Pith. sign in

REVIEW 3 major objections 5 minor 79 references

Parameter estimation and application in two types of uncertain single-index models

T0 review · 3 major / 5 minor · reviewed 2026-07-11 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Uncertain single-index models reduce high-dimensional imprecise data to a single direction and a nonparametric link, estimated by profile least squares.

desk verdict Solid incremental transplant of single-index methods into Liu uncertainty theory; USIC half works cleanly, USIU half is undercut by an untested monotonicity assumption that the authors' own weather fit violates. read the letter →

arxiv 2607.04699 v1 pith:7XECV3YW submitted 2026-07-06 stat.ME stat.AP

Fuguo Wang , Zhiming Li This is my paper · ORCID
classification stat.MEstat.AP MSC 62G0862J0262F03
keywords uncertainsingle-indexmodelestimationmethodshypothesisNadaraya–WatsonkernelB-splineprofileleastsquaresimprecisedata
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

When data are imprecise because of unstable frequencies or expert judgment, classical single-index models that rely on probability can lose information or bias estimates. This paper defines two uncertain single-index models: one with crisp covariates and one with fully uncertain covariates. Both project the covariates onto a single linear index and then apply an unknown univariate link function, so that high-dimensional imprecise data remain interpretable. Estimation proceeds by profile least squares: for crisp covariates a Nadaraya–Watson kernel smoother is used; for uncertain covariates a monotone B-spline is required so that inverse uncertainty distributions stay well-defined. Residuals are then tested with the existing machinery of uncertain hypothesis testing. Simulations recover known coefficients and links, and a Lagos weather series shows humidity as the dominant driver of temperature range under a unimodal nonlinear regime. The practical payoff is a usable semiparametric tool for nonlinear modeling of high-dimensional imprecise observations.

What carries the argument

Profile least-squares estimation under uncertainty: the Nadaraya–Watson kernel (with leave-one-out and Fibonacci bandwidth search) for the USIC model, and monotone B-splines (with explicit inverse-distribution representation via Lemma 2) for the USIU model.

What would settle it

Generate data from a non-monotone link under uncertain covariates and check whether the B-spline profile procedure still recovers the true index coefficients and produces residuals that pass the uncertain hypothesis test; systematic failure would refute the USIU claims.

Watch

Extended reading notes

Core claim

Two uncertain single-index models (USIC for crisp covariates, USIU for uncertain covariates) can be estimated by a two-step profile least-squares procedure that recovers both the index coefficients and the unknown link; the resulting residuals pass uncertain hypothesis tests of model adequacy, so the models supply a practical semiparametric description of high-dimensional imprecise data.

Load-bearing premise

For the model with uncertain covariates the unknown link function must be strictly increasing; without that monotonicity the inverse uncertainty distribution of the residual cannot be written in closed form and the estimation procedure collapses.

Editorial extensions

If this is right

  • High-dimensional imprecise data can be reduced to a single index plus a nonparametric link without forcing a fully parametric form.
  • Uncertain residual analysis and significance tests become available for single-index structure, allowing formal model checks.
  • Weather and other expert-rated series can be fitted with the USIC model to identify dominant drivers and regime switches.
  • The same two-step framework can be reused for partially linear or varying-coefficient extensions once monotonicity or kernel conditions are adapted.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If monotonicity is only local rather than global, a piecewise or constrained-kernel version of the USIU estimator might still be feasible.
  • The same inverse-distribution device used for B-splines could be applied to other monotone bases (e.g., I-splines) for automatic shape control.
  • Once coefficients near zero can be tested, automatic variable selection inside the uncertain single-index model becomes a natural next step.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces two uncertain single-index models under Liu’s uncertainty theory: USIC (crisp covariates, uncertain response) and USIU (uncertain covariates and response). For USIC it develops a profile least-squares procedure that uses a leave-one-out Nadaraya–Watson kernel smoother (kernel taken as the density of the standard normal uncertainty distribution) and a Fibonacci search for bandwidth; for USIU it replaces the kernel by a monotone B-spline approximation and derives the inverse uncertainty distribution of the residual via Lemma 2 under a strict monotonicity assumption on g. Residual expectation/variance estimators, forecast intervals, and uncertain hypothesis tests of model adequacy are supplied. Two Monte-Carlo examples recover the true index coefficients and pass residual tests; a Lagos weather application (USIC only) yields a unimodal link and a significance test for a near-zero coefficient.

Significance. If the procedures are reliable, the paper supplies the first systematic single-index framework for imprecise data under uncertainty theory, filling a gap between existing parametric/nonparametric uncertain regression and classical single-index models. The algebraic reductions (Theorems 1–3) correctly follow from Liu’s lemmas once monotonicity is granted, the residual tests are non-tautological, and the real-data illustration shows that the USIC estimator can recover a scientifically interpretable unimodal link. The absence of asymptotic theory and the restrictive monotonicity assumption for USIU limit immediate impact, but the constructions themselves are a useful contribution to uncertain statistics.

major comments (3)
  1. [Theorem 2, Eqs. (17)–(18), §5] Theorem 2 and Eqs. (17)–(18) require g to be C¹ and strictly increasing so that Lemma 2 yields a closed-form inverse uncertainty distribution of the residual; without it the profile objective cannot be written as the integral that is minimized and the sign-flip map φ_ik is invalid. The paper never supplies a diagnostic or a relaxation; it merely imposes a monotonicity penalty and plots the derivative. Yet the only real-data illustration (Lagos, §5, Fig. 5(b)) produces an explicitly unimodal (hence non-monotone) link, so the USIU theory as stated cannot be applied to the setting the authors advertise. Either a non-monotone extension or a clear statement that USIU is restricted to monotone links is needed.
  2. [§3–§4] No consistency, rate, or asymptotic normality results are proved for either estimator. Bandwidth/knot selection is purely numerical (V-fold CV + Fibonacci/L-BFGS). While finite-sample recovery in the two simulations is encouraging, the central claim that the models are “ready for use” rests on unproved asymptotic properties; at least a sketch of consistency under standard kernel/spline conditions would strengthen the paper.
  3. [§5] The real-data application uses only the USIC model; the USIU procedure is illustrated solely on a 50-observation simulation. Given that the paper’s title and abstract advertise “two types” of models, a genuine USIU application (or an explicit admission that none is available) is required for balance.
minor comments (5)
  1. [Example 2, Figure 7] In Example 2 the rejection region W2 is written with “500” instead of “50”; the same slip appears in the caption of Figure 7 (“i=1,…,500” for n=535).
  2. [Table 2] Table 2 contains a typographical error in row 30 (“L−4.392,−1.988)” missing the opening parenthesis).
  3. [§5.2] The significance-test decision rule in §5.2 is stated as “(ε0,…)/∈W(0) and (ε1,…)∈W(1)” for insignificance, yet the authors conclude significance after finding both residual vectors outside their rejection regions; the logical criterion needs clarification.
  4. [References] Several references list future years (Liu 2026, Zhang & Li 2026, Zhu et al. 2026); if these are preprints the arXiv identifiers should be supplied.
  5. [Algorithm 1] Algorithm 1 line 7 cites “Equation (9)” for CV(h); the CV formula is Eq. (10).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; profile least-squares estimation, kernel/spline approximations, and residual hypothesis tests are self-contained and do not reduce to their inputs by construction.

full rationale

The paper introduces two uncertain single-index models and derives profile least-squares estimators (Theorems 1–3) by rewriting the objective via the standard decomposition E[(ỹ−g)²]=V[ỹ]+(E[ỹ]−g)² and, for the USIU case, applying Liu’s general inverse-distribution lemma under an explicit monotonicity assumption on g. These steps are algebraic identities or direct applications of external uncertainty-theory results; they do not define the target quantities in terms of themselves. Bandwidth/knot selection uses ordinary V-fold CV, and the two-step iteration alternates between a free nonparametric smoother and a free index vector. Residual analysis estimates ê,σ̂ from the fitted residuals and then subjects those same residuals to an uncertain hypothesis test of N(ê,σ̂); this is conventional goodness-of-fit practice, not a definitional loop—the test can reject and is reported as such. Simulations recover known finite-dimensional parameters, and the real-data illustration (USIC) employs an unrestricted nonparametric link. No self-citation supplies a uniqueness theorem that forces the model form, no fitted quantity is renamed a “prediction” of an independent observable, and no ansatz is smuggled in via prior work of the same authors. The derivation chain is therefore free of the listed circularity patterns.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The central claims rest on Liu’s four axioms of uncertainty measure, the existence of regular inverse uncertainty distributions, the classical profile least-squares principle, and the additional modeling restriction that the link is strictly monotone when covariates are uncertain. Free parameters are the usual smoothing choices (bandwidth, number of B-spline bases) selected by cross-validation; no new physical constants are introduced. The only invented entities are the two model classes themselves.

free parameters (3)
  • bandwidth h (USIC)
    Selected by V-fold CV + Fibonacci search; controls bias-variance trade-off of the Nadaraya–Watson estimator and therefore the recovered index coefficients.
  • number of B-spline bases m (USIU)
    Selected by V-fold CV; determines flexibility of the monotone link approximation.
  • knot locations for B-splines
    Placed uniformly on the range of E[βᵀx̃]; updated iteratively with β and therefore data-dependent.
assumptions (4)
  • domain assumption Liu’s four axioms (normality, duality, subadditivity, product) define a valid uncertainty measure
    Invoked throughout §2; all subsequent inverse-distribution calculations rest on them.
  • domain assumption Uncertainty distributions of the observed variables are regular (continuous and strictly increasing)
    Required for the existence of inverse distributions used in Theorems 2–3 and residual calculations.
  • ad hoc to paper Link function g is of class C¹ and strictly increasing (USIU)
    Imposed in Theorem 2 so that Lemma 2 yields an explicit inverse residual distribution; not required for classical single-index models.
  • domain assumption Disturbance terms are independent with finite expectation and variance
    Used to justify residual mean/variance estimators (Eqs. 12, Theorem 3).
invented entities (2)
  • USIC model (uncertain single-index with crisp covariates)
    purpose: Provide a single-index regression structure when only the response is imprecise.
    Defined in Definition 1(i); no independent empirical existence outside the paper.
  • USIU model (uncertain single-index with uncertain covariates)
    purpose: Extend the single-index idea to fully imprecise predictors under monotonicity.
    Defined in Definition 1(ii); likewise paper-internal.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Parameter estimation and application in two types of uncertain single-index models." pith.science (2026). https://pith.science/paper/7XECV3YW

@misc{pith2026260704699,
  author       = {Pith},
  title        = {Pith review of: Parameter estimation and application in two types of uncertain single-index models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7XECV3YW}},
  note         = {Machine review of arXiv:2607.04699}
}
read the original abstract

Uncertain data often arises in complex environments because of frequency instability and subjective judgment. This paper establishes two types of uncertain single-index models to capture the inherent properties of such data. Based on the semiparametric least-squares principle, the Nadaraya-Watson kernel and B-spline methods are used to estimate the unknown coefficients in various scenarios with both crisp and imprecise explanatory variables. Residual analysis and hypothesis testing under uncertainty assess the fit of the proposed models. Furthermore, simulation studies verify the models' validity, and a real-data application demonstrates their effectiveness in practical settings.

Figures

Figures reproduced from arXiv: 2607.04699 by the authors.

Figure 1
Figure 1. Comparison of fitted curves at bandwidths 0.1, 0.3, 0.5, hopt against the true function curve. Based on the model estimated above, we can calculate the estimates of the expectation and variance of the disturbance as: ˆe = 0.0448, ˆσ 2 = 0.0097. Further, we assume that the disturbance follows the normal uncertain distribution N (0.0448, 0.0985). Uncertain hypoth￾esis test (Ye and Liu, 2022) can verify the validity of… view at source ↗
Figure 2
Figure 2. Scatter plots of Ω−1 i (0.05) and Ω−1 i (0.95) for i = 1, . . . , 500 with the boundaries of the test W1. Example 2. Consider both the explanatory variables x1, . . . , xp and the response variable y are imprecise. In the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. The curves of estimated link function in the USIU model (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Scatter plots of Ω−1 i (0.05), Ω −1 i (0.95) for i = 1, . . . , 50 with the boundaries of the test W2. 5 An application In this section, a weather dataset of Lagos, Nigeria, is adopted to illustrate the practi￾cal applicability of the proposed model. The data are colle…
Figure 5
Figure 5. Figure 5: (a) Cross-validation curve for bandwidth selection. (b) Plot of the estimated link function. system shifts from a radiation-dominated state to one controlled by latent heat processes, while the key driving factors remain unchanged. 5.1 Residual analysis Next, we conduc…
Figure 6
Figure 6. Figure 6: Relationships between key meteorological variables and the single-index value in Lagos. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Scatter plots of Ω−1 i (0.05) and Ω−1 i (0.95) for i = 1, . . . , 500 with the boundaries of the test W3. 5.2 Significance test Since the coefficient β2 = 0.0381 is close to zero, we conduct an uncertain significance test by Ye and Liu (2023) to examine whether β2 is s…
Figure 8
Figure 8. Figure 8: Scatter plots of Ω−1 i (0.05) and Ω−1 i (0.95) for i = 1, . . . , 500 with the boundaries of the test W(0). 6 Conclusion This paper proposes two types of uncertain single-index models (USIC and USIU) for im￾precise data in uncertain environments to meet modeling requir…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

79 extracted references

  1. [1]

    Econometrica , volume=

    Consistent Estimation of Scaled Coefficients , author=. Econometrica , volume=

  2. [2]

    1987 , school=

    Estimation of single index models , author=. 1987 , school=

    1987

  3. [3]

    Journal of Econometrics , volume=

    Semiparametric least squares (SLS) and weighted SLS estimation of single-index models , author=. Journal of Econometrics , volume=

  4. [4]

    A goodness-of-fit test for single-index models , AUTHOR =. Statist. Sinica , VOLUME =. 2004 , NUMBER =

    2004

  5. [5]

    Econometrica: Journal of the Econometric Society , volume=

    A semiparametric maximum likelihood estimator , author=. Econometrica: Journal of the Econometric Society , volume=. 1997 , publisher=

    1997

  6. [6]

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=

    Partial least squares estimator for single-index models , author=. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=

  7. [7]

    The Annals of Statistics , volume =

    Marian Hristache and Anatoli Juditsky and Vladimir Spokoiny , title =. The Annals of Statistics , volume =

  8. [8]

    , title =

    Antoniadis, Anestis and Grégoire, Gérard and McKeague, Ian W. , title =. Statistica Sinica , volume =

Show all 79 references
  1. [9]

    Statistica Sinica , volume =

    Wang, Li and Yang, Lijian , title =. Statistica Sinica , volume =

  2. [10]

    The Annals of Statistics

    Cui, Xia and Härdle, Wolfgang Karl and Zhu, Lixing , title =. The Annals of Statistics. , volume =

  3. [11]

    Journal of Applied Statistics and Management , year =

    Xue, Liugen , title =. Journal of Applied Statistics and Management , year =

  4. [12]

    The Annals of Statistics , volume=

    Nonparametric checks for single-index models , author=. The Annals of Statistics , volume=. 2005 , publisher=

    2005

  5. [13]

    The Annals of Statistics , volume=

    Estimation for single-index and partially linear single-index integrated models , author=. The Annals of Statistics , volume=

  6. [14]

    Annals of the Institute of Statistical Mathematics , volume =

    A constructive hypothesis test for the single-index models with two groups , author =. Annals of the Institute of Statistical Mathematics , volume =

  7. [15]

    Electronic Journal of Statistics , volume =

    Tests for high-dimensional single-index models , author =. Electronic Journal of Statistics , volume =

  8. [16]

    Statistics in Medicine , volume=

    Multivariate single index modeling of longitudinal data with multiple responses , author=. Statistics in Medicine , volume=. 2023 , doi=

    2023

  9. [17]

    Journal of Business & Economic Statistics , volume=

    A novel estimation method in generalized single index models , author=. Journal of Business & Economic Statistics , volume=. 2023 , doi=

    2023

  10. [18]

    Journal of Intelligent and Fuzzy Systems , volume=

    Residual and confidence interval for uncertain regression model with imprecise observations , author=. Journal of Intelligent and Fuzzy Systems , volume=

  11. [19]

    Soft computing , volume=

    Uncertain regression analysis: An approach for imprecise observations , author=. Soft computing , volume=

  12. [20]

    , author=

    Least absolute deviations estimation for uncertain regression with imprecise observations. , author=. Fuzzy Optimization and Decision Making , volume=

  13. [21]

    , author=

    Uncertain maximum likelihood estimation with application to uncertain regression analysis. , author=. Soft Computing , volume=

  14. [22]

    , author=

    A modified uncertain maximum likelihood estimation with applications in uncertain statistics. , author=. Communications in Statistics-Theory and Methods , volume=

  15. [23]

    2007 , publisher=

    Liu, Baoding , booktitle=. 2007 , publisher=

    2007

  16. [24]

    2010 , publisher=

    Liu, Baoding , booktitle=. 2010 , publisher=

    2010

  17. [25]

    2023 , publisher=

    Liu, Baoding , booktitle=. 2023 , publisher=

    2023

  18. [26]

    Fuzzy Optimization and Decision Making , year=

    Uncertain hypothesis test with application to uncertain regression analysis , author=. Fuzzy Optimization and Decision Making , year=

  19. [27]

    Communications in Statistics: Theory & Methods , year=

    Ye, Tingqing and Liu, Baoding , title=. Communications in Statistics: Theory & Methods , year=

  20. [28]

    Soft Computing , year=

    Uncertain regression analysis: an approach for imprecise observations , author=. Soft Computing , year=

  21. [29]

    Soft Computing , year=

    Uncertain Johnson–Schumacher growth model with imprecise observations and k-fold cross-validation test , author=. Soft Computing , year=

  22. [30]

    Soft Computing , year=

    Uncertain Gompertz regression model with imprecise observations , author=. Soft Computing , year=

  23. [31]

    Fuzzy Optimization and Decision Making , year=

    Statistical inference on uncertain nonparametric regression model , author=. Fuzzy Optimization and Decision Making , year=

  24. [32]

    Communications in Statistics - Theory and Methods , year =

    Inferences for uncertain nonparametric regression by least absolute deviations , author =. Communications in Statistics - Theory and Methods , year =

  25. [33]

    Bernoulli , year =

    Tang, Jiajun and Dette, Holger , title =. Bernoulli , year =

  26. [34]

    Communications in Statistics: Simulation & Computation , volume=

    Orthogonal series method for uncertain nonparametric regression with application to carbon dioxide emissions , author=. Communications in Statistics: Simulation & Computation , volume=

  27. [35]

    Fuzzy Optimization and Decision Making , year=

    Estimation problems in two types of uncertain varying coefficient models with imprecise observations , author=. Fuzzy Optimization and Decision Making , year=

  28. [36]

    Mathematics and Computers in Simulation , volume=

    Uncertain semi-varying coefficient model with application to housing prices , author=. Mathematics and Computers in Simulation , volume=

  29. [37]

    Journal of Industrial and Management Optimization , year=

    Analysis and applications of an uncertain GTWR model with spatiotemporal and non-stationary data , author=. Journal of Industrial and Management Optimization , year=

  30. [38]

    Statistics and Computing , volume=

    Online learning for high-dimensional single-index model with streaming data , author=. Statistics and Computing , volume=

  31. [39]

    Toxics , volume=

    Modelling Cumulative Effects of Air Pollution on Respiratory Illnesses by Performing Spline Estimation of Constrained, Additive Single-Index Model , author=. Toxics , volume=

  32. [40]

    SCIENTIA SINICA Mathematica , volume=

    A lack-of-fit test for the parametric index model with ultrahigh-dimensional covariates (in Chinese) , author=. SCIENTIA SINICA Mathematica , volume=. 2025 , note=

    2025

  33. [41]

    Applied Soft Computing , volume=

    USVDD: A one-class outlier detection method with imprecisely-observed data , author=. Applied Soft Computing , volume=

  34. [42]

    1984 , address=

    Classification and Regression Trees , author=. 1984 , address=

    1984

  35. [43]

    2009 , publisher=

    The Elements of Statistical Learning: Data Mining, Inference, and Prediction , author=. 2009 , publisher=

    2009

  36. [44]

    Stoker, T. M. (1986). Consistent Estimation of Scaled Coefficients. Econometrica, 54(6), 1461-1481

    1986

  37. [45]

    Ichimura, H. (1987). Estimation of single index models. Massachusetts Institute of Technology

    1987

  38. [46]

    Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics, 54(1), 71-120

    1993

  39. [47]

    Ai, C. (1997). A semiparametric maximum likelihood estimator. Econometrica, 65(4), 933-963

    1997

  40. [48]

    Naik, P., & Prasad, C. L. (2000). Partial least squares estimator for single-index models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 62(4), 763--771

    2000

  41. [49]

    Antoniadis, A., Grégoire, G., & McKeague, I. W. (2004). Bayesian estimation in single-index models. Statistica, 14(4), 1147-1164

    2004

  42. [50]

    Nonparametric checks for single-index models

    Stute, W., & Zhu, L.(2005). Nonparametric checks for single-index models. The Annals of Statistics, 33(3), 1048--1083

    2005

  43. [51]

    Wang, Li., & Yang, L. (2009). Spline estimation of single-index models. Statistica, 19(2), 765-783

    2009

  44. [52]

    K., & Zhu, L.(2011)

    Cui, X., Härdle, W. K., & Zhu, L.(2011). The EFM approach for single-index models. The Annals of Statistics, 39(3), 1658--1688

    2011

  45. [53]

    Tian, Z., & and Qiu, P. (2023). Multivariate single index modeling of longitudinal data with multiple responses. Statistics in Medicine, 42(17), 2982--2998

    2023

  46. [54]

    Zhang, D., Wang, Y., & Liang, H. (2023). A novel estimation method in generalized single index models. Journal of Business & Economic Statistics, 41(2), 399--413

    2023

  47. [55]

    Yao, K., & Liu, B. (2018). Uncertain regression analysis: An approach for imprecise observations. Soft computing, 22(17), 5579-5582

    2018

  48. [56]

    Lio, W., & Liu, B. (2018). Residual and confidence interval for uncertain regression model with imprecise observations. Journal of Intelligent and Fuzzy Systems, 35, 2573–2583

    2018

  49. [57]

    Liu, Z., & Yang, Y. (2020). Least absolute deviations estimation for uncertain regression with imprecise observations. Fuzzy Optimization and Decision Making, 19, 33–52

    2020

  50. [58]

    Lio, W., & Liu, B. (2020). Uncertain maximum likelihood estimation with application to uncertain regression analysis. Soft Computing, 24(4), 6649–6670

    2020

  51. [59]

    Liu, Y., & Liu, B. (2024). A modified uncertain maximum likelihood estimation with applications in uncertain statistics. Communications in Statistics-Theory and Methods, 53(18), 6649–6670

    2024

  52. [60]

    Liu, B. (2007). Uncertainty theory (2nd ed.). Springer

    2007

  53. [61]

    Liu, B. (2010). Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Springer

    2010

  54. [62]

    Liu, B. (2026). Uncertainty theory (6th ed.). Springer

    2026

  55. [63]

    Ye, T., & Liu, B. (2022). Uncertain hypothesis test with application to uncertain regression analysis. Fuzzy Optimization and Decision Making, 21(2), 157-174

    2022

  56. [64]

    Ye, T., & Liu, B. (2023). Uncertain significance test for regression coefficients with application to regional economic analysis. Communications in Statistics: Theory & Methods, 52(20), 7271-7288

    2023

  57. [65]

    Yao, K., & Liu, B. (2018). Uncertain regression analysis: an approach for imprecise observations. Soft Computing, 22(17), 5579-5582

    2018

  58. [66]

    Ding, J., & Zhang, Z. (2021). Statistical inference on uncertain nonparametric regression model. Fuzzy Optimization and Decision Making, 20(4), 451-469

    2021

  59. [67]

    Ding, J., Zhang, H., & Zhang, Z. (2021). Inferences for uncertain nonparametric regression by least absolute deviations. Communications in Statistics - Theory and Methods, 52(16), 5640-5649

    2021

  60. [68]

    Xin, Y., & Gao, J. (2024). Orthogonal series method for uncertain nonparametric regression with application to carbon dioxide emissions. Communications in Statistics: Simulation & Computation, 53(10), 1-10

    2024

  61. [69]

    Zhang, Y., & Li, Z. (2026). Uncertain semi-varying coefficient model with application to housing prices. Mathematics and Computers in Simulation, 243, 270-282

    2026

  62. [70]

    H., Olshen, R

    Breiman, L., Friedman, J. H., Olshen, R. A., & Stone, C. J. (1984). Classification and Regression Trees. Wadsworth

    1984

  63. [71]

    Hastie, T., Tibshirani, R., & Friedman, J. H. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction (2nd ed.). Springer

    2009

  64. [72]

    Xia, Y., Li, W., Tong, H., & Zhang, D. (2004). A goodness-of-fit test for single-index models. Statistica, 14(1), 1-39

    2004

  65. [73]

    A constructive hypothesis test for the single-index models with two groups

    Zhang, J., Feng, Z., & Wang, X.(2018). A constructive hypothesis test for the single-index models with two groups. Annals of the Institute of Statistical Mathematics, 70(5), 1077--1114

    2018

  66. [74]

    Tests for high-dimensional single-index models

    Cai, L., Guo, X., Li, G., & Tan, F.(2023). Tests for high-dimensional single-index models. Electronic Journal of Statistics, 17(1), 429--463

    2023

  67. [75]

    Simultaneous semiparametric inference for single-index models

    Tang, J., & Dette, H.(2025). Simultaneous semiparametric inference for single-index models. Bernoulli, 31(4), 2962--2986

    2025

  68. [76]

    Zhu, Y., Guo, X., & Shi, Y. (2026). Online learning for high-dimensional single-index model with streaming data. Statistics and Computing, 36(3), 117

    2026

  69. [77]

    Zhang, X., Wang, S., Shao, Q., Wang, S., & Wei, Y. (2025). Modelling cumulative effects of air pollution on respiratory illnesses by performing spline estimation of constrained, additive single-index model. Toxics, 13(3), 149

    2025

  70. [78]

    & Liu, J

    Liang, J. & Liu, J. (2025). USVDD: A one-class outlier detection method with imprecisely-observed data. Applied Soft Computing, 175, 113065

    2025

  71. [79]

    Zhou, T., Zhang, Y., & Zhu, L. (2025). A lack-of-fit test for the parametric single index model with ultrahigh-dimensional covariates (in Chinese). Scientia Sinica Mathematica, 55(11), 2049--2070

    2025

Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.