Nonparametric Density Estimation Using Partially Rank-Ordered Set Samples With Application in Estimating the Distribution of Wheat Yield

We study nonparametric estimation of an unknown density function $f$ based on the ranked-based observations obtained from a partially rank-ordered set (PROS) sampling design. PROS sampling design has many applications in environmental, ecological and medical studies where the exact measurement of the variable of interest is costly but a small number of sampling units can be ordered with respect to the variable of interest by any means other than actual measurements and this can be done at low cost. PROS observations involve independent order statistics which are not identically distributed and most of the commonly used nonparametric techniques are not directly applicable to them. We first develop kernel density estimates of $f$ based on an imperfect PROS sampling procedure and study its theoretical properties. Then, we consider the problem when the underlying distribution is assumed to be symmetric and introduce some plug-in kernel density estimators of $f$. We use an EM type algorithm to estimate misplacement probabilities associated with an imperfect PROS design. Finally, we expand on various numerical illustrations of our results via several simulation studies and a case study to estimate the distribution of wheat yield using the total acreage of land which is planted in wheat as an easily obtained auxiliary information. Our results show that the PROS density estimate performs better than its SRS and RSS counterparts.


Introduction
Nonparametric density estimation techniques are widely used to construct an estimate of a density function and to provide valuable information about several features of the underlying population (e.g., skewness, multimodality, etc.) without imposing any parametric assumptions. These methods are very popular in practice and their advantages over histograms or parametric techniques are greatly appreciated. For example, they are widely used for both descriptive and analytic purposes in economics to examine the distribution of income, wages and poverty (e.g., Minoiu and Reddy, 2012); in environmental and ecological studies (e.g., Fieberg , 2007); or in medical research (e.g., Miladinovic et al. ), among others.
Most of these density estimation techniques are based on simple random sampling (SRS) design which involve independent and identically distributed (i.i.d.) samples from the underlying population. There are only a few results available when the sampling design is different (e.g., Buskrik, 1998;Chen, 1999;Gulati, 2004;Lam et al., 2002;Barabesi and Fattorini, 2002;Breunig, 2001Breunig, , 2008Opsomer and Miller, 2005).
The properties of nonparametric kernel density estimation based on i.i.d. samples are well known and extensively studied in the literature (e.g., Wand and Jones, 1995;Silverman, 1986). In many applications, however, the data sets are often generated using more complex sampling designs and they do not meet the i.i.d. assumption. Examples include the rank-based sampling techniques which are typically used when a small number of sampling units can be ordered fairly accurately with respect to a variable of interest without actual measurements on them and this can also be done at low cost. This is a useful property since, quite often, exact measurements of these units can be very tedious and/or expensive. For example, for environmental risks such as radiation (soil contamination and disease clusters) or pollution (water contamination and root disease of crops), exact measurements require substantial scientific processing of materials and a high cost as a result, while the variable of interest from a small number of experimental (sampling) units may easily be ranked. These rank-based sampling designs provide a collection of techniques to obtain more representative samples from the underlying population with the help of the available auxiliary information. The samples obtained from such rank-based sampling designs often involve independent observations based on order statistics which are not identically distributed. So, it is important to develop kernel density estimators of the underlying population using such data sets and study their optimal properties.
In this paper, we study the problem of kernel density estimation based on a partially rank-ordered set (PROS) sampling design. Ozturk (2011) introduced PROS sampling procedure as a generalization of the ranked set sampling (RSS) design. To obtain a ranked set sample of size n one can proceed as follows. A set of n units is drawn from the underlying population. The units are ranked via some mechanism rather than the actual measurements of the variable of interest. Then, only the unit ranked as the smallest is selected for full measurement. Another set of n units is drawn and ranked and only the unit ranked as the second smallest is selected for full measurement. This process is repeated n times until the unit ranked the maximum is selected for the final measurement. See Chen et al. (2003), Wolfe (2004Wolfe ( , 2012 and references therein for more details. In RSS, rankers are forced to assign unique ranks to each observation even if they are not sure about the ranks. PROS design is aimed at reducing the impact of ranking error and the burden on rankers by not requiring them to provide a full ranking of all units in a set. Under PROS sampling technique, rankers have more flexibility by being able to divide the sampling units into subsets of pre-specified sizes based on their partial ranks. Ozturk (2011) obtained unbiased estimators for the population mean and variance using PROS samples. He also showed that PROS sampling has some advantages over RSS. Hatefi and Jafari Jozani (2013a, b) showed that the Fisher information of PROS samples is larger than the Fisher information of RSS samples of the same size. Since 2011, PROS sampling design has been the subject of many studies. Among others, see Gao and Ozturk (2012) for a two-sample distribution-free inference; Ozturk (2012) for quantile estimation; Frey (2012) for nonparametric estimation of the population mean; Arslan and Ozturk (2013) for parametric inference in a location-scale family of distributions, and Hatefi et al. (2013) for finite mixture model analysis based on PROS samples with a fishery application.
The outline of this paper is as follows. In Section 2 we introduce some preliminary results and present a general theory that can be used to obtain nonparametric estimates of some functionals of the underlying distribution based on imperfect PROS samples. In Section 3, we present a nonparametric kernel density estimate of the underlying distribution based on an imperfect PROS sampling scheme and study its properties. We also consider the problem of density estimation when the distribution of population is symmetric. In Section 4, we consider the problem of estimating the misplacement probabilities. To this end, we propose a modified EM-algorithm to estimate the probabilities of subsetting errors. This algorithm is fairly simple to implement and simulation results show that its performance is satisfactory. In Section 5, we compare our PROS density estimate with its RSS and SRS counterparts using simulation studies.
Finally, in Section 6, we illustrate our proposed method by a real example.

Necessary backgrounds and preliminary results
In this section, we first give a formal introduction to PROS sampling design and then present some notations and preliminary results. Also, a general theory is obtained to provide nonparametric estimates of some functionals of the underlying distribution based on PROS samples.

PROS sampling design
To obtain a PROS sample of size N = nL, we choose a set size s and a design parameter D = {d 1 , . . . , d n } that partitions the set {1, . . . , s} into n mutually exclusive subsets, where d j = {(j − 1)m + 1, . . . , jm} and m = s/n. First s units are randomly selected from the underlying population and they are assigned into subsets d j , j = 1, . . . , n, without actual measurement of the variable of interest and only based on visual inspection or judgment, etc. These subsets are partially judgment ordered, i.e. all units in subset d j judged to have smaller ranks than all units in d j , where j < j . Then a unit is selected at random for measurement from the subset d 1 and it is denoted by X [d 1 ]1 . Selecting another s units assigning them into subsets, a unit is randomly drawn from subset d 2 and then it is quantified and denoted by X [d 2 ]1 . This process is repeated until we randomly draw a unit form d n resulting in X [dn]1 . This constitutes one cycle of PROS sampling technique. The cycle is then repeated L times to generate a PROS sample of size N = nL, i.e. X PROS = {X [d j ]i ; j = 1, . . . , n; i = 1, . . . , L}. Table 1 shows the construction of a PROS sample with s = 4, n = 2, L = 2 and the design parameter D = {d 1 , d 2 } = {{1, 2}, {3, 4}}. Each set includes four units assigned into two partially ordered subsets such that units in d 1 have smaller ranks than units in d 2 . In this subsetting process we do not assign any ranks to units within each subset so that they are equally likely to take any place in the subset. One unit, in each set from the bold faced subset, is randomly drawn and is quantified. The fully measured units are denoted by X [d j ]i , j = 1, 2; i = 1, 2.
Note that if all units in d j have actually smaller ranks than all units in d j , j < j , then there is no subsetting error and the PROS sample is perfect. Otherwise, we have subsetting error and this PROS sample is called imperfect. To model an imperfect PROS sampling design, following Arslan and Ozturk (2013) and Hatefi and Jafari Jozani (2013a, b), let α be a double stochastic misplacement probability matrix, where α d j ,d h is the misplacement probability of a unit from subset d h into subset d j with n h=1 α d j ,d h = n j=1 α d j ,d h = 1. Throughout the paper, we use PROS α (n, L, s, D) to denote an imperfect PROS sampling design with subsetting error probability matrix α, the number of subsets n, the number of cycles L, the set size s, and the design parameter D = {d j , j = 1, . . . , n} where d j = {(j − 1)m + 1, . . . , jm}, in which m = s/n is the number of unranked observations in each subset. We note that SRS and imperfect RSS (with ranking error probability matrix α) can be expressed as special cases of the PROS α (n, L, s, D) design when s = 1 and s = n, respectively. For a perfect PROS design, since α d j ,d j = 1 for j = 1, . . . , n and α d j ,d h = 0 for h = j, we use PROS I (n, L, s, D), where I is the identity matrix.

Some notations and preliminary results
In what follows, the probability density function (pdf) and cumulative distribution function (cdf) of the variable of interest are denoted by f and F , respectively. The pdf and cdf of X [d j ]i , for i = 1, . . . , L, are denoted by f [d j ] and F [d j ] , respectively, and the pdf of the r-th order statistic from a SRS of size s is denoted by f (r:s) . We also use i k (g) to denote x k g(x)dx and work with a second-order kernel density function K(·) that is symmetric and satisfies the following conditions where F (x) = 1 − F (x), and consequently Remark 1. For a PROS I (n, L, s, D) design, we have Therefore, a perfect PROS sample is a special case of an imperfect PROS sample and hence the results for a perfect PROS sampling can be obtained as a special case.
We study the method of moments estimate of µ h by using an imperfect PROS sampling procedure, assuming that the required moments of h(X) exist. Note corresponds to the estimation of population moments; h(x) = 1 λ K t−x λ where K(·) is a kernel function and λ is a given constant, corresponds to the kernel estimate of pdf and h(x) = I(x ≤ c), where I(A) is the indicator function of A, corresponds to the estimate of cdf at point c.
The method of moments estimate of µ h based on an imperfect PROS sample of size N is given bŷ The properties ofμ h.PROS is discussed in the following theorem.
from a population with pdf f , and let the method of moments estimator of µ h be defined as in (3). Then sample of comparable size.
(iii)μ h.P ROS is asymptotically distributed as a normal distribution with mean µ h and variance var(μ h.P ROS ) as L → ∞.
Proof. The proof is essentially the same as the one given by Ozturk (2011) for h(x) = x which we present here for the sake of completeness. Part (i) is an immediate consequence of Lemma 1. For part (ii), using Ozturk (2011), we have Note that the equality holds if and only if µ h[d j ] = µ h for all j = 1, . . . , n; i.e. the subsetting process is purely random. For part (iii), note thatμ h.

Kernel density estimate based on PROS samples
In this section, we present a kernel density estimator of f (x) based on an imperfect PROS sample of size N and study some theoretical properties of our proposed estimator. Also, we consider the problem of density estimation when f (·) is assumed to be symmetric.

Main results
To obtain a PROS kernel density estimator of f (·) we first note that from Lemma 1 we have f ( can be estimated by the usual kernel method as followŝ where h is the bandwidth to be determined. We propose a kernel estimate of f (x) aŝ Now, we establish some theoretical properties off PROS (x). To this end, letf be a kernel density estimator of f (x) based on a SRS of size N and note that (e.g., Silverman, 1986) and Theorem 2. Suppose thatf PROS (x) is a kernel density estimator of f (x) based on a PROS α (n, L, s, D) sample of size N = nL and letf SRS (x) denote its corresponding SRS kernel estimator based on a SRS sample of the same size. Then, and variance var(f PROS (x)) for large L.
Proof. The results hold immediately from Theorem 1 by letting Theorem 2 shows thatf PROS (x) has the same expectation asf SRS (x) and a smaller variance than f SRS (x). This implies thatf PROS (x) has a smaller mean integrated square error (MISE) thanf SRS (x), that is In addition, by using part (iii) of Theorem 2, one can construct an asymptotic pointwise 100(1 − ν)% confidence interval for f (x) as followsf where z ν/2 is the 100(1 − ν 2 )-th quantile of the standard normal distribution and given by (4).
Note that our estimatef PROS (x) depends on a bandwidth h which should be determined in practice. We present an asymptotic optimal bandwidth by minimizing the asymptotic expansion of MISE(f PROS ). In this regard, we first present a lemma which is useful for obtaining the asymptotic expansion of MISE(f PROS ).
Lemma 2. Assuming that the underlying density f (·) is sufficiently smooth with desired derivatives and K(·) is a second-order kernel function, for a fixed n, as h → 0, we have Proof. Using (2) and by changing the variable Replacing f (u:s) (hv + x) by its Taylor expansion f (u:s) (x) + hvf (u:s) (x) + (hv) 2 f (u:s) (x) + O(h 2 ), and using the properties of the kernel function K(·), one can easily get Consequently, and it is similarly verified that which completes the proof.
Theorem 3. Suppose that the same bandwidth is used in bothf SRS andf PROS . Then, for large N , Therefore, the result holds.
Theorem 3 shows that the optimal bandwidth which minimizes MISE(f SRS ), asymptotically minimizes MISE(f PROS ) up to order O(N −1 ). That is, one can use the following optimal bandwidth which is obtained by minimizing asymptotical expansion of MISE(f SRS ) Chen (1999)). The optimal bandwidth h opt.SRS depends on f (·) which is unknown and, in practice, a nonparametric version of it can be used (see Silverman (1986)). Theorem 3 also shows that the PROS estimate reduces the MISE of SRS estimate at order O(N −1 ), and the amount of this reduction asymptotically is 1 N ∆(f, n) (note that ∆(f, n) is non-negative). Unfortunately, the value of ∆(f, n) depends on f (·); however, as we show below, one can characterize asymptotic rate of this reduction in a perfect PROS sampling procedure as an upper bound for ∆(f, n). Proof. Using Remark 1, one can easily verify that where Y ∼ B(s − 1, F (x)). Let A j = {(j − 1)m, . . . , jm − 1} for j = 1, . . . , n. Since Z is also distributed as a B(s − 1, F (x)) distribution and it is independent of Y , then n j=1 since A j s constitute a disjoint partition of the set {0, . . . , mn − 1} and this completes the proof.
By Lemma 3, we can derive an asymptotic result which provides more insight into the rate of reduction in MISE in a perfect PROS sampling procedure.
Theorem 4. Under a PROS I (n, L, s, D) sampling design, we have Proof. Note that by the Edgeworth expansion of P{Y = Z} in Lemma 3, we get Consequently, we can write and this completes the proof.
Theorem 4 shows that a perfect PROS density estimate reduces the MISE off SRS at order O(N −1 ) and this reduction is increased by n/m linearly whenever n m δ(f 2 ) − i 0 (f 2 ) is non-negative (a sufficient condition is n ≥ mπ). When m = 1, the result is reduced to the result for perfect RSS density estimate given by Chen (1999). In Section 5, we comparef PROS withf SRS andf RSS in a more general case where the sampling procedure can be either perfect or imperfect.

Density estimation under symmetry assumption
In this section, we consider the problem of kernel density estimation based on an imperfect PROS sample of size N = nL under the assumption that f (·) is symmetric. To this end, suppose that f (x) is symmetric for all j, h = 1, . . . , n. Therefore, based on the sub-sample X [d j ]i , i = 1, . . . , L, it is is given in (4). Consequently, the estimate of f (x) under the symmetry assumption can be defined byf * Now, we consider the mean and the variance off * PROS (x, µ) in the following theorem.
Then, based on an imperfect PROS α (n, L, s, D) sample of size N = nL, we have The result holds by using the Cauchy-Schwartz inequality.

Proof. Part (i) is easily proved by the fact that
Theorem 5 shows thatf * PROS (x, µ) has the same bias asf PROS (x); however, it has smaller variance. Therefore, under the symmetry assumptionf * PROS (x, µ) has smaller MISE and it dominatesf PROS (x). Note also that if the symmetry point µ is unknown, it can be estimated by the PROS sample to obtain a plug-in estimator asf * PROS (x,μ). Based on a PROS sample of size N , several non-parametric estimators of µ can be defined as followŝ . . , L; j = 1, . . . , n , , i, l = 1, . . . , L; j, k = 1, . . . , n , Among these estimatorsμ 1 is the PROS sample mean which is not robust against outliers, whileμ 2 ,μ 3 , andμ 4 are robust estimators of µ. Note thatμ 3 is a Hodges-Lehmann type estimator of the location parameter. In Section 5, we consider the effect of these estimators on the MISE off * PROS (x,μ).

Estimating the misplacement probabilities
So far we assumed that the misplacement probability matrix α defined in (1) is given. In practice, the misplacement probabilities α d j ,d h are unknown and they should always be estimated. This is a very important problem as the performance of our kernel density estimator depends on the estimated values of In this section, we use a modification of the EM algorithm of Arsalan and Ozturk (2013) to estimate α d j ,d h 's. We present the result for a symmetric misplacement probability matrix α with α d j ,d h = α d h ,d j .
However, results for more general α can be obtained by slight modifications of our results. Let and β a,b (·) denotes the pdf of a beta distribution with parameters a and b. Following Arslan and Ozturk (2013) we estimate the misplacement probability matrix α through an iterative method. To this end, we start with an initial estimate of α say α (0) which can be chosen to be a matrix associated with random subsetting with α d j ,d h = 1 n . Then, we use the following iterative method: (i) For a given α (t) at step t of the iterative process, calculate (iii) Maximize Q (t) (α) under the restrictions that the misplacement probabilities are symmetric and doubly stochastic and obtain the new α and call it α (t+1) . This can be done via a Lagrange multipliers method to enforce the constraints as follows where λ = (λ 1 , . . . , λ n ). The details of this process are given in Ozturk (2010) as well as Arslan and Ozturk (2013).
(iv) Repeat Steps (i)-(iii) till the sum of absolute error (SAE) of α (t) and α (t+1) is less than a predetermined value, say δ, that is In practice, to calculate π (t) [d j ,d h ]i , one can replace F (·) by an estimate of F such as the empirical distribution function, i.e.F To investigate the accuracy of our method, we perform a small simulation study when n = m = 3, and L = 4, 10. Following Arslan and Ozturk (2013), we consider three misplacement probability matrices α 1 , α 2 , and α 3 , where We generate PROS samples when the underlying population distributions are the standard Normal and Exponential distributions. For each distribution, the misplacement probabilities are estimated by using our proposed iterative method with the help of the package "Rsolnp" (Ghalanos andTheussl (2012) andYe (1987)) in R with δ = 10 −4 . This process is repeated 100 times and the average of these estimates are used as the estimates of the misplacement probabilities. The values of the estimates and their corresponding standard deviations (given in parentheses) are shown in Table 2. Note that following the properties of α we present the results for α d 1 ,d 1 , α d 1 ,d 2 , α d 1 ,d 3 , α d 2 ,d 2 , and α d 2 ,d 3 . We observe that the estimates are close to the true values and they have satisfactory biases given the fact that our proposed method is a fully nonparametric procedure and the sample size is very small. We observe that our proposed procedure slightly underestimates α d j ,d j , especially for α 1 . This is because the perfect ranking model is at the boundary of the parameter space and as noted by Arslan and Ozturk (2013) the estimates are truncated whenever they exceed 1 due to the constraints on misplacement probabilities. However, the biases and standard deviations get smaller as the cycle size increases.

Simulation Study
In this section, we compare the performance off PROS with its SRS and RSS counterparts. We first discuss the asymptotic reduction rate in the variance (RRV) off SRS andf RSS by using a PROS density estimatê f PROS . We then compare the MISE(f PROS ) with MISE(f SRS ) and MISE(f RSS ). Finally, we consider the effect of estimating the symmetry point on the MISE off * PROS (x,μ) when we assume that the underlying distribution is symmetric.

Comparing the reduction in variances
Using (7), the RRV off PROS overf SRS that measures at what ratef PROS reduces the asymptotic variance off SRS can be defined as We observe that for all values of α 0 the amount of RRV increases symmetrically as p gets away from 0.5 to 0 and 1. This shows that the best performance of the PROS density estimate over its SRS counterpart happens at the tail of the distribution. When n = 2, the PROS and SRS estimates have equal precision at p = 0.5; otherwise, the PROS estimate reduces the variance of SRS estimate. When α 0 = 0, the value of RRV decreases when n increases. This suggests using a small sample size when the misplacement probabilities (ranking errors) are large. We also note that RRV increases as both α 0 and n increase. The best performance of PROS design over SRS design happens when the subsetting is either perfect or it is moderately good, that is when α 0 = 1 or α 0 = 0.7, respectively. Similar results are observed when m = 5 which we do not present here.
To obtain the RRV off PROS overf RSS we first note that (see Chen (1999)) Now, using (7), the RRV off PROS whenf RSS is defined as in which p = F (x) and p rk for r, k = 1, . . . , n are the ranking error probabilities in an imperfect RSS procedure.
(a) α 0 = 0  are obtained as the efficiency off PROS with respect tof RSS andf SRS , respectively. Table 3 shows the values of RP and SP for these distributions with different values of n, L, α 0 = 0, 0.3, 0.5, 0.7, 1, and m = 3. Form Table 3, it is seen that as the misplacement probabilities decrease the efficiency of PROS with respect to RSS and SRS increases and, as we expect, the efficiency with respect to SRS is more than RSS procedure. When the misplacement probabilities are large, α 0 < 0.5, the three estimators have efficiency near one. The efficiency of PROS with respect to RSS and SRS increases slightly as n increases (this increment is faster when m = 4, results in which are not presented here); however, the efficiency decreases as L increases. The amount of efficiency for the Normal distribution is higher than the Gamma and Gumbel distributions. For example, when n = 8, L = 3, and α 0 = 1 the efficiencies of PROS with respect to SRS for the Normal, Gamma, and Gumbel distributions are 145%, 81%, and 94%, respectively. We observe that the main parameter that controls the efficiency is the misplacement probability matrix α or equivalently the ranking error. When the ranking errors are high, there is no substantial difference betweenf PROS , f RSS , andf SRS . However, as the ranking errors decrease, our simulation results show that the PROS density estimate performs better than RSS and SRS density estimates in terms of MISE.

Results under symmetry assumption
To investigate the effect of estimating the symmetry point µ on the MISE off * PROS , we consider four distributions (a) the standard Normal and (b) Logistic distributions as light tail distributions, (c) tstudent with 2 degrees of freedom, and (d) the standard Laplace distributions as heavy tail distributions.
For each distribution, a perfect PROS sample of size N = nL with subset size m are generated and the four symmetry point estimators given in (4) were calculated. Then, MISE(f PROS ) and MISE(f * PROS ) for µ i , i = 1, . . . , 4, were calculated and their ratios are obtained as the efficiency off * PROS (x,μ i )'s with respect tof PROS (x). The results for m = 3, n = 6, 8, and L = 3, 4 are shown in Table 4. The last column shows the efficiency off * PROS with respect tof PROS when the symmetry point is known. We observe thatf * PROS (x,μ 3 ) performs the best in all cases which suggests using the Hodges-Lehmann type estimator,μ 3 , for estimating the symmetry point. For Normal distribution, the efficiencies of f * PROS (x,μ 1 ) andf * PROS (x,μ 3 ) with respect tof PROS (x) are competitive. However, for t(2) distribution, which is a heavy tail distribution, it does not hold. On the other hand, the efficiencies off * PROS (x,μ 4 ) and f * PROS (x,μ 3 ) with respect tof PROS (x) are very close especially in heavy tail distributions. Generally, we recommend usingf * PROS (x,μ 3 ) when it is assumed that the underlying population distribution is symmetric. Comparing the efficiencies off * PROS (x, µ = 0) andf * PROS (x,μ 3 ) with respect tof PROS (x) indicates how much the efficiency reduces when the symmetry point is estimated. This reduction is larger when n = 6 in comparison with n = 8 (the maximum value of reduction is about 11% when n = 6 and L = 3 in Normal distribution and the minimum value is 3% when n = 8 and L = 4 in Laplace distribution).

Real Data Application
In this section, we illustrate our method with a real data set collected by the of policies related to food distribution, import-export decision, etc. We provide kernel density estimates of the distribution of Y = wheat yield (in ton) as the variable of interest by using X = total acreage of the planted land in wheat (in acre) as the auxiliary variable which can be used for the ranking purpose. The correlation coefficient between X and Y is 0.786. For ease of computations, we divided the values of Y by 100,000,000.
In order to estimate the density function of wheat yield, we regarded this data set as a population and extracted PROS, RSS and SRS with replacement samples of size N = nL from the population. For each design, the density estimates are obtained and the asymptotic variance estimates are calculated. Then, this process is repeated M times and the average of density estimates at a fixed point are considered as the density estimates. In addition, for each design, the average of asymptotic variance estimates are also calculated for constructing asymptotic pointwise confidence bounds. We take n = 3, m = 4, and M = 20.
The histogram of 304 records of Y is shown in Figure 3. The PROS density estimate and its 95 percent asymptotic pointwise confidence bounds are shown in third column of Figure 3. The SRS and RSS density estimates and their corresponding 95 percent pointwise confidence bounds are also shown in the 1st and 2nd columns of Figure 3. In all cases, we used Epanechnikov kernel and the bandwidth was determined as in Section 5.2. To obtain the probabilities of subsetting errors, we used the proposed algorithm in The standard deviations of the estimates vary between 0.05 and 0.19. Our estimates show that the probabilities of correct subsetting are much higher than the probabilities of incorrect subsetting and this is due to the fact that X and Y are highly correlated. We observe that the density estimates look similar.
However, the confidence bounds for PROS design are much narrower than the confidence bounds obtained by SRS and RSS designs.