Neutrosophic ratio-type estimators for estimating the population mean
Zaigham Tahir, Hina Khan, Muhammad Aslam, Javid Shabbir, Yasar Mahmood, Florentín Smarandache
Abstract
Abstract All researches, under classical statistics, are based on determinate, crisp data to estimate the mean of the population when auxiliary information is available. Such estimates often are biased. The goal is to find the best estimates for the unknown value of the population mean with minimum mean square error (MSE). The neutrosophic statistics, generalization of classical statistics tackles vague, indeterminate, uncertain information. Thus, for the first time under neutrosophic statistics, to overcome the issues of estimation of the population mean of neutrosophic data, we have developed the neutrosophic ratio-type estimators for estimating the mean of the finite population utilizing auxiliary information. The neutrosophic observation is of the form $${Z}_{N}={Z}_{L}+{Z}_{U}{I}_{N}\, {\rm where}\, {I}_{N}\in \left[{I}_{L}, {I}_{U}\right], {Z}_{N}\in [{Z}_{l}, {Z}_{u}]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>L</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>U</mml:mi> </mml:msub> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mspace/> <mml:mi>where</mml:mi> <mml:mspace/> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:mfenced> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>L</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>U</mml:mi> </mml:msub> </mml:mfenced> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>u</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . The proposed estimators are very helpful to compute results when dealing with ambiguous, vague, and neutrosophic-type data. The results of these estimators are not single-valued but provide an interval form in which our population parameter may have more chance to lie. It increases the efficiency of the estimators, since we have an estimated interval that contains the unknown value of the population mean provided a minimum MSE. The efficiency of the proposed neutrosophic ratio-type estimators is also discussed using neutrosophic data of temperature and also by using simulation. A comparison is also conducted to illustrate the usefulness of Neutrosophic Ratio-type estimators over the classical estimators.