Ranked set sampling (RSS), introduced by McIntyre (1952), is a sampling scheme that can be utilized to potentially increase precision and reduce costs when actual measurements of the observations are costly or/and time-consuming but the ranking of the set of items can be easily done without actual measurements. Such situations normally arise in environmental monitoring and assessment that require observational data. Since the inception of the concept of RSS by McIntyre (1952) and development of its mathematical foundation by Takahasi and Wakimoto (1968), various researchers have investigated the utility of RSS and the conditions under which it may be useful and cost-effective. RSS has been satisfactorily used to estimate pasture yield by McIntyre (1952), forage yields by Halls and Dell (1966), mass herbage in a paddock by Cobby
The selection of a ranked set sample of size
We measure only
For estimating the population mean, it is known that balanced RSS is a more precise method than the simple random sampling (SRS) (Tiwari and Chandra, 2011). For the case of balanced RSS designs, it has been shown by Takahasi and Wakimoto (1968) that the relative precision (RP) of RSS with respect to SRS lies between 1 and (
In this paper, a practical unbalanced RSS design for estimating the population quantiles has been developed. In the next Section, the method of quantile estimation using RSS is discussed. Section 3 provides the estimator of quantiles based on the proposed near-optimal unbalanced RSS model and its asymptotic variance. The AREs under balanced RSS, optimal and proposed near optimal models are given in Section 4. Section 5 discussed about the imperfect ranking cases. The conclusion of the study is given in Section 6.
Suppose the unknown CDF and corresponding probability density function (pdf) are respectively denoted by
It is known that,
The simplified form of
where,
Using these notations, we can write the CDF of
Since
The quantile
The problem of quantile estimation using the method of balanced RSS was first considered by Chen (2000) and found that the RSS substantially improves the efficiency of quantile estimation. This can be regarded as a generalization of sign test given by Hettmansperger (1995). The problem of quantile estimation for any distribution function using unbalanced RSS has already been discussed by Chen (2001) and Zhu and Wang (2005). Chen (2001) proved that the method of unbalanced RSS (optimal choice of unbalanced RSS) outperforms the methods of balanced RSS and SRS in terms of ARE. He used a probability vector
Chen (2001) used at most two rank orders for the optimal unbalanced RSS design. Since the estimator of
where
The
For the optimal RSS design, Chen (2001) given the values of ARE for
Zhu and Wang (2005) suggested a new weighted estimator of
The optimal strategy in the estimator of Zhu and Wang (2005) is to select observations with one fixed rank from different ranked sets. Chen (2001) and Zhu and Wang (2005) also proved that the optimal rank and the gain in relative efficiency are distribution free and depend on the set size and given probability only. This is also seen in the near optimal RSS model discussed in the next Section.
In this Section, we deal with the problem of quantile estimation based on the unbalanced RSS which depends on each rank order. Motivated from the various allocation models of population mean (Tiwari and Chandra, 2011; Kaur
where
In our case, the probability vector
In this Section, the performance of the estimators of population quantiles based on three procedure of balanced, optimum unbalanced RSS (estimator
We now calculated the AREs of three different methods with respect to SRS using
From Tables 3 and 4, it is seen that ARE of proposed method is substantially more than that of balanced RSS for each
Now we wish to see the performance of the proposed methods with the increasing values of sample sizes. For this purpose, we take
From Figure 1, we see that as sample size increases, the performance of proposed estimator is increases for each
Since the RSS procedures perform better in the absence of ranking errors of units, however, this situation is rarely seen in practical situations particularly for the large values of
where,
In the first study of quantile estimation by unbalanced RSS (Chen, 2001), the optimal
The proposed method is shown to be more efficient asymptotically with balanced RSS method and close to optimal allocation design for large sample sizes. As seen in Figure 1, for each set size and desired
Proposed near optimal allocation model for the quantile estimation
Class of | Condition for | Allocation model |
---|---|---|
0 | all | |
0.17 | ||
0.33 | odd | |
even | ||
even |
Proposed allocation model for the quantile estimation at
Rank order ( | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |||
0.01,0.05,0.10 | 2 | 2 | 1 | 3 | ||||||
3 | 4 | 1 | 1 | 6 | ||||||
4 | 7 | 1 | 1 | 1 | 10 | |||||
5 | 11 | 1 | 1 | 1 | 1 | 15 | ||||
6 | 16 | 1 | 1 | 1 | 1 | 1 | 21 | |||
7 | 22 | 1 | 1 | 1 | 1 | 1 | 1 | 28 | ||
8 | 29 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 36 | |
0.25 | 2 | 2 | 1 | 3 | ||||||
3 | 4 | 1 | 1 | 6 | ||||||
4 | 1 | 7 | 1 | 1 | 10 | |||||
5 | 1 | 11 | 1 | 1 | 1 | 15 | ||||
6 | 1 | 16 | 1 | 1 | 1 | 1 | 21 | |||
7 | 1 | 22 | 1 | 1 | 1 | 1 | 1 | 28 | ||
8 | 1 | 1 | 29 | 1 | 1 | 1 | 1 | 1 | 36 | |
0.50 | 2 | 1 | 2 | 3 | ||||||
3 | 1 | 4 | 1 | 6 | ||||||
4 | 1 | 4 | 4 | 1 | 10 | |||||
5 | 1 | 1 | 11 | 1 | 1 | 15 | ||||
6 | 1 | 1 | 9 | 9 | 1 | 1 | 22 | |||
7 | 1 | 1 | 1 | 21 | 1 | 1 | 1 | 28 | ||
8 | 1 | 1 | 1 | 15 | 15 | 1 | 1 | 1 | 36 |
Comparison of three AREs (using Equation (3.3)) at different probability values for
Method | Set size | ||||||||
---|---|---|---|---|---|---|---|---|---|
2 | 3 | 4 | 5 | 6 | 7 | 8 | |||
Balanced | 102.50 | 103.95 | 105.43 | 106.86 | 108.27 | 109.67 | 111.05 | ||
Unbalanced | 201.01 | 299.93 | 397.97 | 494.95 | 590.93 | 685.93 | 779.94 | ||
135.34 | 201.94 | 280.95 | 365.59 | 453.03 | 541.87 | 631.30 | |||
Balanced | 22.52 | 23.89 | 25.20 | 26.44 | 27.63 | 28.77 | 29.87 | ||
Unbalanced | 41.03 | 59.95 | 77.85 | 94.75 | 110.69 | 125.69 | 139.78 | ||
28.69 | 41.92 | 56.79 | 71.98 | 86.96 | 101.46 | 115.35 | |||
Balanced | 12.55 | 13.83 | 15.02 | 16.12 | 17.16 | 18.14 | 19.08 | ||
Unbalanced | 21.05 | 29.89 | 37.69 | 44.50 | 50.41 | 55.46 | 59.72 | ||
15.38 | 21.86 | 28.62 | 35.04 | 40.91 | 46.13 | 50.69 | |||
Balanced | 6.70 | 7.85 | 8.87 | 9.79 | 10.63 | 11.41 | 12.15 | ||
Unbalanced | 9.14 | 11.68 | 14.74 | 19.15 | 22.63 | 25.14 | 28.46 | ||
7.52 | 9.76 | 12.39 | 16.03 | 19.20 | 21.71 | 24.84 | |||
Balanced | 5.33 | 6.43 | 7.37 | 8.21 | 8.96 | 9.66 | 10.32 | ||
Unbalanced | 5.33 | 9.00 | 10.47 | 14.06 | 15.58 | 19.14 | 20.69 | ||
5.33 | 7.71 | 9.23 | 12.11 | 13.78 | 16.68 | 18.38 |
Comparison of three AREs with respect to SRS (using equation 2.1) at different probability values for
Method | Set size | ||||||||
---|---|---|---|---|---|---|---|---|---|
2 | 3 | 4 | 5 | 6 | 7 | 8 | |||
Balanced | 1.0100 | 1.0200 | 1.0300 | 1.0400 | 1.0500 | 1.0599 | 1.0699 | ||
Unbalanced | 1.9899 | 2.9699 | 3.9399 | 4.9000 | 5.8503 | 6.7907 | 7.7214 | ||
1.3366 | 1.9949 | 2.7759 | 3.6132 | 4.4786 | 5.3578 | 6.2431 | |||
Balanced | 1.0499 | 1.0995 | 1.1488 | 1.1976 | 1.2459 | 1.2937 | 1.3408 | ||
Unbalanced | 1.9487 | 2.8475 | 3.6977 | 4.5006 | 5.2577 | 5.9702 | 6.6394 | ||
1.3493 | 1.9727 | 2.6767 | 3.3975 | 4.1087 | 4.7978 | 5.4582 | |||
Balanced | 1.0989 | 1.1959 | 1.2904 | 1.3821 | 1.4708 | 1.5565 | 1.6392 | ||
Unbalanced | 1.8947 | 2.6900 | 3.3917 | 4.0054 | 4.5368 | 4.9915 | 5.3748 | ||
1.3636 | 1.9406 | 2.5470 | 3.1254 | 3.6544 | 4.1258 | 4.5376 | |||
Balanced | 1.2308 | 1.4382 | 1.6248 | 1.7942 | 1.9500 | 2.0947 | 2.2303 | ||
Unbalanced | 1.7143 | 2.1892 | 2.7633 | 3.5904 | 4.2431 | 4.7136 | 5.3369 | ||
1.3901 | 1.8092 | 2.3079 | 2.9904 | 3.5846 | 4.0545 | 4.6459 | |||
Balanced | 1.3333 | 1.6000 | 1.8286 | 2.0317 | 2.2165 | 2.3869 | 2.5461 | ||
Unbalanced | 1.3333 | 2.2500 | 2.6182 | 3.5156 | 3.8961 | 4.7852 | 5.1718 | ||
1.3333 | 1.9231 | 2.3013 | 3.0178 | 3.4360 | 4.1596 | 4.5858 |
Increasing rate of ARE over
Method | Set size | |||||||
---|---|---|---|---|---|---|---|---|
3 | 4 | 5 | 6 | 7 | 8 | |||
Balanced | 1.0099 | 1.0098 | 1.0097 | 1.0096 | 1.0094 | 1.0094 | ||
Unbalanced | 1.4925 | 1.3266 | 1.2437 | 1.1939 | 1.1607 | 1.1371 | ||
1.4925 | 1.3915 | 1.3016 | 1.2395 | 1.1963 | 1.1652 | |||
Balanced | 1.0472 | 1.0448 | 1.0425 | 1.0403 | 1.0384 | 1.0364 | ||
Unbalanced | 1.4612 | 1.2986 | 1.2171 | 1.1682 | 1.1355 | 1.1121 | ||
1.4620 | 1.3569 | 1.2693 | 1.2093 | 1.1677 | 1.1376 | |||
Balanced | 1.0883 | 1.0790 | 1.0711 | 1.0642 | 1.0583 | 1.0531 | ||
Unbalanced | 1.4197 | 1.2609 | 1.1809 | 1.1327 | 1.1002 | 1.0768 | ||
1.4231 | 1.3125 | 1.2271 | 1.1693 | 1.1290 | 1.0998 | |||
Balanced | 1.1685 | 1.1297 | 1.1043 | 1.0868 | 1.0742 | 1.0647 | ||
Unbalanced | 1.2770 | 1.2622 | 1.2913 | 1.1818 | 1.1109 | 1.1322 | ||
1.3015 | 1.2756 | 1.2957 | 1.1987 | 1.1311 | 1.1459 | |||
Balanced | 1.2000 | 1.1429 | 1.1111 | 1.0910 | 1.0769 | 1.0667 | ||
Unbalanced | 1.6875 | 1.1636 | 1.3428 | 1.1082 | 1.2282 | 1.0808 | ||
1.4424 | 1.1967 | 1.3113 | 1.1386 | 1.2106 | 1.1025 |