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 Open Journal of Statistics, 2012, 2, 309-312 http://dx.doi.org/10.4236/ojs.2012.23038 Published Online July 2012 (http://www.SciRP.org/journal/ojs) A Revision of AIC for Normal Error Models Kunio Takezawa Agroinformatics Division, Agricultural Research Center, National Agriculture and Food Research Organization, Graduate School of Life and Environmental Sciences, University of Tsukuba, Tsukuba, Japan Email: [email protected] Received March 21, 2012; revised April 22, 2012; accepted May 4, 2012 ABSTRACT Conventional Akaike’s Information Criterion (AIC) for normal error models uses the maximum-likelihood estimator of error variance. Other estimators of error variance, however, can be employed for defining AIC for normal error models. The maximization of the log-likelihood using an adjustable error variance in light of future data yields a revised version of AIC for normal error models. It also gives a new estimator of error variance, which will be called the “third variance”. If the model is described as a constant plus normal error, which is equivalent to fitting a normal distribution to one-dimensional data, the approximated value of the third variance is obtained by replacing (n − 1) (n is the number of data) of the unbiased estimator of error variance with (n − 4). The existence of the third variance is confirmed by a sim- ple numerical simulation. Keywords: AIC; AICc; Normal Error Models; Third Variance 1. Introduction Akaike’s Information Criterion (AIC) for multiple linear models with normal i.i.d. errors is defined as (e.g., [1,2]) 2 log 2πlog2 4q ˆˆ 2,, 24, j AICnnRSS nn la q Xy 2 11 ˆˆ , ji j i ij a xy 1 i (1) where n is the number of data and q is the number of predictors of the multiple linear model. Hence, the num- ber of regression coefficients in this model is (q + 1) when the error variance is regarded as a regression coef- ficient. X is a design matrix composed of the predictor values in the data. y is the vector composed of values of the target variable in the data. RSS stands for the residual sum of squares: 0 q n RSS a (2) where are the estimators of regression coefficients of a multiple linear model. xij(1 ≤ i ≤ n, 1 ≤ j ≤ q) is an element of X. 01 ˆˆ ˆ ,, , q aa a y in is an element of y. 2 ˆˆ ,, j la Xy is the log-likeli- hood of the regression model in light of the data at hand. It is defined as 22 ˆˆ ˆ ,, log2πlog . 222 j nnn la Xy ˆˆ ,,aa 2 ˆ (3) The multiple linear model for obtaining Equations (1) and (3) contains 01 given by the least squares method (also called the maximum likelihood method for normal errors), and the error variance ( ˆ , q a ) given by the maximum likelihood method. 2 ˆ is derived using 2 ˆ.RSS n (4) 2 ˆ defined above is used as the error variance in AIC because AIC is a statistic based on the maximum-likeli- hood estimator. However, the unbiased error variance shown below rather than the maximum-likelihood esti- mator of error variance is utilized in most statistical cal- culations. 2 ˆ1 ub RSS n q . (5) The maximum-likelihood estimator of error variance may not be the only choice for the error variance for AIC. Hence, in this paper, we discusses the adjustment of error variance to calculate AIC for normal error models after recalling the derivation of conventional AIC for normal error models. Then, this consideration leads to a new estimator of error variance, which will be called the “third variance”. Finally, the existence of the third vari- ance is shown by a simple numerical simulation. 2. Derivation of AIC for Normal Error Models Conventional AIC for normal error models is easily de- rived when the multiple linear model with normal error assumed by an analyst contains the real equation pro- ducing the data as a special case. AIC based on these assumption is an approximation of C opyright © 2012 SciRes. OJS  K. TAKEZAWA 310 *2 ˆˆ 2, , log 2πlog RSS j El a Enn n Xy * RSS , n RSS * y * RSS 2 11 ˆˆ , q ijij ij a x (6) where is a vector comprising the values of the target variable in future data. The design matrix of future data is identical to that of the data at hand (X). is the residual sum of squares when future data are employed: ** 0 n RSSy a (7) where i y iin * y ˆ Ey y IHy y IH ε t is an element of . The expectation of RSS is given by ˆt t t t tt tt ERSS E E E E yy y yIH yIH yε εε εHε (8) where H is the symmetric matrix ( H H 2 ) and idem- potent ( H H y ). Furthermore, it is assumed that if (the values of the target variable with no errors) is em- ployed, H yy ε 2 holds because it is assumed that the regression equation adopted here contains the real equa- tion producing the data as a special case. Since is a normal error (the mean is 0 and the variance is ), the following equation is obtained: 2 1 . ii n n t i EE 22 1 , ij ij q εε (9) The following equation is also derived: 11 trace nn t ij EE H εHε H trace (10) where H is the trace of H . Hence, Equations (8)-(10) give 2 1n q.ERSS (11) Therefore, 2 RSS obeys the 2 distribution with (n – q – 1) degrees of freedom. A similar calculation yields *** ** ** ˆˆ t t tt ERSS E E En yyyy yεHy εyε εεεHε 2 1. q Hy ε (12) Hence, *2 RSS 2 obeys the distribution with (n + q + 1) degrees of freedom. Considering Equations (11) and (12), the content in the third term on the right-hand side of Equation (6) is transformed into []E 2 * 1 1,1 1 2 1 1 ~. 1 nq nq n nq n nRSSn q nF RSSn q 2 (13) 2 where 1nq is a random variable that obeys the distribution with 1nq 2 degrees of freedom. 1nq is a random variable that obeys the 2 distribution with 1nq 1 F degrees of freedom, and 1,nq nq is an F distribution. The first degrees of freedom is 1nq 1nq and the second degrees of freedom is . Hence, the expectation of the random variable given by Equation (13) is 1, 1 111 1112 1. 3 nq nq nqnq nq nEF n nqnq nq nq nnq (14) By substituting this equation into Equation (6) and using Equation (3), the following equation is obtained: *2 2 ˆˆ 2, , 1 log 2π log3 1 ˆˆ 2, ,. 3 j j El a RSSn q nn n nnq nq la nn nq Xy Xy (15) This is AICc for normal error models ([1,3,4]). When n is large, the approximation below holds: 11 113 1 31 3 24 1. qn nqq q nn n nqq nnn q nn (16) By substituting this equation into Equation (6) and using Equation (3), the following equation is obtained: *2 2 ˆˆ 2, , log 2πlog RSS24 ˆˆ 2,, 24. j j El a nnnnq la q Xy Xy 2 AIC (17) This is conventional AIC for normal error models. 3. Adjustment of Error Variance of AIC for Normal Error Models The estimator of error variance is assumed to be adjust- able. That is, error variance ( ) is defined as Copyright © 2012 SciRes. OJS  K. TAKEZAWA 311 2 AIC ,RSS n (18) where is a constant for adjusting error variance. The use of AIC 2 in A c IC a c (Equation (15)) yields A IC (AIC-adjustable): log 2π a c AIC n n log 1. 3 RSS nn nq nq ˆ (19) a c A IC is Then, which minimizes 324. 1q 2 ˆ ˆ1 nq n nq (20) Hence, the following AIC is different from the unbi- ased estimator of error variance: 2 ˆAIC ˆ RSS n. 2 ˆ (21) AIC will be called the “third variance” because the dis- covery of this variance follows those of the maximum- likelihood estimator of error variance and the unbiased estimator of error variance. In particular, when 0q which indicates the fitting of a normal distribution to one- dimensional data. Although 0 or 1 is adopted conventionally, 4 is preferable in terms of log-like- lihood in light of future data. The substitution of Equations (20) and (21) to Equa- tion (15) leads to log 2π log 13 3 1 log 2πlog u c AIC n RSS nnn nq nq n nn nq n RSS n nn nq 1 1 13 1, 3 nq q q qn u (22) where c A IC u c denotes the “ultimate AIC”. Simulation studies show that the model selection characteristics of A IC falls somewhere between A IC and c A IC 1100yi . 4. Numerical Simulation The simulation data consists of (reali- i zations of N(−13.0, 42)) and * 1 100yi 0 i ˆ a2 ˆ (realiza- tions of N(−13.0, 42)). , , , and are expressed as follows: RSS * RSS 2 0 1 1 ˆˆ n i i ay n,,RSSn 2 ** 00 11 ˆˆ ,, nn ii y a (23) 2 ii RSSy aRSS (24) Figure 1. Relationship between and average ˆˆ *2 0 2, i lya . A circle indicates the minimum point of each line. Ten lines reflect 10 repeats of the simulations. 100n where . By altering the seed of random values, 5000 sets of i y and * i y are obtained. Then, 5000 *2 0 ˆˆ 2, i lya values of are obtained and averaged. This procedure is carried out using one of the values 9.8, 9.6, 9.4,,10 as . Figure 1 shows the result of this simulation. Ten lines show that the simulation is repeated 10 times by chang- ing the seed of random values. Each minimum point is located around the 4 point; these ten points appar- ently deviate from the 1 and 0 4 points. This shows that gives a better log-likelihood in light of future data and that the third variance should be con- sidered. 5. Conclusion A The error variance for IC u is adjustable. The optimiza- tion of the errror variance yields c A IC in which the third variance is adopted as the error variance. The third variance is different from both the unbiased estimator of error variance and the maximum-likelihood estimator of error variance. The features and usage of the third vari- ance remains to be elucidated. REFERENCES [1] K. P. Burnham and D. R. Anderson, “Model Selection and Multi-Model Inference A Practical Information- Theoretic Approach,” Springer, Berlin, 2010. [2] S. Konishi and G. Kitagawa, “Information Criteria and Statistical Modeling,” Springer, Berlin, 2007. [3] C. M. Hurvich and C.-L. Tsai, “Regression and Time Series Model Selection in Small Samples,” Biometrika, Vol. 76, No. 2, 1989, pp. 297-307. doi:10.1093/biomet/76.2.297 Copyright © 2012 SciRes. OJS |
