1 Introduction
Abstract

We study nonparametric estimation of the distribution function (DF) of a continuous random variable based on a ranked set sampling design using the exponentially tilted (ET) empirical likelihood method. We propose ET estimators of the DF and use them to construct new resampling algorithms for unbalanced ranked set samples. We explore the properties of the proposed algorithms. For a hypothesis testing problem about the underlying population mean, we show that the bootstrap tests based on the ET estimators of the DF are asymptotically normal and exhibit a small bias of order . We illustrate the methods and evaluate the finite sample performance of the algorithms under both perfect and imperfect ranking schemes using a real data set and several Monte Carlo simulation studies. We compare the performance of the test statistics based on the ET estimators with those based on the empirical likelihood estimators.

Draft of July 3, 2019

Exponentially Titled Empirical Distribution Function

for Ranked Set Samples

Saeid Amiri111Corresponding author: saeid.amiri1@gmail.com, Mohammad Jafari Jozani, and Reza Modarres

Department of Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska, USA

Department of Statistics, University of Manitoba, Winnipeg, MB, CANADA, R3T 2N2

Department of Statistics, The George Washington University, Washington DC, USA

Keywords: Distribution function; Exponential tilting; Imperfect ranking; Ranked set sample.

1 Introduction

Ranked set sampling (RSS) is a powerful and cost-effective data collection technique that is often used to collect more representative samples from the underlying population when a small number of sampling units can be fairly accurately ordered without taking actual measurements on the variable of interest. RSS is most effective when obtaining exact measurement on the variable of interest is very costly, but ranking the sampling units is relatively inexpensive. RSS finds applications in industrial statistics, environmental and ecological studies as well as medical sciences. For recent overviews of the theory and applications of RSS and its variations see Wolfe (2012) and Chen et al. (2004).

Ranked set samples can be either balanced or unbalanced. An unbalanced ranked set sample (URSS) is one in which the ranked order statistics are not quantified the same number of times. To obtain an URSS of size from the underlying population we proceed as follows. Let sets of sampling units, each of size , be randomly chosen from the population using a simple random sampling (SRS) technique. The units of each set are ranked by any means other than the actual quantification of the variable of interest. Finally, one and only one unit in each ordered set with a pre-specified rank is measured. Let be the number of measurements on units with rank , such that . Suppose denotes the measurement on the th unit with rank . The resulting URSS of size from the underlying population is denoted by , where the elements of the th row are independently and identically distributed (i.i.d.) from and is the DF of the th order statistic. Moreover, s are independent for and . Note that if , , then URSS reduces to the balanced RSS. The DF of URSS is

(1)

where and . As it is shown in Chen et al. (2004), when , and , for , we have , where

(2)

One can easily see that is not equal to the underlying DF , unless , , showing that the EDF based on the URSS data does not provide a good estimate of the underlying distribution . The properties of the EDF of the balanced and unbalanced RSS are studied in Stokes and Sager (1988) as well as Chen et al. (2004).

In this paper, we use the empirical likelihood method as a nonparametric approach for estimating . To this end, we propose two methods to estimate using the exponentially tilted (ET) technique. The proposed estimators can be used as standard tools for practitioners to estimate the standard error of any well-defined statistic based on RSS or URSS data and to make inferences about the characteristics of interest of the underlying population. Another interesting problem in this direction is to develop efficient resampling techniques for URSS data, as in many cases the exact or the asymptotic distribution of the statistics based on URSS data are not available or they are very difficult to obtain (e.g., Chen et al., 2004). Akin to the methods of Modarres et al. (2006) and Amiri et al. (2014), the new ET estimators of are used to construct new resampling techniques for URSS data. We study different properties of the proposed algorithms. For a hypothesis testing problem, about the underlying population mean, we show that the bootstrap tests based on the ET estimators are asymptotically normal and exhibit a small bias of order which are desirable properties.

The outline of the paper is as follows. In Section 2, we present ET estimators of based on the URSS data. Section 3 considers two methods for resampling RSS and URSS data based on the ET estimators of . We provide justifications for validity of these methods for a hypothesis testing problem about the population mean. Section 4 describes a simulation study to compare the finite sampling properties of the proposed methods with parametric bootstrap and some existing resampling techniques for testing a hypothesis about the population mean. We consider both perfect and imperfect ranking scenarios, three different distributions and five RSS designs. We compare the performance of our proposed methods with the one based on the empirical likelihood method studied in Liu et al. (2009) as well as Baklizi (2009). In Section 5, we apply our methods for a testing hypothesis problem using a real data set consisting of the birth weight and seven-month weight of 224 lambs along with the mother’s weight at time of mating. Section 6 provides some concluding remarks.

2 Exponential Tilting of DF

Exponential tilting of an empirical likelihood is a powerful technique in nonparametric statistical inference. The impetus of this approach is the use of the estimated DF subject to some constraints rather than the EDF. ET methods find applications in computation of bootstrap tail probabilities (Efron and Tibshirani, 1993), point estimation (Schennach, 2007), estimation of the spatial quantile regression (Kostov, 2012), Bayesian treatment of quantile regression (Schennach, 2005), small area estimation (Chaudhuri and Ghosh, 2011) and Calibration estimation (Kim, 2010), among others.

Let be a generic sample of size from and suppose is the EDF of which places empirical frequencies (weights) on each . Consider an estimator of which assigns weights instead of to each . To obtain the ET estimator of , we minimize an aggregated distance between the empirical weights and subject to some constraints on the ’s. More specifically, one chooses a distance and minimizes subject to and some other constrains such as , using the following Lagrangian multiplier method

(3)

where is often imposed under the null hypothesis in a testing problem or any other conditions that one needs to account for in practice. Note that the minimization in (3) can also be done by minimizing the distance between and any target estimator other than the EDF .

The choice of the discrepancy function for the aggregated loss in (3) leads to different ET estimators of . Since is the nonparametric maximum likelihood estimator of under the Kullback-Leibler distance subject to the restriction , one often uses

We propose two ET estimators of based on URSS data with sample size where is the set size. The ET estimators are then used to propose new bootstrapping algorithms from URSS data.

2.1 Exponential Tilting of All Observations (EAT)

In this section, we propose our first ET estimator of which is later used to resample from within each row of . The idea behind the first ET estimator of , for bootstrapping , is to find an estimator

(4)

subject to the constraints

(5)

where .

Lemma 1.

Let be a URSS sample of size from the underlying population when the set size is and is the -th order statistic in a simple random sample of size from . The optimum values of in (4) under the constraints (5) are given by

(6)

where is obtained from .

Proof.

Using the Lagrange multipliers method, and by minimizing

(7)

with respect to ’s, one can easily obtain the optimum values in (6). ∎

In Section 3, we use for bootstrapping instead of the commonly used empirical DF. It is worth noting that for hypothesis testing problems about the underlying population mean involving the null hypothesis , minimization in (7) is done subject to the condition . Using the optimum weights from the ET estimate of , we also propose to estimate the population variance .

2.2 Exponential Tilting of Rows (EAR)

By the structure of the URSS data, , we observe that are i.i.d. samples from , which is the distribution of the -th order statistic in a simple random sample of size from . Since

the idea behind our next proposed ET estimator of is to estimate each using , and construct an estimator of by averaging over these estimators using suitable weights obtained from the Lagrange multipliers method under some constraints. To this end, we work with an estimator of of the form

(8)

where is the EDF of .

Lemma 2.

Let be a URSS sample of size from where the set size is and are i.i.d. samples from the DF of the -th order statistic of a simple random sample of size from . Then, an optimum estimator of in the form of (8) under the constraints and , where , , is given by

(9)

where is obtained from .

Proof.

The results easily follow using the Lagrange multipliers method and minimizing

(10)

with respect to . ∎

In Section 3, we use and propose a new bootstrapping algorithm to resample from instead of the commonly used empirical DF. Here again for hypothesis testing problems involving where is the population mean, minimization in (10) is done subject to the condition .

Remark 1.

If for the observed URSS data all the s are large enough, then one can use ET estimators of by simply treating ’s as a SRS of size from and constructing the estimator for . Here, and s are obtained subject to constraints and , for , using the following Lagrange multipliers problems:

3 Bootstrapping URSS and RSS

In this section, we propose two new bootstrapping techniques to resample from a balanced or unbalanced ranked set sample of size . The first algorithm is based on the ET estimator of in Lemma 1 to resample the entire URSS while the second one uses the ET estimator of in Lemma 2 to resample from within each row separately. We note that most of the bootstrap methods developed for RSS are based on the EDF and one can easily modify them using ET estimators of . Monte Carlo simulation studies indicate that bootstrapping methods based on the ET estimators of perform better than their counterparts using the EDF.

3.1 Bootstrapping Algorithm: EAT

To resample from the ET estimator of given by

where is defined in (6) we proceed as follows:

  • Assign probability to each element of .

  • Randomly draw from according to probabilities , order them as and retain .

  • Repeat Step 2, for and to generate a bootstrap URSS .

  • Repeat steps 2-3, times to obtain the bootstrap samples.

One can easily validate the use of the ET estimator of for different bootstrapping purposes. For example, suppose we want to carry out a bootstrap test for testing against , where is the unknown parameter of interest. Using Hall (1992), the Edgeworth expansion of the -value for testing against based on a SRS of size from the underlying population with the test statistic , is given by

(11)

where is a quadratic function and and are the standard normal distribution and density functions, respectively. We consider the problem for a balanced RSS case, as the following argument can also be applied to URSS data with some modifications. Let be a balanced ranked set sample of size from the underlying population with mean . We show that the ET bootstrap approximation of the sampling distribution of is in error by only and the -value obtained through the EAT method has the desirable second order accuracy This is similar to results obtained in DiCiccio and Romano (1990). For more details see Efron (1981) and Feuerveger et al. (1999).

Proposition 1.

Suppose is a bootstrap sample generated from the EAT algorithm. Let be the bootstrap test for testing with p-value , where is the mean of the bootstrap sample obtained form the ET estimator of and with . Then,

(12)

where , given by (11), is the p-value of the usual -test based on a simple random sample of comparable size from the underlying population.

Proof.

For simplicity, we write the resampled data as . In order to test , and to ensure that the null hypothesis is incorporated into the ET estimator of , we introduce the Lagrange multipliers for the constraints and , where the weights are obtained as

(13)

and is the coefficient calculated from . One can easily show that s are generated from

(14)

where . To obtain the ET estimator of under the null hypothesis we must have

Therefore, one can use the bootstrap test statistic for testing where is the mean of the bootstrap sample obtained form the ET estimator of and with . Following Hall (1992) and using the Edgeworth expansion, the -value for testing against using the bootstrap test statistic is given by

where is a quadratic function. Now, the results follows from (11). ∎

3.2 Bootstrapping Algorithm EAR

The idea behind this method is to use the ET estimator of given by

where is defined in (9). To this end we proceed as follows:

  • Assign probabilities to each row of , .

  • Select a row randomly using and select an observation randomly from that row.

  • Continue step 2 for times to obtain observations.

  • Order them as and retain

  • Perform Steps 2–4 for and obtain .

  • Perform Steps 2–5 for .

  • Repeat steps 2–6, times to obtain the bootstrap samples.

4 Monte Carlo Study

In this section, we compare the finite sample performance of out nonparametric EAT and EAR resampling methods with a parametric bootstrap (PB) procedure. The PB method uses a parametric test (PT) with an asymptotic normal distribution to test the hypothesis , where is the unknown parameter of interest and is a known constant. The resampling is performed using B=500 resamples and the entire experiment is then replicated 2000 times. We use several RSS and URSS designs with different sample sizes when the set size is chosen to be . We also conducted unreported simulation studies for other values of and we observed similar performance that we summarize below.

The RSS designs that we consider are written as with . For example, the first design is balanced with and observations per stratum, which is denoted by

Similarly, we define the following designs,

We obtain samples from the Normal(0,1), Logistic(1,1) and Exponential(1) distributions.

4.1 Testing a hypothesis about the population mean

We first proceed with the following proposition.

Proposition 2.

Suppose is the DF of the variable of interest in the underlying population with . Let be the EDF of the row of a balanced RSS data and represent the population mean. Then , with , converges in distribution to a multivariate normal distribution with the mean vector zero and the covariance matrix where and .

This proposition suggests to use the following test statistic for testing the hypothesis

(15)

where

(16)

The test statistic , which is approximately Normal for large , is referred to as the PT in the rest of the work. Ahn et al. (2014) consider the Welch-type (WT) approximation to the distribution , where the degree of freedom of the test can be approximated using

(17)

The nonparametric bootstrap tests using the EAT and EAR methods are conducted based on the following steps:

  • Let X be an URSS/RSS sample from .

  • Calculate , given in (15), under the null hypothesis .

  • Apply each of the resampling procedures on to obtain .

  • Calculate , .

  • Obtain the proportion of rejections via to estimate the -value.

We also performed the desired testing hypothesis using PB by generating URSS samples from Normal(0,1), Logistic(1,1) and exponential(1) distributions. To perform PB test we use the following steps (for more details on PB method see Efron and Tibshirani (1993)):

  • Let X be a URSS sample from a distribution where is the unknown parameter and let .

  • Calculate , under the null hypothesis .

  • Estimate from X and take a URSS from , .

  • Calculate .

  • Obtain the proportion of rejections via to estimate the -value.

To conduct the parametric bootstrap we estimated the population mean using the sample mean and used . Subsequently, we generated samples from the N(, 1), Logistic(, 1) and Exponential() distributions. Table 1 displays the observed levels. The parametric bootstrap (PB) method is accurate and the estimated levels are close to the nominal level 0.05. The PT test is liberal and its approximated -value is higher than the nominal level, specially under exponential distribution. We observe that the WT test is a bit conservative under the normal and logistic distributions, i.e., the approximated -values are lower than the nominal level. The observed levels for EAR follow the PB method closely and they are less liberal than the PT under the exponential distribution.

PT WT EAT EAR PB
0.062 0.041 0.056 0.052 0.050
0.078 0.039 0.054 0.056 0.054
N(0, 1) 0.072 0.038 0.046 0.047 0.049
0.071 0.033 0.057 0.058 0.054
0.064 0.039 0.043 0.045 0.047
0.107 0.071 0.081 0.080 0.051
0.133 0.072 0.076 0.079 0.049
Exponential (1) 0.132 0.081 0.089 0.090 0.054
0.131 0.073 0.098 0.094 0.050
0.098 0.074 0.058 0.055 0.053
0.052 0.042 0.05 0.051 0.047
0.076 0.041 0.058 0.059 0.050
Logistic (1, 1) 0.065 0.033 0.048 0.050 0.046
0.068 0.034 0.059 0.057 0.051
0.059 0.034 0.043 0.044 0.041
Table 1: Observed -levels of the proposed tests for testing under the Normal distribution and for the Exponential and Logistic distributions.

Table 2 displays the estimated power values under shift alternatives with . We used 95% percentile bootstrap confidence intervals for , using EAT and EAR to obtain the power of the test statistics at . The entries of these tables are the proportion of times that the bootstrap confidence intervals do not cover zero. Compared with PT, both the EAT and EAR methods lead to high powers, hence they can be nominated to conduct appropriate tests. The results of other simulation studies (not presented here) show similar behavior for other values of such as . We also considered different sample sizes. The better performance of the proposed methods are apparent for small and relatively small sample sizes (which often happens in practice for RSS) and they perform similarly when the sample size gets very large for a fixed set size.

Normal dist. Exponential dist. Logistic dist.
PT WT ETA ETR PB PT WT ETA ETR PB PT WT ETA ETR PB
0.1 0.148 0.097 0.152 0.145 0.138 0.229 0.148 0.222 0.208 0.209 0.076 0.049 0.088 0.088 0.077
0.143 0.069 0.140 0.142 0.139 0.227 0.093 0.216 0.212 0.208 0.116 0.052 0.118 0.125 0.112
0.145 0.061 0.147 0.150 0.142 0.255 0.130 0.255 0.241 0.242 0.122 0.037 0.130 0.128 0.120
0.155 0.057 0.156 0.164 0.149 0.216 0.096 0.216 0.204 0.205 0.112 0.032 0.118 0.120 0.116
0.141 0.064 0.142 0.141 0.142 0.190 0.143 0.187 0.176 0.164 0.108 0.034 0.106 0.102 0.104
0.2 0.389 0.297 0.384 0.388 0.382 0.416 0.304 0.412 0.388 0.380 0.162 0.102 0.177 0.184 0.157
0.340 0.185 0.337 0.344 0.333 0.375 0.180 0.375 0.359 0.347 0.175 0.085 0.183 0.191 0.176
0.333 0.143 0.339 0.335 0.327 0.405 0.235 0.399 0.385 0.386 0.159 0.057 0.174 0.175 0.158
0.336 0.144 0.336 0.337 0.336 0.381 0.172 0.379 0.363 0.360 0.147 0.058 0.155 0.158 0.147
0.308 0.168 0.310 0.315 0.312 0.190 0.286 0.187 0.176 0.164 0.137 0.064 0.141 0.134 0.139
0.3 0.696 0.600 0.698 0.684 0.694 0.644 0.500 0.650 0.618 0.603 0.294 0.215 0.291 0.302 0.282
0.571 0.351 0.571 0.569 0.559 0.553 0.292 0.563 0.538 0.517 0.258 0.145 0.261 0.261 0.252
0.561 0.284 0.564 0.566 0.549 0.604 0.347 0.598 0.581 0.568 0.252 0.093 0.264 0.264 0.249
0.569 0.302 0.566 0.565 0.559 0.524 0.281 0.518 0.520 0.501 0.223 0.102 0.229 0.232 0.227
0.557 0.355 0.549 0.556 0.541 0.640 0.476 0.621 0.592 0.573 0.250 0.129 0.251 0.252 0.243
Table 2: Power comparison for the proposed tests under location shift.
PT ETA ETR IETA IETR PT ETA ETR IETA IETR
Normal Distribution
0.056 0.054 0.054 0.053 0.056 0.069 0.068 0.066 0.066 0.068
0.072 0.072 0.070 0.070 0.073 0.074 0.077 0.081 0.071 0.077
0.067 0.066 0.069 0.067 0.067 0.087 0.081 0.079 0.081 0.077
0.058 0.057 0.057 0.060 0.056 0.068 0.070 0.066 0.060 0.066
0.067 0.063 0.067 0.066 0.065 0.067 0.070 0.069 0.069 0.066
Exponential Distribution
0.073 0.065 0.068 0.068 0.068 0.067 0.059 0.060 0.059 0.056
0.084 0.076 0.079 0.077 0.078 0.083 0.078 0.075 0.078 0.074
0.099 0.094 0.094 0.093 0.092 0.063 0.058 0.063 0.053 0.051
0.103 0.100 0.099 0.096 0.097 0.076 0.082 0.076 0.068 0.070
0.078 0.069 0.070 0.069 0.069 0.071 0.067 0.066 0.059 0.065
Logistic Distribution
0.060 0.061 0.061 0.062 0.064 0.058 0.061 0.061 0.056 0.061
0.071 0.074 0.074 0.070 0.073 0.075 0.076 0.079 0.076 0.079
0.077 0.078 0.079 0.081 0.079 0.071 0.072 0.072 0.067 0.071
0.078 0.080 0.080 0.080 0.078 0.075 0.079 0.075 0.075 0.076
0.068 0.069 0.065 0.067 0.067 0.064 0.060 0.063 0.064 0.063
Table 3: Observed -levels for the proposed tests for testing for normal distribution and for the exponential and logistic distributions, under imperfect ranking.
Normal dist. Exponential dist. Logistic dist.
PT EAT EAR IEAT IEAR PT EAT EAR IEAT IEAR PT EAT EAR IEAT IEAR
0.1 0.162 0.162 0.163 0.168 0.162 0.218 0.212 0.203 0.208 0.202 0.090 0.099 0.101 0.100 0.105
0.163 0.161 0.168 0.170 0.174 0.211 0.208 0.201 0.199 0.195 0.105 0.114 0.110 0.112 0.109
0.143 0.146 0.152 0.143 0.150 0.230 0.223 0.221 0.225 0.225 0.105 0.116 0.116 0.113 0.119
0.149 0.155 0.154 0.154 0.161 0.222 0.215 0.212 0.210 0.208 0.105 0.111 0.116 0.112 0.114
0.158 0.158 0.159 0.155 0.155 0.212 0.190 0.189 0.193 0.190 0.088 0.090 0.093 0.094 0.094
0.2 0.394 0.397 0.399 0.404 0.40 0.413 0.382 0.381 0.388 0.381 0.160 0.171 0.169 0.171 0.166
0.349 0.353 0.355 0.357 0.358 0.379 0.362 0.364 0.355 0.350 0.159 0.171 0.172 0.169 0.172
0.325 0.340 0.337 0.333 0.340 0.394 0.373 0.378 0.374 0.375 0.152 0.157 0.161 0.156 0.162
0.332 0.327 0.333 0.328 0.338 0.373 0.354 0.358 0.352 0.351 0.169 0.171 0.176 0.179 0.181
0.326 0.322 0.328 0.328 0.334 0.412 0.374 0.374 0.372 0.367 0.148 0.156 0.151 0.152 0.153
0.3 0.709 0.708 0.704 0.709 0.706 0.643 0.615 0.609 0.607 0.607 0.303 0.310 0.309 0.309 0.312
0.584 0.588 0.586 0.584 0.588 0.517 0.498 0.498 0.493 0.482 0.259 0.269 0.273 0.277 0.275
0.556 0.563 0.561 0.557 0.556 0.594 0.571 0.570 0.569 0.566 0.238 0.249 0.254 0.250 0.255
0.571 0.570 0.565 0.565 0.569 0.530 0.516 0.518 0.508 0.507 0.247 0.249 0.249 0.248 0.257
0.558 0.563 0.555 0.556 0.561 0.619 0.572 0.574 0.568 0.563 0.247 0.255 0.252 0.255 0.251
Table 4: Power comparison for the proposed tests under location shift and imperfect ranking with .

4.2 Imperfect ranking

In this section, we compare the finite sample performance of our proposed bootstrapping techniques with the PB under imperfect ranking cases. In order to produce the imperfect URSS/RSS samples, we use the model proposed by Dell and Clutter (1972). Let and denote the judgment and true order statistics, respectively. Suppose

where and are independent.

Using imperfect URSS with and 1, we report the observed significance levels for testing against for different methods in Table 3. These choices of resulted in the observed correlation coefficients of 0.89 and 0.70 between the ranking variable and the variable of interest, respectively. As compared with the results under the perfect ranking assumption, the proposed methods seem to be robust with respect to imperfect ranking. It was shown that the test under exponential distribution for the imperfect sampling is a bit liberal. We also observe that imperfect ranking affects the power of the tests since, as it is shown in Table 4, by adding errors in ranking, the power of the proposed tests decreases. The importance of accurate ranking in RSS designs has been mentioned in several works. Frey, Ozturk and Deshpande (2007) considered nonparametric tests for the perfect judgment ranking. Li and Balakrishnan (2008) proposed several nonparametric tests to investigate perfect ranking assumption. Vock and Balakrishnan (2011) suggested a Jonckheere-Terpstra type test statistic for perfect ranking in balanced RSS. These tests are further studied by Frey and Wang (2013) and compared with the most powerful test.

In order to derive the theoretical results under the imperfect ranking assumption, one can proceed as follow. First, note that under the imperfect ranking the density function of characteristic of interest for the unit judged to be ranked is no longer . We denote this density with . One approach to derive the CDF of the th judgmental order statistic is to use the following model

(18)

where is the probability that the th order statistic is judged to have rank , with .

Lemma 3.

Suppose the imperfect ranking in the RSS design is such that

For the resampling technique EAT (or EAR) under the imperfect ranking assumption, which is denoted by IEAR (or IEAR), we have

where is the EDF of the resulting bootstrap sample.

Proof.

We first note that using the IEAT (or IEAR), we have

where and are the EDF of the resulting bootstrap samples under the IEAT (or IEAR) and EAT (or EAR), respectively. One can easily show that

Hence, we have

and this completes the proof. ∎

4.3 Comparison with the empirical likelihood method

In this section, we compare the performance of the bootstrap tests based on ET estimators of with the one based on the empirical likelihood estimator of which is already studied in the literature by Baklizi (2009) and Liu et al. (2009). Empirical likelihood is an estimation method based on likelihood functions without having to specify a parametric family for the observed data. Empirical likelihood methodology has become a powerful and widely applicable tool for non-parametric statistical inference and it has been used under different sampling designs. For a comprehensive review of the empirical likelihood method and some of its variations see Owen (2001). For testing the null hypothesis using the empirical likelihood estimator of based on a balanced RSS sample, Baklizi (2009) showed that under the finite variance assumption

(19)

where