A New Class of L-Moments Based Calibration Variance Estimators

: Variance is one of the most important measures of descriptive statis-tics and commonly used for statistical analysis. The traditional second-order central moment based variance estimation is a widely utilized methodology. However, traditional variance estimator is highly affected in the presence of extreme values. So this paper initially, proposes two classes of calibration estimators based on an adaptation of the estimators recently proposed by Koyuncu and then presents a new class of L-Moments based calibration variance estimators utilizing L-Moments characteristics (L-location, L-scale, L-CV) and auxiliary information. It is demonstrated that the proposed L-Moments based calibration variance estimators are more efficient than adapted ones. Artificial data is considered for assessing the performance of the proposed estimators. We also demonstrated an application related to apple fruit for purposes of the article. Using artificial and real data sets, percentage relative efficiency (PRE) of the proposed class of estimators with respect to adapted ones are calculated. The PRE results indicate to the superiority of the proposed class over adapted ones in the presence of extreme values. In this manner, the proposed class of estimators could be applied over an expansive range of survey sampling whenever auxiliary information is available in the presence of extreme values.

see [1][2][3][4][5]. In some cases when some population parameters, as the mean, coefficient of variation, the standard deviation of the auxiliary variable have been known many authors including [6,7] imagined an enormous number of improved estimators for the population mean of the variate of interest. This urge researchers to utilize the characteristics of the auxiliary variable (mean, coefficient of variation, standard deviation, median and quartiles) in order to improve the population variance estimate of the study variable.
Variations are available in our life. For example, a doctor needs a full comprehension of varieties in the level of human circulatory strain, internal heat level, and heartbeat rate for satisfactory medicine. A maker needs steady information on the degree of varieties in individuals' response to his item to have the option to realize whether to decrease or increment his cost or improve the nature of his item. An agriculturist requests a sufficient comprehension of the varieties in climatic factors particularly all around (or from time to time) to have the option to anticipate when, where, and how to plant his harvest. A lot of circumstances can be experienced where the population variance estimate of the study variable assumes a huge significance. Consequently, different authors, for example, [8,9] have given their consideration to the improved estimation of population variance of the study variable with the utilization of auxiliary information. However, all these quoted estimators based on traditional second-order central moment based variance estimation which is highly affected in presence of extreme values. So, in the current article, we have characterized a class of estimators for the population variance of the study variable based on L-Moments and calibration approach that is less effective to extreme values.
An experimental investigation utilizing apple production data sets was led, and we got good outcomes, numerically. The analysis of apple fruit production has great importance in food sciences. In view of the applications here, this study well provide a significant premise for future food sciences, medical researches, and many engineering applications.
The rest of the article is arranged in the following major sections. In Section 2, we present some classes of calibration based variance estimators according to the traditional second-order central moment. In Section 3, we provide L-Moments characteristics (L-location, L-scale, L-CV) and, propose a new class of calibration based variance estimators under stratified random sampling scheme. Results discussion is documented in Section 4. Finally, some conclusions are given in Section 5.

Adapted Classes of Estimators
Consider a finite population Ω a = {v a1 , v a2 , . . . , v an } of size N. Suppose (Y , X ) be the study and auxiliary variables associated with Ω a . Let Ω a is stratified into L strata with the hth stratum containing N h units, where h = 1, 2, . . . , L, such that L h=1 N h = N. A simple random sample (srs) of size n h is drawn without replacement from the hth stratum such that L h=1 n h = n. Further, (x hi , y hi ) are the observed values of Y and X in hth stratum, where i = 1, 2, . . . , N h and h = 1, 2, . . ., L. In light of this stratified sampling design, taking motivation from [1], we present the first calibrated class of variance estimators as given below where s 2 yh is the sample variance of the study variable in hth stratum. Further, Φ SH h are calibrated weights that are selected in such a way that the sum of the chi-square type distance measure is minimum, subject to the calibration constraints Note that W h = N h N denoting the traditional stratum weight, Δ xh , Δ xh are the sample and population characteristics of the auxiliary variable in hth stratum and, Ψ h are appropriately selected weights that express the form of the estimator. Some suitable choices for Ψ h are provided in Tab. 1.
Minimization of Eq. (2), subject to the calibration constraints set out in Eqs. (3) and (4), the optimum weights shall be calculated by and thus wherê The whole class T (SH a ) is provided in Tab. 1.
Taking motivation from [10], we adapt the following calibration estimator of the population variance given by Minimization of Eq. (8), subject to the calibration constraints set out in Eqs. (9) and (10), the optimum weights shall be calculated by and thus The whole class

L-Moments and Propose Class of Estimators
The occurrence of extreme values, for example, in agriculture engineering, hydrology, food sciences. Meteorology and climatology, among others, observed nowadays in various parts of the world may influence adversely on human culture. Therefore, it is essential to appraise the accurate measurements of the numerical data in the presence of extreme values. As we have depicted before that variance is one of the significant measures of data description. Variance estimation dependent on traditional moments which is profoundly influenced by extreme values. Hence, a quantifiable procedure is necessitated that will catch the extreme values in the assessment, however less influenced by their quality. An elective strategy that has the vital ability to settle this issue is the L-Moments. Because L-Moments are more robust than traditional moments in the presence of extreme values [11]. The general form of the population L-Moments for the auxiliary variable with respect to hth stratum are as given below Corresponding to these, we can write the sample based L-Moments such asλ 1xh ,λ 2xh ,λ 3xh , λ 4xh , based on sample observations, respectively. Similarly, we can write the general form of the L-Moments for study variable by replacing x with y. For more details regarding L-Moments, interested readers may refer to [12]. Some notations for upcoming proposed work in light of L-Moments with respect to h th stratum are   The calibration estimation is a method of adjusting the original design weights to expand the accuracy of estimates by utilizing auxiliary information. Calibration based estimators utilize calibrated/adjusted weights that are planned to minimize a given measure of distance to the original weights while fulfilling a group of constraints with the auxiliary information. The authors in [13] are pioneer of calibration estimation. After that much of the work has been done in this filed such as [1,10,14]. These results motivated us to define the following calibrated estimator of variance as where s 2 ymh is the sample variance of the study variable in hth stratum. Further, Φ Pai h are calibrated weights that are selected in such a way that the sum of the chi-square type distance measure is minimum, subject to the calibration constraints Note that W h = N h N denotes the traditional stratum weight, Δ xmh , Δ xmh are the sample and population characteristics of the auxiliary variable in h th stratum and, Ψ h are appropriately selected weights that express the form of the estimator. In most situations, Ψ h = 1. Minimization of Eq. (13), subject to the calibration constraints set out in Eqs. (14) and (15), the optimum weights shall be calculated by and thus wherê

Results and Discussion
In this section, the performance of the suggested estimators is investigated through a simulation study.

Simulation Design
The simulation design is organized as follows: A random variable X h and random variable Y h are defined as follows:   Further, we assume that p = 1.6, v = 5, R = 2 and ε has a standard normal distribution in Eq. (19). We consider the population of size N = 1000 for hth stratum. The overall sample of size n h = 400 is selected for purposes of the article. Using equal allocation of a sample of size 100 is selected from h th stratum. A scatter plot for each stratum is provided (see Figs. 1-4). These figures clearly show the presence of extreme values and hence suitable for our proposed estimators.   The steps of simulation study can be carried out as: Step-1: A sample of size n h is selected through SRSWOR from the hth stratum.
Step-4: The mean square error (MSE) of the estimators is computed as Step

Real Life Data
To illustrate the behavior of the proposed estimators in this article, we consider a data set of apple fruit, used by [15], where X = number of apple trees (1 unit = 100 trees), Y = level of apple production (1 unit = 100 tonnes).
It is worth mentioning that we consider 477 villages in 4 strata, respectively (like 1: Marmarian, 2: Agean, 3: Mediterranean, and 4: Central Anatolia) in 1999. We draw a scatter plot for each stratum (see Figs. [5][6][7][8]. The scatter plot of each stratum representing extreme values. Some important characteristics of the data are provided in Tab. 6. The PRE of the estimators is calculated using defined five steps and available in Tabs. 7 and 8. However, this investigation can assist with revealing insight into the value of the proposed class. Actually, it gives a reasonable sign that more mind-boggling options than the traditional moments based variance estimators can be helpful when the data is contaminated with extreme values. Because each PRE result of the proposed estimator is greater than 100. Which means that the proposed class beats the adapted ones thoroughly. Based on real and artificial numerical illustration, we feel confident that the same could hold in other contexts of food sciences and agricultural engineering.

Conclusions
Usually, in test reviews, it uses traditional moments-based techniques to obtain improved designs and more accurate estimators. However, in the presence of extreme values, traditional moments-based techniques may provide misleading results. In this paper, a calibration approach is utilized and developed two adapted families of estimators. Then a new class of L-Moments based variance estimators has been proposed which provides a better estimate of variance in the presence of extreme values. The performance of adapted and proposed estimators has been assessed through artificial and apple fruit data set. Based on satisfactory results, we recommend survey practitioners utilize the proposed class of variance estimators in variou environmental researches.