Статті в журналах: "Ratio estimators"

1

Ekiz, Meltem, and Osman Ufuk Ekiz. "Modelling Tap Water Consumer Ratio." Mathematics 8, no. 9 (September 10, 2020): 1557. http://dx.doi.org/10.3390/math8091557.

Повний текст джерела

Анотація:

Increasing population and the rising air temperatures are known as factors that cause water depletion in the watersheds. Therefore, it is important to accurately predict the future ratios of tap water consumers using the same watershed to the population living in the specified area, to produce better water policies and to take the necessary measures. Predictions can be made by a growth curve model (GCM). Parameter estimations of the GCM are usually based on the ordinary least square (OLS) estimator. However, the outlier presence affects the estimations and the predictions, which are obtained by using the estimated model. The present article attempts to construct first- and third-order GCMs with robust least median square (LMS) and M estimators to make short-term predictions of ratios of tap water consumers. According to the findings, parameter estimations of the models, the outliers, and the predictions vary with respect to the estimators. The M estimator for short-term predictions is suggested for use, due to its robustness against outlier points.

Стилі APA, Harvard, Vancouver, ISO та ін.

2

Ijaz, Muhammad, Syed Muhammad Asim, Atta ullah, and Ibrahim Mahariq. "Flexible Robust Regression-Ratio Type Estimators and Its Applications." Mathematical Problems in Engineering 2022 (September 28, 2022): 1–6. http://dx.doi.org/10.1155/2022/8977392.

Повний текст джерела

Анотація:

In real-world situations, the data set under examination may contain uncommon noisy measurements that unreasonably affect the data’s outcome and produce incorrect model estimates. Practitioners employed robust-type estimators to reduce the weight of the noisy measurements in a data set in such a scenario. Using auxiliary information that will produce reliable estimates, we have looked at a few flexible robust-type estimators in this study. In order to estimate the population mean, this study presents unique flexible robust regression type ratio estimators that take into account the data from the midrange and interdecile range of the auxiliary variables. Up to the first order of approximate computation, the bias and mean square were calculated. In order to compare the flexibility of the proposed estimator to those of the existing estimators, theoretical conditions were also obtained. We took into account data sets containing outliers for empirical computation, and it was found that the suggested estimators produce results with higher precision than the existing estimators.

Стилі APA, Harvard, Vancouver, ISO та ін.

3

Chadyšas, V., and D. Krapavickaitė. "Investigation of Accuracy of a Calibrated Estimator of a Ratio by Modelling." Nonlinear Analysis: Modelling and Control 10, no. 4 (October 25, 2005): 333–42. http://dx.doi.org/10.15388/na.2005.10.4.15113.

Повний текст джерела

Анотація:

Estimator of finite population parameter – ratio of totals of two variables – is investigated by modelling in the case of simple random sampling. Traditional estimator of the ratio is compared with the calibrated estimator of the ratio introduced by Plikusas [1]. The Taylor series expansion of the estimators are used for the expressions of approximate biases and approximate variances [2]. Some estimator of bias is introduced in this paper. Using data of artificial population the accuracy of two estimators of the ratio is compared by modelling. Dependence of the estimates of mean square error of the estimators of the ratio on the correlation coefficient of variables which are used in the numerator and denominator, is also shown in the modelling.

Стилі APA, Harvard, Vancouver, ISO та ін.

4

Malik, Sachin, and Rajesh Singh. "Some Improved Multivariate-Ratio-Type Estimators Using Geometric and Harmonic Means in Stratified Random Sampling." ISRN Probability and Statistics 2012 (August 26, 2012): 1–7. http://dx.doi.org/10.5402/2012/509186.

Повний текст джерела

Анотація:

Auxiliary variable is commonly used in survey sampling to improve the precision of estimates. Whenever there is auxiliary information available, we want to utilize it in the method of estimation to obtain the most efficient estimator. In this paper using multiauxiliary information we have proposed estimators based on geometric and harmonic mean. It was also shown that estimators based on harmonic mean and geometric mean are less biased than Olkin (1958) and Singh (1967) estimators under certain conditions. However, the MSE of Olkin (1958) estimator and geometric and harmonic estimators are same up to the first order of approximations.

Стилі APA, Harvard, Vancouver, ISO та ін.

5

Chami, Peter S., Bernd Sing, and Doneal Thomas. "A Two-Parameter Ratio-Product-Ratio Estimator Using Auxiliary Information." ISRN Probability and Statistics 2012 (October 14, 2012): 1–15. http://dx.doi.org/10.5402/2012/103860.

Повний текст джерела

Анотація:

We propose a two-parameter ratio-product-ratio estimator for a finite population mean in a simple random sample without replacement following the methodology in the studies of Ray and Sahai (1980), Sahai and Ray (1980), A. Sahai and A. Sahai (1985), and Singh and Espejo (2003).The bias and mean squared error of our proposed estimator are obtained to the first degree of approximation. We derive conditions for the parameters under which the proposed estimator has smaller mean squared error than the sample mean, ratio, and product estimators. We carry out an application showing that the proposed estimator outperforms the traditional estimators using groundwater data taken from a geological site in the state of Florida.

Стилі APA, Harvard, Vancouver, ISO та ін.

6

Cahalan, Jennifer A., Jason Gasper, and Jennifer Mondragon. "Catch estimation in the federal trawl fisheries off Alaska: a simulation approach to compare the statistical properties of three trip-specific catch estimators." Canadian Journal of Fisheries and Aquatic Sciences 72, no. 7 (July 2015): 1024–36. http://dx.doi.org/10.1139/cjfas-2014-0347.

Повний текст джерела

Анотація:

Quantifying catch has been recognized worldwide as a critical component in fisheries management. Assessment of discard is challenging because of the requirement for at-sea observation, which is both logistically difficult and costly to fishery agencies. Statistical estimators using robust sampling methods may yield accurate and imprecise estimates given the variability associated with many at-sea discard species and inability for agencies to obtain high sampling fractions. However, biased estimates occur if an inappropriate estimator is used. Using Alaska trawl fisheries as an example, we investigated the statistical properties and implementation issues for three commonly used estimators: the ratio estimator; a simple mean estimator; and a deterministic imputation method currently in use in federal fisheries off Alaska. We used a simulation approach to evaluate the performance of these estimators to estimate trip-specific catch. Several statistical properties were evaluated: bias of the estimators, variability of the estimators, and accuracy of the variance estimators. The simple mean estimator had the best performance for vessels landing catch at shoreside processors. The choice of estimator was less clear for vessels processing catch, owing to sensitivity associated with species composition and implementation issues for the simple mean and ratio estimators.

Стилі APA, Harvard, Vancouver, ISO та ін.

7

Shaheen, Nazia, Muhammad Nouman Qureshi, Osama Abdulaziz Alamri, and Muhammad Hanif. "Optimized inferences of finite population mean using robust parameters in systematic sampling." PLOS ONE 18, no. 1 (January 23, 2023): e0278619. http://dx.doi.org/10.1371/journal.pone.0278619.

Повний текст джерела

Анотація:

In this article, we have proposed a generalized estimator for mean estimation by combining the ratio and regression methods of estimation in the presence of auxiliary information using systematic sampling. We incorporated some robust parameters of the auxiliary variable to obtain precise estimates of the proposed estimator. The mathematical expressions for bias and mean square error of proposed the estimator are derived under large sample approximation. Many other generalized ratio and product-type estimators are obtained from the proposed estimator using different choices of scalar constants. Some special cases are also discussed in which the proposed generalized estimator reduces to the usual mean, classical ratio, product, and regression type estimators. Mathematical conditions are obtained for which the proposed estimator will perform more precisely than the challenging estimators mentioned in this article. The efficiency of the proposed estimator is evaluated using four populations. Results showed that the proposed estimator is efficient and useful for survey sampling in comparison to the other existing estimators.

Стилі APA, Harvard, Vancouver, ISO та ін.

8

Subramani, J., and G. Kumarapandiyan. "A Class of Modified Ratio Estimators for Estimation of Population Variance." Journal of Applied Mathematics, Statistics and Informatics 11, no. 1 (May 1, 2015): 91–114. http://dx.doi.org/10.1515/jamsi-2015-0006.

Повний текст джерела

Анотація:

Abstract In this paper we have proposed a class of modified ratio type variance estimators for estimation of population variance of the study variable using the known parameters of the auxiliary variable. The bias and mean squared error of the proposed estimators are obtained and also derived the conditions for which the proposed estimators perform better than the traditional ratio type variance estimator and existing modified ratio type variance estimators. Further we have compared the proposed estimators with that of the traditional ratio type variance estimator and existing modified ratio type variance estimators for certain natural populations.

Стилі APA, Harvard, Vancouver, ISO та ін.

9

Rather, Khalid Ul Islam, Eda Gizem Koçyiğit, Ronald Onyango, and Cem Kadilar. "Improved regression in ratio type estimators based on robust M-estimation." PLOS ONE 17, no. 12 (December 12, 2022): e0278868. http://dx.doi.org/10.1371/journal.pone.0278868.

Повний текст джерела

Анотація:

In this article, a new robust ratio type estimator using the Uk’s redescending M-estimator is proposed for the estimation of the finite population mean in the simple random sampling (SRS) when there are outliers in the dataset. The mean square error (MSE) equation of the proposed estimator is obtained using the first order of approximation and it has been compared with the traditional ratio-type estimators in the literature, robust regression estimators, and other existing redescending M-estimators. A real-life data and simulation study are used to justify the efficiency of the proposed estimators. It has been shown that the proposed estimator is more efficient than other estimators in the literature on both simulation and real data studies.

Стилі APA, Harvard, Vancouver, ISO та ін.

10

Payandeh, Bijan, and Alan R. Ek. "Distance methods and density estimators." Canadian Journal of Forest Research 16, no. 5 (October 1, 1986): 918–24. http://dx.doi.org/10.1139/x86-163.

Повний текст джерела

Анотація:

The relative performance of five distance–density estimators was evaluated for the n-tree circular, semicircular, and strip plots on several simulated and natural tree populations. Results indicate that for the n-tree circular plot, the ratio estimator performed very well for most populations examined and for n > 10. The performance of both the maximum likelihood and first moment estimators was affected to a great degree by the spatial pattern of the populations, but they performed satisfactorily for the random and uniform populations and for large n values (i.e., n > 10). Smaltschinski's estimator resulted in nearly bias-free estimates for nonaggregated populations and for all n, but performed poorly otherwise. The generalized Prodan estimator performed well for the random population, but overestimated the density otherwise. The relative performance of all estimators for the n-tree semicircular plot was quite similar to that of estimators for the n-tree circular plot, except that the former tended to produce lower density estimates. For n-tree strip plots, the generalized Prodan and the ratio estimator performed very well for the nonaggregated populations and for n > 10. All other estimators resulted in density estimates lower than those for n-tree circular and semicircular plots.

Стилі APA, Harvard, Vancouver, ISO та ін.

11

Agbebia, C. S., and E. I. Enang. "Calibration Approach Ratio Estimators of Population Median in Stratified Random Sampling Design." Asian Journal of Probability and Statistics 21, no. 4 (March 29, 2023): 64–77. http://dx.doi.org/10.9734/ajpas/2023/v21i4472.

Повний текст джерела

Анотація:

Median as a measure of location gives a more robust estimate than the mean when dealing with heavy tailed or skewed distributions. It can also be used in cases of qualitative variables and open end intervals. Calibration, an approach that adjusts the original design weight by incorporating auxiliary information is employed using the chi square distance measure on a ratio median estimator under stratified random sampling to propose some estimators of population median. These proposed estimators are: the regression and ratio-type calibrated estimators with one constraint and the regression and ratio-type calibrated estimators with two constraints. The estimators of variance of these proposed estimators are also obtained. Empirical investigations on the performance of these estimators are carried out using R software simulated data set under underlying distributional assumptions of Cauchy and Lognormal, for sample sizes of 10%, 20% and 25%. The results showed that the proposed regression and ratio-type calibrated estimators with one constraint and the regression-type calibrated estimator with two constraints were more efficient than the existing ratio estimator and the proposed ratio-type calibrated estimator under two constraints for both Cauchy and the lognormal distributions.

Стилі APA, Harvard, Vancouver, ISO та ін.

12

Shabbir, Javid, Shakeel Ahmed, Aamir Sanaullah, and Ronald Onyango. "Measuring Performance of Ratio-Exponential-Log Type General Class of Estimators Using Two Auxiliary Variables." Mathematical Problems in Engineering 2021 (October 1, 2021): 1–12. http://dx.doi.org/10.1155/2021/5245621.

Повний текст джерела

Анотація:

In this paper, a ratio-exponential-log type general class of estimators is proposed in estimating the finite population mean using two auxiliary variables when population parameters of the auxiliary variables are known. From the proposed estimator, some special estimators are identified as members of the proposed general class of estimators. The mean square error (MSE) expressions are obtained up to the first order of approximation. This study finds that the proposed general class of estimators outperforms as compared to the conventional mean estimator, usual ratio estimators, exponential-ratio estimators, log-ratio type estimators, and many other competitor regression type estimators. Four real-life applications are used for efficiency comparison.

Стилі APA, Harvard, Vancouver, ISO та ін.

13

Suleiman, Sikiru A., and Amos A. Adewara. "Improved Modified Ratio Estimation of Population Mean Using Information on Size of the Sample." Tanzania Journal of Science 47, no. 5 (December 29, 2021): 1753–65. http://dx.doi.org/10.4314/tjs.v47i5.22.

Повний текст джерела

Анотація:

In sample surveys, auxiliary information is used for estimation to improve the eﬃciency of estimators. Increased precision can be obtained when the variable under study is highly correlated with auxiliary information. In this study, the sample size has been used as information for improved estimation of population mean of the main variable under study. A new modified generalized ratio type estimator of population mean has been proposed and the efficiency was examined using Murthy (1967) and Mukhopadhyay (2009) dataset. The large sample properties, the bias and the mean squared error of the newly proposed modified ratio estimator were obtained up to first order of approximation. The optimum value of the characterizing scalar which minimizes the mean squared error was obtained and the minimum value of the mean squared error of the proposed modified ratio estimator for this optimum value was also obtained. A theoretical comparison of the proposed modified ratio estimators was made with the other existing related estimators of population mean using auxiliary information. The conditions under which the proposed modified ratio estimators perform better than the other existing estimators of population mean are given. A numerical study was also carried out to see the performances of the proposed modified ratio estimators and some existing related ratio estimators of population mean and verify the conditions under which the proposed modified ratio estimators are better than some other existing related ratio estimators considered. It was shown that the proposed modified ratio estimators perform better than some existing related ratio estimators as they are having lower mean squared errors. Keywords: Ratio Estimator, Sample size, Bias, Mean Squared Error, Efficiency

Стилі APA, Harvard, Vancouver, ISO та ін.

14

Eyo, Effiong Eyo, and E. I. Enang. "Calibration Ratio Estimators of Population Mean Using Median of Auxiliary Variable." Journal of Modeling and Simulation of Materials 5, no. 1 (December 29, 2022): 21–30. http://dx.doi.org/10.21467/jmsm.5.1.21-30.

Повний текст джерела

Анотація:

In this study we propose a calibration ratio estimator and a calibration separate ratio-product estimator of population mean of study variable under stratified sampling using the median of auxiliary variable. The calibration estimator used calibrated weight determined to minimize a chi-square distance measure subject to a set of constraint related to the auxiliary variable in other to increase precision of the estimators. The median of the auxiliary variable was used in defining the calibration constraints. The variances of the proposed estimators were also obtained. An empirical study to ascertain the performance of these estimators using simulated data under underlying distribution assumption of Student-T distribution, Cauchy distribution, Lognormal distribution, and Standard normal distribution with varying sample sizes of 10%, 20%, and 25% were carried out. The result of simulation reveals that when the underlying distribution is Student-T, at 10% sample size, the efficiency performance of the proposed calibration separate ratio-product estimator is better than other competing estimators. As the sample size is increased to 20% and 25%, the efficiency performance of the existing stratified ratio estimator and existing calibration ratio estimator respectively become better than the other estimators. Under the skewed distributions (Cauchy and Lognormal) and the standard normal distribution, it is observed that the proposed calibration ratio estimator is better than other competing estimators in terms of efficiency, consistency and reliability. The result also reveals that under the lognormal distribution, the conventional stratified ratio estimator and the conventional calibration ratio estimator give the same result.

Стилі APA, Harvard, Vancouver, ISO та ін.

15

Mittal, Alisha, and Manoj Kumar. "Two new estimators of finite population variance under random non-response: An application." Model Assisted Statistics and Applications 18, no. 1 (March 24, 2023): 73–84. http://dx.doi.org/10.3233/mas-221361.

Повний текст джерела

Анотація:

In this paper, an attempt has been made to study the estimation of finite population variance in simple random sampling without replacement (SRSWOR) in presence of non-response by proposing two estimators. The two estimators (dual to ratio and ratio cum dual to ratio type) have been proposed on the basis of availability and non-availability of auxiliary information. The properties such as bias and mean square error of the proposed estimators have also been studied up to first order of approximation. The proposed estimators are compared with some existing estimators and are also mutually compared. An empirical study, based on both vegetable crop data and simulated data, has been performed to find out the best estimator. It has been seen that out of the two proposed estimators, the ratio cum dual to ratio type estimator performed better than dual to ratio estimator.

Стилі APA, Harvard, Vancouver, ISO та ін.

16

Raghav, Yashpal Singh. "Neutrosophic Generalized Exponential Robust Ratio Type Estimators." International Journal of Analysis and Applications 21 (May 1, 2023): 41. http://dx.doi.org/10.28924/2291-8639-21-2023-41.

Повний текст джерела

Анотація:

Estimators proposed under classical statistics fail if data are vague or indeterminate. Neutrosophic Statistics are the only alternative because its deal with indeterminacy. Extensive reserch has been conducted in this field because of its wide applicability. This study aimed to further develop the theory of neutosophic simple random sampling without replacement. In this study, a generalized neutrosophic exponential robust ratio-type estimator was proposed, and five of its member neutrosophic estimators were developed. Derivations of the bias and Mean Square Error were provided up to the first-order approximation. To demonstrate the high efficiency of the proposed neutrosophic estimators an empirical study on the stock price of Moderna and four simulation studies have been conducted, and the results show that the proposed neutrosophic estimators are more efficient than similar existing ratio type estimators discussed in this paper in neutrosophic as well as classical forms.

Стилі APA, Harvard, Vancouver, ISO та ін.

17

Pal, Surya K., Sagir A. Mahmud, Madan M. Gupta, Housila P. Singh, and Ramkrishna S. Solanki. "Estimation of finite population mean in sample surveys: A new estimator." Journal of Information & Optimization Sciences 44, no. 1 (2023): 157–69. http://dx.doi.org/10.47974/jios-1304.

Повний текст джерела

Анотація:

Utilizing supplementary information in simple random sampling, this research paper discussed a new method for finding the finite population mean of a predictive variable, and the properties of the suggested method have been investigated. The suggested estimator’s advantages over traditional estimators are demonstrated using theoretical asymptotic techniques and empirical analysis. The recommended estimator outperforms the customary unbiased, ratio, product, and regression estimators, as well as many other known population mean estimators.

Стилі APA, Harvard, Vancouver, ISO та ін.

18

Shirley, Angela, Ashok Sahai, and Isaac Dialsingh. "On Improving Ratio/Product Estimator by Ratio/Product-cum-Mean-per-Unit Estimator Targeting More Efficient Use of Auxiliary Information." Journal of Probability and Statistics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/360549.

Повний текст джерела

Анотація:

To achieve a more efficient use of auxiliary information we propose single-parameter ratio/product-cum-mean-per-unit estimators for a finite population mean in a simple random sample without replacement when the magnitude of the correlation coefficient is not very high (less than or equal to 0.7). The first order large sample approximation to the bias and the mean square error of our proposed estimators are obtained. We use simulation to compare our estimators with the well-known sample mean, ratio, and product estimators, as well as the classical linear regression estimator for efficient use of auxiliary information. The results are conforming to our motivating aim behind our proposition.

Стилі APA, Harvard, Vancouver, ISO та ін.

19

Topolewski, Marcin. "Analysis of Possibility of Application of Higher-Order Hierarchical Credibility Estimators in Automobile Insurance." Przegląd Statystyczny 62, no. 2 (June 30, 2015): 199–217. http://dx.doi.org/10.5604/01.3001.0014.1742.

Повний текст джерела

Анотація:

The work concerns the applicability of credibility estimators of order higher than one, called hierarchical estimators, to posterior estimates of claim ratio in motor insurance systems based on the history of the insured. The first part presents the idea of the credibility estimation method. In the second part the linear credibility estimator is discussed. The third part introduces appropriate for risk models in the form of claims distributions, which are used to derive the corresponding hierarchical estimates of the first order. The fourth section expands the discussion of the hierarchical estimators of the order two. In the empirical fifth part previously presented solutions are used to estimate the loss ratio for the sample insurance portfolio, results are discussed and conclusions concerning the possibility of application of second order hierarchical estimators into motor insurance are formulated.

Стилі APA, Harvard, Vancouver, ISO та ін.

20

Hone, J. "A Test of the Accuracy of Line and Strip Transect Estimators in Aerial Survey." Wildlife Research 15, no. 5 (1988): 493. http://dx.doi.org/10.1071/wr9880493.

Повний текст джерела

Анотація:

The accuracy and precision of eight line transect estimators and one strip transect estimator were examined by helicopter aerial survey. Carcasses of feral pigs were counted in an area of treeless floodplain and Eucalyptus woodland. The ratio and Cox's methods, Fourier series, exponential power series, half-normal, exponential polynomial, negative exponential, hermite polynomial and hazard rate estimators gave accurate estimates. Using the survey method described, most estimators were of similar accuracy and precision, but the Fourier series estimator was the most accurate.

Стилі APA, Harvard, Vancouver, ISO та ін.

21

IJAZ, MUHAMMAD, ATTA ULLAH, and TOLGA ZAMAN. "ON THE DEVELOPMENT OF RATIO TYPE ESTIMATORS USING AUXILIARY INFORMATION." Journal of Science and Arts 21, no. 1 (March 30, 2021): 163–70. http://dx.doi.org/10.46939/j.sci.arts-21.1-a14.

Повний текст джерела

Анотація:

The paper produces some new modified forms of the ratio estimators using the auxiliary information. The large sample properties, that is, the bias and mean squared error up to the first order of approximation are determined. The comparison is made with other existing estimators by using an applied data. It has been observed that the proposed estimators have a fewer mean squared error and leads to the efficient results as compared to the classical ratio estimator, Sisodia and Dwivedi, Singh and Kakran, Upadhyaya and Singh estimators.

Стилі APA, Harvard, Vancouver, ISO та ін.

22

Subzar, Mir, Showkat Ahmad Lone, Emmanuel J. Ekpenyong, Abdul Salam, Muhammad Aslam, T. A. Raja, and Salmeh A. Almutlak. "Efficient class of ratio cum median estimators for estimating the population median." PLOS ONE 18, no. 2 (February 9, 2023): e0274690. http://dx.doi.org/10.1371/journal.pone.0274690.

Повний текст джерела

Анотація:

In estimation theory, the use of auxiliary information significantly improves precision while estimating population parameters. In this paper, an efficient class of ratio cum median estimators of the population median is suggested using simple random sampling without replacement. The expressions for bias and mean square error of the proposed class are derived theoretically. The condition for the asymptotic optimum estimator is obtained with its bias and mean square error expressions. Under certain realistic conditions, the asymptotic optimum estimator is more proficient, based on analytical and numerical comparisons with some existing estimators that are members of the suggested class of estimators. The superiority of the proposed ratio cum median estimators is shown through real data applications. Such a new proposed estimator will be useful in the future for data analysis and making decisions.

Стилі APA, Harvard, Vancouver, ISO та ін.

23

Magnussen, Steen, Erik Næsset, and Terje Gobakken. "An Estimator of Variance for Two-Stage Ratio Regression Estimators." Forest Science 60, no. 4 (August 28, 2014): 663–76. http://dx.doi.org/10.5849/forsci.12-163.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

24

Kumari, Chandni, and Ratan Kumar Thakur. "Improved Ratio Type Estimators Using Auxiliary Attribute for Population Variance." International Journal of Science and Research (IJSR) 9, no. 4 (April 5, 2020): 1491–98. http://dx.doi.org/10.21275/sr20426180318.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

25

Muili, Jamiu Olasunkanmi, R. V. K. Singh, G. I. Onwuka, and A. Audu. "NEW CALIBRATION OF FINITE POPULATION MEAN OF COMBINED RATIO ESTIMATORS IN STRATIFIED RANDOM SAMPLING." FUDMA JOURNAL OF SCIENCES 6, no. 4 (August 18, 2022): 37–44. http://dx.doi.org/10.33003/fjs-2022-0604-920.

Повний текст джерела

Анотація:

This study deals with using calibration estimation approaches to modified the combined ratio estimator in stratified random sampling. Calibration distance measures with their associate constraints were used to modify combined ratio estimator. In stratified random sampling, new sets of optimum calibration weights are created and used to obtain new calibration estimators of population mean. Empirical study through simulation was conducted to look into the efficiency of the suggested estimators obtained. The suggested calibration estimators are more efficient than other existing estimators investigated in the study, according to the findings.

Стилі APA, Harvard, Vancouver, ISO та ін.

26

Vishwakarma, Gajendra Kumar, and Sayed Mohammed Zeeshan. "Generalized Ratio-cum-Product Estimator for Finite Population Mean under Two-Phase Sampling Scheme." Journal of Modern Applied Statistical Methods 19, no. 1 (June 8, 2021): 2–16. http://dx.doi.org/10.22237/jmasm/1608553320.

Повний текст джерела

Анотація:

A method to lower the MSE of a proposed estimator relative to the MSE of the linear regression estimator under two-phase sampling scheme is developed. Estimators are developed to estimate the mean of the variate under study with the help of auxiliary variate (which are unknown but it can be accessed conveniently and economically). The mean square errors equations are obtained for the proposed estimators. In addition, optimal sample sizes are obtained under the given cost function. The comparison study has been done to set up conditions for which developed estimators are more effective than other estimators with novelty. The empirical study is also performed to supplement the claim that the developed estimators are more efficient.

Стилі APA, Harvard, Vancouver, ISO та ін.

27

Bahl, Shashi, and Meenu Nain. "Difference-cum-ratio type estimators." Journal of Statistics and Management Systems 2, no. 1 (March 1999): 1–8. http://dx.doi.org/10.1080/09720510.1999.10700986.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

28

Tankou, Victoire, and Sudhakar Dharmadhikari. "Improvement of Ratio-Type Estimators." Biometrical Journal 31, no. 7 (1989): 795–802. http://dx.doi.org/10.1002/bimj.4710310705.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

29

Izunobi, Chinyeaka Hostensia, and Aloysius Chijioke Onyeka. "Logarithmic Ratio and Product-type Estimators of Population Mean." International Journal of Advanced Statistics and Probability 7, no. 2 (November 21, 2019): 47. http://dx.doi.org/10.14419/ijasp.v7i2.26943.

Повний текст джерела

Анотація:

Based on the natural logarithm of known population mean of an auxiliaryvariable, x, the study introduces logarithmic ratio and product-type estimatorsof the population mean of the study variable, y, in simple random samplingwithout replacement (SRSWOR) scheme. Part of the eciency conditions forthe proposed logarithmic estimators to be more ecient than the existing ex-ponential ratio and product-type estimators, as well as the customary ratio andproduct-type estimators, is that the natural logarithm of the known populationmean of the auxiliary variable, x, must be greater than 2. Generally, there is ahigh tendency for the proposed logarithmic estimators to be more ecient thanexisting customary and exponential ratio and product-type estimators whenthe natural logarithm of the auxiliary variable population mean is greater than2. The theoretical results are illustrated and conrmed using some numericaldatasets.

Стилі APA, Harvard, Vancouver, ISO та ін.

30

Zaman, Tolga, Emre Dünder, Ahmed Audu, David Anekeya Alilah, Usman Shahzad, and Muhammad Hanif. "Robust Regression-Ratio-Type Estimators of the Mean Utilizing Two Auxiliary Variables: A Simulation Study." Mathematical Problems in Engineering 2021 (September 3, 2021): 1–9. http://dx.doi.org/10.1155/2021/6383927.

Повний текст джерела

Анотація:

Many authors defined the modified version of the mean estimator by using two auxiliary variables. These proposed estimators highly depend on the calculated regression coefficients. In the presence of outliers, these estimators do not give satisfactory results. In this study, we improve the suggested estimators using several robust regression techniques while obtaining the regression coefficients. We compared the efficiencies between the suggested estimators and the estimators presented in the literature. We used two numerical examples and a simulation study to support these theoretical results. Empirical results show that the modified ratio estimator performs well in the presence of outliers when adopting robust regression techniques.

Стилі APA, Harvard, Vancouver, ISO та ін.

31

Onyango, Ronald, Brian Oduor, and Francis Odundo. "Mean Estimation of a Sensitive Variable under Nonresponse Using Three-Stage RRT Model in Stratified Two-Phase Sampling." Journal of Probability and Statistics 2022 (April 22, 2022): 1–14. http://dx.doi.org/10.1155/2022/4530120.

Повний текст джерела

Анотація:

The present study addresses the problems of mean estimation and nonresponse under the three-stage RRT model. Auxiliary information on an attribute and variable is used to propose a generalized class of exponential ratio-type estimators. Expressions for the bias, mean squared error, and minimum mean squared error for the proposed estimator are derived up to the first degree of approximation. The efficiency of the proposed estimator is studied theoretically and numerically using two real datasets. From the numerical analysis, the proposed generalized class of exponential ratio-type estimators outperforms ordinary mean estimators, usual ratio estimators, and exponential ratio-type estimators. Furthermore, the efficiencies of the mean estimators are observed to decrease with an increase in the sensitivity level of the survey question. As the inverse sampling rate and nonresponse rate go up, so does the efficiency of the mean estimators, which makes them more accurate.

Стилі APA, Harvard, Vancouver, ISO та ін.

32

Meng, Xiao-Li. "On the absolute bias ratio of ratio estimators." Statistics & Probability Letters 18, no. 5 (December 1993): 345–48. http://dx.doi.org/10.1016/0167-7152(93)90026-f.

Повний текст джерела

Стилі APA, Harvard, Vancouver, ISO та ін.

33

Yunusa, Mojeed Abiodun, Jamiu Olasunkanmi Muili, Ahmed Audu, and Ran Vijay Kumar Singh. "A Sine Type Median Based Estimator for the Estimation of Population Mean." Oriental Journal of Physical Sciences 8, no. 1 (July 15, 2023): 21–26. http://dx.doi.org/10.13005/ojps08.01.05.

Повний текст джерела

Анотація:

In the literature, there are numerous estimators for estimating population means when auxiliary information is provided. Subramani suggested ratio median based estimator when the median of the study variable is available and the regression estimator was shown to be significantly less efficient than the estimator. In this research, we suggested an estimator for the population mean of the studied variable based on a sine type median. Using Taylor series expansion, the bias and mean square error of the estimator were obtained up to the first order of approximation. The condition under which the proposed estimator is more efficient than the existing estimators was established. An empirical investigation was done to compare the suggested estimator's efficiency to that of the existing estimators, and the numerical findings showed that the proposed estimator is more efficient.

Стилі APA, Harvard, Vancouver, ISO та ін.

34

Khan, Mursala, and Rajesh Singh. "Estimation of Population Mean in Chain Ratio-Type Estimator under Systematic Sampling." Journal of Probability and Statistics 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/248374.

Повний текст джерела

Анотація:

A chain ratio-type estimator is proposed for the estimation of finite population mean under systematic sampling scheme using two auxiliary variables. The mean square error of the proposed estimator is derived up to the first order of approximation and is compared with other relevant existing estimators. To illustrate the performances of the different estimators in comparison with the usual simple estimator, we have taken a real data set from the literature of survey sampling.

Стилі APA, Harvard, Vancouver, ISO та ін.

35

Chutiman, Nipaporn, and Monchaya Chiangpradit. "Ratio Estimator in Adaptive Cluster Sampling without Replacement of Networks." Journal of Probability and Statistics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/726398.

Повний текст джерела

Анотація:

In this paper, we study the estimators of the population total in adaptive cluster sampling by using the information of the auxiliary variable. The numerical examples showed that the ratio estimator in adaptive cluster sampling without replacement of networks is more efficient than the ratio estimators in adaptive cluster sampling without replacement of units.

Стилі APA, Harvard, Vancouver, ISO та ін.

36

Singh, H. P., and A. Yadav. "An Improved Family of Estimators of Finite Population Mean Using Information on an Auxiliary Variable in Sample Surveys." Journal of Applied Mathematics, Statistics and Informatics 13, no. 2 (December 20, 2017): 77–108. http://dx.doi.org/10.1515/jamsi-2017-0011.

Повний текст джерела

Анотація:

Abstract In this paper we have suggested a family of estimators of the population mean using auxiliary information in sample surveys. The bias and mean squared error of the proposed class of estimators have been obtained under large sample approximation. We have derived the conditions for the parameters under which the proposed class of estimators has smaller mean squared error than the sample mean, ratio, product, regression estimator and the two parameter ratio-product-ratio estimators envisaged by Chami et al (2012). An empirical study is carried out to demonstrate the performance of the proposed class of estimators over other existing estimators.

Стилі APA, Harvard, Vancouver, ISO та ін.

37

Magnussen, Steen, and Thomas Nord-Larsen. "A Jackknife Estimator of Variance for a Random Tessellated Stratified Sampling Design." Forest Science 65, no. 5 (March 12, 2019): 543–47. http://dx.doi.org/10.1093/forsci/fxy070.

Повний текст джерела

Анотація:

Abstract Semisystematic sampling designs—in which a population area frame is tessellated into cells, and a randomly located sample is taken from each cell—affords random tessellated stratified (RTS) Horvitz–Thompson-type estimators. Forest inventory applications with RTS estimators are rare, possibly because of computational complexities with the estimation of variance. To reduce this challenge, we propose a jackknife estimator of variance for RTS designs. We demonstrate an application with a model-assisted ratio of totals estimator and data from the Danish National Forest Inventory. RTS estimators of standard error were, as a rule, smaller than comparable estimates obtained under the assumption of simple random sampling. The proposed jackknife estimator performed well.

Стилі APA, Harvard, Vancouver, ISO та ін.

38

RATHER, KHALID UL ISLAM, M. IQBAL JEELANI, M. YOUNIS SHAH, AFSHAN TABASSUM, and S. E. H. RIZVI. "DIFFERENCE-CUM-EXPONENTIAL EFFICIENT ESTIMATOR OF POPULATION VARIANCE." Journal of Science and Arts 22, no. 2 (June 30, 2022): 367–74. http://dx.doi.org/10.46939/j.sci.arts-22.2-a10.

Повний текст джерела

Анотація:

In the current investigation, we have suggested a Difference Cum-Exponential Type Efficient Estimator of Population variance of the study variable using information on the auxiliary variable. Up to the first order of approximation, the proposed estimator's bias and mean square error (MSE) expressions are derived and suggested optimum estimator is also found, with its optimal qualities are investigated. The suggested estimator is proven to be more competent than sample variance, classic ratio estimators based on Isaki , Singh et al. and Kadilar and Cingi estimators in [1-3]. Numerical study is also carried out by using real data sets.

Стилі APA, Harvard, Vancouver, ISO та ін.

39

Kamal, Asifa, Nimra Amir, and Huma Dastagir. "Some Exponential Type Predictive Estimators of Finite Population Mean in Two-Phase Sampling." STATISTICS, COMPUTING AND INTERDISCIPLINARY RESEARCH 2, no. 1 (June 30, 2020): 51–57. http://dx.doi.org/10.52700/scir.v2i1.10.

Повний текст джерела

Анотація:

This study is designed for predictive estimation of finite population mean in two phase sampling using two auxiliary variables. Two phase exponential ratio type estimator and exponential chain ratio type estimator are proposed under predictive approach suggested by Bahl & Tuteja (1991). The expressions of bias and mean square error of both suggested estimators have been carried out for theoretical comparison. The numerical study on real life data sets as well as simulation study has been conducted to examine the performance of suggested estimators. Finally, it is demonstrated that the suggested exponential ratio estimators are more efficient than competitive estimators in support of numerical and simulation study as well.

Стилі APA, Harvard, Vancouver, ISO та ін.

40

Bhushan, Shashi, Anoop Kumar, Amer Ibrahim Al-Omari, and Ghadah A. Alomani. "Mean Estimation for Time-Based Surveys Using Memory-Type Logarithmic Estimators." Mathematics 11, no. 9 (April 30, 2023): 2125. http://dx.doi.org/10.3390/math11092125.

Повний текст джерела

Анотація:

This article examines the issue of population mean estimation utilizing past and present data in the form of an exponentially weighted moving average (EWMA) statistic under simple random sampling (SRS). We suggest memory-type logarithmic estimators and derive their properties, such as mean-square error (MSE) and bias up to a first-order approximation. Using the EWMA statistic, the conventional and novel memory-type estimators are compared. Real and artificial populations are used as examples to illustrate the theoretical findings. According to the empirical findings, memory-type logarithmic estimators are superior to the conventional mean estimator, ratio estimator, product estimator, logarithmic-type estimator, and memory-type ratio and product estimators.

Стилі APA, Harvard, Vancouver, ISO та ін.

41

Olasunkanmi Muili, Jamiu, and A. Audu. "Estimation of Finite Population Mean of Median Based Using Power Transformation." Oriental Journal of Physical Sciences 6, no. 1-2 (February 28, 2022): 26–31. http://dx.doi.org/10.13005/ojps06.01-02.05.

Повний текст джерела

Анотація:

This paper deals with the assessment of finite population mean. An estimator is suggested for estimation of finite population mean of study variable. The purpose of this study is to evolve a ratio-type estimator to enhance the proficiency of the existing estimators considered in the study in sample random sampling without replacement using information of auxiliary variable. Expressions of the bias and mean square error (MSE) of the proposed estimator was derived by Taylor series method. The efficiency conditions under which the proposed ratio-type estimator is better than sample mean, ratio estimator, and other estimators considered in this study have been established. Theoretical and empirical findings are incentive and brace the robustness of the proposed estimator for mean estimation. The empirical results shown that the suggested estimator is more efficient than the sample mean, ratio estimator and other estimators.

Стилі APA, Harvard, Vancouver, ISO та ін.

42

Villanueva, Beatriz, and Javier Moro. "Variance and efficiency of the combined estimator in incomplete block designs of use in forest genetics: a numerical study." Canadian Journal of Forest Research 31, no. 1 (January 1, 2001): 71–77. http://dx.doi.org/10.1139/x00-138.

Повний текст джерела

Анотація:

The efficiency of combined interblock-intrablock and intrablock analysis for the estimation of treatment contrasts in alpha designs is compared using Monte-Carlo simulation. The combined estimator considers treatments and replications as fixed effects and blocks as random effects, whereas the intrablock estimator considers treatments, replications, and blocks as fixed effects. The variances of the estimators are used as the criterion for comparison. The combined estimator yields more accurate estimates than the intrablock estimator when the ratio of the block to the error variance is small, especially for designs with the fewest degrees of freedom. The accuracy of both estimators is similar when the ratio of variances is large. The variance of the combined estimator is very close to that of the best linear unbiased estimator except for designs with small number of replicates and families or provenances. Approximations commonly used for the variance of the combined estimator when variances of the random effects are unknown are studied. The downward or negative bias in the estimates of the variance given by the standard approximation used in statistical packages is largest under the conditions in which the combined estimator is more efficient than the intrablock estimator. Estimates of the relative efficiency of combined estimators have an upward bias that can exceed 10% of the true value in small- and middle-sized designs with two or three replicates. In designs with four or more replicates, often used in forest genetics, the bias is negligible.

Стилі APA, Harvard, Vancouver, ISO та ін.

43

Dar, Amber Yousaf, Nadia Saeed, Moustafa Omar Ahmed Abu-Shawiesh, Saman Hanif Shahbaz, and Muhammad Qaiser Shahbaz. "A new class of ratio type estimators in single- and two-phase sampling." AIMS Mathematics 7, no. 8 (2022): 14208–26. http://dx.doi.org/10.3934/math.2022783.

Повний текст джерела

Анотація:

<abstract> <p>The estimation of a certain population characteristics is required for several situations. The estimates are built so that the error of estimation is minimized. In several situations estimation of the population mean is required. Different estimators for the mean are available but, there is still room for improvement. In this paper, a new class of ratio-type estimators is proposed for the estimation of the population mean. The estimators are proposed for single- and two-phase sampling schemes. The expressions for bias and mean square error are obtained for single-phase and two-phase sampling estimators. Mathematical comparison of the proposed estimators has been achieved by using some existing single-phase and two-phase sampling estimators. Extensive simulations have been conducted to compare the proposed estimators with some available single- and two-phase sampling estimators. It has been observed that the proposed estimators are better than the existing estimators. Consequently, the proposed ratio estimators are recommended for use by the practitioners in various fields of industry, engineering and medical and physical sciences.</p> </abstract>

Стилі APA, Harvard, Vancouver, ISO та ін.

44

Rabiu, Adam, Abubakar Yahaya, and Muhammad Abdulkarim. "MODIFICATION OF SEPARATE RATIO TYPE EXPONENTIAL ESTIMATOR: A POST-STRATIFICATION APPROACH." FUDMA JOURNAL OF SCIENCES 5, no. 2 (July 10, 2021): 404–12. http://dx.doi.org/10.33003/fjs-2021-0502-629.

Повний текст джерела

Анотація:

In this research, modification of separate ratio type exponential estimator introduced in an earlier study is proposed. Expressions for the bias and mean square error (MSE) of the proposed estimator up to first degree of approximation are derived. The optimum value of the constant which minimize the MSE of the suggested estimator is also obtained. In the same vein, efficiency comparisons between the proposed estimator and some related existing ones under the case of post-stratification is conducted. Empirical studies have been conducted to demonstrate the efficiencies of the suggested estimators over other considered estimators. The proposed MSE and Percentage Relative Efficiency (PRE) were used to evaluate the achievement of the modified estimator.

Стилі APA, Harvard, Vancouver, ISO та ін.

45

Mehta, Nitu, and V. L. Mandowara. "Some Efficient Methods to Remove Bias in Ratio and Product Types Estimators in Ranked Set Sampling." International Journal of Bio-resource and Stress Management 13, no. 3 (March 31, 2022): 276–82. http://dx.doi.org/10.23910/1.2022.2771a.

Повний текст джерела

Анотація:

Ranked set sampling is one method to potentially increase precision and reduce costs by using quantitative or qualitative information to obtain a more representative sample. Use of auxiliary information has shown its significance in improvement of efficiency of estimators of unknown population parameters. Ratio estimator is used when auxiliary information in the form of population mean of auxiliary variable at estimation stage for the estimation of population parameters when study and auxiliary variable are positively correlated. In case of negative correlation between study variable and auxiliary variable, Product estimator is defined for the estimation of population mean. This paper proposed the problem of reducing the bias of the ratio and product estimators of the population mean in ranked set sampling (RSS). This paper suggested several type unbiased estimators of the finite population mean using information on known population parameters of the auxiliary variable in ranked set sampling. An important objective in any statistical estimation procedure is to obtain the estimators of parameters of interest with more precision. The Variance of the proposed unbiased ratio and product estimators are obtained up to first degree of approximation. Theoretically, it is shown that these suggested estimators are more efficient than the unbiased estimators in Simple random sampling. A numerical illustration is also carried out to demonstrate the merits of the proposed estimators using RSS over the usual estimators in SRS.

Стилі APA, Harvard, Vancouver, ISO та ін.

46

Tahir, Muhammad, Bu Yude, Saima Bashir, Sardar Hussain, and Tahir Munir. "A new improved estimator for the population mean using twofold auxiliary information under simple random sampling." Management Science Letters 13, no. 4 (2023): 265–76. http://dx.doi.org/10.5267/j.msl.2023.5.001.

Повний текст джерела

Анотація:

In this manuscript, the mean of the study and the auxiliary variable, as well as the rank of the auxiliary variable, were needed to develop a new, improved ratio-in-regression type estimator for population mean. Up to the first order of approximation, expressions for the bias and mean square error of the existing and proposed estimators are computed. The effectiveness and stability of our new, enhanced estimator are evaluated using simulation and two actual data sets. The suggested estimator's superior performance to all other considered estimators is shown both conceptually and numerically. The mean square error is the lowest, and PREs out-performs other known estimators by a factor of more than one hundred. Overall, we draw the conclusion that the suggested new improved estimator outperforms all its predecessors.

Стилі APA, Harvard, Vancouver, ISO та ін.

47

Niilo-Rämä, Mikko, Salme Kärkkäinen, Dario Gasbarra, and Timo Lappalainen. "INCLUSION RATIO BASED ESTIMATOR FOR THE MEAN LENGTH OF THE BOOLEAN LINE SEGMENT MODEL WITH AN APPLICATION TO NANOCRYSTALLINE CELLULOSE." Image Analysis & Stereology 33, no. 2 (June 19, 2014): 147. http://dx.doi.org/10.5566/ias.v33.p147-155.

Повний текст джерела

Анотація:

A novel estimator for estimating the mean length of fibres is proposed for censored data observed in square shaped windows. Instead of observing the fibre lengths, we observe the ratio between the intensity estimates of minus-sampling and plus-sampling. It is well-known that both intensity estimators are biased. In the current work, we derive the ratio of these biases as a function of the mean length assuming a Boolean line segment model with exponentially distributed lengths and uniformly distributed directions. Having the observed ratio of the intensity estimators, the inverse of the derived function is suggested as a new estimator for the mean length. For this estimator, an approximation of its variance is derived. The accuracies of the approximations are evaluated by means of simulation experiments. The novel method is compared to other methods and applied to real-world industrial data from nanocellulose crystalline.

Стилі APA, Harvard, Vancouver, ISO та ін.

48

Naz, Farah, Tahir Nawaz, Tianxiao Pang, and Muhammad Abid. "Use of Nonconventional Dispersion Measures to Improve the Efficiency of Ratio-Type Estimators of Variance in the Presence of Outliers." Symmetry 12, no. 1 (December 19, 2019): 16. http://dx.doi.org/10.3390/sym12010016.

Повний текст джерела

Анотація:

The use of auxiliary information in survey sampling to enhance the efficiency of the estimators of population parameters is a common phenomenon. Generally, the ratio and regression estimators are developed by using the known information on conventional parameters of the auxiliary variables, such as variance, coefficient of variation, coefficient of skewness, coefficient of kurtosis, or correlation between the study and auxiliary variable. The efficiency of these estimators is dubious in the presence of outliers in the data and a nonsymmetrical population. This study presents improved variance estimators under simple random sampling without replacement with the assumption that the information on some nonconventional dispersion measures of the auxiliary variable is readily available. These auxiliary variables can be the inter-decile range, sample inter-quartile range, probability-weighted moment estimator, Gini mean difference estimator, Downton’s estimator, median absolute deviation from the median, and so forth. The algebraic expressions for the bias and mean square error of the proposed estimators are obtained and the efficiency conditions are derived to compare with the existing estimators. The percentage relative efficiencies are used to numerically compare the results of the proposed estimators with the existing estimators by using real datasets, indicating the supremacy of the suggested estimators.

Стилі APA, Harvard, Vancouver, ISO та ін.

49

Ahmad, Sohaib, Sardar Hussain, Uzma Yasmeen, Muhammad Aamir, Javid Shabbir, M. El-Morshedy, Afrah Al-Bossly, and Zubair Ahmad. "A simulation study: Using dual ancillary variable to estimate population mean under stratified random sampling." PLOS ONE 17, no. 11 (November 28, 2022): e0275875. http://dx.doi.org/10.1371/journal.pone.0275875.

Повний текст джерела

Анотація:

In this paper, we propose an improved ratio-in-regression type estimator for the finite population mean under stratified random sampling, by using the ancillary varaible as well as rank of the ancillary varaible. Expressions of the bias and mean square error of the estimators are derived up to the first order of approximation. The present work focused on proper use of the ancillary variable, and it was discussed how ancillary variable can improve the precision of the estimates. Two real data sets as well as simulation study are carried out to observe the performances of the estimators. We demonstrate theoretically and numerically that proposed estimator performs well as compared to all existing estimators.

Стилі APA, Harvard, Vancouver, ISO та ін.

50

Yunusa, Mojeed Abiodun, Ahmed Audu, and Awwal Adejumobi. "Logarithmic-Product-Cum-Ratio Type Estimator for Estimating Finite Population Coefficient of Variation." Oriental Journal of Physical Sciences 7, no. 2 (January 10, 2023): 82–87. http://dx.doi.org/10.13005/ojps07.02.05.

Повний текст джерела

Анотація:

Estimation of population parameters have been a challenging aspect in sample survey for sometimes and many efforts have been made to enhance the exactness of the parameters of these estimators. We suggested the logarithmic-product-cum-ratio type estimator. Expression of the MSE of the intended estimator originated using the Taylor series technique. A numerical illustration was conducted and the results revealed that the modified work is ameliorated than sample means with other estimators observed.

Стилі APA, Harvard, Vancouver, ISO та ін.

Статті в журналах з теми "Ratio estimators"

Оформте джерело за APA, MLA, Chicago, Harvard та іншими стилями