Truncation-based median ranked set sampling with applications to breast cancer data

1 Introduction

Measurement is often the most expensive part of a sampling study. In ecology, agriculture, environmental monitoring and, increasingly, the biomedical sciences, obtaining an accurate value for the variable of interest can be slow, costly or destructive, even when a few units can be placed in rough order almost for free, whether by eye or through a correlated auxiliary variable. Ranked set sampling (RSS), introduced by McIntyre [1] for estimating mean pasture yield, was designed to turn that imbalance to advantage. One ranks small sets of units by judgment, measures only one unit from each set, and in doing so spreads the measured observations across the population, so that every measurement carries more information than it would under simple random sampling (SRS). Takahasi and Wakimoto [2] supplied the theory: the RSS estimator of the mean is unbiased, its variance never exceeds that of the SRS mean, and its relative precision is at most (m+1)/2 for sets of size m. Ranking is rarely flawless, but Dell and Clutter [3] showed that errors in it do not bias the RSS mean, and that robustness goes some way toward explaining the method's appeal.

Most of the work that followed turns on one question: once the sets are ranked, which order statistics should be measured? Muttlak [4] measured only the median of each set, his median RSS (MRSS) cutting the variance sharply for symmetric populations. Al-Odat and Al-Saleh [5] took the opposite tack, measuring the extremes in their moving-extreme scheme, while the quartile RSS of Muttlak [6] and the L-ranked set sampling of Al-Nasser [7], which discards the extreme ranks in favor of their neighbors, sit between the two. A different idea improves precision by adding ranking stages instead of changing the measured ranks, as in the double RSS of Al-Saleh and Al-Kadiri [8] and the multistage design of Al-Saleh and Al-Omari [9]; yet another ranks on an auxiliary variable, in the tradition of Stokes [10]. In every case the set of measured ranks acts as a design lever, trading variance, bias and robustness against the difficulty and reliability of the ranking. For further sampling designs see, for example, Obeidat et al.[11, 12] and the references therein. The usefulness of ranked set sampling is not confined to estimation of the mean. The design has been carried into a wide range of inferential problems, including entropy-based goodness-of-fit testing, where Mahdizadeh and Arghami [13] developed an improved test of uniformity from ranked set samples and Mahdizadeh [14] an entropy-based test of exponentiality; the estimation of stress–strength type reliability [27]; and the estimation of diagnostic accuracy, where Yin et al. [16] proposed rank-based kernel estimators of the area under the ROC curve. This breadth is one reason why the choice of which order statistics to measure, the question addressed here, has consequences well beyond the estimation of a population mean.

That design choice admits two opposing strategies. Measuring the extremes widens the spread of the sample and tends to favor estimation of the mean; measuring the central ranks steadies the estimator and protects it from outliers. Truncation-based RSS (TBRSS), proposed by Al-Omari and Raqab [17], makes the outward pull adjustable, replacing the first k and last k diagonal selections of an RSS set by the set minimum and maximum so that the design slides from RSS toward extreme RSS as k increases. The result is excellent for uniform and other light-tailed parents, but relying on the most variable order statistics backfires for heavier tails and skewed shapes: for the exponential the relative precision of TBRSS can drop below one, below even SRS, when k is large, and its median estimator cannot match MRSS. The inward pull has attracted more recent interest. Aldrabseh and Ismail [15] introduced except-extreme RSS (EERSS), which enlarges each set to m+2 units and measures ranks 2 through m+1, leaving out the two most variable order statistics; the same idea was later applied to variance estimation [18]. Closest to the present work, Aldrabseh et al. [19] proposed median-augmented RSS (MARSS), filling the truncated tail positions with the median rather than the extremes, so that the estimator is unbiased under symmetry, more precise than RSS for unimodal symmetric parents, and considerably more resistant to outliers.

Different as they are, these designs all slide the measured ranks along one center-to-tail axis, and nearly all of the analysis is for the mean. TBRSS pulls toward the tails; EERSS and MARSS pull toward the center. Each stops just short of the same combination. MARSS centralizes only the truncated positions, keeping the ordinary RSS diagonal elsewhere and the set size at m, so most of its sample is not median-based. EERSS drops the extremes but does not build the sample on the median. TBRSS does treat the median, but gives up median efficiency to its outward truncation. None of them anchors the sample on the median, trims the tails gently by measuring the second-smallest and second-largest order statistics of slightly enlarged sets in place of the raw extremes, and controls the balance between the two with a single coefficient that works for the mean and the median alike. The body-health problems behind several of these papers, where something like body-fat percentage or tumor morphology is expensive to measure but a correlated size variable is easy to rank, are exactly the settings such a design is meant to serve.

This paper proposes a design that fills this gap, truncation-based median ranked set sampling (TBMRSS). It builds the measured sample around the median, replaces the raw extremes of the tails by their nearest neighbors in slightly enlarged comparison sets, and uses a single truncation coefficient to control the balance between the two. Because the design is the main contribution of the paper, it is not developed here: its formal definition, the associated estimators of the mean and the median, and their basic properties are set out in Section 4, which is devoted entirely to the proposal.

Our contribution is fourfold. First, we establish unbiasedness of the mean estimator under symmetry for both parities of the set size, and derive closed-form variances and relative precisions for the uniform and exponential parents together with integral expressions for the normal. Second, we obtain a distribution-free asymptotic relative efficiency for the median estimator and show that it is maximized by the median-only member of the family. Third, we compare the design with simple random sampling and with existing ranked set sampling schemes by simulation, for both the mean and the median, across eight symmetric and skewed distributions and set sizes m = 3 to 8, under perfect ranking and under the fraction-of-random-rankings model of imperfect ranking. Fourth, we illustrate the design on the Wisconsin Diagnostic Breast Cancer data of Mangasarian et al. [20], where a costly marker of malignancy is estimated by ranking on an easily judged measure of nuclear size. A single thread runs through the results: the truncation coefficient is estimand-dependent, a moderate value serving the mean and a value near zero serving the median, where TBMRSS improves on the classical MRSS estimator.

Section 3 reviews the ranked-set designs most closely related to our proposal. Section 4 defines TBMRSS and its estimators and establishes their basic properties, and Section 5 derives the closed-form variances and efficiencies for the uniform, exponential and normal parents. Sections 6 and 7 report the simulation studies for the mean and the median, the latter also establishing a distribution-free asymptotic efficiency for the median estimator. Section 8 examines robustness to imperfect ranking, Section 9 considers the additional ranking effort and the resulting cost trade-off, and Section 10 presents the breast-cancer application. Section 11 concludes. The tables and figures are collected in Appendices A and B; Appendix C gives a detailed derivation of the closed forms for the uniform parent, and Appendix D collects the full per-distribution simulation tables.

2 Notation

Table 1 collects the notation used throughout the paper. Two conventions are worth stating at the outset. A colon subscript always records the size of the set from which an order statistic is taken, so that X(i:s) and X(i:m+2) refer to sets of size s and m+2 respectively; and square brackets distinguish judgment ranks, which may be in error, from the round brackets used for true order statistics.

Table 1. Notation used throughout the paper.

3 Related RSS designs

Before introducing the proposed design we summarize the four ranked-set schemes most closely related to it. All of them share the same first step—ranking m comparison sets—and differ only in which order statistics are quantified once the sets have been ordered. Throughout, X_(*i*:*s*) denotes the ith order statistic of a set of size s, and Algorithms 1–4 describe one cycle of each design; repeating the procedure over c cycles multiplies the realized sample size by c.

RSS, introduced by McIntyre [1], measures the diagonal of the ranked sets (Algorithm 1); its estimator μ^RSS=1m∑i=1mX(i:m) is unbiased with variance no larger than σ²/m [2].

Algorithm 1: RSS 1: Draw m independent simple random sets, each of size m, from the population. 2: Rank the units within each set by judgment or an inexpensive auxiliary variable, without quantification. 3: From the ith set, quantify the ith smallest unit X_(*i*:*m*), for i = 1, …, m. 4: Repeat Steps 1–3 over c cycles to obtain n = mc measured units; estimate the mean by their average.

Extreme ranked set sampling [21] measures only the smallest and largest ranks, with the central set contributing the median when m is odd (Algorithm 2). Identifying the extremes is the easiest possible ranking task, but the heavy reliance on tail order statistics inflates the variance for heavy-tailed and skewed parents.

Algorithm 2: Extreme ranked set sampling (ERSS) Text-based diagram describing an experimental procedure: draw m independent sets of size m, rank units, and measure specific values based on whether m is even or odd, then repeat for c cycles to obtain n equals m c measured units.

Except-extreme ranked set sampling [15] enlarges each set to m+2 units and measures the shifted diagonal of ranks 2, …, m+1 (Algorithm 3), so the two extremes are never observed; the estimator is μ^EERSS=1m∑i=2m+1X(i:m+2), and the construction was later extended to variance estimation by Aldrabseh et al. [18].

Algorithm 3: Except-extreme ranked set sampling (EERSS) 1: Draw m independent sets, each of size m+2, and rank the units within each set. 2: From the ith set, quantify the (i+1)th smallest unit X_{(*i*+1:*m*+2)}, for i = 1, …, m; equivalently, measure ranks 2, 3, …, m+1 and never the two extremes. 3: Repeat over c cycles to obtain n = mc measured units.

Truncation-based ranked set sampling [17] introduces a coefficient k that truncates the RSS diagonal toward the extremes (Algorithm 4), reducing to RSS at k = 0 and approaching ERSS as k grows. It is efficient for light-tailed populations, but, as noted in Section 1, its emphasis on the extremes is detrimental for skewed parents.

Algorithm 4: TBRSS 1: Draw m independent sets, each of size m, and rank the units within each set. 2: Fix a truncation coefficient k = ⌊αm⌋ with 0 ≤ α < 1/2. 3: Measure the minimum X_(1:*m*) from each of the first k sets, the maximum X_(*m*:*m*) from each of the last k sets, and the diagonal element X_(*i*:*m*) from each remaining set i = k+1, …, m−k. 4: Repeat over c cycles to obtain n = mc measured units.

Appendix Table 2 lists the order statistics selected by these four designs for m = 4 and m = 5. Because TBRSS interpolates between RSS and ERSS, for such small set sizes its admissible coefficients realize only those two endpoints—reproducing RSS at m = 4 and ERSS at m = 5; genuinely intermediate TBRSS selections first appear at m≥6.

4 The proposed design: TBMRSS

Let X have pdf f, CDF F, mean μ, variance σ², and let X(i:m+2) be the ith order statistic of a set of size m+2, with mean μ_(*i*:*m*+2) and variance σ(i:m+2)2. The proposed truncation-based median ranked set sampling (TBMRSS) design, governed by a coefficient k∈{0, …, ⌊m/2⌋}, is summarized in Algorithm 5.

Algorithm 5: Truncation-based median ranked set sampling (TBMRSS) 1: Draw m independent sets, each of size m+2, from the population, and rank the units within each set. 2: Fix a coefficient k∈{0, 1, …, ⌊m/2⌋}. 3: From each of the first k sets measure the second-smallest unit X(2:m+2); from each of the last k sets measure the second-largest unit X(m+1:m+2); and from each of the remaining m−2k central sets measure the median—the single central rank when m is odd, and the two central ranks split equally when m is even (detailed below). 4: Repeat Steps 1–3 over c cycles to obtain n = mc measured units; estimate the population mean (or median) by the mean (or median) of the measured units.

The design measures k+(m−2k)+k = m units per cycle, the same number as RSS, MRSS and TBRSS, so that all schemes are compared on an equal measurement budget.

Two features distinguish it from existing designs. The enlarged comparison sets of size m+2 are taken from EERSS [15], which avoids the two extreme ranks altogether, but whereas EERSS measures the whole shifted diagonal, TBMRSS organizes the sample around the median in the manner of MRSS [4] and trims only at the two ends. The coefficient k interpolates between these two ideas: at k = 0 the design is purely median-based, measuring the median of every set, and at k = ⌊m/2⌋ it becomes trimmed-extreme, with the measured units concentrated at the second-smallest and second-largest ranks. It differs from TBRSS [17] in both respects, since TBRSS truncates toward the raw minimum and maximum within sets of size m.

The relationship with EERSS is in fact closer than a shared set size. For the two smallest set sizes considered here the member k = 1 of the proposed family reproduces EERSS exactly: at m = 3 it measures the ranks 2, 3, 4 of a set of size five, and at m = 4 the ranks 2, 3, 4, 5 of a set of size six, which are in each case precisely the ranks quantified by EERSS. The two designs separate from m = 5 onwards, because TBMRSS concentrates the m−2k untruncated sets on the central rank or ranks whereas EERSS continues to spread them along the shifted diagonal; at m = 8, for instance, TBMRSS with k = 1 measures the ranks 2, 5, 5, 5, 6, 6, 6, 9 against the ranks 2, ⋯ , 9 of EERSS. EERSS may therefore be regarded as a boundary member of the proposed family at small set sizes, and the family as a genuine generalization of it thereafter. This also explains the equality of the EERSS and k = 1 columns in the rows m = 3 and m = 4 of Appendix D, which is an identity rather than a numerical coincidence.

The mean estimator under TBMRSS is Because the measured ranks are not the balanced diagonal of RSS, but repeat the two trimmed ranks and the central rank with multiplicities k, k and m−2k, TBMRSS is an unbalanced ranked-set design. The efficiency attainable by unbalanced schemes is itself constrained, and Frey [22] derives bounds on how far such designs can improve on their balanced counterparts; the gains reported in Sections 5–7 should be read against that background.

X̄TBMRSS(k,m)=1m(∑i=1kX(2:m+2)[i]+∑i=k+1m-k(median)[i]+∑i=m-k+1mX(m+1:m+2)[i]).(1)

4.1 Median for even and odd m

If m is odd, m+2 is odd and the median is the single order statistic of rank d=m+32. If m is even, m+2 is even and there is no single middle rank; following the usual MRSS convention we measure the lower central rank a=m+22 from half of the m−2k middle sets and the upper central rank b=m+42 from the other half (an equal split, possible because m−2k is even whenever m is even). The ranks 2 and m+1 are mirror images about the center of the size-(m+2) set, and so are a and b; this mirror pairing is what preserves unbiasedness for symmetric F. Appendix Table 3 illustrates the construction for an even and an odd set size: m = 6, k = 1, where the even size splits the central selections between ranks 4 and 5, and m = 7, k = 2, where the odd size uses the single central rank 5.

4.2 General moments

The m comparison sets are drawn independently of one another and exactly one unit is measured in each, so the m quantified observations are mutually independent. All covariance terms in Var(∑iX[i])=∑iVar(X[i])+∑i≠jCov(X[i],X[j]) therefore vanish, and the variance of the estimator reduces to m⁻² times the sum of the individual variances. It is worth noting that this is a property of the design rather than an assumption: within a single set the order statistics are strongly dependent, but only one of them is measured, so no two quantified units come from the same set. Writing the center contribution generically, where the center sums are Sμ=(m-2k)μ(d:m+2), Sσ=(m-2k)σ(d:m+2)2 for odd m, and Sμ=m-2k2 μ(a:m+2)+μ(b:m+2) , Sσ=m-2k2(σ(a:m+2)2+σ(b:m+2)2) for even m. With Bias=𝔼(X̄)-μ, MSE = Var+Bias²,

𝔼(X̄)=1m[k(μ(2:m+2)+μ(m+1:m+2))+Sμ],Var(X̄)=1m2[k(σ(2:m+2)2+σ(m+1:m+2)2)+Sσ],(2)

eff(X̄TBMRSS(k,m),X̄SRS)=σ2/mMSE(X̄TBMRSS(k,m)).(3)

Theorem 1. Let F be a continuous distribution with a finite first moment, 𝔼|X| < ∞, and suppose F is symmetric about μ. Then X̄TBMRSS(k,m) is an unbiased estimator of μ for every admissible k and for both parities of m.

Proof. The moment condition guarantees that every order-statistic mean μ_(*i*:*m*+2) exists and is finite, since 𝔼|X(i:s)|≤∑j=1s𝔼|Xj|=s𝔼|X|<∞, so 𝔼(X̄) in Equation 2 is well defined and the terms below may be rearranged. For symmetric F, μ_(*j*:*m*+2)+μ_{(*m*+3−*j*:*m*+2)} = 2μ. Ranks 2 and m+1 are mirror images, so μ_(2:*m*+2)+μ_{(*m*+1:*m*+2)} = 2μ. For odd m, d is self-mirror, μ_(*d*:*m*+2) = μ; for even m, a and b are mirror images, μ_(*a*:*m*+2)+μ_(*b*:*m*+2) = 2μ. Substituting in Equation 2 gives 𝔼(X̄)=1m[2kμ+(m-2k)μ]=μ in either case.

For symmetric F also σ(2:m+2)2=σ(m+1:m+2)2 and (even m) σ(a:m+2)2=σ(b:m+2)2, so

Var(X̄)=1m2[2kσ(2:m+2)2+(m-2k)σ(c:m+2)2],c={m+32m oddm+22m even.(4)

5 Closed-form variance and efficiency

5.1 Uniform distribution

For X~U(0, 1), μ=12, σ2=112 and μ(i:m+2)=im+3, σ(i:m+2)2=i(m+3-i)(m+3)2(m+4) with set size m+2. From Equations 4 and 3: Both reduce at k = 0 to the MRSS efficiency for size-(m+2) sets ((m+4)/3 odd, (m+3)²/[3(m+2)] even). Appendix Table 4 lists exact values.

(m odd) Var=m(m+3)2-2k(m-1)24m2(m+3)2(m+4), eff=m(m+3)2(m+4)3[m(m+3)2-2k(m-1)2];(5)

(m even)Var=(m+2)(m+4)-2k(m-2)4m(m+3)2(m+4), eff=(m+3)2(m+4)3[(m+2)(m+4)-2k(m-2)].(6)

5.2 Exponential distribution

For X~Exp(1), μ = 1, σ² = 1 and μ(i:m+2)=∑r=m+3-im+21r, σ(i:m+2)2=∑r=m+3-im+21r2. Here Hn=∑r=1n1r is the nth harmonic number and Hn(2)=∑r=1n1r2 the nth generalized harmonic number of order two, with the convention H0=H0(2)=0; the latter is the partial sum of the series whose limit is π²/6. These quantities arise here because the exponential order statistics admit the Rényi representation X(i:s)=d∑r=s-i+1sZr/r with Z₁, …, Z_*s* independent standard exponential variables, so that the mean and variance of X(i:s) are the corresponding partial sums of 1/r and 1/r² ([22], Sec. 4.6). With this notation and set size m+2, and the center moments are, for the two parities, These feed Equations 2, 3 directly.

μ(2:m+2)=1m+1+1m+2, μ(m+1:m+2)=Hm+2-1,σ(2:m+2)2=1(m+1)2+1(m+2)2, σ(m+1:m+2)2=Hm+2(2)-1,

m odd(d=m+32):μ(d:m+2)=Hm+2-Hm+12, σ(d:m+2)2=Hm+2(2)-Hm+12(2);m even(a=m+22,b=m+42):μ(a:m+2)=Hm+2-Hm+22, μ(b:m+2)=Hm+2-Hm2, σ(a:m+2)2=Hm+2(2)-Hm+22(2), σ(b:m+2)2=Hm+2(2)-Hm2(2).

Appendix Table 5 reports the resulting efficiencies, with the bias shown in parentheses; the two worked examples below trace the computation for one odd and one even set size.

5.2.1 Worked examples

(odd) m = 9, k = 3: 𝔼(X̄)=40854158=0.98244, Bias=-734158=-0.01756, Var = 0.024841, MSE = 0.025150, eff=16465680037269517=4.4180. (even) m = 6, k = 1: center ranks a = 4, b = 5, μ_(4:8) = H₈−H₄ = 0.634524, μ_(5:8) = H₈−H₃ = 0.884524, 𝔼(X̄)=16[(μ(2:8)+μ(7:8))+2(μ(4:8)+μ(5:8))]=0.83729, Bias = −0.16271, Var = 0.0307, MSE = 0.0571, eff = 2.9174.

5.3 Normal distribution

For X~N(0, 1), μ = 0, σ² = 1. Normal order-statistic moments have no elementary form; they are By symmetry the estimator is unbiased (Theorem 1) and Equation 4 gives eff=m/[2kσ(2:m+2)2+(m-2k)σ(c:m+2)2]. The quadrature inputs and the resulting efficiencies are in Appendix Tables 6 and 7.

σ(i:m+2)2=∫x2ψi:m+2(x)dx-(∫xψi:m+2(x)dx)2,ψi:m+2(x)=(m+2) (m+1i-1)Φ(x)i-1 [1-Φ(x)]m+2-iϕ(x).

5.3.1 Numerical procedure

So that Appendix Tables 6 and 7 can be reproduced independently, we record the computation in full. The first and second moments ∫xpψi:m+2(x)dx, p = 1, 2, were evaluated in Python 3.12 with SciPy 1.17 and NumPy 2.4, using scipy.integrate.quad, which calls the adaptive QUADPACK routine QAGI for integrals over (−∞, ∞); the infinite range was passed directly rather than truncated. The absolute and relative error tolerances were both set to 10⁻¹² and the maximum number of subintervals to 400. The binomial factor in ψ_*i*:*m*+2 was evaluated through log-gamma functions to avoid overflow at the larger set sizes. Variances were then formed as σ(i:m+2)2=∫x2ψi:m+2-(∫xψi:m+2)2.

Convergence was assessed from the error estimate returned by the routine, which did not exceed 9 × 10⁻¹³ for any entry of Appendix Table 6; the tabulated values are therefore accurate well beyond the six decimal places shown, and the efficiencies of Appendix Table 7 to the four places shown. As an independent check the same variances were recomputed by a 2000-node Gauss–Legendre rule on the truncated range [−12, 12], which agrees with the adaptive values to all six tabulated decimals, and were compared with Monte-Carlo estimates. Since the integrand is a bounded smooth function decaying like the normal density, and the quantities are computed to near machine precision, the reported values are not sensitive to the choice of tolerance or routine.

6 Mean estimation: simulation study

6.1 Design

We compare four ranked-set schemes for estimating the population mean against simple random sampling (SRS): ordinary RSS, except-extreme RSS (EERSS), the truncationbased RSS of Al-Omari–Raqab (TBRSS), and the proposed truncation-based median RSS (TBMRSS), and the proposed truncation-based median RSS (TBMRSS). All schemes measure m units per cycle. RSS and TBRSS use sets of size m; TBMRSS uses sets of size m+2. The selection coefficient ranges over k = 0, 1, …, ⌊(m−1)/2⌋ for TBRSS (with k ≤ 1 reproducing RSS) and k = 0, 1, …, ⌊m/2⌋ for TBMRSS.

Efficiency is eff=Var(X̄SRS)/MSE(μ^)=(σ2/m)/MSE(μ^), with the SRS variance taken at its exact value σ²/m and the competing MSE estimated from N = 120, 000 Monte-Carlo replications (so for skewed parents the reported efficiency already incorporates estimator bias). Sample sizes are m = 3, 4, 5, 6, 7, 8 and the eight parent distributions are those of the original TBRSS study. In each row the largest efficiency among all the designs compared is shown in bold.

6.2 Summary at m = 8

Appendix Table 8 reports, for each distribution at m = 8, the best efficiency attainable by each method (optimal k in parentheses) and the relative gain of the proposed method over the better of RSS/TBRSS. The full per-distribution results, giving every design, every truncation coefficient and every set size, are collected in Appendix D; the findings below summarize them. They confirm the superiority of the proposed TBMRSS.

Two points about Appendix Table 8 deserve emphasis. First, EERSS is included as a competitor, since it is the design closest to the proposal and, for six of the eight parents, the most efficient of the existing schemes; the gains reported here are therefore measured against a considerably stronger benchmark than RSS alone. Second, the member k = 0 is admitted on the same footing as the others. It is sometimes described as a median-type design, but it is not classical MRSS: it takes the median of a set of size m+2 rather than of size m, and is correspondingly more efficient, so there is no reason to exclude it from the family. With both changes in place the proposed design remains the most efficient scheme for every parent distribution, by margins of 5 to 32 per cent.

6.3 Enlargement versus selection: decomposing the gain

Because the proposed design draws comparison sets of size m+2 while RSS, MRSS and TBRSS use sets of size m, it is natural to ask how much of its advantage follows from the enlarged sets and how much from the selection rule itself. The two effects can be separated exactly, because EERSS provides a control: like TBMRSS it draws m sets of size m+2 and measures m units, so the two designs have identical set size, sampling effort, ranking effort and measurement budget, and differ only in which ranks are measured. The step from RSS to EERSS therefore isolates the effect of enlarging the sets, and the step from EERSS to TBMRSS isolates the effect of the selection rule. Appendix Table 9 reports the decomposition.

Both components are real, and their relative sizes vary with the parent. For the bell-shaped and mildly skewed parents the enlargement accounts for the larger share of the gain, reaching 91% for the gamma at m = 6: much of the benefit there comes from not measuring the two extreme ranks, which is what EERSS already does. The ordering reverses for the short-tailed uniform parent, where the selection rule contributes 70 to 85% of the gain, and for the exponential at m = 8, where it contributes 70%; in these cases the value lies in how the measured ranks are distributed rather than in avoiding the extremes.

We draw two conclusions and state them plainly. First, a substantial part of the improvement over RSS is attributable to the larger comparison sets, and it would be misleading to present the whole of the gain as a property of the selection rule alone. Second, the selection rule nevertheless contributes a genuine improvement in its own right: at identical set size, sampling effort and ranking effort, TBMRSS is more efficient than EERSS for every parent and every set size in Appendix Table 9, by between 3% and 39%. The proposal is therefore not simply a larger-set version of an existing design, although part of its advantage does come from that source.

6.4 Findings

• Skewed and heavy-tailed parents: TBMRSS dominates decisively. For the exponential, gamma, logistic and Student-t families the proposed method is far more efficient than both competitors at every m. The exponential is the sharpest case: TBRSS collapses as k grows (efficiency 0.88 at m = 8, k = 3, i.e. worse than SRS), whereas TBMRSS reaches 4.14—a more than four-fold gain over SRS and roughly 40% above RSS.
• TBRSS is only competitive for the uniform. For the normal, logistic, Student-t, exponential and gamma parents TBRSS is worse than plain RSS (its extremes inflate variance/bias); it improves on RSS only for the light-tailed uniform and the mildly skewed Weibull families.
• Uniform is the one regime where TBRSS rivals TBMRSS. Here efficiency grows with k for both; TBRSS leads at moderate m, while TBMRSS overtakes it at m = 8 (6.73 at k = 4 vs. 6.13 for TBRSS) thanks to its wider k-range.
• Optimal k depends on tail weight and skewness. Symmetric parents with moderate or heavy tails favor the median-centered end k = 0; strongly skewed parents favor a moderate k; and the light-tailed uniform favors the largest admissible k. No single fixed value serves all three regimes, and Section 6.5 gives a rule based on pilot skewness and kurtosis together with the cost of getting the choice wrong.
• Robustness. Across the eight parents and the six set sizes the proposed family attains the highest efficiency in every case except the uniform at m = 7, where TBRSS at its largest admissible coefficient is slightly ahead, making TBMRSS the most broadly reliable of the schemes considered.

6.5 Choosing the truncation coefficient in practice

The two studies show that the best value of k depends on the estimand and on the shape of the parent, which raises the practical question of how to choose it when that shape is unknown. For the median the question does not arise: the asymptotic efficiency of Section 7.2 is distribution-free and strictly decreasing in k, so k = 0 is optimal for every continuous parent, whatever its shape, and no pilot information is needed.

For the mean the choice does matter, and it is worth quantifying how much. Taking a single fixed value across the eight parents and m = 4, …, 8, the worst-case loss relative to the best attainable efficiency is 61% for k = 0, 38% for k = 1 and 31% for k = 2. No fixed coefficient is therefore safe. The pattern behind these losses is, however, interpretable, and it depends on two features of shape rather than on the identity of the distribution. Heavy or moderate tails favor k = 0, because the median-only design already suppresses the most variable order statistics: this is the optimum for the normal, logistic, Student-t and both Weibull parents. Short tails favor the largest admissible k, since with little tail to protect against the value lies in spreading the measured ranks, as for the uniform. Pronounced skewness favors an intermediate k, trimming rather than centring being what controls a long tail, as for the gamma and the exponential.

Both features are properties of shape rather than of location or scale, so they can be estimated from a modest pilot sample, or from the concomitant variable when ranking is performed on one. Writing γ^1 for the sample skewness and γ^2 for the sample kurtosis of such a pilot, we suggest Applied to the eight parents at m = 4, …, 8, the rule Equation 7 selects the optimal coefficient in 38 of the 40 cases. In the two exceptions, the exponential at m = 6 and the gamma at m = 5, it loses 2.1% and 2.3% of the attainable efficiency respectively, against the worst-case losses of 31 to 61% incurred by the fixed rules.

k={⌊m/2⌋,|γ^1|≤1 and γ^2<2.5 (symmetric, short-tailed),0,|γ^1|≤1 and γ^2≥2.5 (symmetric, moderate to heavy tails),min{⌊m/3⌋,⌊m/2⌋},1<|γ^1|≤1.5 (moderately skewed),min{⌊m/3⌋+1,⌊m/2⌋},|γ^1|>1.5 (strongly skewed).(7)

Three remarks temper the need for precision here. The efficiency surface is flat near its optimum for the symmetric parents, so a misclassification between neighboring coefficients is inexpensive; the tables of Section 6 show, for instance, that at m = 8 the normal loses under 10% in moving from k = 0 to k = 1. Every admissible k leaves the design more efficient than SRS for every parent examined, so a poor choice forfeits part of the gain but never makes the design harmful. And when no pilot information at all is available and the mean is the target, we recommend k = 1, which is never the worst choice for any parent in our study and whose worst-case loss is appreciably smaller than that of the median-only setting.

7 Median estimation: simulation study

The study of Section 6 is now repeated for the population median. Each scheme selects m units exactly as before, but the estimator is the median of those units, and efficiency is defined as with both mean-squared errors estimated from N = 120, 000 replications (the SRS sample median is no longer exact, so its MSE is simulated too). We add classic median ranked set sampling (MRSS, sets of size m) as the benchmark, since it was the scheme that outperformed TBRSS for the median in the original study.

eff=MSE(θ^SRS)/MSE(θ^),

7.1 Monte-Carlo error

Unlike the efficiencies of Section 6, which are computed exactly from the moments of the order statistics and therefore carry no simulation error, the median efficiencies reported here are Monte-Carlo estimates and their precision should be read accordingly. Each entry is a ratio MSE^SRS/MSE^ of two mean squared errors estimated from independent runs of N replications, so writing se(·) for the standard error of a sample mean, the delta method gives With N = 120, 000 this gives relative standard errors between 0.5% and 0.9% across the designs and parents examined, that is roughly ±0.02 on an efficiency of 3 and ±0.05 on an efficiency of 7; the corresponding 95% intervals are about ±2 units in the third decimal place. Two consequences follow for reading the tables. Differences smaller than about 0.05 between neighboring entries are within simulation noise and should not be interpreted, whereas the systematic patterns discussed below are an order of magnitude larger. And because the same random numbers are used across designs within a given distribution and set size, the differences between competing schemes are estimated more precisely than the individual entries, so the ordering in each row is considerably better determined than the entries themselves. Note that the proposed family at k = 0 is a median-of-medians on the enlarged sets of size m+2. The full per-distribution results are collected in Appendix D; in each row of those tables the most efficient scheme is in bold.

se(eff^)≈eff^(se(MSE^SRS)MSE^SRS)2+(se(MSE^)MSE^)2.

From the tables of Appendix D we observe the following.

• The proposed family is the most efficient median estimator. At k = 0 TBMRSS is uniformly best across all eight distributions and all sizes, beating classic MRSS by 22–26% at m = 8 (e.g. normal 6.45 vs. 5.19; exponential 7.29 vs. 5.84). The gain comes from taking medians of the larger size-(m+2) sets.
• Efficiency decreases in k — the mirror image of the mean. For the median the trimmed extremes add dispersion that inflates the sample median's error, so k = 0 (all medians) is optimal and efficiency falls monotonically as k grows; even k = 1 still matches or slightly exceeds classic MRSS, but TBRSS (which truncates toward the extremes) is the weakest scheme for the median.
• A single tuning rule. Combining the two studies, the practitioner tunes the one coefficient k to the estimand: increase k (moderate truncation) for the mean, set k = 0–1 (median-centered) for the median. In both regimes TBMRSS dominates RSS, TBRSS and MRSS.

7.2 Asymptotic efficiency of the median estimator

The stability of the median gains across distributions is not accidental; it follows from a distribution-free efficiency formula. Let θ^ be the TBMRSS median estimator, the sample median of the n = cm selected units pooled over the cycles, and let θ be the population median. The selected units are independent but not identically distributed: a proportion 1/m of them come from each measured rank r_*j* of a set of size s = m+2. For such a mixture the sample-quantile central limit theorem [23] gives, as c → ∞, where G_*j* and g_*j* are the CDF and pdf of the r_*j*th order statistic of a size-s set. Because the measured ranks are placed symmetrically about the center of the set, the pooled selection has its median at θ for every continuous parent, so θ^ is consistent for the population median, and median-unbiased when F is symmetric.

cm(θ^-θ) →d N(0, 1m∑j=1mGj(θ){1-Gj(θ)}{1m∑j=1mgj(θ)}2),

The simplification occurs at the median itself. Since F(θ)=12, the CDF and density of the rth order statistic at θ are both free of F, and f(θ) cancels between the design and SRS. The asymptotic relative efficiency of the median estimator is therefore distribution-free, depending only on m and k. This is precisely why the simulated median efficiencies hardly move from one distribution to the next.

G(r:s)(θ)=Pr{Bin(s,12)≥r}, g(r:s)(θ)f(θ)=s(s-1r-1)21-s,

ARE(k,m)={s21-sm∑j(s-1rj-1)}24·1m∑jG(rj:s)(θ){1-G(rj:s)(θ)}, s=m+2,

Two consequences follow. The density of the rth order statistic at θ is proportional to s-1r-1, which is largest at the central rank; raising k trades central selections for the near-extreme ranks 2 and m+1, whose binomial coefficients are far smaller, so ARE(k, m) is strictly decreasing in k and the median-only design k = 0 is asymptotically optimal. This mirrors the behavior of the mean for the short-tailed and skewed parents, whose efficiency rises with k. However, for symmetric parents with moderate or heavy tails the proposed mean estimator is also most efficient at k = 0. Moreover, since TBMRSS takes the median of a size-(m+2) set rather than the size-m set of MRSS, its advantage over MRSS is close to the ratio of set sizes, (m+2)/m.

Appendix Table 10 reports these efficiencies. They match the simulation of this section closely: at m = 8 the predicted 24.6% gain of k = 0 over MRSS lies inside the 22–26% observed across the eight distributions, and the monotone decrease in k reproduces the simulated ordering.

8 Robustness to imperfect ranking

In practice the comparison sets are ranked by eye or through an inexpensive concomitant variable, so the ranking is not perfect. We adopt the fraction-of-random-rankings model of Frey [24], in which a judgment rank is correct with probability p and is assigned at random with probability 1−p, independently across sets. The effect of ranking error has been studied from several directions; in a related contribution, Frey [25] examines how imperfect ranking affects the Fisher information carried by a ranked-set sample, complementing the efficiency-based treatment adopted here. Writing X_[*r*:*s*] for the unit measured at judgment rank r of a set of size s, the model states that a two-point mixture of the correct order-statistic distribution and the parent itself. The single parameter has a direct interpretation as the proportion of rankings that are correct: p = 1 is perfect ranking, and p = 0 makes every measured unit an ordinary draw from F, so that all of the designs reduce to SRS. This model is simpler than a concomitant-variable formulation, in that it involves no auxiliary variable, no copula, and no assumption about the joint distribution of the ranking variable and the variable of interest, and its parameter is interpretable without reference to any such structure.

F[r:s]=pF(r:s)+(1-p)F, p∈[0,1],(8)

8.1 Moments under imperfect ranking

From Equation 8, the measured unit at judgment rank r has where τ_(*r*:*s*) = μ_(*r*:*s*)−μ. The sets remain independent, so the measured units are independent and the variance of any of the estimators considered here is the average of the individual variances. Two consequences follow immediately, and they make the model particularly convenient for the present design.

𝔼(X[r:s])=pμ(r:s)+(1-p)μ,Var(X[r:s])=pσ(r:s)2+(1-p)σ2+p(1-p)τ(r:s)2,(9)

Theorem 2. Let F satisfy the conditions of Theorem 1. Then under the model Equation 8 the estimator X̄TBMRSS(k,m) is unbiased for every p∈[0, 1].

Proof. By Equation 9, 𝔼(X̄)=pm∑jμ(rj:m+2)+(1-p)μ, where r₁, …, r_*m* are the measured ranks. Theorem 1 shows that the mirror pairing of the TBMRSS ranks gives 1m∑jμ(rj:m+2)=μ under symmetry, whence 𝔼(X̄)=pμ+(1-p)μ=μ.

For a skewed parent the bias is not removed but is attenuated in proportion to the ranking quality, since Equation 9 gives Bias(p) = pBias(1): imperfect ranking scales the perfect-ranking bias by p and therefore never increases it.

8.2 Exact efficiency

A further advantage of Equation 8 is that it requires no simulation. Summing Equation 9 over the measured ranks gives the variance of the estimator in closed form, and with the bias above,

eff(p)=σ2/m1m2[p∑jσ(rj:m+2)2+m(1-p)σ2+p(1-p)∑jτ(rj:m+2)2]+p2Bias(1)2.(10)

Formula 10 applies to any of the designs compared here, with r_*j* and the set size changed accordingly. It returns exactly 1 at p = 0, as it must, and reduces to the perfect-ranking efficiency of Sections 5–6 at p = 1. Appendix Table 11 evaluates it for m = 6; the entries are exact to the digits shown and were confirmed by independent simulation, agreeing to within Monte-Carlo error in every case. Throughout this section TBRSS is shown with k = 2 and only for m≥5: its coefficient is restricted to k ≤ ⌊(m−1)/2⌋, so at m = 4 the only admissible values are k = 0 and k = 1, both of which reproduce RSS exactly, and no proper TBRSS truncation exists at that set size.

Three features stand out, the third of which is the most consequential for the choice of design. First, the proposed design retains a clear advantage over RSS across the whole range of ranking quality for every parent except the uniform. The uniform is the exception in both directions: TBMRSS with the large truncation coefficient that suits this parent is in fact slightly below RSS while the ranking is poor, by about two per cent at p = 0.25 and p = 0.5, overtaking it only at around p = 0.7 and then rising well above it. Second, the advantage is proportionally largest for the heavy-tailed logistic and Student-t parents, where TBMRSS at p = 0.5 already exceeds the efficiency that RSS attains at p = 0.75.

Third, and unlike the other designs considered, TBMRSS is never worse than simple random sampling. This is not automatic. The variance in Equation 9 contains the mixture term p(1-p)τ(r:s)2, which is the penalty for mixing a rank whose mean is far from μ with the parent distribution itself, and it is largest at intermediate p where the two components are most nearly balanced. For a design that measures raw extremes this penalty can outweigh the gain: TBRSS with k = 2 at m = 6 falls below SRS for every one of the eight parents at low p, its efficiency dipping to 0.96 for the normal and to 0.93 for the exponential before recovering, so that its efficiency curve is not even monotone in p. The proposed design escapes this because it trims to the second-smallest and second-largest ranks of an enlarged set rather than to the raw extremes, and these lie much closer to the center: for the normal parent τ(1:6)2=1.606 against τ(2:8)2=0.726, less than half. Over all eight parents, all admissible k and all p, the smallest TBMRSS efficiency we observe is 0.996, at the uniform parent with the largest admissible k; in every other case it is at least one. Poor ranking therefore costs the proposed design part of its gain, but, unlike extreme-based truncation, it does not turn the design into a liability.

8.3 Median estimation

The distribution-free property established in Section 7.2 survives the introduction of ranking error. At the population median, F(θ)=12, so Equation 8 gives both of which are free of F, and f(θ) again cancels between the design and SRS. The asymptotic relative efficiency of the median estimator therefore depends only on m, k and p, for every continuous parent. Appendix Table 12 reports it. The ordering found under perfect ranking is preserved at every level of ranking quality: the median-only design k = 0 is best for all p, and its advantage over MRSS persists, narrowing only slightly as the ranking deteriorates.

G[r:s](θ)=pPr{Bin(s,12)≥r}+1-p2,g[r:s](θ)f(θ)=ps(s-1r-1)21-s+(1-p),

The same comparisons are displayed graphically in Appendix Figures 1–4 for the eight parent distributions and set sizes m = 4, 5, 6, with the proportion of correct rankings on the horizontal axis and efficiency with respect to SRS on the vertical axis; all curves are based on N = 60, 000 replications with common random numbers across p.

Appendix Figure 1 shows the mean estimator for the four symmetric parents, with the proposed design plotted at k = 0, 1, 2 so that the coefficient identified as best in Appendix Table 11 appears in the figure. The three bell-shaped parents behave alike: the ordering is k = 0 above k = 1 above k = 2 at every p, and all three lie above RSS as soon as the ranking carries any information, the margin widening as p grows. The uniform panels are the exception and should be read separately. There the ordering is reversed, k = 2 above k = 1 above k = 0, because with short tails the value lies in spreading the measured ranks rather than concentrating them at the center; the median-only member is in fact slightly less efficient than RSS for this parent (3.375 against 3.500 at m = 6 under perfect ranking), and the truncated members overtake RSS only once p is moderately large. Appendix Figure 2 shows the same comparison for the four skewed parents, where the moderately truncated members are again preferred. In both figures the dip of TBRSS below the SRS reference line at small p, discussed above, is visible in the panels for m≥5; TBRSS is absent from the top row because no proper truncation exists at m = 4.

Appendix Figures 3, 4 repeat the exercise for the median estimator, with classical MRSS as the benchmark, for the symmetric and the skewed parents respectively. They confirm the pattern established analytically in Appendix Table 12: the median-only design k = 0 is the most efficient member at every p and for every parent, its margin over MRSS is maintained as the ranking deteriorates, and, in accordance with the distribution-free result, the curves for a given m and k are almost indistinguishable from one panel to the next.

Taken together, the two tables and the four figures give a consistent picture. Ranking error erodes the advantage of every design in an orderly way, the proposed design retains its lead for both estimands across essentially the whole range of ranking quality, and the truncation coefficient continues to play the estimand-dependent role identified under perfect ranking, a moderate value serving the mean for skewed parents and a value at or near zero serving both the mean for symmetric parents and the median throughout.

9 Ranking effort and cost efficiency

The comparisons made so far follow the convention of the ranked-set literature and equalize the number of measured units: every design considered quantifies m units per cycle. The designs do not, however, rank equal numbers. RSS, MRSS and TBRSS identify and rank m² units per cycle, whereas EERSS and the proposed TBMRSS rank m(m+2), a relative increase of 2/m that is largest at small set sizes. Ranking is assumed to be inexpensive, which is the premise on which ranked set sampling rests: from McIntyre [1] onwards the cost of such a design has been reckoned in quantified units, because the variable of interest is expensive, slow or destructive to measure while a rough ordering can be obtained by eye or from a cheap concomitant variable. RSS itself ranks m² units in order to measure m and is nonetheless compared with SRS on m measured units. Inexpensive is not the same as free, however, and this section examines what the additional ranking costs and when it ceases to be worth paying.

9.1 Comparison at matched ranking effort

A clean comparison is available because of the identity m(m+2) = (m+1)²−1. One cycle of TBMRSS with comparison sets of size m+2 ranks exactly one unit fewer than one cycle of ordinary RSS with sets of size m+1. The two designs are therefore matched almost exactly in ranking effort, and since efficiency is defined relative to SRS with the same number of measured units in each case, their efficiencies may be compared directly. Appendix Table 13 reports this comparison. The proposed design is more efficient at matched ranking effort for every parent distribution and every set size considered, and the optimal truncation coefficient is the same one identified under the measured-unit comparison. The margin is narrowest for the uniform parent, where the spread discarded by trimming is most valuable, and widest for the heavy-tailed logistic and Student-t parents, where it exceeds a factor of two.

A second comparison at identical effort is already available in Section 6.3. EERSS draws m sets of size m+2 and measures m units, so its ranking effort is not merely comparable to that of TBMRSS but identical to it; the two designs differ only in which ranks are quantified. TBMRSS is more efficient than EERSS for every parent and every set size in Appendix Table 9. Neither comparison can therefore be attributed to the proposal having ranked more units.

9.2 Cost-adjusted efficiency

A more flexible accounting charges the two operations separately, following Dell and Clutter [3]. Let c_*m* be the cost of measuring one unit, c_*r* the cost of ranking one, and γ = c_*r*/c_*m* their ratio, so that γ = 0 is the idealization of costless ranking and larger γ describes settings in which ranking is itself burdensome. A design that measures m units and ranks R of them costs mc_*m*+Rc_*r* per cycle, so its efficiency per unit of cost relative to SRS is which is eff/(1+mγ) for RSS, MRSS and TBRSS and eff/{1+(m+2)γ} for EERSS and TBMRSS. The proposal pays the larger penalty, but the penalty is bounded. The ratio of the two denominators, (1+mγ)/{1+(m+2)γ}, decreases in γ from 1 at γ = 0 to m/(m+2) as γ → ∞, so TBMRSS remains preferable to RSS at every cost ratio whenever its raw efficiency advantage exceeds the limiting factor, The threshold is 1.500 at m = 4, 1.333 at m = 6 and 1.250 at m = 8, and it falls toward one as m grows, so the extra ranking matters least where set sizes are largest. Evaluating the left-hand side of Equation 12 at the optimal coefficient, the condition holds for every parent at m = 6 and m = 8: the ratios range from 1.321 (gamma at m = 8) to 2.374 (Student-t at m = 8), and are 1.404 and 1.494 for the exponential and uniform at m = 8. For these set sizes the ordering is therefore unaffected by the ranking cost, however large it may be.

effcost=eff1+(R/m)γ,(11)

effTBMRSSeffRSS>m+2m.(12)

The exception is instructive and we state it plainly. At m = 4 the uniform parent gives a ratio of 1.307 against a threshold of 1.500, so the ordering reverses once ranking becomes sufficiently expensive; solving Equation 11 for the crossing point, RSS overtakes the proposal at γ≈0.40, that is when ranking a unit costs about two-fifths of measuring one. This is the combination of circumstances least favorable to the design: a small set size, where the relative increase in ranking is greatest at 2/m = 50%, and a short-tailed parent, for which the efficiency gain is smallest. Where ranking is genuinely costly and the parent is known to be light-tailed, the simpler RSS design is to be preferred at small m.

In summary, the additional ranking is real but modest, amounting to 2/m of the ranking workload and diminishing as m grows. It is fully compensated at matched effort, it plays no part in the comparison with EERSS, and under explicit cost accounting it leaves the ordering unchanged for all cost ratios in every case examined except the uniform parent at the smallest set size. The design is nevertheless best suited, as ranked set sampling generally is, to applications in which measurement dominates the budget and ranking is cheap.

10 Application: Wisconsin Diagnostic Breast Cancer data

We illustrate the schemes on the Wisconsin Diagnostic Breast Cancer (WDBC) data of Mangasarian et al. [20], see also Hanandeh et al. [26]: n = 569 fine-needle-aspirate samples (212 malignant, 357 benign), each described by 30 features of the cell nuclei. The 569 records are treated as the finite population; Appendix Figure 5 shows the distribution of the variable of interest and its relationship to the ranking variable. The variable of interest is mean concave points, the single most diagnostic of the mean features (AUC = 0.96 for malignancy) and costly to quantify, as it requires counting concave segments of each nuclear boundary. Ranking is done by mean radius, a nucleus size that is cheap to judge by eye; its association with the target is strong but imperfect (Pearson r = 0.82, Spearman 0.76), see Appendix Figure 5, so this is a genuine imperfect-ranking application in the sense of Section 8. The population values are μ = 0.04892 and θ = 0.03350.

For each m we draw sets by resampling patients from the population, rank each set by mean radius, apply every scheme, and estimate the mean and median of mean concave points; efficiencies relative to SRS are averaged over N = 100, 000 replications; by the same delta-method calculation as in Section 7, the relative standard error of each tabulated efficiency is below 1%, so differences of less than about 0.05 should not be over-interpreted. Appendix Tables 14, 15 report the results (best in bold); TBRSS(2) is undefined at m = 4.

The application reproduces the simulation conclusions under realistic imperfect ranking. For the mean, TBMRSS(k = 2) is the most efficient scheme at every m, reaching 2.06 × SRS; for the median, TBMRSS(k = 0) is best at every m, reaching 3.14 × SRS and exceeding classic MRSS by about 10%. The efficiencies are smaller than the perfect-ranking values of Sections 6 and 7 precisely because the radius-based ranking is imperfect, consistent with the curves of Section 8. The practical rule is reaffirmed on real data: a single coefficient k, raised for the mean and lowered for the median, makes TBMRSS the most efficient of the five schemes.

10.1 Why the design suits biomedical sampling

The efficiency gains reported above are not the only reason the design is attractive in a setting of this kind, and it is worth setting out the features that make a median-centered, mildly trimmed selection well suited to biomedical work.

The first is the cost structure. In the WDBC study the variable of interest, mean concave points, is obtained by tracing the boundary of each cell nucleus and counting concave segments, which requires a trained reader and image analysis, while mean radius is a size measurement that can be judged quickly and cheaply. This asymmetry is exactly what ranked set sampling exists to exploit, and it recurs throughout the biomedical literature: histological grading, immunohistochemical scoring, genomic assays and imaging biomarkers are all expensive per specimen, whereas a correlated size, weight or intensity measure is usually available at negligible cost. In such studies the binding constraint is the assay budget rather than the number of specimens that can be examined superficially, which is precisely the regime in which measuring m units out of m(m+2) ranked ones is a good bargain.

The second is the shape of biomedical data. Morphological and biomarker distributions are commonly right-skewed with a long upper tail, as mean concave points is here, and they frequently contain genuine extreme values arising from a subpopulation, in this case the malignant cases. Designs that deliberately quantify the sample minimum and maximum, such as ERSS and TBRSS at large k, are exposed to exactly these observations; this is the mechanism behind the breakdown of TBRSS for the exponential parent in Section 6, and behind its falling below simple random sampling under poor ranking in Section 8. By quantifying the second-smallest and second-largest ranks rather than the extremes, and by placing the bulk of the sample at the median, the proposed design limits the influence of an aberrant specimen without discarding information about the tails. The same property makes the median estimator, for which k = 0 is optimal, a natural summary when a typical value rather than an average is wanted, as it often is for skewed clinical markers.

The third is robustness to the ranking itself. Judgment ranking in a clinical setting is performed by a human reader and is inevitably imperfect; here the ranking variable correlates with the target at r = 0.82, which is realistic rather than favorable. The results of Section 8 show that the design degrades gracefully as the proportion of correctly ranked sets falls and never becomes worse than simple random sampling, so a study that adopts it cannot be harmed by ranking errors, only prevented from benefiting fully. This matters more in medicine than in settings where the ranking variable is instrumental, because reader disagreement is expected and must be tolerated rather than eliminated.

Finally, ranking on an auxiliary variable that is already recorded, as size measurements typically are in pathology and radiology archives, means the design can be applied retrospectively to specimens that have been cataloged but not yet assayed, and that the ranking step adds no burden to the clinical workflow.

11 Conclusions

We have proposed TBMRSS, a one-parameter family of ranked-set designs built on comparison sets of size m+2. A single coefficient k governs the design: the first and last k sets contribute the second-smallest and second-largest order statistics, while the remaining m−2k central sets contribute the median. The family spans a median-only scheme at k = 0 and a trimmed-extreme scheme at k = ⌊m/2⌋, combining the variance reduction of median-based sampling with a mild, symmetric trimming of the tails on enlarged sets.

The estimator is unbiased for symmetric parents at both parities of m, and we obtained closed-form variances and efficiencies for the uniform and exponential distributions together with integral expressions for the normal. Monte-Carlo studies over eight symmetric and skewed distributions and set sizes m = 3 to 8 showed that, for the population mean, moderate truncation makes TBMRSS the most efficient of the competing schemes and removes the breakdown that TBRSS suffers for the exponential parent, whereas for the population median the median-centered setting k = 0 dominates classical MRSS by roughly 22–26%. The practical message is that the same coefficient adapts the design to the target: it should be raised to estimate the mean and lowered toward zero to estimate the median. These gains persist under the fraction-of-random-rankings model of imperfect ranking, degrading gracefully toward simple random sampling as the proportion of correctly ranked sets falls, and they are reproduced on the Wisconsin Diagnostic Breast Cancer data, where a cheaply ranked nuclear-size feature is used to estimate a costly morphological marker of malignancy.

Two limitations temper these conclusions. As with other truncation designs, the mean estimator acquires a bias for skewed parents, although the accompanying variance reduction leaves it more precise than its competitors in mean-squared error; and the enlarged sets require ranking m(m+2) rather than m² units per cycle. The latter is examined directly in Section 9: the proposal remains the more efficient design when the schemes are matched on ranking effort rather than on measured units, and under explicit cost accounting the ordering is unchanged at every cost ratio except for the light-tailed uniform parent at the smallest set size. The design is nevertheless best suited, as ranked set sampling generally is, to settings in which measurement dominates the budget. Several extensions are natural. The same selection rule could be used to estimate the distribution function, quantiles, the variance, or the parameters of specific families, and it could be embedded within double or multistage procedures. A fuller comparison against the closely related except-extreme and median-augmented designs, an explicit treatment of robustness to outliers, exact relative-precision calculations for the tabulated distributions, and an analysis of cost efficiency under ranking cost are promising directions for further work. The same selection rule could also be turned to inferential problems other than point estimation of a mean or a median, such as the estimation of stress–strength reliability [27] or entropy-based goodness-of-fit testing [13, 14], where the choice of measured ranks is likely to matter in a similar way.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fams.2026.1924702/full#supplementary-material

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Footnotes

Edited by: Ceylan Yozgatligil, Middle East Technical University, Türkiye

Reviewed by: Mahdi Mahdizadeh, Hakim Sabzevari University, Iran

Ravi Bhatt, Ananya College of Medicine and Research, India