# Two classes of almost unbiased type principal component estimators in linear regression model.

1. Introduction Consider the following multiple linear regression model: y = X[beta] + [epsilon], (1) where y is an n x 1 vector of responses, X is an n x p known design matrix of rank p, [beta] is a p x 1 vector of unknown parameters, [epsilon] is an n x 1 vector of disturbances assumed to be distributed with mean vector 0 and variance covariance matrix [[sigma].sup.2][I.sub.n], and [I.sub.n] is an identity matrix of order n.According to the Gauss-Markov theorem, the ordinary least squares estimate (OLSE) of (1) is obtained as follows:

[??] = [(X' X).sup.-1] X' y. (2)

It has been treated as the best estimator for a long time. However, many results have proved that the OLSE is no longer a good estimator when the multicollinearity is present. To overcome this problem, many new biased estimators have been proposed, such as principal components regression estimator (PCRE) [1], ridge estimator [2], Liu estimator [3], almost unbiased ridge estimator [4], and the almost unbiased Liu estimator [5].

To hope that the combination of two different estimators might inherit the advantages of both estimators, Kaciranlar et al. [6] improved Liu's approach and introduced the restricted Liu estimator. Akdeniz and Erol [7] compared some biased estimators in linear regression in the mean squared error matrix (MSEM) sense. By combining the mixed estimator and Liu estimator, Hubert and Wijekoon [8] obtained the two-parameter estimator which is a general estimator including the OLSE, ridge estimator, and Liu estimator. Baye and Parker [9] proposed the r - k class estimator which includes as special cases the PCRE, the RE, and the OLSE. Then, Kaciranlar and Sakallioglu [10] proposed the r - d estimator which is a generalization of the OLSE, PCRE, and Liu estimator. Based on the r-k estimator and r-d estimator, Xu and Yang [11] considered the restricted r - k estimator and restricted r - d estimator and Wu and Yang [12] introduced the stochastic restricted r - k estimator and the stochastic restricted r - d estimator, respectively.

The primary aim in this paper is to introduce two new classes of estimators where one includes the OLSE, PCRE, and AURE as special cases and the other one includes the OLSE, PCRE, and AULE as special cases and provide some alternative methods to overcome multicollinearity in linear regression.

The paper is organized as follows. In Section 2, the new estimators are introduced. In Section 3, some properties of the new estimator are discussed. Then we give a Monte Carlo simulation in Section 4. Finally, some conclusions are given in Section 5.

2. The New Estimators

In the linear model given by (1), the almost unbiased ridge estimator (AURE) proposed by Singh et al. [4] and the almost unbiased Liu estimator (AULE) proposed by Akdeniz and Kaciranlar [5] are defined as

[[??].sub.AU](k) = [(I - [k.sup.2](S + kI).sup.-2])[??], (3)

[[??].sub.AULE(d)] = (I - [(1 - d).sup.2][(S + I).sup.-2])[??], (4)

respectively, where k > 0, 0 < d < 1, S = X' X.

Now consider the spectral decomposition of the matrix given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [[LAMBDA].sub.r] = diag([[lambda].sub.1], ..., [[lambda].sub.r)], [[LAMBDA].sub.p-r] = diag([[lambda].sub.r+1], ..., [[lambda].sub.p-r]) and [[lambda].sub.1] [greater than or equal to] [[lambda].sub.2] [greater than or equal to] ... [greater than or equal to] [[lambda].sub.p] > 0 are the ordered eigenvalues of S. The matrix T = [([T.sub.r]: [T.sub.p-r]).sub.pxp] is orthogonal with [T.sub.r] = ([t.sub.1], ..., [t.sub.r]) consisting of its first r columns and [T.sub.p-r] = ([t.sub.r+1], ..., [t.sub.p]) consisting of the remaining p - r columns of the matrix T. Then [T'.sub.r][ST.sub.r] = [[LAMBDA].sub.r]; the PCRE of [beta] can be written as

[[??].sub.r] = [T.sub.r][([T'.sub.r][ST.sub.r]).sup.-1][T.sub.r]X' y = [T.sub.r][[LAMBDA].sup.-1.sub.r][T.sub.r]X' y. (6)

The r - k class estimator proposed by Baye and Parker [9] and the r - d class estimator proposed by Kaciranlar and Sakallioglu [10] are defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Followed by Xu and Yang [11], the r - k class estimator and r - d class estimator can be rewritten as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [??](k) = T[([LAMBDA] + kI).sup.-1]T'X' y = [(S + kI).sup.-1] X' y is the ridge estimator by Hoerl and Kennard [2] and [??](d) = T[([LAMBDA] + I).sup.-1](I + d[[LAMBDA].sup.-1])T'X' y = [(S +I).sup.-1] (I + d[S.sup.-1])X' y is the

Liu estimator proposed by Liu [3].

Now, we are to propose two new estimator classes by combining the PCRE with the AURE and AULE, that is, the almost unbiased ridge principal components estimator (AURPCE) and the almost unbiased Liu estimator principal component estimator (AULPCE), as follows:

[[??].sub.AU](r,k) = [T.sub.r][T'.sub.r](I - [k.sup.2][(S + kI).sup.-2]) [??] = [T.sub.r][T'.sub.r][G.sub.k][??], (9)

[[??].sub.AU](r,d) = [T.sub.r][T'.sub.r](I - [(1 - d).sup.2][(S + I).sup.-2] [??] = [T.sub.r][T'.sub.r][H.sub.d][??], (10)

respectively, where [G.sub.k] = I - [k.sup.2][(S + kI).sup.-2], [H.sub.d] = I - [(1 - d).sup.2] [(S + I).sup.-2].

From the definition of the AURPCE, we can easily obtain the following.

If r = p, then [[??].sub.AU](r, k) = [[??].sub.AU](k), AURE.

If k = 0, r = p, then [[??].sub.SRAU](r, k) = [??], OLSE.

If k = 0, then [[??].sub.AU](r, k) = [??](r) = [T.sub.r][T'.sub.r][??], PCRE.

From the definition of the SRAULPCE, we can similarly obtain the following.

If r = p, then [[??].sub.AU](r, d) = [[??].sub.AU](d), AULE.

If d = 0, r = p, then [[??].sub.AU](r, d) = [??], OLSE.

If d = 0, then [[??].sub.AU](r, d) = [T.sub.r][T'.sub.r][??], PCRE.

So the [[??].sub.AU](r,k) could be regarded as a generalization of PCRE, OLSE, and AURE, while [[??].sub.AU](r, d) could be regarded as a generalization of PCRE, OLSE, and AULE.

Furthermore, we can compute that the bias, dispersion matrix, and mean squared error matrix of the new estimators [[??].sub.AU](r,k) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

MSEM ([[??].sub.AU](r,k)) = [[sigma].sup.2][T.sub.r][T'.sub.r][G.sub.k] [S.sup.-1][G'.sub.k][T.sub.r][T'.sub.r] + ([T.sub.r][T'.sub.r][G.sub.k] - I)[beta][beta]'([T.sub.r][T'.sub.r][G.sub.k] - I)', (12) respectively.

In a similar way, we can get the MSEM of the [[??].sub.AU](r, d) as follows:

MSEM ([[??].sub.AU](r, d)) = [[sigma].sup.2][T.sub.r][T'.sub.r][H.sub.d] [S.sup.-1][H'.sub.d][T.sub.r][T'.sub.r] + ([T.sub.r][T'.sub.r][H.sub.d] -I)[beta][beta]'([T.sub.r][T'.sub.r][h.sub.d] - I)' (13)

In particular, if we let r = p in (12) and (13), then we can get the MSEM of the AURE and AULE as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

3. Superiority of the Proposed Estimators

For the sake of convenience, we first list some notations, definitions, and lemmas needed in the following discussion. For a matrix M, M', [M.sup.+], rank(M), R{M), and N(M) stand for the transpose, Moore-Penrose inverse, rank, column space, and null space, respectively. M [greater than or equal to] 0 means that M is nonnegative definite and symmetric.

Lemma 1. Let [C.sub.nxp] be the set of n x p complex matrices, let [H.sub.nxn] be the subset of [C.sub.nxp] consisting of Hermitian matrices, and L [member of] [C.sub.nxp], [L.sup.*], M(L), and J(D) stand for the conjugate transpose, the range, and the set of all generalized inverses, respectively. Let D [member of] [H.sub.nxn], [a.sub.1] and [a.sub.2] [member of] [C.sub.nx1] be linearly independent, [f.sub.ij] = [a.sup.*.sub.i] [D.sup.-] [a.sub.j], i, j = 1, 2, and if [a.sub.2] [not member of] M(D), let s = [[a.sup.*.sub.1][(I - D[D.sup.-]).sup.*] (I - D[D.sup.-])[a.sub.2]]/[[a.sup.*.sub.1][(I - D[D.sup.-]).sup.*](I - D[D.sup.-])[a.sub.1]].

Then D + [a.sub.1][a.sup.*.sub.1] - [a.sub.2][a.sup.*.sub.2] [greater than or equal to] 0 if and only if one of the following sets of conditions holds:

(a) D [greater than or equal to] > 0, [a.sub.i] [member of] M(D), i = 1, 2, ([f.sub.11] + 1)([f.sub.22] - 1) [less than or equal to] [[absolute value of [f.sub.12].sup.2];

(b) D [greater than or equal to] 0, [a.sub.1] [not member of] M(D), [a.sub.2] [member of] M(D[??][a.sub.1]), [([a.sub.2] - [sa.sub.1]).sup.*][D.sup.-] ([a.sub.2] - [sa.sub.1]) [less than or equal to] 1 - [[absolute value of s].sup.2];

(c) D = U[DELTA][U.sup.*]-[lambda]v[v.sup.*], [a.sub.i] [member of] M(D), i = 1, 2, [v.sup.*] [a.sub.1] [not equal to] 0, [f.sub.11] + 1 [less than or equal to] 0, [f.sub.22] - 1 [less than or equal to] 0, ([f.sub.11] + 1)([f.sub.22] - 1) [less than or equal to] [[absolute value of [f.sub.12].sup.2],

where (U[??]v) is a subunitary matrix (U possibly absent), [DELTA] a positive-definite diagonal matrix (occurring when U is present), and X a positive scalar. Further, all expressions in (a), (b), and (c) are independent of the choice of [D.sup.-] [member of] J(D).

Proof. Lemma 1 is due to Baksalary and Trenkler [13].

Let us consider the comparison between the AURPCE and AURE and the AULPCE and AULE, respectively. From (12)-(14), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where [D.sub.1] = [[sigma].sup.2]([G.sub.k][S.sup.-1] [G'.sub.k] - [T.sub.r][T'.sub.r][G.sub.k] [S.sup.-1][G'.sub.k][T.sub.r][T'.sub.r]), [D.sub.2] = [[sigma].sup.2] ([H.sub.d][S.sup.-1][H'.sub.d] - [T.sub.r][T'.sub.r][H.sub.d] [S.sup.-1][H'.sub.d][T.sub.r][T'.sub.r]) and [b.sub.1] = ([G.sub.k] - I)[beta], [b.sub.2] = ([T.sub.r][T'.sub.r][G.sub.k] - I) [beta], [b.sub.3] = ([H.sub.d] - I)[beta], [b.sub.2] = ([T.sub.r][T'.sub.r][H.sub.d] - I)[beta].

Now, we will use Lemma 1 to discuss the differences [[DELTA].sub.1] and [[DELTA].sub.2] following Sarkar [14] and Xu and Yang [11]. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

we assume that [T'.sub.r][S.sup.-1][T.sub.p-r] = 0 and [T'.sub.p-r] [S.sup.-1][T.sub.p-r] is invertible; then

[S.sup.-1] = [T.sub.r][T'.sub.r][S.sup.-1][T.sub.r][T'.sub.r] + [T.sub.p-r][T'.sub.p-r][S.sup.-1][T.sub.p-r][T'.sub.p-r]. (17)

Meanwhile, it is noted that the assumptions are reasonable which is equivalent to the partitioned matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], that is, a block diagonal matrix and the second main diagonal being invertible.

Theorem 2. Suppose that [T'.sub.r][S.sup.-1][T.sub.p-r] = 0 and [T'.sub.p-r][S.sup.-1][T.sub.p-r] is invertible; then the AURPCE is superior to the AURE if and only if [beta] [member of] N(F), where F = [[sigma].sup.-1] [([T'.sub.p-r][S.sup.-1][T.sub.p-r]).sup.-1/2] [T'.sub.p-r].

Proof. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

And the Moore-Penrose inverse [D.sup.+.sub.1] of [D.sub.1] is

[D.sup.+.sub.1] = [[sigma].sup.-2][T.sub.p-r](I - [k.sup.2][([[LAMBDA].sub.p-r] + kI).sup.-2])([T'.sub.p-r][S.sup.-1][T.sub.p-r] x (I - [k.sup.2] [([[LAMBDA].sub.p-r] + kI).sup.-2])[T'.sub.p-r]. (20)

Note that [D.sub.1][D.sup.+.sub.1] = [T.sub.p-r][T'.sub.p-r] = I - [T.sub.r][T'.sub.r], I - [k.sup.2][([[LAMBDA].sub.p-r] + kI).sup.-2], is a positive definition matrix since [[LAMBDA].sub.p-r] supposed to be invertible and [D.sub.1][D.sup.+.sub.1][a.sub.1] [not equal to] [a.sub.1], so [a.sub.1] [not member of] M(D). Moreover,

[b.sub.2] - [b.sub.1] = -[T.sub.p-r][I - [k.sup.2][([[LAMBDA].sub.p-r] + kI).sup.-2])[T'.sub.p-r][beta] = [D.sub.1][[eta].sub.1], (21)

where [[eta].sub.1] = -[[sigma].sup.-2][T.sub.p-r] [(I -[k.sup.2][([[LAMBDA].sub.p-r] + kI).sup.-2]).sup.-1] [([T'.sub.p-r][S.sup.- 1][T.sub.p-r]).sup.-1][T'.sub.p-r][beta]. This implies that [b.sub.2] [member of] M([D.sub.1][??][b.sub.1]): So the conditions of part (b) in Lemma 1 can be employed. Since (I - [DD.sup.-])' (I - [DD.sup.-]) = [T.sub.r][T'.sub.r][T.sub.r][T'.sub.r] = [T.sub.r] [T'.sub.r] and [T'.sub.r][b.sub.2] = [T'.sub.r][b.sub.1], it is concluded that s = 1 in our case. Thus, it follows from Lemma 1 that the [[??].sub.AU](r, k) is superior to [[??].sub.AU](k) in the MSEM sense if and only if ([b.sub.2] - [b.sub.1])'[D.sup.-.sub.1]([b.sub.2] - [b.sub.1]) = [[eta]'[D'.sub.1][D.sup.-.sub.1][D.sub.1][[eta].sub.n] = [[eta]'.sub.1][D.sub.1][[eta].sub.1] [less than or equal to] 0.

Observing that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

where F = [[sigma].sup.-1][([T'.sub.p-r][S.sup.-1][T.sub.p-r]).sup.-1/2] [T'.sub.p-r], thus the necessary and sufficient condition turns out to be [beta] [member of] N(F).

Theorem 3. Suppose that [T'.sub.r][S.sup.-1][T.sub.p-r] = 0 and [T'.sub.r][S.sup.-1][T.sub.p-r] is invertible; then the new estimator AULPCE is superior to the AULE if and only if [beta] [member of] N(F), where F = [[sigma].sup.-1] [([T'.sub.r][S.sup.- 1][T.sub.p-r].sup.-1/2][T'.sub.p-r].

Proof. In order to apply Lemma 1, we can similarly compute that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Therefore, the Moore-Penrose inverse [D.sup.+.sub.2] of [D.sub.2] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Since = [D.sub.2][D.sup.+.sub.2] = [T.sub.p-r][T'.sub.p-r] then [b.sub.3] [not member of] M(D). Moreover,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where [[eta].sub.1] = -[[sigma].sup.-2][T.sub.p-r][(I - [(1 - d).sup.2][([[LAMBDA].sub.p-r] + I).sup.-2]).sup.-1] [([T'.sub.p-r][S.sup.-1][T.sub.p-r]).sup.-1][T'.sub.p-r][beta]. This implies that [b.sub.4] [member of] m([D.sub.2][??][b.sub.3]). So the conditions of part (b) in Lemma 1 can be employed. Since (I - [DD.sup.-])'(I - [DD.sup.-]) = [T.sub.r][T'.sub.r][T.sub.r][T'.sub.r] = [T.sub.r][T'.sub.r] and [T'.sub.r][b.sub.4] = [T'.sub.r][b.sub.3], it is concluded that s = 1 in our case. Thus, it follows from Lemma 1 that the [[??].sub.AU](r, d) is superior to [[??].sub.AU](d) in the MSEM sense if and only if ([b.sub.4] - [b.sub.3])'[D.sup.- .sub.2]([b.sub.4] - [b.sub.4]) = [[eta]'.sub.2][D'.sub.2][[D.sup.-.sub.2][D.sub.2][[eta].sub.2] = [[eta]'.sub.2] [[D'.sub.2][[eta].sub.2] [less than or equal to] 0: Observing that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

where F = [[sigma].sub.-1][([T'.sub.p-r][S.sup.-1][T.sub.p-r]).sup.-1/2] [T'.sub.p-r], thus the necessary and sufficient condition turns out to be [beta] [member of] N(F).

4. Monte Carlo Simulation

In order to illustrate the behaviour of the AURPCE and AULPCE, we perform a Monte Carlo simulation study. Following the way of Li and Yang [15], the explanatory variables and the observations on the dependent variable are generated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where [[omega].sub.ij] are independent standard normal pseudorandom numbers and [gamma] is specified so that the correlation between any two explanatory variables is given by [[gamma].sub.2]. In this experiment, we choose r = 2 and [[sigma].sup.2] = 1. Let us consider the AURPCE, AULPCE, AURE, AULE, PCRE, and OLSE and compute their respective estimated MSE values with the different levels of multicollinearity, namely, [gamma] = 0.7, 0.85, 0.9, 0.999 to show the weakly, strong, and severely collinear relationships between the explanatory variables (see Tables 1 and 2). Furthermore, for the convenience of comparison, we plot the estimated MSE values of the estimators when [gamma] = 0.999 in Figure 1.

From the simulation results shown in Tables 1 and 2 and the estimated MSE values of these estimators, we can see that for most cases, the AURPCE and AULPCE have smaller estimated MSE values than those of the AURE, AULE, PCRE, and OLSE, respectively, which agree with our theoretical findings. From Figure 1, the AURPCE and AULPCE also have more stable and smaller estimated MSE values. We can see that our estimator is meaningful in practice.

5. Conclusion

In this paper, we introduce two classes of new biased estimators to provide an alternative method of dealing with multicollinearity in the linear model. We also show that our new estimators are superior to the competitors in the MSEM criterion under some conditions. Finally, a Monte Carlo simulation study is given to illustrate the better performance of the proposed estimators.

http://dx.doi.org/10.1155/2014/639070

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11201505) and the Fundamental Research Funds for the Central Universities (no. 0208005205012):

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Yalian Li and Hu Yang

Department of Statistics and Actuarial Science, Chongqing University, Chongqing 401331, China

Correspondence should be addressed to Yalian Li; yaliancn@gmail.com

Received 15 January 2014; Accepted 8 March 2014; Published 2 April 2014

Academic Editor: Li Weili

TABLE 1: MSE values of the OLSE, PCRE, AURE, and AURPCE. k 0.00 0.10 0.30 0.40 0.50 [gamma] = 0.7 OLSE 0.0619 0.0619 0.0619 0.0619 0.0619 PCRE 0.0285 0.0285 0.0285 0.0285 0.0285 AURE 0.0619 0.0619 0.0619 0.0619 0.0619 AURPCE 0.0285 0.0285 0.0285 0.0285 0.0285 [gamma] = 0.85 OLSE 0.1085 0.1085 0.1085 0.1085 0.1085 PCRE 0.0384 0.0384 0.0384 0.0384 0.0384 AURE 0.1085 0.1085 0.1085 0.1085 0.1085 AURPCE 0.0384 0.0384 0.0384 0.0383 0.0383 [gamma] = 0.99 OLSE 1.4636 1.4636 1.4636 1.4636 1.4636 PCRE 0.3522 0.3522 0.3522 0.3522 0.3522 AURE 1.4636 1.4565 1.4116 1.3797 1.3441 AURPCE 0.3522 0.3515 0.3464 0.3426 0.3381 [gamma] = 0.999 OLSE 14.5437 14.5437 14.5437 14.5437 14.5437 PCRE 3.3903 3.3903 3.3903 3.3903 3.3903 AURE 14.5437 1.4399 6.0117 4.5727 3.5858 AURPCE 3.3903 2.9735 1.8963 1.5285 1.2518 k 0.80 0.90 1.00 OLSE 0.0619 0.0619 0.0619 PCRE 0.0285 0.0285 0.0285 AURE 0.0619 0.0619 0.0618 AURPCE 0.0285 0.0285 0.0285 OLSE 0.1085 0.1085 0.1085 PCRE 0.0384 0.0384 0.0384 AURE 0.1084 0.1084 0.1083 AURPCE 0.0383 0.0383 0.0383 OLSE 1.4636 1.4636 1.4636 PCRE 0.3522 0.3522 0.3522 AURE 1.2281 1.1889 1.1502 AURPCE 0.3220 0.3161 0.3101 OLSE 14.5437 14.5437 14.5437 PCRE 3.3903 3.3903 3.3903 AURE 1.9800 1.6797 1.4430 AURPCE 0.7514 0.6496 0.5673 TABLE 2: MSE values of the OLSE, PCRE, AULE, and AULPCE. d 0.00 0.10 0.20 0.40 [gamma] = 0.7 OLSE 0.0709 0.0709 0.0709 0.0709 PCRE 0.0303 0.0303 0.0303 0.0303 AULE 0.0709 0.0709 0.0709 0.0709 AULPCE 0.0303 0.0303 0.0303 0.0303 [gamma] = 0.85 OLSE 0.1085 0.1085 0.1085 0.1085 PCRE 0.0384 0.0384 0.0384 0.0384 AULE 0.1083 0.1083 0.1084 0.1084 AULPCE 0.0383 0.0383 0.0383 0.0383 [gamma] = 0.99 OLSE 1.4636 1.4636 1.4636 1.4636 PCRE 0.3522 0.3522 0.3522 0.3522 AULE 1.1502 1.2066 1.2583 1.3461 AULPCE 0.3101 0.3179 0.3249 0.3367 [gamma] = 0.999 OLSE 14.5437 14.5437 14.5437 14.5437 PCRE 3.3903 3.3903 3.3903 3.3903 AULE 1.4430 2.8578 4.5509 8.2191 AULPCE 0.5673 0.9193 1.3076 2.0980 d 0.50 0.70 0.90 1.00 OLSE 0.0709 0.0709 0.0709 0.0709 PCRE 0.0303 0.0303 0.0303 0.0303 AULE 0.0709 0.0709 0.0709 0.0709 AULPCE 0.0303 0.0303 0.0303 0.0303 OLSE 0.1085 0.1085 0.1085 0.1085 PCRE 0.0384 0.0384 0.0384 0.0384 AULE 0.1085 0.1085 0.1085 0.1085 AULPCE 0.0383 0.0384 0.0384 0.0384 OLSE 1.4636 1.4636 1.4636 1.4636 PCRE 0.3522 0.3522 0.3522 0.3522 AULE 1.3814 1.4337 1.4603 1.4636 AULPCE 0.3414 1.3483 0.3518 0.3522 OLSE 14.5437 14.5437 14.5437 14.5437 PCRE 3.3903 3.3903 3.3903 3.3903 AULE 9.9597 12.7929 14.3436 14.5437 AULPCE 2.4599 3.0381 3.3502 3.3903

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Title Annotation: | Research Article |
---|---|

Author: | Li, Yalian; Yang, Hu |

Publication: | Journal of Applied Mathematics |

Article Type: | Report |

Date: | Jan 1, 2014 |

Words: | 4294 |

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