On ridge parameter estimators under stochastic subspace hypothesis. (English) Zbl 07191984
Summary: This paper considers several estimators for estimating the restricted ridge parameter estimators. A simulation study has been conducted to compare the performance of these estimators. Based on the simulation study we found that, increasing the correlation between the independent variables has positive effect on the mean square error (MSE). However, increasing the value of \(\rho\) has negative effect on MSE. When the sample size increases, the MSE decreases even when the correlation between the independent variables is large. Two real life examples have been considered to illustrate the performance of the estimators.
MSC:
| 62J05 | Linear regression; mixed models |
| 62J07 | Ridge regression; shrinkage estimators (Lasso) |
| 62F10 | Point estimation |
References:
| [1] | Ehsanes Saleh AKMd. Theory of preliminary test and stein-type estimation with applications. New York: John Wiley; 2006. [Crossref], [Google Scholar] · Zbl 1094.62024 |
| [2] | Tabatabaey SMM, Ehsanes Saleh AKMd, Kibria BMG. Simultaneous estimation of regression parameters with spherically symmetric errors under possible stochastic constrains. Int J Statist Sci. 2004;3:1-20. [Google Scholar] |
| [3] | Arashi M, Tabatabaey SMM. Stein-type improvement under stochastic constraints: use of multivariate Student-t model in regression. Statist Probab Lett. 2008;78:2142-2153. doi: 10.1016/j.spl.2008.02.003[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1283.62140 |
| [4] | Arashi M, Tabatabaey SMM, Iranmanesh A. Improved estimation in stochastic linear models under elliptical symmetry. J Appl Prob Statist. 2010;5(2):145-160. [Google Scholar] |
| [5] | Theil H, Goldberger AS. On pure and mixed estimation in econometrics. Internat Econom Rev. 1961;2:65-78. doi: 10.2307/2525589[Crossref], [Web of Science ®], [Google Scholar] |
| [6] | Theil H. On the use of incomplete prior information in regression analysis. J Amer Statist Assoc. 1963;58:401-414. doi: 10.1080/01621459.1963.10500854[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0129.11401 |
| [7] | Nagar AL, Kakwani NC. Note on the use of prior information in statistical estimation of economic relations. Sankhyã. 1965;105-112. [Google Scholar] · Zbl 0138.13602 |
| [8] | Ozkale MR. A stochastic restricted ridge regression estimator. J Multivariate Anal. 2009a;100:1706-1716. doi: 10.1016/j.jmva.2009.02.005[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1163.62052 |
| [9] | Liu K. A new class of biased estimate in linear regression. Comm Statist Theory Methods. 1993;22:393-402. doi: 10.1080/03610929308831027[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0784.62065 |
| [10] | Akdeniz F, Kaciranlar S. On the almost unbiased generalized Liu estimator and unbiased estimation of the bias and MSE. Comm Statist Theory Methods. 1995;24:1789-1797. doi: 10.1080/03610929508831585[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0937.62612 |
| [11] | Liu K. Using Liu-type estimator to combat collinearity. Comm Statist Theory Methods. 2003;32:1009-1020. doi: 10.1081/STA-120019959[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1107.62345 |
| [12] | Rao CR, Toutenburg H. Linear models: least squares and alternatives. 2nd ed.New York: Springer; 1995. [Crossref], [Google Scholar] · Zbl 0846.62049 |
| [13] | Swindel BF. Good ridge estimators based on prior information. Comm Statist Theory Methods. 1976;5(11):1065-1075. doi: 10.1080/03610927608827423[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0342.62035 |
| [14] | Hoerl AE, Kennard RW. Ridge regression: biased estimation for non-orthogonal problems. Thechnometrics. 1970;12:69-82. doi: 10.1080/00401706.1970.10488635[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0202.17206 |
| [15] | Hassanzadeh Bashtian M, Arashi M, Tabatabaey SMM. Ridge estimation under the stochastic restriction. Comm Statist Theory Methods. 2011a;40:3711-3747. doi: 10.1080/03610926.2010.494809[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1277.62177 |
| [16] | Hassanzadeh Bashtian M, Arashi M, Tabatabaey SMM. Using improved estimation strategies to combat multicollinearity. J Stat Comput Simul. 2011;81(12):1773-1797. doi: 10.1080/00949655.2010.505925[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1365.62278 |
| [17] | Najarian S, Arashi M, Golam Kibria BM. A simulation study on some restricted ridge regression estimators. Comm Statist Simulation Comput. 2013;42(4):871-890. doi: 10.1080/03610918.2012.659953[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1347.62144 |
| [18] | Swamy PAVB, Mehta JS, Roppoport PN. Two methods of evaluating Hoerl and Kennard’s ridge regression. Comm Statist Theory Methods. 1978;12:1133-1155. doi: 10.1080/03610927808827697[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0393.62027 |
| [19] | Swamy PAVB, Mehta JS. A note on minimum average risk estimator for coefficients in linear models. Comm Statist Theory Methods. 1977;6:1161-1186. doi: 10.1080/03610927708827561[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0385.62044 |
| [20] | Zhong Z, Yang Hu. Ridge estimation to the restricted linear model. Comm Statist Theory Methods. 2007;36(11):2099-2115. doi: 10.1080/03610920601144095[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1124.62041 |
| [21] | Hocking RR, Speed FM, Lynn MJ. A class of biased estimator in linear restriction in linear regression model with non-normal disturbance. Comm Statist A. 1976;25:2349-2369. [Google Scholar] · Zbl 0887.62077 |
| [22] | Hoerl AE, Kennard RW, Baldwin KF. Ridge regression: some simulation. Comm Statist. 1975;4:105-123. doi: 10.1080/03610927508827232[Taylor & Francis Online], [Google Scholar] · Zbl 0296.62062 |
| [23] | Lawless JF, Wang P. A simulation study of ridge and other regression estimators. Comm Statist Theory Methods. 1976;5:307-323. doi: 10.1080/03610927608827353[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0336.62056 |
| [24] | Kibria BMG. Performance of some new ridge estimation. Comm Statist Simulation Comput. 2003;32(2):419-435. doi: 10.1081/SAC-120017499[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1075.62588 |
| [25] | Roozbeh M, Arashi M. Feasible ridge estimator in seemingly unrelated semiparametric models. Comm Statist Simulation Comput. 2014;43(10):2593-2613. doi: 10.1080/03610918.2012.758740[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1462.62428 |
| [26] | Kibria BMG, Banik S. Some ridge regression estimators and their performances. J Mod Appl Statist Methods. 2016;15(1):206-238. [Google Scholar] |
| [27] | McDonald GC, Galarneau DI. A monte carlo evaluation of some Ridge- type estimators. J Amer Statist Assoc. 1975;70:407-416. doi: 10.1080/01621459.1975.10479882[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0319.62049 |
| [28] | Gibbons DG. A simulation study of some ridge estimators. J Amer Statist Assoc. 1981;76:131-139. doi: 10.1080/01621459.1981.10477619[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 0452.62055 |
| [29] | Wood H, Steinnour HH, Starke HR. Effect of composition of portland cement on heat evolved during hardening. Ind Eng Chem. 1932;24:1207-1241. doi: 10.1021/ie50275a002[Crossref], [Google Scholar] |
| [30] | Kaciranalar S, Sakallioglu S, Akdeniz F, et al. A new biased estimator in linear regression and a detailed analysis of the widely-analyzed data set on Portland cement. Sankhya. 1999;61:443-456. [Google Scholar] |
| [31] | Akdeniz F, Erol H. Mean squared error matrix comparisons of some biased estimators in linear regression. Comm Statist Theory Methods. 2003;32:2389-2413. doi: 10.1081/STA-120025385[Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1028.62054 |
| [32] | Gruber MHJ. Improving efficiency by shrinkage. New York: Marcel Dekker; 1998. [Google Scholar] · Zbl 0920.62085 |
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