On Extended Neoteric Ranked Set Sampling Plan: Likelihood Function Derivation and Parameter Estimation

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Introduction
Ranked set sampling (RSS) plans were proposed to provide estimators that are more efficient than those derived under simple random sampling (SRS) plans.RSS plans were first proposed by McIntyre in 1952, to find efficient estimates of the mean pasture yields.ese plans assume that there are no errors in ranking the units concerning the variable of interest.In most practical applications, imperfect ranking exists and there will be an efficiency loss in the estimators [1].To reduce such losses, several modifications to the RSS procedure were proposed.e main purpose was to allow for the achievement of higher statistical efficiency and probably a lower operating effort.e first modification of RSS was the extreme ranked set sampling (ERSS) plan introduced by Samwai et al. [2].Muttlak [3] proposed the median ranked set sampling (MRSS) plan, Al-Odat and Al-Saleh [4] proposed the moving extreme ranked set sampling (MERSS) plan, Al-Saleh and Al-Omari [5] proposed the multistage ranked set sampling (MSRSS), and others proposed the multistage ranked set sampling (MSRSS).Recently, Zamanzade E, Al-Omari [1] proposed a new ranked set sampling plan based on a dependent scheme, namely, the neoteric ranked set sampling (NRSS) plan which showed relative improvement in the efficiency of the population mean and variance estimates.Moreover, Taconeli and Cabral [6] proposed several modifications to the NRSS plan.One of these plans is the extended neoteric ranked set sampling (ENRSS) plan.ey showed that the ENRSS plan is superior to NRSS and other plans.Unlike RSS and ERSS plans, NRSS and ENRSS are classified as dependent RSS plans as the resulting samples have a dependence structure.
In 2020, Sabry and Shabaan proposed the likelihood function of the NRSS plan and used it to estimate the parameters of the inverse Weibull distribution.Chen et al. [7] obtained RSS for efficient estimation of a population proportion.Terpstra and Liudahl [8] constructed concomitantbased rank set sampling proportion estimates.Mahdizadeh [9] discussed entropy-based test of exponentiality in ranked set sampling.Mahdizadeh and Strzalkowska-Kominiak [10] discussed resampling-based inference for a distribution function using censored ranked set samples.Akhter et al. [11] discussed RSS for generalized Bilal distribution.Strzalkowska-Kominiak and Mahdizadeh [12] discussed Kaplan-Meier estimator based on ranked set samples.Aljohani et al. [13] discussed ranked set sampling with an application of modified Kies exponential distribution.Sabry et al. [14] used a hybrid approach to evaluate the performance of some ranked set sampling strategies.Sabry and Almetwally [15] used under-ranked and double-ranked set sampling designs to estimate the parameters of the exponential Pareto distribution.e results showed similar results to Zamanzade and Al-Omari [16] and Taconeli and Cabral [6] results when estimating the population means and variance.
is paper aims to compute the joint order distribution of an ENRSS sample and consequently propose the associated likelihood function.Also, to use the proposed likelihood function to estimate the parameters of the inverted Nadarajah-Haghighi distribution and to conduct Monte Carlo simulations to assess the performance of the ENRSS plan and compare the results with RSS, ERSS, and NRSS plans.
e inverted Nadarajah-Haghighi [INH(λ, α)] distribution was proposed by Taher and co-authors (2018).e cumulative distribution function (CDF), probability density function (PDF), and quantile function used are as follows: x where λ, α > 0, and u have a uniform U(0, 1) distribution.Figure 1 illustrates different PDF plots for the INH distribution.e remainder of the paper is laid out as follows: the second section is devoted to a brief overview of the various RSS plans discussed in this study.In Section 3, maximum likelihood analysis for the INH distribution is considered for all plans, and in Section 4, an extensive simulation study is conducted and the different plans are compared and the results are reported.Finally, in Section 5, the paper is concluded.

Ranked Set Sampling Designs
e ranked set sampling plans covered in this study are discussed in this section, with the number of cycles of the RSS plans assumed to be one for simplicity.

RSS Design.
e RSS design according to Wolfe [17] can be described as follows: Step 1: select m 2 units randomly from the target population with CDF F(x; θ) and PDF f(x; θ) Step 2: allocate the m 2 selected units as randomly as possible into m sets, each of size m Step 3: rank the units within each set without yet knowing any values for a variable of interest Step 4: choose a sample for real quantification by selecting the smallest ranked unit from the first set, the second smallest ranked unit from the second set, and so on until the largest ranked unit from the last set is chosen Step 5: to get a sample of size n � mr, repeat steps 1 through 4 for r cycles Figure 2 describes the process for choosing an RSS sample from one cycle where X (11) is the lowest observation in the first raw, X (22) is the second-lowest observation from the second raw, and finally the greatest observation from the previous raw is X (mm) .
Let y is , i � 1, 2, . . ., m, s � 1, 2, . . ., r and − ∞ < x (i)  < ∞} denote a ranked set sample derived from a distribution with PDF f(x; θ) and CDF F(x; θ), where m denotes the set size, r denotes the number of cycles, and θ is the parameter space.e probability function for this design is as follows:   Complexity (4)

ERSS Design.
It is the first variation of RSS proposed by Samawi et al. [2] which only uses a maximum or minimum ranked unit from each set to determine the population mean.e following approach is used to estimate based on ERSS.
Step 1: repeat Steps 1 through 3 in the RSS design.
Step 2: the selection mechanism can be altered depending on whether the set size is even or odd.Select the lowest-ranked unit of each set from the first m/2 sets and the highest-ranked unit of each set from the other m/2 sets if the set size m is even.If the number of units in the set is odd, choose the lowest-ranked unit from the first m − 1/2 sets, the highest-ranked unit from the second m − 1/2 sets, and the median from the remaining set.
Step 3: to get a sample of size n � mr, repeat the previous procedures r times.
e process for one cycle, as well as the cases of m � 4 and m � 5, is as shown in Figure 3.
Let z i , i � 1, 2, . . ., m   be a ranked set sample (RSS) generated from a distribution with pdf g(x; θ) and cdf G(x; θ), where the set size is m and the parameter space is θ.Let g � m/2, h � m − 1/2, and u i � m − i + 1, then the likelihood function of the ERSS sample is given by Case 1. m odd: ( Case 2. m even:

Neoteric Ranked Set Sampling (NRSS) Design.
e NRSS sampling plan presented by Zamanzade E and Al-Omari [1] is described in the following steps: Step 1: select m 2 random units from the target population.
Step 2: rank the m 2 sample units according to some predetermined criteria.
Figure 4 displays the step for establishing an NRSS sample in one cycle when m � 3 be a neoteric ranked set sample obtained from a distribution with PDF h(u; θ) and CDF H(u; θ), and let u be a random sample of size n from a continuous population, where m is the set size, r is the number of cycles, θ is the parameter space, and n � mr.en, the likelihood function of NRSS samples is given by where and (see Sabry and Shabaan [18]). □

Extended Neoteric Ranked Set Sampling (ENRSS) Plan.
In ENRSS, Taconeli and Cabral's [6] proposal is based on a single ranking stage, where m 3 sample units are observed instead m 2 sample units as in all one-stage ranked set sampling designs.ese m 3 sample units are arranged in a single set and ordered utilizing some inexpensive ranking criteria.e units that will make up the final sample must next be chosen from (almost) evenly spaced spots.e ENRSS technique is described in the steps as follows: Step 1: choose m 3 elements from the target population and combine them into a single ranking set.
Step 3: to obtain a sample of size n � mr, repeat steps 1-2r times.
To pick an ENRSS sample with a set size of m � 3, a single set of m 3 � 27 sample units should be taken, and the sample units ranked in positions 5, 14, and 23 should be chosen to create the final sample.If you want a sample of set size m � 4, use m 3 � 64, and the desired ENRSS sample will constitute the ranked units in positions 9, 24, 41, and 56 (see [6]).
Since to get an ENRSS sample X (i j )   of size m, m 3 units are selected and ranked (ordered).erefore and according to [9], the joint PDF of an ENRSS sample of size m is given by If the ENRSS design is repeated r time to get a sample of size mr, the likelihood function of an ENRSS sample with set size m and number of cycles r will be given by

Complexity
where ) � ∞ and i t is defined in (8). is completes the proof.For more elaborations, we can see in Figure 6. □

Estimation of the Inverted Nadarajah-Haghighi Distribution Parameters
e parameters of INH (λ, α) distribution are estimated using the maximum likelihood estimation (MLE) method.
e estimation process is taking place when samples are drawn according to SRS, RSS, ERSS, NRSS, and ERSS sampling plans as illustrated in section 2.

Estimation
Based on SRS.Let (x i , i � 1, 2, . . ., m) be a SRS with CDF and PDF given in ( 1) and ( 2), respectively, the likelihood function of the SRS samples from INH (λ, α) distribution is e log-likelihood function is thus given by and the associated likelihood equations are therefore identified as

Estimation Based on RSS.
Let y is , i � 1, 2,  . . ., m; s � 1, 2, . . ., r} be a ranked set sample with the CDF and PDF provided in (1) and ( 2), respectively, where m is the set size, r is the number of cycles, and n � mr.Assuming r � 1 and according to (4), the likelihood function of the RSS samples from INH (λ, α) distribution is given by where e following is a direct derivation of the associated log-likelihood function: and the normal likelihood equations is as follows: v (11) v (20) z (1) v (10) v (19) 6 Complexity

Estimation Based on ERSS.
Let z i , i � 1, 2, . . ., m   be a ERSS drawn from a distribution with CDF and PDF as in ( 1) and ( 2), respectively, where m is the set size.Let g � m/2, h � m − 1/2, and u i � m − i + 1, then the likelihood function of the ERSS representative sample from INH (λ, α) according to (3) and ( 4) is given by Case 1. m odd: e log-likelihood function follows directly as erefore, the associated likelihood normal equations will be Figure 6: Order statistics of a subset of complete order statistics corresponding to the random sample x i , i � 1, 2, . . ., n  . Complexity Case 2. m even: Similarly, the log-likelihood function is given directly as us, the associated likelihood normal equations will be given as 8 Complexity and k i is defined as in ( 6)} be a neoteric ranked set sample, where m is the set size.According to (5), the likelihood function of NRSS samples drawn from INH (λ, α) for one cycle is given by e log-likelihood associated with this design is then given by where e associated normal equations are directly derived as   1 and 2. From the table, some important conclusions can be observed from the results.
As expected, the efficiency of all RSS-based designs increases as the sample size increases and as the shape of the distribution is near symmetry.
(i) It is clear that the proposed likelihood function for the ENRSS plan is working effectively and provides efficient estimators similar to the results reported by Taconeli and Cabral [6] (ii) e ENRSS plan estimators outperform the onestage RSS plans when the process does not include ranking errors (iii) e NRSS plan estimators outperform the one-stage ERSS and RSS and SRS plans (iv) e RSS plan estimators outperform the one-stage SRS plans (v) Biases are almost negligible when the shape of the distribution is near symmetry  imperfect ranking cases and study their performance when making parameter estimations for several symmetric and asymmetric distributions [19][20][21].

Figure 2 :
Figure 2: Display of m 2 observations in one cycle and the selected RSS sample of size m � 3.

Figure 3 :
Figure 3: Display of m 2 observations in one cycle and the selected ERSS sample of size: (a)m � 4 and (b)m � 5.

Figure 4 :
Figure 4: Display of m 2 observations in one cycle and the selected NRSS sample of size m � 3.

Figure 5 :
Figure 5: Display of m 3 observations in one cycle and the selected ENRSS sample of size m � 3.
SeeFigure5which shows m 3 observations in one cycle and the selected ENRSS sample of size m � 3.
Proof.Let x i , i � 1, 2, . . ., n   be a random sample from a continuous distribution with order statistics x (i) , i �  1, 2, . . ., n}.Let x (i j ) , i � 1, 2, . . ., n   be a subset of the order statistics corresponding to x i  .en, the joint PDF of

Table 1 :
Relative efficiency for RSS-based estimators for different sampling plans and INH distribution with scale parameter λ� 1.

Table 2 :
Relative bias for RSS-based estimators for different sampling plans and INH distribution with scale parameter λ� 1.