| Title:
|
Bias of LS estimators in nonlinear regression models with constraints. Part II: Biadditive models (English) |
| Author:
|
Denis, Jean-Baptiste |
| Author:
|
Pázman, Andrej |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
44 |
| Issue:
|
5 |
| Year:
|
1999 |
| Pages:
|
375-403 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
General results giving approximate bias for nonlinear models with constrained parameters are applied to bilinear models in anova framework, called biadditive models. Known results on the information matrix and the asymptotic variance matrix of the parameters are summarized, and the Jacobians and Hessians of the response and of the constraints are derived. These intermediate results are the basis for any subsequent second order study of the model. Despite the large number of parameters involved, bias formulæ turn out to be quite simple due to the orthogonal structure of the model. In particular, the response estimators are shown to be approximately unbiased. Some simulations assess the validity of the approximations. (English) |
| Keyword:
|
asymptotic variance |
| Keyword:
|
bilinear model |
| Keyword:
|
nonlinear least squares |
| Keyword:
|
response function |
| Keyword:
|
second order approximation |
| MSC:
|
62F12 |
| MSC:
|
62F30 |
| MSC:
|
62J02 |
| idZBL:
|
Zbl 1060.62527 |
| idMR:
|
MR1709502 |
| DOI:
|
10.1023/A:1023045028073 |
| . |
| Date available:
|
2009-09-22T18:01:23Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133892 |
| . |
| Related article:
|
http://dml.cz/handle/10338.dmlcz/133891 |
| . |
| Reference:
|
[CD91] J. Chadœuf and J.-B. Denis: Asymptotic variances for the multiplicative interaction model.Journal of Applied Statistics 18 (1991), 331–353. 10.1080/02664769100000032 |
| Reference:
|
[DG92] J.-B. Denis and J.C. Gower: Biadditive models.Technical report, Unité de biométrie, Versailles, 1992. |
| Reference:
|
[DG94a] J.-B. Denis and J.C. Gower: Biadditive models.Biometrics 50 (1994), 310–311. |
| Reference:
|
[DG94b] J.-B. Denis and J.C. Gower: Asymptotic covariances for the parameters of biadditive models.Utilitas Mathematica 46 (1994), 193–205. MR 1301307 |
| Reference:
|
[DG96] J.-B. Denis and J.C. Gower: Asymptotic confidence regions for biadditive models: interpreting genotype-environment interactions.Applied Statistics 45 (1996), 479–493. 10.2307/2986069 |
| Reference:
|
[DP98] J.-B. Denis and A. Pázman: Biadditive ANOVA models: reminders and asymptotical bias.Technical report n$^{\circ }$4, Unité de biométrie INRA, Versailles (1998). |
| Reference:
|
[DM93] K.M.M. Dorkenoo and J.-R. Mathieu: Etude d’un modèle factoriel d’analyse de la variance comme modèle linéaire généralisé.Revue de Statistique Appliquée 41 (1993), 43–57. MR 1253515 |
| Reference:
|
[EY36] C. Eckart and G. Young: The approximation of one matrix by another of lower rank.Psychometrika 1 (1936), 211–219. 10.1007/BF02288367 |
| Reference:
|
[E96] F.A. vanEeuwijk: Between and beyond additivity and non-additivity; the statistical modelling of genotype by environment interaction in plant breeding.Thesis, Agricultural University, Wageningen, 1996. |
| Reference:
|
[FM23] R.A. Fisher and W.A. Mackenzie: Studies in crop variation, II. The manurial response of different potato varieties.Journal of Agricultural Science XIII (1923), 311–320. 10.1017/S0021859600003592 |
| Reference:
|
[G92] H.D. Gauch: Statistical analysis of regional trials: AMMI analysis of factorial designs.Elsevier, Amsterdam, 1992. |
| Reference:
|
[G63] N. Gilbert: Non-additive combining abilities.Genetical Research 4 (1963), 65–73. 10.1017/S0016672300003438 |
| Reference:
|
[G68] H.F. Gollob: A statistical model which combines features of factor analytic and analysis of variance techniques.Psychometrika 33 (1968), 73–115. Zbl 0167.48601, MR 0221658, 10.1007/BF02289676 |
| Reference:
|
[GH90] L.A. Goodman and S.J. Haberman: The analysis of non-additivity in two-way analysis of variance.Journal of the American Statistical Association 85 (1990), 139–145. MR 1137360, 10.1080/01621459.1990.10475317 |
| Reference:
|
[JG72] D.E. Johnson and F.A. Graybill: An analysis of a two-way model with interaction and no replication.Journal of the American Statistical Association 67 (1972), 862–868. MR 0400566, 10.1080/01621459.1972.10481307 |
| Reference:
|
[M71] J. Mandel: A new analysis of variance model for non-additive data.Technometrics 13 (1971), 1–18. Zbl 0216.48104, 10.1080/00401706.1971.10488751 |
| Reference:
|
[PD??] A. Pázman and J.-B. Denis: Bias of L.S. estimators in nonlinear regression models with constraints. Part I: the general case.Appl. Math. 44 (1999), 359–374. MR 1709501, 10.1023/A:1023092911235 |
| Reference:
|
[SC92] M.S. Seyedsadr and P.L. Cornelius: Shifted multiplicative models for nonadditive two way tables.Comm. Stat. B Simul. Comp. 21 (1992), 807–832. MR 1185174, 10.1080/03610919208813051 |
| Reference:
|
[S79] S.D. Silvey: Statistical Inference, 3rd edition.Chapman and Hall, London, 1979. MR 0500810 |
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