligned}$$

since $C<1$ according to Condition (C2), one obtains that $\min _{1\le s\le S}CW_{s}^{-1}w_{s}<1$. The Lemma 1 is proved. $\square$

1.2 A.2 Proof of Theorem 2

For $s=1,\ldots ,S$, combining (8) in Theorem 1 with Skorohod’s Representation Theorem shown in Theorem 6.7 of Billingsley (1999), there exits a version $\tilde{B}_{sk}\left( \cdot \right)$ of Brownian bridge $B_{s}\left( \cdot \right)$ that satisfies $\tilde{B} _{sk}\left( F_{s}( x) \right) \overset{d}{\rightarrow } B_{s}\left( F_{s}( x) \right)$ as $k\rightarrow \infty$ such that

$$\begin{aligned} \sup \nolimits _{x\in \mathbb {R}}\left| \lambda _{sk}\left\{ F_{n_{sk}}( x) -F_{N_{sk}}(x) \right\} -\tilde{B} _{sk}\left( F_{s}( x) \right) \right| \rightarrow 0,\,a.s., \end{aligned}$$

which implies that

$$\begin{aligned} F_{n_{sk}}( x) -F_{N_{sk}}(x) =\lambda _{sk}^{-1} \tilde{B}_{sk}\left( F_{s}( x) \right) +u_{a.s.}\left( \lambda _{sk}^{-1}\right) \text {.} \end{aligned}$$

Recalling the definitions of $F_{N_{k}}( x)$ and $F_{n_{k}}( x)$ given in (1) and (4), one has

$$\begin{aligned} \lambda _{k}\left\{ F_{n_{k}}( x) -F_{N_{k}}( x) \right\}= & {} \lambda _{k}\left\{ \sum \limits _{s=1}^{S}W_{sk}F_{n_{sk}}( x) -\sum \limits _{s=1}^{S}W_{sk}F_{N_{sk}}( x) \right\} \\= & {} \lambda _{k}\sum \limits _{s=1}^{S}W_{sk}\left\{ F_{n_{sk}}( x) -F_{N_{sk}}( x) \right\} \\= & {} \lambda _{k}\sum \limits _{s=1}^{S}W_{sk}\left\{ \lambda _{sk}^{-1}\tilde{ B}_{sk}\left( F_{s}( x) \right) +u_{a.s.}\left( \lambda _{sk}^{-1}\right) \right\} . \end{aligned}$$

According to Condition (C2), as $k\rightarrow \infty$,

$$\begin{aligned} \frac{n_{sk}}{N_{sk}}=\frac{n_{sk}}{n_{k}}\cdot \frac{n_{k}}{N_{k}}\cdot \frac{N_{k}}{N_{sk}}\rightarrow _{p} w_{s}CW_{s}^{-1}, \end{aligned}$$

and

$$\begin{aligned} \lambda _{k}W_{sk}\lambda _{sk}^{-1}=\frac{N_{sk}}{N_{k}}\sqrt{\frac{n_{k}}{ n_{sk}}\cdot \frac{1-n_{sk}/N_{sk}}{1-n_{k}/N_{k}}}\rightarrow _{p} W_{s}\sqrt{ w_{s}^{-1}\frac{1-w_{s}CW_{s}^{-1}}{1-C}}. \end{aligned}$$

(A.1)

Hence,

$$\begin{aligned} \lambda _{k}\left\{ F_{n_{k}}( x) -F_{N_{k}}( x) \right\} \overset{d}{\rightarrow }\sum \limits _{s=1}^{S}W_{s}\sqrt{\left( w_{s}^{-1}-CW_{s}^{-1}\right) /\left( 1-C\right) }B_{s}\left\{ F_{s}( x) \right\} .\ \ \end{aligned}$$

The proof of Theorem 2 is completed. $\square$

1.3 A.3 Proof of Theorem 3

Note that $\lambda _{k}N_{k}^{-1/2}=\left( n_{k}^{-1}-N_{k}^{-1}\right) ^{-1/2}N_{k}^{-1/2}=\left( n_{k}/N_{k}\right) ^{1/2}\left( 1-n_{k}/N_{k}\right) ^{-1/2}$$\rightarrow 0$ when $n_{k}/N_{k}\rightarrow C\equiv 0$ as $k\rightarrow \infty$. Because of a sequence of populations $\left\{ \pi _{k}\right\} _{k=1}^{\infty }$ as i.i.d. random samples generated from F(x) , Donsker’s Theorem entails that $N_{k}^{1/2}\left\{ F_{N_{k}}( x) -F(x) \right\} \overset{d}{\rightarrow }B\left\{ F( x) \right\}$. Hence, as $k\rightarrow \infty$,

$$\begin{aligned} \lambda _{k}M( F_{N_{k}},F) =\lambda _{k}O_{p}\left( N_{k}^{-1/2}\right) =o_{p}\left( 1\right) . \end{aligned}$$

Then Theorem 3 follows by Theorem 2 and Slutsky’s Theorem. $\square$

1.4 A.4 Proof of Theorem 4

According to the definitions of $F_{n_{k}}( x)$ and $\hat{F} _{k}( x)$ given in (4) and (6), one has

$$\begin{aligned} \lambda _{k}\left\{ F_{n_{k}}( x) -\hat{F}_{k}( x) \right\} =\lambda _{k}\left\{ \sum \limits _{s=1}^{S}W_{sk}F_{n_{sk}}( x) -\sum \limits _{s=1}^{S}W_{sk}\hat{F}_{sk}( x) \right\} . \end{aligned}$$

Then (9) and (A.1) imply that

$$\begin{aligned} \lambda _{k}M( \hat{F}_{k},F_{N_{k}})= & {} \lambda _{k}\sup \nolimits _{x\in \mathbb {R}}\left| \hat{F}_{k}( x) -F_{n_{k}}( x) \right| \\\le & {} \lambda _{k}\sum \limits _{s=1}^{S}W_{sk}\sup \nolimits _{x\in \mathbb {R }}\left| \hat{F}_{sk}( x) -F_{n_{sk}}( x) \right| \\= & {} \lambda _{k}\sum \limits _{s=1}^{S}W_{sk}\lambda ^{-1} _{sk}\times o_{p}\left( 1\right) =o_{p}\left( 1\right) . \end{aligned}$$

Applying Theorems 2 and 3 and Slutsky’s Theorem, Theorem 4 is proved. $\square$