Interval estimation of $P(X<Y)$ in ranked set sampling

Abstract

This article deals with constructing a confidence interval for the reliability parameter using ranked set sampling. Some asymptotic and resampling-based intervals are suggested, and compared with their simple random sampling counterparts using Monte Carlo simulations. Finally, the methods are applied on a real data set in the context of agriculture.

Appendix

In this section, we first provide some results about the two-sample U-statistics, and then present proofs of Propositions 1 and 2.

Suppose that $h(x_1,\ldots ,x_p;y_1,\ldots ,y_q)$ is a symmetric kernel of degree (p, q) for the parameter $\theta =E\left( h \right)$. For independent simple random samples $X_1,\ldots ,X_{mk}$ from F, and $Y_1,\ldots ,Y_{n\ell }$ from G, the corresponding two-sample U-statistic for $\theta$ is given by

$$\begin{aligned} U_{\text {SRS}}= & {} U(X_1,\ldots ,X_{mk};Y_1,\ldots ,Y_{n\ell }) \\= & {} \frac{1}{{mk \atopwithdelims ()p}{n\ell \atopwithdelims ()q}} \sum _{{\varvec{\alpha }} \in \mathcal {A}} \sum _{{\varvec{\beta }} \in \mathcal {B}} h(X_{\alpha _1},\ldots ,X_{\alpha _p};Y_{\beta _1},\ldots ,Y_{\beta _q}), \end{aligned}$$

where ${\varvec{\alpha }}=(\alpha _1,\ldots ,\alpha _p)$, ${\varvec{\beta }}=(\beta _1,\ldots ,\beta _q)$, and $\mathcal {A}$ ($\mathcal {B}$) is the collection of all subsets of size p (q) chosen from integers $1,\ldots ,mk$ ($1,\ldots ,n\ell$). Let

$$\begin{aligned} h_{10}(x)=E\left( h(x,X_2,\ldots ,X_p;Y_1,\ldots ,Y_q) \right) , \end{aligned}$$

and

$$\begin{aligned} h_{01}(y)=E\left( h(X_1,\ldots ,X_p;y,Y_2,\ldots ,Y_q) \right) . \end{aligned}$$

Also, in connection with the above functions, define $\zeta _{10}=Var\left( h_{10}(X_1) \right)$ and $\zeta _{01}=Var\left( h_{01}(Y_1) \right)$.

Let $\{X_{[r]i}: r=1,\ldots ,k\,;i=1,\ldots ,m \}$ and $\{Y_{[s]j}: s=1,\ldots ,\ell \,;j=1,\ldots ,n \}$ be independent ranked set samples from two populations with the distribution functions F and G, respectively. The two-sample U-statistic for $\theta$ in the RSS is given by

$$\begin{aligned} U_{\text {RSS}}=U({\mathbf {X}}_1,\ldots ,{\mathbf {X}}_m;\mathbf {Y}_1,\ldots ,\mathbf {Y}_n), \end{aligned}$$

where ${\mathbf {X}}_i=(X_{[1]i},\ldots ,X_{[k]i})$ ($i=1,\ldots ,m$), and $\mathbf {Y}_j=(Y_{[1]j},\ldots ,Y_{[\ell ]j})$ ($j=1,\ldots ,n$). In addition, suppose $\gamma _{r 0}=E\left( h_{10}(X_{[r]1}) \right)$, $\gamma _{0 s}=E\left( h_{01}(Y_{[s]1}) \right)$, $\xi _{r 0}=Var\left( h_{10}(X_{[r]1}) \right)$, and $\xi _{0 s}=Var\left( h_{01}(Y_{[s]1}) \right)$.

Assume that $N=mk+n\ell$, and $(mk)/N \rightarrow \lambda \in (0,1)$ as $m,n \rightarrow \infty$. Then, according to Theorem 2 in Presnell and Bohn (1999),

$$\begin{aligned} \sqrt{N}(U_{\text {RSS}}-\theta ) {\mathop {\rightarrow }\limits ^{d}} N\left( 0,\frac{p^2 \phi }{\lambda }+\frac{q^2 \varphi }{1-\lambda }\right) , \end{aligned}$$

where

$$\begin{aligned} \phi =\zeta _{10}-\frac{1}{k} \sum _{r=1}^k \left( \gamma _{r 0} -\theta \right) ^2, \end{aligned}$$

and

$$\begin{aligned} \varphi =\zeta _{01}-\frac{1}{\ell } \sum _{s=1}^\ell \left( \gamma _{0 s} -\theta \right) ^2. \end{aligned}$$

If we set $h(x,y)=I(x<y)$, which is a kernel of degree (1, 1), then $\theta =P(X<Y)$ and Proposition 1 follows.

Also, from Corollary 2 in Presnell and Bohn (1999), the asymptotic relative efficiency of $U_{\text {RSS}}$ to $U_{\text {SRS}}$ is

$$\begin{aligned} ARE(U_{\text {RSS}},U_{\text {SRS}})=1+\frac{\frac{(1-\lambda ) p^2}{k}\sum _{r=1}^k \left( \gamma _{r 0}-\theta \right) ^2 + \frac{\lambda q^2}{\ell }\sum _{s=1}^\ell \left( \gamma _{0 s}-\theta \right) ^2 }{\frac{(1-\lambda ) p^2}{k}\sum _{r=1}^k \xi _{r 0} + \frac{\lambda q^2}{\ell }\sum _{s=1}^\ell \xi _{0 s}}. \end{aligned}$$

Again, by the same choice of the kernel mentioned above, Proposition 2 is concluded.