Sains Malaysiana 55(6)(2026): 1077-1087

http://doi.org/10.17576/jsm-2026-5506-12

 

Ratio Estimators under Stratified Inverse Random Sampling

(Penganggar Nisbah di bawah Pensampelan Rawak Songsang Berstrata)

 

PRAYAD SANGNGAM¹, WIPAWAN LAOARUN^1,*, rn Pathom, grant number SRIFJRGPANPHARISA KHONGTHIP¹ & SUREEPORN SUNGSUWAN²

 

¹Department of Statistics, Faculty of Science, Silpakorn University, Nakhon Pathom, 73000, Thailand

²Department of Mathematics, Faculty of Science, Mahanakorn University of Technology,

Nong Chok, Bangkok, 10530, Thailand

 

Diserahkan: 20 Januari 2026/Diterima: 5 Jun 2026

 

Abstract

Stratified inverse random sampling (SIRS) is an effective technique for studying rare populations, as it guarantees a pre-specified number of interested units within each stratum. Although ratio estimation has been extensively studied under various sampling designs, its application under SIRS remains largely unexplored. This study addresses this gap by developing two novel ratio estimators for the population mean under SIRS: the separate ratio estimator and the combined ratio estimator. The contributions of this work are threefold. First, first-order approximations for the bias and mean squared error (MSE) of each estimator are rigorously derived using Taylor series expansions under large-sample assumptions, and consistent sample-based estimators of the MSE are developed. Second, analytical conditions, expressed in terms of population parameters, are established under which the proposed estimators outperform the conventional unbiased estimator. Third, an extensive Monte Carlo simulation study with 50,000 replications is conducted to assess finite-sample performance. The simulation results demonstrate that the separate ratio estimator consistently attains the smallest MSE across all scenarios, achieving efficiency gains of up to 108.9% relative to the unbiased estimator. Moreover, all proposed estimators exhibit a decrease in absolute bias, MSE, skewness, and kurtosis as the pre-specified number of interested units in the sample increases.

Keywords: Bias; combined ratio estimator; mean square error; rare population; separate ratio estimator

Abstrak

Pensampelan rawak songsang berstrata (SIRS) merupakan teknik yang berkesan untuk mengkaji populasi yang jarang ditemui kerana ia menjamin bilangan unit yang berminat yang telah ditentukan terlebih dahulu dalam setiap stratum. Walaupun anggaran nisbah telah dikaji secara meluas di bawah pelbagai reka bentuk pensampelan, aplikasinya di bawah SIRS masih belum diterokai sepenuhnya. Kajian ini menangani jurang ini dengan membangunkan dua penganggar nisbah baharu untuk min populasi di bawah SIRS: penganggar nisbah berasingan dan penganggar nisbah gabungan. Sumbangan kerja ini adalah tiga kali ganda. Pertama, anggaran tertib pertama untuk bias dan ralat kuasa dua min (MSE) bagi setiap penganggar diterbitkan secara teliti menggunakan pengembangan siri Taylor di bawah andaian sampel besar dan penganggar berasaskan sampel yang tekal bagi MSE dibangunkan. Kedua, keadaan analitikal yang dinyatakan dalam bentuk parameter populasi diwujudkan dengan penganggar yang dicadangkan mengatasi penganggar tidak berat sebelah konvensional. Ketiga, kajian simulasi Monte Carlo yang meluas dengan 50,000 ulangan dijalankan untuk menilai prestasi sampel terhingga. Keputusan simulasi menunjukkan bahawa penganggar nisbah berasingan secara tekal mencapai MSE terkecil merentasi semua senario, mencapai peningkatan kecekapan sehingga 108.9% berbanding dengan penganggar tidak berat sebelah. Selain itu, semua penganggar yang dicadangkan menunjukkan penurunan dalam bias mutlak, MSE, kecondongan dan kurtosis apabila bilangan unit yang berminat yang telah ditentukan dalam sampel meningkat.

Kata kunci: Bias; penganggar nisbah berasingan; penganggar nisbah gabungan; populasi yang jarang berlaku; ralat min kuasa dua

 

RUJUKAN

Chao, C. 2004. Ratio estimation on adaptive cluster sampling. Journal of the Chinese Statistical Association 42: 307-327.

Christman, M.C. & Lan, F. 2001. Inverse adaptive cluster sampling. Biometrics 57(4): 1096-1105. https://doi.org/10.1111/j.0006-341X.2001.01096.x

Cochran, W.G. 1977. Sampling Techniques. 3rd ed. New York: John Wiley & Sons.

Finney, D.J. 1949. On a method of estimating frequencies. Biometrika 36(1-2): 233-234. https://doi.org/10.1093/biomet/36.1-2.233

Greco, L. & Naddeo, S. 2007. Inverse sampling with unequal selection probabilities. Communications in Statistics - Theory and Methods 36(5): 1039-1048. https://doi.org/10.1080/03610920601033926

Haldane, J.B.S. 1945. On a method of estimating frequencies. Biometrika 33(3): 222-225. https://doi.org/10.2307/2332299

Kadilar, C. & Cingi, H. 2004. Ratio estimators in simple random sampling. Applied Mathematics and Computation 151(3): 893-902. https://doi.org/10.1016/S0096-3003(03)00803-8

Murthy, M.N. 1964. Product method of estimation. Sankhyā: The Indian Journal of Statistics Series A 26: 69-74.

Salehi, M.M. & Seber, G.A.F. 2001. A new proof of Murthy’s estimator which applies to sequential sampling. Australian & New Zealand Journal of Statistics 43(3): 281-286. https://doi.org/10.1111/1467-842X.00174

Sangngam, P. & Laoarun, W. 2025. Ratio estimators under inverse sampling. Songklanakarin Journal of Science and Technology 47(3): 192-199.

Sangngam, P. & Suwattee, P. 2012. Stratified inverse sampling for rare populations. Thailand Statistician 10(1): 69-86. https://ph02.tci-thaijo.org/index.php/thaistat/article/view/34233

Singh, H.P. & Nigam, P. 2020. Ratio-ratio-type exponential estimator of finite population mean in double sampling for stratification. International Journal of Agricultural Statistics Sciences 16(1): 251-257.

Sisodia, B.V.S. & Dwivedi, V.K. 1981. A modified ratio estimator using coefficient of variation of auxiliary variable. Journal of the Indian Society of Agricultural Statistics 33: 13-18.

Subramani, J. 2013. A new modified ratio estimator for estimation of population mean when median of the auxiliary variable is known. Pakistan Journal of Statistics and Operation Research 9(2): 137-145(2). https://doi.org/10.18187/pjsor.v9i2.486

Sungsuwan, S. & Suwattee, P. 2014. Model-assisted estimation in inverse sampling. Chiang Mai Journal of Science 41(3): 704-713.

 

*Pengarang untuk surat-menyurat; email: laoarun_w@su.ac.th