Variable selection for joint mean and dispersion models of the inverse Gaussian distribution. (English) Zbl 1410.62132
Summary: The choice of distribution is often made on the basis of how well the data appear to be fitted by the distribution. The inverse Gaussian distribution is one of the basic models for describing positively skewed data which arise in a variety of applications. In this paper, the problem of interest is simultaneously parameter estimation and variable selection for joint mean and dispersion models of the inverse Gaussian distribution. We propose a unified procedure which can simultaneously select significant variables in mean and dispersion model. With appropriate selection of the tuning parameters, we establish the consistency of this procedure and the oracle property of the regularized estimators. Simulation studies and a real example are used to illustrate the proposed methodologies.
MSC:
| 62J07 | Ridge regression; shrinkage estimators (Lasso) |
| 62G20 | Asymptotic properties of nonparametric inference |
Keywords:
joint mean and dispersion models of the inverse Gaussian distribution; Lasso; penalized maximum likelihood; SCAD; variable selectionSoftware:
GLIMReferences:
| [1] | Aitkin M (1987) Modelling variance heterogeneity in normal regression using GLIM. J R Stat Soc C-Appl 36: 332–339 |
| [2] | Antoniadis A (1997) Wavelets in statistics: a review (with discussion). J Italian Stat Assoc 6: 97–144 · doi:10.1007/BF03178905 |
| [3] | Carroll RJ (1987) The effect of variance function estimating on prediction and calibration: an example. In: Berger JO, Gupta SS (eds) Statistical decision theory and related topics IV, vol II. Springer, Heidelberg |
| [4] | Carroll RJ, Rupert D (1988) Transforming and weighting in regression. Chapman and Hall, London |
| [5] | Chhikara RS, Folks JL (1989) The inverse Gaussian distribution. Marcel Dekker, New York |
| [6] | Cook RD, Weisberg S (1983) Diagnostics for heteroscedasticity in regression. Biometrika 70: 1–10 · Zbl 0502.62063 · doi:10.1093/biomet/70.1.1 |
| [7] | Fan JQ, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96: 1348–1360 · Zbl 1073.62547 · doi:10.1198/016214501753382273 |
| [8] | Fan JQ, Lv JC (2010) A selective overview of variable selection in high dimensional feature space. Stat Sin 20: 101–148 · Zbl 1180.62080 |
| [9] | Harvey AC (1976) Estimating regression models with multiplicative heteroscedasticity. Econometrica 44: 460–465 · Zbl 0333.62040 · doi:10.2307/1913974 |
| [10] | Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1. Wiley, New York |
| [11] | Lee Y, Nelder JA (1998) Generalized linear models for the analysis of quality improvement experiments. Can J Stat 26: 95–105 · Zbl 0899.62088 · doi:10.2307/3315676 |
| [12] | Li R, Liang H (2008) Variable selection in semiparametric regression modeling. Ann Stat 36: 261–286 · Zbl 1132.62027 · doi:10.1214/009053607000000604 |
| [13] | Lin JG, Wei BC, Zhang NS (2004) Varying dispersion diagnostics for inverse Gaussian regression models. J Appl Stat 31: 1157–1170 · Zbl 1121.62426 · doi:10.1080/0266476042000285512 |
| [14] | Nelder JA, Lee Y (1991) Generalized linear models for the analysis of Taguchi-type experiments. Appl Stoch Model Data Anal 7: 107–120 · doi:10.1002/asm.3150070110 |
| [15] | Park RE (1966) Estimation with heteroscedastic error terms. Econometrica 34: 888 · doi:10.2307/1910108 |
| [16] | Seshadri V (1993) The inverse Gaussian distribution: a case study in exponential families. Claredon, New York |
| [17] | Seshadri V (1999) The inverse Gaussian distribution: statistical theory and applications. Springer, New York · Zbl 0942.62011 |
| [18] | Smyth GK (1989) Generalized linear models with varying dispersion. J R Stat Soc B 51: 47–60 |
| [19] | Smyth GK, Verbyla AP (1999) Adjusted likelihood methods for modelling dispersion in generalized linear models. Environmetrics 10: 696–709 · doi:10.1002/(SICI)1099-095X(199911/12)10:6<695::AID-ENV385>3.0.CO;2-M |
| [20] | Taylor JT, Verbyla AP (2004) Joint modelling of location and scale parameters of the t distribution. Stat Model 4: 91–112 · Zbl 1112.62010 · doi:10.1191/1471082X04st068oa |
| [21] | Tibshirani R (1996) Regression shrinkage and selection via the LASSO. J R Stat Soc B 58: 267–288 · Zbl 0850.62538 |
| [22] | Verbyla AP (1993) Variance heterogeneity: residual maximum likelihood and diagnostics. J R Stat Soc B 52: 493–508 · Zbl 0783.62051 |
| [23] | Wang DR, Zhang ZZ (2009) Variable selection in joint generalized linear models. Chin J Appl Probab Stat 25: 245–256 · Zbl 1211.62121 |
| [24] | Wang H, Li R, Tsai C (2007) Tuning parameter selectors for the smoothly clipped absolute deviation method. Biometrika 94: 553–568 · Zbl 1135.62058 · doi:10.1093/biomet/asm053 |
| [25] | Zhao PX, Xue LG (2010) Variable selection for semiparametric varying coefficient partially linear errors-in-variables models. J Multivar Anal 101: 1872–1883 · Zbl 1190.62090 · doi:10.1016/j.jmva.2010.03.005 |
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