On estimation of $P\left( X > Y \right)$ based on judgement post stratification

Abstract

We propose an unbiased estimator for $P\left( X>Y\right)$ and obtain an exact expression for its variance, based on judgement post stratification (JPS) sampling scheme. We then prove that the introduced estimator is consistent and establish its asymptotic normality. We show that the proposed estimator is at least as efficient asymptotically as its counterpart in simple random sampling (SRS), regardless of the quality of the rankings. For finite sample sizes, a Monte Carlo simulation study and a real data set are employed to show the preference of the JPS estimator to its SRS competitor in a wide range of settings.

A Appendix

Let the random variables U and S be as defined in Remark 1. Then, they can be rewritten as

$$\begin{aligned} U&= E\left[ {\bar{F}}_{n;JPS}\left( Y \right) - {\bar{F}} \left( Y \right) \left| \left( \mathbf {X,}{{{\mathbf {R}}}^{{\mathbf {x}}}} \right) =\left( \mathbf {x,}{{{\mathbf {r}}}^{{\mathbf {x}}}} \right) \right. \right] \nonumber \\&\quad + E\left[ G_{m;JPS}^{-}\left( X \right) - G^{-} \left( X \right) \left| \left( \mathbf {Y,}{{{\mathbf {R}}}^{{\mathbf {y}}}} \right) =\left( \mathbf {y,}{{{\mathbf {r}}}^{{\mathbf {y}}}} \right) \right. \right] \nonumber \\&=\left( \sum _{i=1}^{n}{W_i^x G^{-}\left( X_i \right) } - \theta \right) + \left( \sum _{j=1}^{m}{W_j^y {\bar{F}}\left( Y_j \right) } - \theta \right) , \end{aligned}$$

(7)

and

$$\begin{aligned} S= & {} \sum _{j=1}^{m}{W_j^y \left( {\bar{F}}_{n;JPS}\left( Y_j \right) - {\bar{F}}\left( Y_j \right) \right) } \nonumber \\&- E\left[ {\bar{F}}_{n;JPS}\left( Y \right) - {\bar{F}} \left( Y \right) \left| \left( \mathbf {X,}{{{\mathbf {R}}}^{{\mathbf {x}}}} \right) =\left( \mathbf {x,}{{{\mathbf {r}}}^{{\mathbf {x}}}} \right) \right. \right] . \end{aligned}$$

(8)

The following lemma provides some of moment properties of U and V.

Lemma A. 1

Let U and S be random variables which are defined in Remark 1. Then,

• (i) $E \left( U \right) =E \left( S \right) =0$.

• (ii)
  $$\begin{aligned} V \left( U \right)= & {} c_1^x V \left( G^{-} \left( X \right) \right) + c_2^x \sum _{h_1=1}^{H_1}{V \left( G^{-} \left( X_{\left[ h_1 \right] } \right) \right) } \\&+\, c_1^y V \left( {\bar{F}} \left( Y \right) \right) + c_2^y \sum _{h_2=1}^{H_2}{V \left( {\bar{F}} \left( Y_{\left[ h_2 \right] } \right) \right) }. \end{aligned}$$
• (iii)
  $$\begin{aligned} V \left( S \right)= & {} c_1^x c_1^y \left( E \left( {\bar{F}} \left( Y \right) F \left( Y \right) \right) - V \left( G^{-} \left( X \right) \right) \right) \\&+\, c_2^x c_1^y \sum _{h_1=1}^{H_1}{\left( E \left( {\bar{F}}_{\left[ h_1 \right] } \left( Y \right) F_{\left[ h_1 \right] } \left( Y \right) \right) - V \left( G^{-} \left( X_{\left[ h_1 \right] } \right) \right) \right) } \\&+ \,c_1^x c_2^y \sum _{h_2=1}^{H_2}{\left( E \left( {\bar{F}} \left( Y_{\left[ h_2 \right] } \right) F \left( Y_{\left[ h_2 \right] } \right) \right) - V \left( G^{-}_{\left[ h_2 \right] } \left( X \right) \right) \right) } \\&+ \,c_2^x c_2^y \sum _{h_1=1}^{H_1}{\sum _{h_2=1}^{H_2}{\left( E \left( {\bar{F}}_{\left[ h_1 \right] } \left( Y_{\left[ h_2 \right] } \right) F_{\left[ h_1 \right] } \left( Y_{\left[ h_2 \right] } \right) \right) - V \left( G^{-}_{\left[ h_2 \right] } \left( X_{\left[ h_1 \right] } \right) \right) \right) }}. \end{aligned}$$

• (iv) $COV \left( U,S \right) =0$.

Proof

• (i) It follows from Theorem 2.1 that $E \left( U \right) =0$. Also,

  $$\begin{aligned} E \left( S \left| \left( \mathbf {X,}{{{\mathbf {R}}}^{{\mathbf {x}}}} \right) =\left( \mathbf {x,}{{{\mathbf {r}}}^{{\mathbf {x}}}} \right) \right. \right) =0. \end{aligned}$$

  (9)

  And this proves the result.

• (ii) By using the fact that $\sum _{i=1}^{n}{W_i^x G^{-}\left( X_i \right) }$ and $\sum _{j=1}^{m}{W_j^y {\bar{F}}\left( Y_j \right) }$ are independent random variables, we have: $V \left( U \right) =V \left( \sum _{i=1}^{n}{W_i^x G^{-}\left( X_i \right) } \right) + V \left( \sum _{j=1}^{m}{W_j^y {\bar{F}}\left( Y_j \right) } \right)$. Thus, Theorem 2.1 completes the proof.

• (iii) We can write

  $$\begin{aligned} V \left( S \right)&= E \left[ V \left( S \left| \left( \mathbf {X,}{{{\mathbf {R}}}^{{\mathbf {x}}}} \right) \right. \right) \right] \\&= c_1^y E \left[ V \left( {\bar{F}}_{n;JPS} \left( Y \right) - {\bar{F}} \left( Y \right) \left| \left( \mathbf {X,}{{{\mathbf {R}}}^{{\mathbf {x}}}} \right) \right. \right) \right] \\&\quad + c_2^y \sum _{h_2=1}^{H_2}{E \left[ V \left( {\bar{F}}_{n;JPS} \left( Y_{\left[ h_2 \right] } \right) - {\bar{F}} \left( Y_{\left[ h_2 \right] } \right) \left| \left( \mathbf {X,}{{{\mathbf {R}}}^{{\mathbf {x}}}} \right) \right. \right) \right] }, \end{aligned}$$

  where

  $$\begin{aligned}&E \left[ V \left( {\bar{F}}_{n;JPS} \left( Y \right) - {\bar{F}} \left( Y \right) \left| \left( \mathbf {X,}{{{\mathbf {R}}}^{{\mathbf {x}}}} \right) \right. \right) \right] = V \left( {\bar{F}}_{n;JPS} \left( Y \right) - {\bar{F}} \left( Y \right) \right) \\&\qquad - V \left[ E \left( {\bar{F}}_{n;JPS} \left( Y \right) - {\bar{F}} \left( Y \right) \left| \left( \mathbf {X,}{{{\mathbf {R}}}^{{\mathbf {x}}}} \right) \right. \right) \right] \\&\quad = E \left[ V \left( {\bar{F}}_{n;JPS} \left( Y \right) - {\bar{F}} \left( Y \right) \left| Y \right. \right) \right] - V \left( \sum _{i=1}^{n}{W_i^x G^{-}\left( X_i \right) } \right) \\&\quad = c_1^x E \left( {\bar{F}} \left( Y \right) F \left( Y \right) \right) + c_2^x \sum _{h_1=1}^{H_1}{ E \left( {\bar{F}}_{\left[ h_1 \right] } \left( Y \right) F_{\left[ h_1 \right] } \left( Y \right) \right) } \\&\qquad - c_1^x V \left( G^{-} \left( X \right) \right) - c_2^x \sum _{h_1=1}^{H_1}{ V \left( G^{-} \left( X_{\left[ h_1 \right] } \right) \right) }. \end{aligned}$$

  Also, $E \left[ V \left( {\bar{F}}_{n;JPS} \left( Y_{\left[ h_2 \right] } \right) - {\bar{F}} \left( Y_{\left[ h_2 \right] } \right) \left| \left( \mathbf {X,}{{{\mathbf {R}}}^{{\mathbf {x}}}} \right) \right. \right) \right]$ can be computed in a similar manner, and this completes the proof.

• (iv) We can write $COV \left( U,S \right) =COV \left( U,E \left( S \left| \left( \mathbf {X,}{{{\mathbf {R}}}^{{\mathbf {x}}}} \right) \right. \right) \right)$. And the equality (7) completes the proof.$\square$

The following lemma establishes the asymptotic behaviours of U and V.

Lemma A. 2

Let U and S be random variables which are defined in Remark 1. If $\nu =min\left( n,m\right)$ approaches to infinity, then

• (i) $\sqrt{\nu } \left( U - \theta \right)$ converges to a normal distribution with mean zero and variance $\sigma _{ \left[ JPS\right] }^2=\lambda _1 \sigma _1^2 + \lambda _2 \sigma _2^2$, where

  $$\begin{aligned} \sigma _1^2=\frac{1}{H_1} \sum _{h_1=1}^{H_1} V\left( G^{-}\left( X_{[h_1]} \right) \right) ,\\ \sigma _2^2=\frac{1}{H_2} \sum _{h_2=1}^{H_2} V\left( F\left( Y_{[h_2]} \right) \right) ,\end{aligned}$$

  and $\lambda _1=\lim _{\nu \rightarrow +\infty } \frac{\nu }{n}$, $\lambda _2=\lim _{\nu \rightarrow +\infty } \frac{\nu }{m}$.

• (ii) $\sqrt{\nu } S$ converges in probability to zero.

Proof

• (i) By using Theorem 2.2, we have

  $$\begin{aligned} \sqrt{\nu } \left( \sum _{i=1}^{n}{W_i^x G^{-}\left( X_i \right) } - \theta \right) \overset{D}{ \rightarrow } \sigma _1 Z_1, \end{aligned}$$

  and

  $$\begin{aligned} \sqrt{\nu } \left( \sum _{j=1}^{m}{W_j^y {\bar{F}}\left( Y_j \right) } - \theta \right) \overset{D}{ \rightarrow } \sigma _2 Z_2, \end{aligned}$$

  where $Z_1$ and $Z_2$ are the standard normal random variables. So the theorem is proved by noting that $\sum _{i=1}^{n}{W_i^x G^{-}\left( X_i \right) }$ and $\sum _{j=1}^{m}{W_j^y {\bar{F}}\left( Y_j \right) }$ are independent random variables.

• (ii) It is clear that $E \left( \sqrt{\nu } S \right) =0$. So, it is enough to show that $\lim _{\nu \rightarrow +\infty } {V \left( \sqrt{\nu } S \right) }=0$. The result is simply verified by noting that $n c_1^x \rightarrow 0$, $n H_1 c_2^x \rightarrow 1$ as $n \rightarrow + \infty$ and $m c_1^y \rightarrow 0$, $m H_2 c_2^y \rightarrow 1$ as $m \rightarrow + \infty$ (see Dastbaravarde et al. (2016)).$\square$