In this study, we extend Neyman smooth-type goodness-of-fit tests to complex survey settings involving categorical data, by incorporating design-consistent estimators under the survey framework. This extension is implemented through data-driven, nonparametric order selection methods. We examine the asymptotic properties of the proposed estimators and demonstrate, through simulations, that our methods improve statistical power while maintaining strong control over Type I error, particularly in detecting subtle yet systematic differences across categories. We also illustrate the practical utility of our approach using data from the National Youth Tobacco Survey (NYTS).

References

• Arfken, G. (1985). Mathematical Methods for Physicists (3rd ed.). Academic Press.
• Bedrick, E. J. (1983). Adjusted chi-squared tests for cross-classified tables of survey data. Biometrika, 70(3), 591–595. https://doi.org/10.1093/biomet/70.3.591
• Eubank, R. L. (1997). Testing goodness of fit with multinomial data. Journal of the American Statistical Association, 92(439), 1084–1093. https://doi.org/10.1080/01621459.1997.10474064
• Eubank, R. L. (1999). Nonparametric Regression and Spline Smoothing (2nd ed.). CRC Press.
• Eubank, R. L., & Hart, J. D. (1992). Testing goodness of fit in regression via order selection criteria. The Annals of Statistics, 20(3), 1412–1425. https://doi.org/10.1214/aos/1176348775
• Fay, R. E. (1979). On adjusting the Pearson chi-square statistic for clustered sampling. In Proceedings of the American Statistical Association, Social Statistics Section (pp. 402–406). American Statistical Association.
• Fay, R. E. (1985). A jackknifed chi-squared test for complex samples. Journal of the American Statistical Association, 80(389), 370–375.
• Isaki, C. T., & Fuller, W. A. (1982). Survey design under the regression superpopulation model. Journal of the American Statistical Association, 77(377), 89–96. https://doi.org/10.1080/01621459.1982.10477770
• Jamil, H., Moustaki, I., & Skinner, C. J. (2025). Pairwise likelihood estimation and limited information goodness-of fit test statistics for binary factor analysis models under complex survey sampling. British Journal of Mathematical and Statistical Psychology, 78(1), 258–285. https://doi.org/10.1111/bmsp.v78.1
• Kim, J., Rao, J. N. K., & Wang, Z. (2019). Hypotheses Testing from Complex Survey Data Using Bootstrap Weights: A Unified Approach (Technical Paper No. 265). Iowa State University.
• Kish, L. (1965). Survey Sampling. John Wiley & Sons, Inc.
• Lancaster, H. O. (n.d.). The Chi-squared Distribution. Wiley.
• Lindsay, B. G. (1988). Composite likelihood methods. Statistical Inference from Stochastic Processes, 80, 221–239. https://doi.org/10.1090/conm/080
• Lohr, S. L. (2021). Sampling: Design and Analysis (3rd ed.). Chapman and Hall/CRC.
• Lu, Y. (2014). Chi-squared tests in dual frame surveys. Survey Methodology, 40(2), 323–334.
• Muthén, B. O., & Satorra, A. (1995). Complex sample data in structural equation modeling. Sociological Methodology, 25, 267–316. https://doi.org/10.2307/271070
• Neyman, J. (1937). Smooth test for goodness of fit. Skandinavisk Aktuarietidskrift, 20(3–4), 149–199.
• Office on Smoking and Health (2014). 2014 National Youth Tobacco Survey: Methodology Report (Technical Paper). Department of Health and Human Services, Centers for Disease Control and Prevention, National Center for Chronic Disease Prevention and Health Promotion, Office on Smoking and Health.
• Rao, J. N. K., & Scott, A. J. (1981). The analysis of categorical data from complex sample surveys: Chi-squared tests for goodness of fit and independence in two-way tables. Journal of the American Statistical Association, 76(374), 221–230. https://doi.org/10.1080/01621459.1981.10477633
• Rao, J. N. K., & Scott, A. J. (1984). On chi-squared tests for multiway contingency tables with cell proportions estimated from survey data. The Annals of Statistics, 12(1), 46–60. https://doi.org/10.1214/aos/1176346391
• Rao, J. N. K., & Scott, A. J. (1987). On simple adjustments to chi-square tests with sample survey data. The Annals of Statistics, 15(1), 385–397. https://doi.org/10.1214/aos/1176350273
• Rayner, J. C. W., & Best, D. J. (1986). Neyman-type smooth tests for location-scale families. Biometrika, 73(2), 437–446. https://doi.org/10.1093/biomet/73.2.437
• Rayner, J. C. W., & Best, D. J. (1989). Smooth Tests of Goodness of Fit. Oxford University Press.
• Rayner, J. C. W., & Best, D. J. (1990). Smooth tests of goodness of fit: An overview. International Statistical Review, 58(1), 9–17. https://doi.org/10.2307/1403470
• Rayner, J. C. W., Best, D. J., & Dodds, K. G. (1985). The construction of the simple x2 and Neyman smooth goodness of fit tests. Statistica Neerlandica, 39(1), 35–50. https://doi.org/10.1111/stan.1985.39.issue-1
• Rayner, J. C. W., Thas, O., & Best, D. J. (2009). Smooth Tests of Goodness of Fit (2nd ed.). Wiley.
• Sárndal, C.-E. (2003). Model Assisted Survey Sampling. Springer.
• Skinner, C. J. (1989). Domain means, regression and multivariate analysis. In C. J. Skinner, D. Holt & T. M. F. Smith (Eds.), Analysis of complex surveys (pp. 59–75). Wiley.
• Skinner, C. J. (2019). Analysis of categorical data for complex surveys. International Statistical Review, 87(S1), 64–78. https://doi.org/10.1111/insr.v87.S1
• Skinner, C. J., & Rao, J. N. K. (1996). Estimation in dual frame surveys with complex designs. Journal of the American Statistical Association, 91(433), 349–356. https://doi.org/10.1080/01621459.1996.10476695
• Varin, C. (2008). On composite marginal likelihoods. AStA Advances in Statistical Analysis, 92(1), 1–28. https://doi.org/10.1007/s10182-008-0060-7
• Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society, 54(3), 426–482. https://doi.org/10.1090/tran/1943-054-03
• Zhang, P. (1992). On the distributional properties of model selection criteria. Journal of the American Statistical Association, 87(418), 732–737. https://doi.org/10.1080/01621459.1992.10475275
• Zhou, L. (2016). Neyman Smooth-type Goodness of Fit Tests in Complex Surveys [Unpublished doctoral dissertation]. University of New Mexico.

To cite this article: Yan Lu, Lang Zhou, Guoyi Zhang & Ronald Christensen (2026) Neyman smooth-type goodness-of-fit tests in complex surveys, Statistical Theory and Related Fields, 10:3, 341-359, DOI: 10.1080/24754269.2026.2616882

To link to this article: https://doi.org/10.1080/24754269.2026.2616882