{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T23:43:40Z","timestamp":1778543020073,"version":"3.51.4"},"reference-count":42,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T00:00:00Z","timestamp":1761696000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Accurate estimation of population distribution characteristics is a fundamental task in survey sampling and statistical inference. This paper introduces a new family of estimators for the cumulative distribution function (CDF) under probability proportional to size (PPS) sampling, incorporating auxiliary information to enhance efficiency. The proposed approach employs dual auxiliary variables in the estimation phase, while the sampling design relies on a single auxiliary variable. Theoretical properties, including bias and mean squared error (MSE), are rigorously derived to establish the efficiency of the new class. An extensive empirical evaluation using three distinct populations\u2014fisheries data, wine chemistry data, and demographic records\u2014demonstrates the superiority of the proposed estimators. In terms of accuracy, the best-performing proposed estimator achieves an MSE of 0.0012, compared to 0.0127 for the widely used GK estimator. Percentage relative efficiency (PRE) values further underscore these improvements, with gains ranging from 123% to over 328% across the three populations. Graphical comparisons confirm these trends, illustrating that the proposed estimators consistently dominate conventional approaches. Overall, the findings highlight both the theoretical soundness and practical utility of the proposed family, offering robust and computationally efficient improvements for CDF estimation in complex survey designs.<\/jats:p>","DOI":"10.3390\/axioms14110796","type":"journal-article","created":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T03:44:39Z","timestamp":1761795879000},"page":"796","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Novel Family of CDF Estimators Under PPS Sampling: Computational, Theoretical, and Applied Perspectives"],"prefix":"10.3390","volume":"14","author":[{"given":"Salman","family":"Shah","sequence":"first","affiliation":[{"name":"Department of Statistics, Yazd University, Yazd 89158-18411, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eisa","family":"Mahmoudi","sequence":"additional","affiliation":[{"name":"Department of Statistics, Yazd University, Yazd 89158-18411, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8533-5410","authenticated-orcid":false,"given":"Hasnain","family":"Iftikhar","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Peshawar, Peshawar 25120, Pakistan"},{"name":"Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1248-9910","authenticated-orcid":false,"given":"Paulo Canas","family":"Rodrigues","sequence":"additional","affiliation":[{"name":"Department of Statistics, Federal University of Bahia, Salvador 40170-110, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ronny Ivan","family":"Gonzales Medina","sequence":"additional","affiliation":[{"name":"Facultad de Ciencias e Ingenier\u00edas F\u00edsicas y Formales, Universidad Cat\u00f3lica de Santa Mar\u00eda, Arequipa 04013, Peru"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0847-0552","authenticated-orcid":false,"given":"Javier Linkolk","family":"L\u00f3pez-Gonzales","sequence":"additional","affiliation":[{"name":"Escuela de Posgrado, Universidad Peruana Uni\u00f3n, Lima 15024, Peru"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1080\/01621459.1952.10483446","article-title":"A Generalization of Sampling Without Replacement from a Finite Universe","volume":"47","author":"Horvitz","year":"1952","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1214\/aoms\/1177700375","article-title":"Asymptotic Theory of Rejective Sampling with Varying Probabilities from a Finite Population","volume":"35","year":"1964","journal-title":"Ann. Math. Stat."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1080\/01621459.1988.10478591","article-title":"Resampling Inference with Complex Survey Data","volume":"83","author":"Rao","year":"1988","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_4","unstructured":"Food and Agriculture Organization of the United Nations (1998). Sampling Manual for Forest and Range Inventory, FAO. FAO Forestry Paper 82."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Abbas, N., Shabbir, J., and Hanif, M. (2025). Estimation of cumulative distribution function under complex survey designs using auxiliary information. PLoS ONE, 20.","DOI":"10.1371\/journal.pone.0322660"},{"key":"ref_6","unstructured":"Flood, J., and Mostafa, S. (2024). Survey Data Integration for Distribution Function Estimation. arXiv."},{"key":"ref_7","unstructured":"Dagdoug, A., Goga, C., and Haziza, D. (2020). Model-assisted estimation in survey sampling using random forests. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5366","DOI":"10.1109\/TNNLS.2022.3165627","article-title":"Metalearning-Based Alternating Minimization Algorithm for Nonconvex Optimization","volume":"34","author":"Xia","year":"2023","journal-title":"IEEE Trans. Neural Networks Learn. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"23429","DOI":"10.3934\/math.20251041","article-title":"Efficient Estimators of Finite Population Variance Using Raw Moments under Two- and Three-Stage Cluster Sampling Schemes","volume":"10","author":"Abbas","year":"2025","journal-title":"AIMS Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.3934\/math.2025051","article-title":"An improved family of unbiased ratio estimators for a population distribution function","volume":"10","author":"Ahmad","year":"2025","journal-title":"AIMS Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Hussain, S., Ahmad, S., Akhtar, S., Javed, A., and Yasmeen, U. (2020). Estimation of finite population distribution function with dual use of auxiliary information under non-response. PLoS ONE, 15.","DOI":"10.1371\/journal.pone.0243584"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3584663","article-title":"The Design of Fast and Lightweight Resemblance Detection for Efficient Post-Deduplication Delta Compression","volume":"19","author":"Xia","year":"2023","journal-title":"ACM Trans. Storage"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/s10440-008-9243-1","article-title":"A family of estimators of finite-population distribution function using auxiliary information","volume":"104","author":"Singh","year":"2008","journal-title":"Acta Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2498","DOI":"10.1080\/03610918.2022.2078492","article-title":"Estimation of population distribution function in the presence of non-response using stratified random sampling","volume":"53","author":"Yaqub","year":"2024","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.53555\/ks.v12i5.3508","article-title":"Novel Methods For Estimation Of Population Mean Using Auxiliary Information Under PPS Sampling: Application With Real And Simulated Data Sets","volume":"12","author":"Mustafa","year":"2024","journal-title":"Kurd. Stud."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.21015\/vtm.v12i2.1942","article-title":"Improved estimator for estimation of population mean using predictive approach under PPS sampling","volume":"12","author":"Khan","year":"2024","journal-title":"VFAST Trans. Math."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Wang, J., Ahmad, S., Arslan, M., Lone, S.A., Abd Ellah, A., Aldahlan, M.A., and Elgarhy, M. (2023). Estimation of finite population mean using double sampling under probability proportional to size sampling in the presence of extreme values. Heliyon, 9.","DOI":"10.1016\/j.heliyon.2023.e21418"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"12235","DOI":"10.1109\/TPEL.2024.3388024","article-title":"A General Modeling and Analysis of Impacts of Unbalanced Inductance on PWM Schemes for Two-Parallel Interleaved Power Converters","volume":"39","author":"Zeng","year":"2024","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_19","first-page":"99","article-title":"Use of extreme values to estimate finite population mean under pps sampling scheme","volume":"11","author":"Ahmad","year":"2018","journal-title":"J. Reliab. Stat. Stud."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Azeem, M., Iftikhar, S., Ijaz, M., Salahuddin, N., and Ilyas, M. (2025). An improved estimator of population mean under PPS sampling with application to radiation data sets. J. Radiat. Res. Appl. Sci., 18.","DOI":"10.1016\/j.jrras.2025.101543"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"17","DOI":"10.17713\/ajs.v53i2.1622","article-title":"An efficient variant of ranked set sampling, probability proportional to size with application to economic data","volume":"53","author":"Amiri","year":"2024","journal-title":"Austrian J. Stat."},{"key":"ref_22","first-page":"323","article-title":"Homogeneity Pursuit in the Functional-Coefficient Quantile Regression Model for Panel Data with Censored Data","volume":"29","author":"Li","year":"2025","journal-title":"Stud. Nonlinear Dyn. Econom."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4255","DOI":"10.14778\/3749646.3749691","article-title":"GraphCSR: A Degree-Equalized CSR Format for Large-scale Graph Processing","volume":"18","author":"Gan","year":"2025","journal-title":"Proc. VLDB Endow."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1392","DOI":"10.1109\/TETCI.2025.3537918","article-title":"Towards the Explanation Consistency of Citizen Groups in Happiness Prediction via Factor Decorrelation","volume":"9","author":"Wu","year":"2025","journal-title":"IEEE Trans. Emerg. Top. Comput. Intell."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"104391","DOI":"10.1016\/j.artint.2025.104391","article-title":"Regression-based conditional independence test with adaptive kernels","volume":"347","author":"Ren","year":"2025","journal-title":"Artif. Intell."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"045024","DOI":"10.1088\/1361-6420\/ad2cf7","article-title":"Adaptive tempered reversible jump algorithm for Bayesian curve fitting","volume":"40","author":"Tian","year":"2024","journal-title":"Inverse Probl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"025136","DOI":"10.1088\/1361-6501\/ad0e3b","article-title":"Multisource information fusion considering the weight of focal element\u2019s beliefs: A Gaussian kernel similarity approach","volume":"35","author":"Yang","year":"2024","journal-title":"Meas. Sci. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2755","DOI":"10.1109\/TFUZZ.2025.3576588","article-title":"Belief-based fuzzy and imprecise clustering for arbitrary data distributions","volume":"33","author":"Zhang","year":"2025","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"4089","DOI":"10.1109\/TCYB.2025.3577745","article-title":"Understanding the Dimensional Need of Noncontrastive Learning","volume":"55","author":"Cao","year":"2025","journal-title":"IEEE Trans. Cybern."},{"key":"ref_30","first-page":"1","article-title":"Research on the Expansion of Deposit Insurance Pricing Model Based on the Merton Option Pricing Framework","volume":"8","author":"Yang","year":"2025","journal-title":"Comput. Econ."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1017\/S0021859600048012","article-title":"The estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce","volume":"30","author":"Cochran","year":"1940","journal-title":"J. Agric. Sci."},{"key":"ref_32","first-page":"159","article-title":"Ratio and product type exponential estimators","volume":"12","author":"Bahl","year":"1991","journal-title":"J. Inf. Optim. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3325","DOI":"10.1080\/03610929108830705","article-title":"On certail methods of improving ration and regression estimators","volume":"20","author":"Rao","year":"1991","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1552","DOI":"10.1080\/03610918.2012.736579","article-title":"A generalized class of ratio type exponential estimators of population mean under linear transformation of auxiliary variable","volume":"43","author":"Grover","year":"2014","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Singh, S. (2003). Advanced Sampling Theory with Applications: How Michael \u201cSelected\u201d Amy, Springer Science & Business Media.","DOI":"10.1007\/978-94-007-0789-4"},{"key":"ref_36","unstructured":"Herbert, A. (2025, January 20). Wine Dataset. UCI Machine Learning Repository. Available online: http:\/\/archive.ics.uci.edu\/ml\/datasets\/wine."},{"key":"ref_37","first-page":"241","article-title":"Methods for estimating the precision of survey estimates when imputation has been used","volume":"18","year":"1992","journal-title":"Surv. Methodol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/LGRS.2025.3617522","article-title":"High-Precision Deblending of 3-D Simultaneous Source Data Based on Prior Information Constraint","volume":"22","author":"Zhang","year":"2025","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Cai, X., and Zhang, C. (2025). An Innovative Differentiated Creative Search Based on Collaborative Development and Population Evaluation. Biomimetics, 10.","DOI":"10.3390\/biomimetics10050260"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"16049","DOI":"10.1109\/TNNLS.2025.3563769","article-title":"Learning Self-Growth Maps for Fast and Accurate Imbalanced Streaming Data Clustering","volume":"36","author":"Zhang","year":"2025","journal-title":"IEEE Trans. Neural Networks Learn. Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"4471","DOI":"10.1109\/TKDE.2025.3566038","article-title":"Cost-Sensitive Neighborhood Granularity Selection for Hierarchical Classification","volume":"37","author":"Li","year":"2025","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"10259","DOI":"10.1109\/TPAMI.2024.3435503","article-title":"Towards Effective Causal Partitioning by Edge Cutting of Adjoint Graph","volume":"46","author":"Zhang","year":"2024","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/11\/796\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T04:01:52Z","timestamp":1761796912000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/11\/796"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,29]]},"references-count":42,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["axioms14110796"],"URL":"https:\/\/doi.org\/10.3390\/axioms14110796","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,29]]}}}