# Two-Parameter Modified Ridge-Type M-Estimator for Linear Regression Model

Authors: Adewale F. Lukman; Kayode Ayinde; B. M. Golam Kibria; Segun L. Jegede
Journal: The Scientific World Journal
 (2020)

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## Abstract

The general linear regression model has been one of the most frequently used models over the years, with the ordinary least squares estimator (OLS) used to estimate its parameter. The problems of the OLS estimator for linear regression analysis include that of multicollinearity and outliers, which lead to unfavourable results. This study proposed a two-parameter ridge-type modified M-estimator (RTMME) based on the M-estimator to deal with the combined problem resulting from multicollinearity and outliers. Through theoretical proofs, Monte Carlo simulation, and a numerical example, the proposed estimator outperforms the modified ridge-type estimator and some other considered existing estimators.


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## Body

## 1. Introduction

A multiple linear regression model can be defined mathematically as(1)y=Xβ+ε,where y is an n × 1 vector of observations referred to as the dependent variable; X is a known full column rank of n × pstandardized and centered explanatory variable matrix; β is an p × 1 vector of unknown parameters; and ε is an n × 1 vector of disturbances with Eε=0 and dispersion matrix Covε=σ2I, I is the n × n identity matrix. The ordinary least squares estimator (OLSE) of β is given as(2)β^=X′X−1X′y.According to Gauss–Markov theorem, OLS estimator is the best linear unbiased estimator (BLUE) possessing minimum variance in the class of all unbiased linear estimators [1, 2]. However, the performance of the estimator is imprecise in the presence of multicollinearity [3]. Biased estimators such as ridge regression estimator [4], Liu estimator [5] and Stein estimator [6], principal component estimator [7], modified ridge regression estimator [8], and others are often employed to tackle this problem. Another factor whose presence can negatively influence the regression coefficients of the OLS estimator is the outlier. The general practice in the literature is that one adopts robust estimators as an alternative to the OLS estimator. The M-estimator is popularly used to handle outlier in the y-direction [9].Hoerl and Kennard [4] defined the ridge estimator (RE) as(3)β^k=X′X+kI−1X′y=Rkβ^,where Rk=X′X+kI−1X′X and k>0. However, RE can be sensitive to outliers in the y-direction; a remedial measure is the ridge M-estimator (RME) suggested by Silvapulle [10] as(4)β^Mk=Rkβ^M,where β^M is the M-estimator of β [11].Dorugade [12] modified the ridge estimator and defined it as(5)β^Dk=Rkd1β^,whereRkd1=X′X+kdI−1X′X and d introduced as an additional basing parameter.Following Dorugade [12], Lukman et al. [13] modified the ridge estimator in (3) and called it the modified ridge-type estimator (MRT). The estimator is defined as(6)β^MRTk,d=X′X+k1+dI−1X′Y=Rkdβ^,whereRkd=X′X+k1+dI−1X′X.The organization of the paper is as follows. We proposed the new estimator in Section2 and provided a theoretical comparison among the estimators in Section 3. We discussed the robust choice of the biasing parameters in Section 4 and conducted simulation studies in Section 5 to evaluate the performance of the proposed estimator. A real-life data set is analyzed in Section 6 to illustrate the findings in the paper, and Section 7 ends with some concluding remarks.


## 2. A New Estimator

The presence of outliers in they-direction affects the performance of the MRT estimator. Therefore, we suggest a ridge-type modified M-estimator (RTMME). This is defined as(7)β^RTMMEk,d=Rkdβ^M,where k>0,0<d<1. It appears that β^RTMMEk,d is a general estimator, which includes β^M and β^Mk:(8)β^RTMME0,0=β^M,β^RTMMEk,0=β^Mk.The canonical form of model (1) is written as(9)y=zα+ε,where Z=XT,α=T′β, and T is the orthogonal matrix whose columns contain the eigenvectors of X′X. Then,(10)Z′Z=T′X′XT=Λ=diagλ1,λ2,…,λp,where λ1,λ2,…,λp>0 are the ordered eigenvalues of X′X.Letα^M be M-defined by the solution of the M-estimating equations ∑​φei/szi=0, where ei=yi−ziα^M,s is an estimator of scale for the errors and φ⋅ is some suitably chosen function [14]. The estimators presented in equations (2)–(7) can be written in canonical form as follows:(11)α^=Λ−1Z′y,α^k=Λ+kI−1Z′y=Rk∗α^,α^Mk=Rk∗α^M,α^MRTk,d=Λ+k1+dI−1Z′y,α^MRTk,d=Rkdα^,α^RTMMEk,d=Rkdα^M,where Rkd=ΛΛ+k1+dI−1, Rk∗=ΛΛ+kI−1, and k>0.


## 3. Superiority of the New Estimator

The mean square error (MSE) criterion is used to compare the performance of the estimators. The following conditions are imposed to present our main theorems:(i)
φ is skew-symmetric and nondecreasing(ii)
The errors are symmetric(iii)
Ω is finiteNote that any estimator ofα has a corresponding relation β˜=T′α˜ such that MSEβ˜=MSEα˜. Thus, it is sufficient to consider the canonical form only. The MSEs of the aforementioned estimators are derived to be(12)MSEα^=σ2∑i=1p1λi,MSEα^M=∑i=1pΩii,MSEα^k=σ2∑i=1pλiλi+k2+∑i=1pk2αi2λi+k2,MSEα^Mk=∑i=1pλi2λi+k2Ωii+∑i=1pk2αi2λi+k2,MSEα^MRTk,d=σ2∑i=1pλiλi+k1+d2+∑i=1pk21+d2αi2λi+k1+d2,MSEα^RTMMEk,d=∑i=1pλi2λi+k1+d2Ωii+∑i=1pk21+d2αi2λi+k1+d2,where Ωii=Covα^M.Theorem 1.
ForMSEα^RTMMEk,d<MSEα^MRTk,d, then ∑i=1pΩii<∑i=1pσ2λi−1 for every k > 0 and i = 1, 2, …, p, where Ωii are the diagonal elements of Ω.Proof.
After some algebraic manipulation, the difference betweenMSEα^RTMMEk,d−MSEα^MRTk,d gives(13)Δ1=MSEα^RTMMEk,d−MSEα^MRTk,d,(14)Δ1=∑i=1pλi2Ωii−σ2λiλi+k1+d2.
For equation (14) to be less than zero, we should have Ωii−σ2λi−1<0, which also implies ∑i=1pΩii<∑i=1pσ2λi−1 for k>0,0<d<1 and for i = 1, 2, …, p.Theorem 2.
WhenMSEα^RTMMEk,d<MSEα^Mk, there exists a positive constant k>ki>0, where(15)k1i=λid+2αi2−Ωii2+8αi2Ωiid+1−d+2αi2−Ωii4αi2d+1.Proof.
The  difference MSEα^RTMMEk,d−MSEα^Mkis:(16)Δ2=∑i=1pλi2Ωii+k21+d2αi2λi+k1+d2−∑i=1pλi2Ωii+k2αi2λi+k2.
The difference is strictly less than zero if and only if, after simplification, the following expression holds:(17)k22αi2λid+kλi2dd+2αi2−Ωii−2Ωiiλi3d<0.
Solving inequality (17) for k, we get(18)k1i=λid+2αi2−Ωii2+8αi2Ωiid+1−d+2αi2−Ωii4αi2d+1.
Notice that ifλid+2αi2−Ωii2+8αi2Ωiid+1−d+2αi2−Ωii>0;k1i>0, and there is, therefore, a positive constant k>k1i>0.Theorem 3.
A necessary condition forMSEα^RTMMEk,d<MSEα^M is(19)∑i=1pΩii>∑i=1pk1+dαi22λi+k1+d.Proof.
The difference MSEα^RTMMEk,d−MSEα^Mis(20)Δ3=∑i=1pλi2Ωii+k21+d2αi2λi+k1+d2−∑i=1pΩii,(21)Δ3=−Ωii2λik1+d+k21+d2+k21+d2αi2.
To obtainMSEα^RTMMEk,d<MSEα^M,(22)∑i=1pΩii<∑i=1pk1+dαi22λi+k1+d.
Theorem2 provided k>k1i such that MSEα^RTMMEk,d<MSEα^M. Besides this, in the Theorem of Silvapulle [10] (part (i), pg. 321), it is indicated that there exists k>0 such that MSEα^Mk−MSEα^M<0. Thus, we obtain the corollary as follows.Corollary.
There existsk > 0 such thatMSEα^RTMMEk,d−MSEα^M.


## 4. Robust Choice ofk and d for α^RTMMEk,d

For the robust biasing parametersk and d for the modified two-parameter estimator, the optimal values can be determined by minimizing equation (23) with respect to each of the parameters:(23)fk,d=∑i=1pλi2λi+k1+d2Ωii+∑i=1pk21+d2αi2λi+k1+d2.This can be obtained by solving(24)∂fk,d∂d=0,∂fk,d∂k=0.By doing this, we have(25)d=λiΩiikαi2−1,(26)k=λiΩiiαi21+d.We substituteΩii and αi2 into equations (24) and (25) with their corresponding estimates. We assume that α^M is normally distributed with mean α and covariance matrix A2Λ−1. This assumption holds since n1/2α^M2−α⟶N0,A2Λ−1, where(27)A2=so2Eφ2ε/soEφ′ε/so2,with the scale estimate so. Thus, the estimate of αi2isα^Mi2, and the unbiased estimator of Ωii is asymptotically A^2/λi, where A^2 is given by Huber [9] as(28)A^2=s2n−p−1∑i=1pφei/s2∑i=1p1/nφ′ei/s2.We get the optimal estimator ofd and k as(29)d=A^2kα^Mi2−1,i=1,2,…,p,(30)k=A^21+dα^Mi2,i=1,2,…,p.Following Kibria [15], the arithmetic and geometric mean version of k is obtained, respectively, as(31)k˜GMR=∏i=1pA^21+dα^Mi21/p,(32)k˜AMR=1p∑i=1pA^21+dα^Mi2The harmonic mean version is generally preferred to other versions [3]. Hence, the robust harmonic mean version of the proposed d and k from (31) and (32) is obtained as(33)k˜HMR=pA^2∑i=1p1+dα^Mi2,(34)d˜HMR=p∑i=1p1/d.The selection of the estimators of the parametersd and k can be obtained iteratively as follows:Step 1: used^=minA^2/α^Mi2 to obtain an initial estimate for dStep 2: from (33), get k˜HMR using d in Step 1Step 3: calculated˜HMR in (34) by using k˜HMR in Step 2Step 4: used^ in Step 1 if d˜HMR < 0


## 5. Monte Carlo Simulation Study

We adopted the simulation design by McDonald and Galarneau [16], Kibria [15], and Lukman et al. [17]. The explanatory variables are generated using the following equation:(35)xij=1−ρ21/2zij+ρzi,p+1,i=1,2,…,n,j=1,2,…,p,where ρ2 denotes the correlation between explanatory variables and zij are pseudo-random numbers from the standard normal distribution. The coefficients β1,β2,…,βp are selected as the normalized eigenvectors corresponding to the largest eigenvalue of X′X so that we have β′β=1, which is a common restriction in simulation studies of this type ([3]). The dependent variable is then determined using(36)yi=β0+β1xi1+β2xi2+⋯+βpxip+εi,i=1,2,…,n,where the error term εi′s is generated with mean and variance 0 and σ2, respectively. We fixed the number of explanatory variables to three and seven (p = 3, 7), and other parameters such as ρ, σ, and n were varied; their values considered in this study are given as follows:ρ=0.7,0.8,0.9 and 0.99The standard deviation (σ) of the error term in this simulation study is 1,5,10n=20,50,100We considered three different cases in this study:Case I: no outlierCase II: one outlierCase III: two outliersIn the case of no outliers, equation (36) is taken into consideration. For the case of one outlier, the tenth observation is changed as y10∗=y10+20σ. For the case of two outliers, the fifth and the tenth observations are changed as y5∗=y5+20σ and y10∗=y10−20σ, respectively. The experiment is replicated 2,000 times by generating new pseudo-random numbers, and the estimated MSE is calculated as(37)mseα^=12000∑j=12000α^ij−αi′α^ij−αi.The results of the simulation are presented in Tables1–18. As expected, the OLSE is observed to have the least performance. The following observations are also made:(i)
As the error standard deviation (σ) and the degree of multicollinearity (ρ) increase, the MSEs of the estimators (α,α^M,α^k,α^Mk,α^MRT,andα^RTMME) increase.(ii)
As the biasing parametersk and d increase, the MSEs of the estimators (α,α^M,α^k,α^Mk,α^MRT,andα^RTMME) also decrease.(iii)
The MSEs of the estimators (α,α^M,α^k,α^Mk,α^MRT, and α^RTMME) decrease as the sample size increases. However, as the number of outliers increases, the MSEs also increase.(iv)
Finally, just as in the outcome of Lukman et al. [13], the MRT estimator outperforms other estimators considered in the case of no outlier. However, when outliers were introduced, the RTMME outperforms other considered estimators.Table 1
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 20 and p = 3 when there is no outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.78.9734.8988.2084.4838.0694.4088.9734.8988.2084.4837.8674.2998.9734.8988.2084.4837.6754.1950.813.3747.51811.6646.54411.3666.37513.3747.51811.6646.54410.9436.13513.3747.51811.6646.54410.5475.9110.90.9450.9800.9300.9660.6810.7070.9450.9800.9300.9660.6350.6590.9450.9800.9300.9660.5950.6180.9911.19711.8069.35110.0001.3441.40411.19711.8069.35110.0001.0501.09411.19711.8069.35110.0000.8580.89250.70.7890.6230.7850.6200.7840.6190.7890.6230.7850.6200.7830.6180.7890.6230.7850.6200.7820.6170.81.1610.8931.1520.8861.1510.8841.1610.8931.1520.8861.1480.8821.1610.8931.1520.8861.1450.8800.90.9450.9800.9440.9790.9310.9650.9450.9800.9440.9790.9270.9620.9450.9800.9440.9790.9240.9580.9911.19711.80611.11211.7239.3779.88211.19711.80611.11211.7238.9989.48111.19711.80611.11211.7238.6449.107100.70.1790.1150.1790.1150.1790.1150.1790.1150.1790.1150.1790.1140.1790.1150.1790.1150.1790.1140.80.2590.1700.2590.1700.2590.1700.2590.1700.2590.1700.2580.1700.2590.1700.2590.1700.2580.1700.90.9450.9800.9440.9800.9410.9760.9450.9800.9440.9800.9400.9750.9450.9800.9440.9800.9390.9740.9911.19711.80611.17611.78510.69011.27011.19711.80611.17611.78510.56911.14211.19711.80611.17611.78510.45011.0160.710.78.9734.8987.3524.0207.0893.8788.9734.8987.3524.0206.7243.6828.9734.8987.3524.0206.3913.5030.813.3747.5189.9005.5459.3885.25613.3747.5189.9005.5458.7004.86913.3747.5189.9005.5458.0954.5290.90.9450.9800.9300.9660.4930.5110.9450.9800.9300.9660.4410.4570.9450.9800.9300.9660.4000.4150.9911.19711.8069.35110.0000.5420.55911.19711.8069.35110.0000.4440.45511.19711.8069.35110.0000.3860.39550.70.7890.6230.7800.6160.7780.6140.7890.6230.7800.6160.7750.6120.7890.6230.7800.6160.7720.6100.81.1610.8931.1410.8761.1370.8731.1610.8931.1410.8761.1310.8681.1610.8931.1410.8761.1250.8640.90.9450.9800.9440.9790.9120.9470.9450.9800.9440.9790.9050.9390.9450.9800.9440.9790.8970.9310.9911.19711.80611.11211.7237.6178.02211.19711.80611.11211.7237.0137.38411.19711.80611.11211.7236.4846.826100.70.1790.1150.1780.1140.1780.1140.1790.1150.1780.1140.1780.1140.1790.1150.1780.1140.1780.1140.80.2590.1700.2580.1700.2580.1690.2590.1700.2580.1700.2570.1690.2590.1700.2580.1700.2570.1690.90.9450.9800.9440.9800.9360.9710.9450.9800.9440.9800.9340.9690.9450.9800.9440.9800.9320.9670.9911.19711.80611.17611.78510.07110.61511.19711.80611.17611.7859.81810.34811.19711.80611.17611.7859.57610.0930.910.78.9734.8986.9813.8206.6743.6558.9734.8986.9813.8206.2573.4318.9734.8986.9813.8205.8853.2320.813.3747.5189.1825.1408.6094.81713.3747.5189.1825.1407.8584.39613.3747.5189.1825.1407.2134.0350.90.9450.9800.9300.9660.4340.4500.9450.9800.9300.9660.3850.3990.9450.9800.9300.9660.3480.3610.9911.19711.8069.35110.0000.4340.44511.19711.8069.35110.0000.3680.37611.19711.8069.35110.0000.3310.33750.70.7890.6230.7770.6140.7750.6120.7890.6230.7770.6140.7710.6090.7890.6230.7770.6140.7670.6060.81.1610.8931.1350.8721.1300.8681.1610.8931.1350.8721.1220.8621.1610.8931.1350.8721.1150.8560.90.9450.9800.9440.9790.9040.9380.9450.9800.9440.9790.8940.9270.9450.9800.9440.9790.8840.9180.9911.19711.80611.11211.7236.9337.29911.19711.80611.11211.7236.2786.60811.19711.80611.11211.7235.7206.019100.70.1790.1150.1780.1140.1780.1140.1790.1150.1780.1140.1780.1140.1790.1150.1780.1140.1780.1140.80.2590.1700.2580.1690.2570.1690.2590.1700.2580.1690.2570.1690.2590.1700.2580.1690.2570.1690.90.9450.9800.9440.9800.9340.9690.9450.9800.9440.9800.9310.9660.9450.9800.9440.9800.9290.9640.9911.19711.80611.17611.7859.78310.31111.19711.80611.17611.7859.4769.98611.19711.80611.17611.7859.1849.678Table 2
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 20 and p = 3 when there is one outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.714.0156.60312.8426.06612.6285.96814.0156.60312.8426.06612.3195.82714.0156.60312.8426.06612.0235.6910.821.0729.88818.3528.63317.8798.41421.0729.88818.3528.63317.2078.10421.0729.88818.3528.63316.5767.8130.9608.8181.152592.0461.140410.6940.853608.8181.152592.0461.140376.5930.800608.8181.152592.0461.140346.6580.7530.998184.09914.1325874.69512.202574.3201.6018184.09914.1325874.69512.202411.0881.2398184.09914.1325874.69512.202308.9361.00450.71.1250.9241.1200.9201.1190.9191.1250.9241.1200.9201.1180.9181.1250.9241.1200.9201.1160.9170.81.6391.2921.6281.2841.6261.2821.6391.2921.6281.2841.6231.2791.6391.2921.6281.2841.6201.2770.9122.5611.152122.4251.152120.4761.136122.5611.152122.4251.152119.9641.133122.5611.152122.4251.152119.4541.1290.991646.04914.1291622.59214.0431331.11711.7841646.04914.1291622.59214.0431266.59911.2961646.04914.1291622.59214.0431206.68810.841100.70.2030.1270.2030.1270.2030.1270.2030.1270.2030.1270.2030.1270.2030.1270.2030.1270.2030.1270.80.3060.1900.3060.1900.3060.1900.3060.1900.3060.1900.3050.1900.3060.1900.3060.1900.3050.1900.961.7681.15261.7511.15261.5061.14861.7681.15261.7511.15261.4411.14761.7681.15261.7511.15261.3761.1460.99828.72414.128825.75514.106784.25513.474828.72414.128825.75514.106773.70113.318828.72414.128825.75514.106763.36113.1650.710.714.0156.60311.5265.46411.1205.27914.0156.60311.5265.46410.5585.02214.0156.60311.5265.46410.0444.7870.821.0729.88815.5497.33914.7356.96321.0729.88815.5497.33913.6446.45921.0729.88815.5497.33912.6846.0160.9608.8181.152592.0461.140269.4080.631608.8181.152592.0461.140230.0190.568608.8181.152592.0461.140198.8520.5170.998184.09914.1325874.69512.202148.7370.6168184.09914.1325874.69512.202101.1430.4968184.09914.1325874.69512.20273.3640.42550.71.1250.9241.1140.9141.1120.9121.1250.9241.1140.9141.1080.9101.1250.9241.1140.9141.1050.9070.81.6391.2921.6141.2721.6101.2681.6391.2921.6141.2721.6031.2621.6391.2921.6141.2721.5961.2560.9122.5611.152122.4251.152117.7811.116122.5611.152122.4251.152116.6311.108122.5611.152122.4251.152115.4981.0990.991646.04914.1291622.59214.0431035.1019.5231646.04914.1291622.59214.043935.9958.7501646.04914.1291622.59214.043850.5508.075100.70.2030.1270.2030.1270.2030.1270.2030.1270.2030.1270.2030.1270.2030.1270.2030.1270.2020.1270.80.3060.1900.3050.1900.3050.1900.3060.1900.3050.1900.3040.1890.3060.1900.3050.1900.3040.1890.961.7681.15261.7511.15261.1601.14361.7681.15261.7511.15261.0091.14161.7681.15261.7511.15260.8591.1380.99828.72414.128825.75514.106730.36012.676828.72414.128825.75514.106708.52512.351828.72414.128825.75514.106687.66312.0400.910.714.0156.60310.9545.20310.4814.98714.0156.60310.9545.2039.8374.69214.0156.60310.9545.2039.2604.4290.821.0729.88814.4086.81213.4996.39221.0729.88814.4086.81212.3075.84221.0729.88814.4086.81211.2855.3700.9608.8181.152592.0461.140225.1220.560608.8181.152592.0461.140187.4510.499608.8181.152592.0461.140158.6950.4520.998184.09914.1325874.69512.20296.2780.4848184.09914.1325874.69512.20264.8980.4038184.09914.1325874.69512.20246.8360.35850.71.1250.9241.1110.9121.1080.9091.1250.9241.1110.9121.1040.9051.1250.9241.1110.9121.1000.9020.81.6391.2921.6081.2661.6021.2611.6391.2921.6081.2661.5931.2541.6391.2921.6081.2661.5841.2460.9122.5611.152122.4251.152116.4681.106122.5611.152122.4251.152115.0181.095122.5611.152122.4251.152113.5961.0850.991646.04914.1291622.59214.043923.0158.6481646.04914.1291622.59214.043817.4977.8121646.04914.1291622.59214.043729.1907.100100.70.2030.1270.2030.1270.2030.1270.2030.1270.2030.1270.2020.1270.2030.1270.2030.1270.2020.1260.80.3060.1900.3050.1900.3040.1890.3060.1900.3050.1900.3040.1890.3060.1900.3050.1900.3030.1890.961.7681.15261.7511.15260.9881.14061.7681.15261.7511.15260.7951.13761.7681.15261.7511.15260.6031.1340.99828.72414.128825.75514.106705.48712.306828.72414.128825.75514.106679.00511.910828.72414.128825.75514.106653.99811.535Table 3
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 20 and p = 3 when there are two outliers.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.721.04116.79819.13415.12218.78614.81921.04116.79819.13415.12218.28214.38121.04116.79819.13415.12217.80013.9640.829.86624.38926.04420.91325.37420.31229.86624.38926.04420.91324.42019.45929.86624.38926.04420.91323.52218.6610.9875.7952.113832.4432.065697.9501.534875.7952.113832.4432.065662.4291.430875.7952.113832.4432.065629.6861.3360.998864.95625.8225151.31619.5811588.5792.6848864.95625.8225151.31619.5811227.8301.9848864.95625.8225151.31619.581978.5041.53050.70.9090.8930.9040.8890.9040.8880.9090.8930.9040.8890.9020.8860.9090.8930.9040.8890.9010.8850.81.2761.2421.2671.2331.2651.2311.2761.2421.2671.2331.2631.2291.2761.2421.2671.2331.2601.2260.9176.0022.111175.6372.109174.2882.081176.0022.111175.6372.109173.8642.074176.0022.111175.6372.109173.4412.0670.991779.07125.8121732.19125.5031575.74321.5961779.07125.8121732.19125.5031531.29520.7161779.07125.8121732.19125.5031489.00919.893100.70.2180.1460.2180.1460.2180.1460.2180.1460.2180.1460.2180.1460.2180.1460.2180.1460.2180.1460.80.3320.2200.3320.2190.3320.2190.3320.2200.3320.2190.3320.2190.3320.2200.3320.2190.3310.2190.988.5062.11188.4602.11088.2882.10388.5062.11188.4602.11088.2342.10188.5062.11188.4602.11088.1792.1000.99894.05625.809888.02825.731866.22324.636894.05625.809888.02825.731859.50924.356894.05625.809888.02825.731852.88824.0820.710.721.04116.79816.99213.26716.33112.70121.04116.79816.99213.26715.41511.92121.04116.79816.99213.26714.57711.2140.829.86624.38922.05017.36520.87916.34329.86624.38922.05017.36519.29914.97729.86624.38922.05017.36517.89913.7820.9875.7952.113832.4432.065537.0791.089875.7952.113832.4432.065484.3070.958875.7952.113832.4432.065439.1890.8510.998864.95625.8225151.31619.581533.8810.7918864.95625.8225151.31619.581383.3110.5698864.95625.8225151.31619.581289.6370.44250.70.9090.8930.8990.8830.8970.8811.1250.9241.1140.9141.1080.9100.9090.8930.8990.8830.8910.8750.81.2761.2421.2551.2211.2511.2171.6391.2921.6141.2721.6031.2621.2761.2421.2551.2211.2401.2050.9176.0022.111175.6372.109172.0452.043176.0022.111175.6372.109171.0782.026176.0022.111175.6372.109170.1192.0100.991779.07125.8121732.19125.5031361.73917.4991779.07125.8121732.19125.5031283.34816.0891779.07125.8121732.19125.5031212.31614.853100.70.2180.1460.2180.1460.2180.1460.2180.1460.2180.1460.2180.1460.2180.1460.2180.1460.2180.1450.80.3320.2200.3310.2190.3310.2190.3320.2200.3310.2190.3310.2190.3320.2200.3310.2190.3300.2180.988.5062.11188.4602.11087.9992.09388.5062.11188.4602.11087.8722.08988.5062.11188.4602.11087.7472.0850.99894.05625.809888.02825.731831.46523.202894.05625.809888.02825.731817.03522.618894.05625.809888.02825.731803.04522.0560.910.721.04116.79816.06112.47015.29011.81621.04116.79816.06112.47014.24010.93021.04116.79816.06112.47013.30010.1450.829.86624.38920.40715.93319.08914.79729.86624.38920.40715.93317.34813.31529.86624.38920.40715.93315.84512.0540.9875.7952.113832.4432.065477.4310.941875.7952.113832.4432.065421.8250.812875.7952.113832.4432.065375.6650.7110.998864.95625.8225151.31619.581367.2500.5468864.95625.8225151.31619.581260.0490.4048864.95625.8225151.31619.581195.0020.32650.70.9090.8930.8960.8800.8940.8770.9090.8930.8960.8800.8900.8740.9090.8930.8960.8800.8870.8700.81.2761.2421.2501.2151.2451.2101.2761.2421.2501.2151.2371.2021.2761.2421.2501.2151.2301.1950.9176.0022.111175.6372.109170.9402.024176.0022.111175.6372.109169.7112.003176.0022.111175.6372.109168.4961.9830.991779.07125.8121732.19125.5031272.77615.9021779.07125.8121732.19125.5031183.87114.3701779.07125.8121732.19125.5031104.87513.061100.70.2180.1460.2180.1460.2180.1450.2180.1460.2180.1460.2180.1450.2180.1460.2180.1460.2170.1450.80.3320.2200.3310.2190.3310.2190.3320.2200.3310.2190.3300.2180.3320.2200.3310.2190.3300.2180.988.5062.11188.4602.11087.8542.08888.5062.11188.4602.11087.6932.08388.5062.11188.4602.11087.5312.0770.99894.05625.809888.02825.731815.01022.536894.05625.809888.02825.731797.17921.823894.05625.809888.02825.731780.02821.146Table 4
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 50 and p = 3 when there is no outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.74.8892.7534.7352.6714.7062.6554.8892.7534.7352.6714.6622.6314.8892.7534.7352.6714.6192.6080.87.2824.2476.9284.0466.8604.0087.2824.2476.9284.0466.7623.9527.2824.2476.9284.0466.6663.8970.90.3390.3560.3370.3540.2960.3110.3390.3560.3370.3540.2870.3010.3390.3560.3370.3540.2780.2920.994.0004.2473.7143.9681.2471.3224.0004.2473.7143.9681.0461.1084.0004.2473.7143.9680.8980.95050.70.2420.1560.2410.1560.2410.1560.2420.1560.2410.1560.2410.1560.2420.1560.2410.1560.2410.1560.80.3620.2330.3610.2320.3610.2320.3620.2330.3610.2320.3610.2320.3620.2330.3610.2320.3610.2320.90.3390.3560.3390.3560.3370.3540.3390.3560.3390.3560.3370.3540.3390.3560.3390.3560.3360.3530.994.0004.2473.9884.2353.7193.9494.0004.2473.9884.2353.6543.8794.0004.2473.9884.2353.5913.813100.70.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.80.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.90.3390.3560.3390.3560.3390.3560.3390.3560.3390.3560.3380.3550.3390.3560.3390.3560.3380.3550.994.0004.2473.9974.2443.9264.1694.0004.2473.9974.2443.9084.1504.0004.2473.9974.2443.8904.1310.710.74.8892.7534.5442.5674.4802.5334.8892.7534.5442.5674.3882.4834.8892.7534.5442.5674.2992.4350.87.2824.2476.5013.8036.3633.7247.2824.2476.5013.8036.1653.6127.2824.2476.5013.8035.9773.5050.90.3390.3560.3370.3540.2520.2650.3390.3560.3370.3540.2370.2490.3390.3560.3370.3540.2240.2350.994.0004.2473.7143.9680.6080.6414.0004.2473.7143.9680.4980.5244.0004.2473.7143.9680.4260.44650.70.2420.1560.2410.1560.2410.1550.2420.1560.2410.1560.2410.1550.2420.1560.2410.1560.2400.1550.80.3620.2330.3600.2320.3600.2310.3620.2330.3600.2320.3590.2310.3620.2330.3600.2320.3590.2310.90.3390.3560.3390.3560.3350.3510.3390.3560.3390.3560.3330.3500.3390.3560.3390.3560.3320.3490.994.0004.2473.9884.2353.3943.6034.0004.2473.9884.2353.2673.4684.0004.2473.9884.2353.1483.341100.70.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.80.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.90.3390.3560.3390.3560.3380.3550.3390.3560.3390.3560.3380.3550.3390.3560.3390.3560.3370.3540.994.0004.2473.9974.2443.8324.0684.0004.2473.9974.2443.7914.0264.0004.2473.9974.2443.7523.9840.910.74.8892.7534.4542.5194.3752.4764.8892.7534.4542.5194.2612.4154.8892.7534.4542.5194.1532.3560.87.2824.2476.3053.6916.1373.5967.2824.2476.3053.6915.9003.4617.2824.2476.3053.6915.6793.3350.90.3390.3560.3370.3540.2350.2470.3390.3560.3370.3540.2190.2290.3390.3560.3370.3540.2040.2140.994.0004.2473.7143.9680.4860.5114.0004.2473.7143.9680.4020.4214.0004.2473.7143.9680.3490.36450.70.2420.1560.2410.1550.2400.1550.2420.1560.2410.1550.2400.1550.2420.1560.2410.1550.2400.1550.80.3620.2330.3600.2310.3590.2310.3620.2330.3600.2310.3590.2300.3620.2330.3600.2310.3580.2300.90.3390.3560.3390.3560.3330.3500.3390.3560.3390.3560.3320.3490.3390.3560.3390.3560.3300.3470.994.0004.2473.9884.2353.2493.4494.0004.2473.9884.2353.0993.2904.0004.2473.9884.2352.9603.142100.70.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.0350.0190.80.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.0530.0300.90.3390.3560.3390.3560.3380.3550.3390.3560.3390.3560.3370.3540.3390.3560.3390.3560.3300.3470.994.0004.2473.9974.2443.7864.0204.0004.2473.9974.2443.7353.9664.0004.2473.9884.2352.9603.142Table 5
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 50 and p = 3 when there is one outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.75.2402.8795.0812.7955.0512.7795.2402.8795.0812.7955.0052.7555.2402.8795.0812.7954.9612.7310.87.9554.4877.5744.2777.5014.2377.9554.4877.5744.2777.3954.1797.9554.4877.5744.2777.2924.1220.937.7250.36237.3650.36032.7090.31837.7250.36237.3650.36031.6290.30937.7250.36237.3650.36030.6090.3000.99450.8764.394398.7854.144127.3161.381450.8764.394398.7854.144104.5151.161450.8764.394398.7854.14487.8501.00050.70.3320.1850.3320.1850.3320.1850.3320.1850.3320.1850.3320.1850.3550.2010.3550.2010.3540.2010.80.5010.2800.5000.2790.5000.2790.5010.2800.5000.2790.4990.2790.5200.3000.5190.3000.5180.2990.97.7830.3627.7810.3627.7390.3607.7830.3627.7810.3627.7280.3597.7830.3627.7810.3627.7170.3590.9994.6234.39494.1434.38387.5904.08594.6234.39494.1434.38385.9674.01494.6234.39494.1434.38384.3933.945100.70.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.80.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.94.0490.3624.0490.3624.0430.3614.0490.3624.0490.3624.0420.3614.0490.3624.0490.3624.0410.3610.9949.7794.39449.7164.39148.8134.31349.7794.39449.7164.39148.5764.29349.7794.39449.7164.39148.3414.2730.710.75.2402.8794.8832.6904.8182.6555.2402.8794.8832.6904.7222.6045.2402.8794.8832.6904.6292.5550.87.9554.4877.1154.0246.9663.9427.9554.4877.1154.0246.7533.8247.9554.4877.1154.0246.5513.7130.937.7250.36237.3650.36027.5790.27437.7250.36237.3650.36025.7490.25837.7250.36237.3650.36024.1170.2450.99450.8764.394398.7854.14455.2190.684450.8764.394398.7854.14442.7450.563450.8764.394398.7854.14434.3390.48450.70.3320.1850.3310.1850.3310.1850.3320.1850.3310.1850.3310.1850.3320.1850.3310.1850.3300.1840.80.5010.2800.4990.2780.4980.2780.5010.2800.4990.2780.4970.2780.5010.2800.4990.2780.4970.2770.97.7830.3627.7810.3627.6810.3577.7830.3627.7810.3627.6550.3567.7830.3627.7810.3627.6300.3550.9994.6234.39494.1434.38379.4783.72894.6234.39494.1434.38376.3153.58994.6234.39494.1434.38373.3573.458100.70.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.80.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.94.0490.3624.0490.3624.0360.3614.0490.3624.0490.3624.0330.3614.0490.3624.0490.3624.0290.3600.9949.7794.39449.7164.39147.5724.20949.7794.39449.7164.39147.0464.16549.7794.39449.7164.39146.5294.1220.910.75.2402.8794.7902.6404.7082.5975.2402.8794.7902.6404.5912.5345.2402.8794.7902.6404.4792.4750.87.9554.4876.9043.9086.7233.8087.9554.4876.9043.9086.4683.6687.9554.4876.9043.9086.2313.5360.937.7250.36237.3650.36025.5050.25637.7250.36237.3650.36023.4720.23937.7250.36237.3650.36021.7020.2250.99450.8764.394398.7854.14441.3460.550450.8764.394398.7854.14431.5400.457450.8764.394398.7854.14425.0610.39850.70.3320.1850.3310.1850.3310.1850.3320.1850.3310.1850.3300.1840.3320.1850.3310.1850.3300.1840.80.5010.2800.4980.2780.4970.2780.5010.2800.4980.2780.4960.2770.5010.2800.4980.2780.4960.2770.97.7830.3627.7810.3627.6520.3567.7830.3627.7810.3627.6190.3557.7830.3627.7810.3627.5870.3530.9994.6234.39494.1434.38375.8803.57094.6234.39494.1434.38372.1473.40594.6234.39494.1434.38368.7133.253100.70.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.0390.0200.80.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.0590.0320.94.0490.3624.0490.3624.0320.3604.0490.3624.0490.3624.0280.3607.7830.3627.7810.3627.5870.3530.9949.7794.39449.7164.39146.9714.15949.7794.39449.7164.39146.3104.10394.6234.39494.1434.38368.7133.253Table 6
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 50 and p = 3 when there are two outliers.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.75.8963.4085.7003.2985.6623.2775.8963.4085.7003.2985.6063.2465.8963.4085.7003.2985.5513.2150.88.4795.1278.0494.8737.9684.8258.4795.1278.0494.8737.8484.7548.4795.1278.0494.8737.7324.6850.914.0660.39913.8830.39613.2400.35014.0660.39913.8830.39613.0470.34014.0660.39913.8830.39612.8600.3300.99136.9744.789118.9974.26478.2161.486136.9744.789118.9974.26469.9211.249136.9744.789118.9974.26462.9561.07550.70.3550.2010.3550.2010.3540.2010.3550.2010.3550.2010.3540.2010.3550.2010.3550.2010.3540.2010.80.5200.3000.5190.3000.5190.2990.5200.3000.5190.3000.5190.2990.5200.3000.5190.3000.5180.2990.93.0850.3993.0830.3993.0770.3973.0850.3993.0830.3993.0760.3963.0850.3993.0830.3993.0740.3960.9930.9214.78930.7104.76629.9554.44930.9214.78930.7104.76629.7244.37030.9214.78930.7104.76629.4984.294100.70.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.80.0760.0380.0760.0380.0760.0380.0760.0380.0760.0380.0760.0380.0760.0380.0760.0380.0760.0380.91.7120.3991.7120.3991.7110.3981.7120.3991.7120.3991.7110.3981.7120.3991.7120.3991.7100.3980.9917.5834.78917.5494.78317.4234.70017.5834.78917.5494.78317.3844.67817.5834.78917.5494.78317.3454.6560.710.75.8963.4085.4563.1625.3753.1175.8963.4085.4563.1625.2573.0515.8963.4085.4563.1625.1442.9880.88.4795.1277.5334.5677.3664.4688.4795.1277.5334.5677.1274.3278.4795.1277.5334.5676.9024.1930.914.0660.39913.8830.39612.2690.30114.0660.39913.8830.39611.8840.28414.0660.39913.8830.39611.5210.2690.99136.9744.789118.9974.26446.2380.735136.9744.789118.9974.26438.3680.606136.9744.789118.9974.26432.4400.52150.70.3550.2010.3540.2010.3540.2010.3550.2010.3540.2010.3540.2010.3550.2010.3540.2010.3530.2010.80.5200.3000.5180.2990.5170.2980.5200.3000.5180.2990.5170.2980.5200.3000.5180.2990.5160.2980.93.0850.3993.0830.3993.0680.3943.0850.3993.0830.3993.0640.3933.0850.3993.0830.3993.0600.3910.9930.9214.78930.7104.76628.7704.05630.9214.78930.7104.76628.2843.90230.9214.78930.7104.76627.8163.758100.70.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.80.0760.0380.0760.0380.0760.0380.0760.0380.0760.0380.0760.0380.0760.0380.0760.0380.0750.0380.91.7120.3991.7120.3991.7090.3981.7120.3991.7120.3991.7090.3971.7120.3991.7120.3991.7080.3970.9917.5834.78917.5494.78317.2164.58517.5834.78917.5494.78317.1284.53717.5834.78917.5494.78317.0404.4890.910.75.8963.4085.3413.0985.2413.0425.8963.4085.3413.0985.0972.9625.8963.4085.3413.0984.9602.8850.88.4795.1277.2964.4277.0944.3078.4795.1277.2964.4276.8094.1398.4795.1277.2964.4276.5443.9810.914.0660.39913.8830.39611.8310.28214.0660.39913.8830.39611.3710.26314.0660.39913.8830.39610.9430.2470.99136.9744.789118.9974.26437.4190.592136.9744.789118.9974.26430.3370.493136.9744.789118.9974.26425.1850.42950.70.3550.2010.3540.2010.3540.2010.3550.2010.3540.2010.3530.2000.3550.2010.3540.2010.3530.2000.80.5200.3000.5170.2980.5170.2980.5200.3000.5170.2980.5160.2970.5200.3000.5170.2980.5150.2970.93.0850.3993.0830.3993.0630.3933.0850.3993.0830.3993.0580.3913.0850.3993.0830.3993.0530.3890.9930.9214.78930.7104.76628.2163.88130.9214.78930.7104.76627.6203.69930.9214.78930.7104.76627.0513.532100.70.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.0510.0250.80.0760.0380.0760.0380.0750.0380.0760.0380.0760.0380.0750.0380.0760.0380.0760.0380.0750.0380.91.7120.3991.7120.3991.7090.3971.7120.3991.7120.3991.7080.3973.0850.3993.0830.3993.0530.3890.9917.5834.78917.5494.78317.1154.53017.5834.78917.5494.78317.0034.46930.9214.78930.7104.76627.0513.532Table 7
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 100 and p = 3 when there is no outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.73.3502.2313.2892.1883.2772.1803.3502.2313.2892.1883.2592.1683.3502.2313.2892.1883.2412.1550.84.7563.2004.6213.1064.5943.0884.7563.2004.6213.1064.5553.0614.7563.2004.6213.1064.5173.0350.90.1890.2000.1880.1990.1760.1860.1890.2000.1880.1990.1730.1830.1890.2000.1880.1990.1700.1800.992.0792.2102.0052.1371.0341.1002.0792.2102.0052.1370.9080.9662.0792.2102.0052.1370.8070.85850.70.0940.0460.0940.0460.0940.0460.0940.0460.0940.0460.0940.0460.0940.0460.0940.0460.0940.0460.80.1340.0650.1330.0650.1330.0650.1340.0650.1330.0650.1330.0650.1340.0650.1330.0650.1330.0650.90.1890.2000.1890.2000.1880.1990.1890.2000.1890.2000.1880.1990.1890.2000.1890.2000.1880.1990.992.0792.2102.0762.2072.0062.1322.0792.2102.0762.2071.9892.1132.0792.2102.0762.2071.9722.095100.70.0220.0120.0220.0120.0220.0120.0220.0120.0220.0120.0220.0120.0220.0120.0220.0120.0220.0120.80.0320.0180.0320.0180.0320.0180.0320.0180.0320.0180.0320.0180.0320.0180.0320.0180.0320.0180.90.1890.2000.1890.2000.1890.1990.1890.2000.1890.2000.1890.1990.1890.2000.1890.2000.1890.1990.992.0792.2102.0792.2092.0612.1902.0792.2102.0792.2092.0562.1852.0792.2102.0792.2092.0512.1800.710.73.3502.2313.2102.1343.1832.1153.3502.2313.2102.1343.1432.0883.3502.2313.2892.1883.2412.1550.84.7563.2004.4502.9884.3922.9484.7563.2004.4502.9884.3082.8914.7563.2004.6213.1064.5173.0350.90.1890.2000.1880.1990.1610.1700.1890.2000.1880.1990.1560.1640.1890.2000.1880.1990.1500.1590.992.0792.2102.0052.1370.5830.6202.0792.2102.0052.1370.4870.5172.0792.2102.0052.1370.4190.44550.70.0940.0460.0940.0460.0940.0460.0940.0460.0940.0460.0940.0460.0940.0460.0940.0460.0940.0460.80.1340.0650.1330.0640.1330.0640.1340.0650.1330.0640.1330.0640.1340.0650.1330.0650.1330.0650.90.1890.2000.1890.2000.1880.1980.1890.2000.1890.2000.1870.1980.1890.2000.1890.2000.1870.1980.992.0792.2102.0762.2071.9162.0362.0792.2102.0762.2071.8781.9962.0792.2102.0762.2071.8421.957100.70.0220.0120.0220.0120.0220.0120.0220.0120.0220.0120.0220.0120.0220.0120.0220.0120.0220.0120.80.0320.0180.0320.0180.0320.0180.0320.0180.0320.0180.0320.0180.0320.0180.0320.0180.0320.0180.90.1890.2000.1890.2000.1890.1990.1890.2000.1890.2000.1880.1990.1890.2000.1890.2000.1880.1990.992.0792.2102.0792.2092.0362.1642.0792.2102.0792.2092.0262.1532.0792.2102.0792.2092.0152.1420.910.70.1750.1850.1650.1750.1630.1730.1750.1850.1650.1750.1610.1700.1750.1850.1650.1750.1590.1680.80.2760.2930.2510.2660.2470.2620.2760.2930.2510.2660.2410.2550.2760.2930.2510.2660.2360.2500.90.1890.2000.1880.1990.1550.1640.1890.2000.1880.1990.1480.1560.1890.2000.1880.1990.1420.1500.992.0792.2102.0052.1370.4760.5052.0792.2102.0052.1370.3960.4202.0792.2102.0052.1370.3420.36150.70.1500.1580.1490.1580.1490.1580.1500.1580.1490.1580.1490.1580.1500.1580.1490.1580.1490.1570.80.2290.2420.2280.2410.2280.2410.2290.2420.2280.2410.2280.2400.2290.2420.2280.2410.2270.2400.90.1890.2000.1890.2000.1870.1980.1890.2000.1890.2000.1870.1970.1890.2000.1890.2000.1860.1970.992.0792.2102.0762.2071.8731.9902.0792.2102.0762.2071.8271.9412.0792.2102.0762.2071.7821.894100.70.1560.1640.1560.1640.1550.1630.1560.1640.1560.1640.1550.1630.1560.1640.1560.1640.1550.1630.80.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.90.1890.2000.1890.2000.1880.1990.1890.2000.1890.2000.1880.1990.1890.2000.1890.2000.1880.1990.992.0792.2102.0792.2092.0242.1512.0792.2102.0792.2092.0112.1372.0792.2102.0792.2091.9982.123Table 8
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 100 and p = 3 when there is one outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.73.4042.2523.3432.2103.3312.2013.4042.2523.3432.2103.3132.1893.4042.2523.3432.2103.2962.1770.84.8213.2304.6863.1364.6603.1184.8213.2304.6863.1364.6213.0914.8213.2304.6863.1364.5833.0640.95.5320.2015.5160.2005.2870.1875.5320.2015.5160.2005.2290.1845.5320.2015.5160.2005.1730.1810.9969.9362.22466.7952.16139.8341.11469.9362.22466.7952.16135.8150.98069.9362.22466.7952.16132.4820.87350.70.1000.0470.1000.0470.1000.0470.1000.0470.1000.0470.1000.0470.1000.0470.1000.0470.1000.0470.80.1440.0680.1440.0670.1440.0670.1440.0680.1440.0670.1440.0670.1440.0680.1440.0670.1440.0670.91.2610.2011.2610.2011.2590.2001.2610.2011.2610.2011.2580.2001.2610.2011.2610.2011.2580.2000.9915.7252.22315.6952.22115.2692.14615.7252.22315.6952.22115.1582.12715.7252.22315.6952.22115.0502.109100.70.0300.0140.0300.0140.0300.0140.0300.0140.0300.0140.0300.0140.0300.0140.0300.0140.0300.0140.80.0450.0210.0450.0210.0450.0210.0450.0210.0450.0210.0450.0210.0450.0210.0450.0210.0450.0210.90.7260.2010.7260.2010.7260.2010.7260.2010.7260.2010.7260.2000.7260.2010.7260.2010.7260.2000.998.9302.2238.9262.2238.8622.2048.9302.2238.9262.2238.8462.1998.9302.2238.9262.2238.8292.1940.710.73.4042.2523.2652.1553.2382.1373.4042.2523.2652.1553.1992.1103.4042.2523.3432.2103.2962.1770.84.8213.2304.5163.0184.4592.9784.8213.2304.5163.0184.3762.9204.8213.2304.6863.1364.5833.0640.95.5320.2015.5160.2004.9940.1725.5320.2015.5160.2004.8770.1675.5320.2015.5160.2004.7650.1610.9969.9362.22466.7952.16124.5670.63769.9362.22466.7952.16120.8380.53569.9362.22466.7952.16118.0010.46250.70.1000.0470.1000.0470.1000.0470.1000.0470.1000.0470.1000.0470.1000.0470.1000.0470.1000.0470.80.1440.0680.1430.0670.1430.0670.1440.0680.1430.0670.1430.0670.1440.0680.1440.0670.1440.0670.91.2610.2011.2610.2011.2560.1991.2610.2011.2610.2011.2550.1991.2610.2011.2610.2011.2530.1990.9915.7252.22315.6952.22114.6982.04915.7252.22315.6952.22114.4612.00915.7252.22315.6952.22114.2321.971100.70.0300.0140.0300.0140.0300.0140.0300.0140.0300.0140.0300.0140.0300.0140.0300.0140.0300.0140.80.0450.0210.0450.0210.0450.0210.0450.0210.0450.0210.0450.0210.0450.0210.0450.0210.0450.0210.90.7260.2010.7260.2010.7260.2000.7260.2010.7260.2010.7250.2000.7260.2010.7260.2010.7250.2000.998.9302.2238.9262.2238.7742.1788.9302.2238.9262.2238.7352.1668.9302.2238.9262.2238.6972.1550.910.70.4140.1870.3950.1770.3910.1750.4140.1870.3950.1770.3870.1720.4140.1870.3950.1770.3820.1700.80.6060.2950.5640.2680.5560.2640.6060.2950.5640.2680.5460.2580.6060.2950.5640.2680.5360.2520.95.5320.2015.5160.2004.8600.1665.5320.2015.5160.2004.7190.1595.5320.2015.5160.2004.5860.1530.9969.9362.22466.7952.16120.3860.5235.5322.22466.7952.16116.9840.43869.9362.22466.7952.16114.4630.38050.70.2110.1600.2110.1600.2110.1605.5320.1600.2110.1600.2100.1600.2110.1600.2110.1600.2100.1600.80.3160.2460.3150.2450.3150.2455.5320.2460.3150.2450.3140.2440.3160.2460.3150.2450.3140.2440.91.2610.2011.2610.2011.2550.1995.5320.2011.2610.2011.2530.1991.2610.2011.2610.2011.2510.1980.9915.7252.22315.6952.22114.4282.0045.5322.22315.6952.22114.1361.95515.7252.22315.6952.22113.8541.907100.70.1990.1680.1990.1680.1990.1685.5320.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.80.2950.2490.2950.2480.2950.2485.5320.2490.2950.2480.2940.2480.2950.2490.2950.2480.2940.2480.90.7260.2010.7260.2010.7250.2005.5320.2010.7260.2010.7250.2000.7260.2010.7260.2010.7250.2000.998.9302.2238.9262.2238.7302.1655.5322.2238.9262.2238.6812.1518.9302.2238.9262.2238.6332.136Table 9
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 100 and p = 3 when there are two outliers.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.73.5292.3603.4632.3163.4502.3073.5292.3603.4632.3163.4312.2943.5292.3603.4632.3163.4122.2810.85.0463.3884.9013.2894.8733.2695.0463.3884.9013.2894.8313.2415.0463.3884.9013.2894.7903.2120.92.7370.2072.7060.2062.5960.1942.7370.2072.7060.2062.5630.1912.7370.2072.7060.2062.5310.1880.9933.7392.27328.7762.15118.2461.16133.7392.27328.7762.15116.2741.02633.7392.27328.7762.15114.6610.91850.70.0990.0490.0990.0490.0990.0490.0990.0490.0990.0490.0990.0490.0990.0490.0990.0490.0990.0490.80.1400.0700.1400.0700.1400.0700.1400.0700.1400.0700.1400.0700.1400.0700.1400.0700.1400.0700.90.7020.2070.7020.2070.7010.2060.7020.2070.7020.2070.7000.2060.7020.2070.7020.2070.7000.2060.998.4712.2738.4132.2678.2012.1958.4712.2738.4132.2678.1352.1768.4712.2738.4132.2678.0712.158100.70.0240.0150.0240.0150.0240.0150.0240.0150.0240.0150.0240.0150.0240.0150.0240.0150.0240.0150.80.0340.0210.0340.0210.0340.0210.0340.0210.0340.0210.0340.0210.0340.0210.0340.0210.0340.0210.90.4470.2070.4470.2070.4470.2070.4470.2070.4470.2070.4470.2070.4470.2070.4470.2070.4470.2070.995.2982.2735.2892.2715.2542.2535.2982.2735.2892.2715.2432.2485.2982.2735.2892.2715.2322.2430.710.73.5292.3603.3782.2583.3492.2383.5292.3603.3782.2583.3072.2093.5292.3603.4632.3163.4122.2810.85.0463.3884.7183.1634.6563.1215.0463.3884.7183.1634.5663.0595.0463.3884.9013.2894.7903.2120.92.7370.2072.7060.2062.4310.1792.7370.2072.7060.2062.3650.1732.7370.2072.7060.2062.3040.1680.9933.7392.27328.7762.15110.9210.67833.7392.27328.7762.1519.2070.57433.7392.27328.7762.1517.9240.49950.70.0990.0490.0990.0490.0990.0490.0990.0490.0990.0490.0990.0490.0990.0490.0990.0490.0990.0490.80.1400.0700.1400.0700.1400.0700.1400.0700.1400.0700.1400.0700.1400.0700.1400.0700.1400.0700.90.7020.2070.7020.2070.6990.2060.7020.2070.7020.2070.6980.2050.7020.2070.7020.2070.6970.2050.998.4712.2738.4132.2677.8642.0998.4712.2738.4132.2677.7242.0598.4712.2738.4132.2677.5892.020100.70.0240.0150.0240.0150.0240.0150.0240.0150.0240.0150.0240.0150.0240.0150.0240.0150.0240.0150.80.0340.0210.0340.0210.0340.0210.0340.0210.0340.0210.0340.0210.0340.0210.0340.0210.0340.0210.90.4470.2070.4470.2070.4460.2070.4470.2070.4470.2070.4460.2070.4470.2070.4470.2070.4460.2060.995.2982.2735.2892.2715.1962.2275.2982.2735.2892.2715.1712.2165.2982.2735.2892.2715.1462.2040.910.70.8350.1920.8110.1820.8070.1800.8350.1920.8110.1820.8000.1780.8350.1920.8110.1820.7940.1760.81.2200.3021.1630.2761.1530.2711.2200.3021.1630.2761.1380.2651.2200.3021.1630.2761.1230.2600.92.7370.2072.7060.2062.3560.1722.7370.2072.7060.2062.2780.1662.7370.2072.7060.2062.2050.1590.9933.7392.27328.7762.1519.0010.56133.7392.27328.7762.1517.4690.47333.7392.27328.7762.1516.3510.41350.70.2640.1630.2640.1620.2630.1620.2640.1630.2640.1620.2630.1620.2640.1630.2640.1620.2630.1620.80.4010.2510.3990.2490.3980.2490.4010.2510.3990.2490.3980.2490.4010.2510.3990.2490.3970.2490.90.7020.2070.7020.2070.6980.2050.7020.2070.7020.2070.6970.2050.7020.2070.7020.2070.6960.2050.998.4712.2738.4132.2677.7042.0538.4712.2738.4132.2677.5322.0048.4712.2738.4132.2677.3671.957100.70.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.80.4630.2650.4620.2650.4620.2650.4630.2650.4620.2650.4620.2650.4630.2650.4620.2650.4620.2650.90.4470.2070.4470.2070.4460.2070.4470.2070.4470.2070.4460.2060.4470.2070.4470.2070.4460.2060.995.2982.2735.2892.2715.1682.2145.2982.2735.2892.2715.1362.2005.2982.2735.2892.2715.1052.186Table 10
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 20 and p = 7 when there is no outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.71.4301.5431.1401.2281.0981.1821.4301.5431.1401.2281.0411.1201.4301.5431.1401.2280.9911.0660.82.1602.3201.5291.6411.4491.5552.1602.3201.5291.6411.3451.4432.1602.3201.5291.6411.2581.3490.94.4694.8852.3352.5342.1452.3264.4694.8852.3352.5341.9182.0764.4694.8852.3352.5341.7391.8800.9956.15759.8564.3104.5753.5803.79856.15759.8564.3104.5752.8292.99856.15759.8564.3104.5752.3232.45850.71.3891.4921.3741.4761.3721.4731.3891.4921.3741.4761.3671.4681.3891.4921.3741.4761.3631.4640.82.0932.2672.0592.2292.0522.2222.0932.2672.0592.2292.0422.2112.0932.2672.0592.2292.0322.2000.94.3164.6374.1574.4664.1274.4344.3164.6374.1574.4664.0824.3864.3164.6374.1574.4664.0384.3390.9948.68753.00931.91934.60529.91832.41448.68753.00931.91934.60527.38029.63648.68753.00931.91934.60525.27127.331100.71.0341.1071.0321.1051.0321.1051.0341.1071.0321.1051.0311.1041.0341.1071.0321.1051.0311.1040.81.5611.6791.5581.6751.5571.6741.5611.6791.5581.6751.5561.6721.5611.6791.5581.6751.5551.6710.93.1513.4043.1343.3853.1303.3813.1513.4043.1343.3853.1253.3753.1513.4043.1343.3853.1203.3700.9934.93137.51132.40234.79731.94234.30434.93137.51132.40234.79731.27933.59234.93137.51132.40234.79730.64432.9110.710.71.4301.5430.9150.9830.8600.9231.4301.5430.9150.9830.7920.8491.4301.5430.9150.9830.7380.7900.82.1602.3201.1321.2131.0441.1182.1602.3201.1321.2130.9401.0052.1602.3201.1321.2130.8590.9170.94.4694.8851.4991.6171.3421.4454.4694.8851.4991.6171.1661.2524.4694.8851.4991.6171.0361.1100.9956.15759.8561.7511.8481.4391.51656.15759.8561.7511.8481.1451.20156.15759.8561.7511.8480.9631.00650.71.3891.4921.3561.4561.3491.4491.3891.4921.3561.4561.3401.4391.3891.4921.3561.4561.3301.4290.82.0932.2672.0142.1811.9992.1642.0932.2672.0142.1811.9772.1402.0932.2672.0142.1811.9552.1160.94.3164.6373.9634.2583.8994.1904.3164.6373.9634.2583.8084.0924.3164.6373.9634.2583.7213.9980.9948.68753.00922.28724.07320.24521.84748.68753.00922.28724.07317.85019.24248.68753.00922.28724.07316.00217.234100.71.0341.1071.0301.1031.0291.1021.0341.1071.0301.1031.0281.1011.0341.1071.0301.1031.0271.1000.81.5611.6791.5531.6691.5511.6671.5611.6791.5531.6691.5481.6641.5611.6791.5531.6691.5461.6620.93.1513.4043.1113.3603.1033.3513.1513.4043.1113.3603.0913.3383.1513.4043.1113.3603.0793.3250.9934.93137.51129.58331.77128.71730.84234.93137.51129.58331.77127.51829.55534.93137.51129.58331.77126.42428.3800.910.71.4301.5430.8390.9010.7840.8401.4301.5430.8390.9010.7170.7671.4301.5430.8390.9010.6650.7100.82.1602.3201.0111.0820.9270.9912.1602.3201.0111.0820.8290.8852.1602.3201.0111.0820.7550.8040.94.4694.8851.2861.3841.1451.2294.4694.8851.2861.3840.9901.0594.4694.8851.2861.3840.8770.9360.9956.15759.8561.3391.4091.1131.16756.15759.8561.3391.4090.9060.94556.15759.8561.3391.4090.7820.81150.71.3891.4921.3471.4461.3381.4371.3891.4921.3471.4461.3261.4241.3891.4921.3471.4461.3151.4120.82.0932.2671.9932.1571.9742.1362.0932.2671.9932.1571.9462.1062.0932.2671.9932.1571.9192.0770.94.3164.6373.8734.1613.7954.0784.3164.6373.8734.1613.6843.9594.3164.6373.8734.1613.5813.8480.9948.68753.00919.49021.02617.55818.92448.68753.00919.49021.02615.33216.50748.68753.00919.49021.02613.64014.673100.71.0341.1071.0291.1021.0281.1011.0341.1071.0291.1021.0271.0991.0341.1071.0291.1021.0251.0980.81.5611.6791.5501.6661.5481.6641.5611.6791.5501.6661.5451.6601.5611.6791.5501.6661.5411.6570.93.1513.4043.0993.3473.0893.3363.1513.4043.0993.3473.0743.3203.1513.4043.0993.3473.0593.3040.9934.93137.51128.36330.46227.35629.38034.93137.51128.36330.46225.98327.90734.93137.51128.36330.46224.75426.587Table 11
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 20 and p = 7 when there is one outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.743.4181.97530.6221.59028.7981.53243.4181.97530.6221.59026.3781.45443.4181.97530.6221.59024.2771.3860.866.9702.99539.5372.12236.2312.01066.9702.99539.5372.12232.0581.86466.9702.99539.5372.12228.6231.7400.9143.7906.31252.3393.19445.3982.921143.7906.31252.3393.19437.5102.594143.7906.31252.3393.19431.6842.3380.991790.87376.05136.8215.13129.2994.2711790.87376.05136.8215.13122.1913.3901790.87376.05136.8215.13117.7182.79850.76.6791.9246.6201.9056.6081.9016.6791.9246.6201.9056.5911.8966.6791.9246.6201.9056.5731.8900.89.5072.8389.3832.7939.3582.7849.5072.8389.3832.7939.3222.7719.5072.8389.3832.7939.2852.7580.917.8375.64317.3775.45017.2885.41317.8375.64317.3775.45017.1575.35917.8375.64317.3775.45017.0275.3060.99177.29963.383138.85843.305133.29040.828177.29963.383138.85843.305125.77337.648177.29963.383138.85843.305119.08734.970100.72.8371.4052.8321.4032.8311.4032.8371.4052.8321.4032.8301.4022.8371.4052.8321.4032.8281.4010.84.3702.1504.3582.1444.3552.1434.3702.1504.3582.1444.3522.1414.3702.1504.3582.1444.3482.1390.98.9344.3608.8824.3348.8714.3298.9344.3608.8824.3348.8554.3228.9344.3608.8824.3348.8404.3140.99103.74048.97096.08845.34394.69944.685103.74048.97096.08845.34392.69543.735103.74048.97096.08845.34390.77942.8270.710.743.4181.97521.1661.28118.9551.20443.4181.97521.1661.28116.2971.10843.4181.97521.1661.28114.2181.0290.866.9702.99523.8501.56120.6741.43666.9702.99523.8501.56117.0861.28766.9702.99523.8501.56114.4511.1710.9143.7906.31224.4761.99820.2001.778143.7906.31224.4761.99815.8221.534143.7906.31224.4761.99812.8911.3550.991790.87376.05112.9032.12710.3541.7601790.87376.05112.9032.1277.9351.4091790.87376.05112.9032.1276.4061.18950.76.6791.9246.5421.8806.5151.8726.6791.9246.5421.8806.4751.8596.6791.9246.5421.8806.4361.8460.89.5072.8389.2212.7369.1662.7169.5072.8389.2212.7369.0852.6879.5072.8389.2212.7369.0052.6590.917.8375.64316.8015.21416.6095.13617.8375.64316.8015.21416.3295.02417.8375.64316.8015.21416.0584.9160.99177.29963.383108.81631.115101.15828.426177.29963.383108.81631.11591.43325.213177.29963.383108.81631.11583.32422.686100.72.8371.4052.8251.4002.8231.3992.8371.4052.8251.4002.8201.3972.8371.4052.8251.4002.8161.3950.84.3702.1504.3422.1364.3362.1344.3702.1504.3422.1364.3282.1304.3702.1504.3422.1364.3202.1260.98.9344.3608.8124.3008.7884.2898.9344.3608.8124.3008.7524.2718.9344.3608.8124.3008.7164.2540.99103.74048.97087.57541.30884.96540.070103.74048.97087.57541.30881.35138.357103.74048.97087.57541.30878.05636.7950.910.72.8371.4052.8221.3982.8191.39743.4181.97518.1241.17413.4631.00043.4181.97518.1241.17411.5710.9240.84.3702.1504.3342.1324.3272.12966.9702.99519.5261.39013.5301.12966.9702.99519.5261.39011.3051.0220.98.9344.3608.7784.2848.7474.269143.7906.31218.7481.70011.9191.292143.7906.31218.7481.7009.6811.1390.99103.74048.97083.89739.56480.86138.1251790.87376.0519.5321.6415.9131.1191790.87376.0519.5321.6414.8010.96450.76.6791.9246.5041.8686.4701.8576.6791.9246.5041.8686.4191.8416.6791.9246.5041.8686.3691.8250.89.5072.8389.1432.7089.0732.6839.5072.8389.1432.7088.9712.6479.5072.8389.1432.7088.8702.6120.917.8375.64316.5285.10316.2895.00817.8375.64316.5285.10315.9444.87217.8375.64316.5285.10315.6134.7430.99177.29963.38398.18627.42190.18624.816177.29963.38398.18627.42180.24421.759177.29963.38398.18627.42172.14619.393100.72.8371.4052.8221.3982.8191.3972.8371.4052.8221.3982.8151.3952.8371.4052.8221.3982.8101.3930.84.3702.1504.3342.1324.3272.1294.3702.1504.3342.1324.3162.1244.3702.1504.3342.1324.3062.1190.98.9344.3608.7784.2848.7474.2698.9344.3608.7784.2848.7014.2468.9344.3608.7784.2848.6564.2240.99103.74048.97083.89739.56480.86138.125103.74048.97083.89739.56476.73136.167103.74048.97083.89739.56473.03634.415Table 12
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 20 and p = 7 when there are two outliers.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.7105.9453.88478.6463.16274.6873.052105.9453.88478.6463.16269.3992.902105.9453.88478.6463.16264.7682.7680.8156.5215.766100.7744.20593.8603.998156.5215.766100.7744.20585.0273.727156.5215.766100.7744.20577.6503.4940.9315.66011.802140.7346.437126.3445.937315.66011.802140.7346.437109.4845.329315.66011.802140.7346.43796.5474.8430.993440.840126.654194.47311.626157.6639.5993440.840126.654194.47311.626120.1147.4843440.840126.654194.47311.62694.9816.04150.712.3252.26512.2182.24312.1972.23812.3252.26512.2182.24312.1662.23212.3252.26512.2182.24312.1342.2250.817.6323.30417.4123.25417.3683.24417.6323.30417.4123.25417.3033.23017.6323.30417.4123.25417.2393.2150.933.2186.51532.4136.30432.2576.26433.2186.51532.4136.30432.0256.20433.2186.51532.4136.30431.7976.1450.99327.37571.474261.24050.147251.25747.452327.37571.474261.24050.147237.62743.963327.37571.474261.24050.147225.35940.997100.77.1523.4807.1393.4747.1373.4737.1523.4807.1393.4747.1333.4717.1523.4807.1393.4747.1293.4700.810.6555.15710.6265.14410.6215.14110.6555.15710.6265.14410.6125.13710.6555.15710.6265.14410.6035.1330.921.14310.22221.02210.16520.99810.15421.14310.22221.02210.16520.96210.13721.14310.22221.02210.16520.92610.1210.99222.411106.264207.28299.298204.53098.029222.411106.264207.28299.298200.55396.195222.411106.264207.28299.298196.74994.4390.710.7105.9453.88457.8272.56252.8142.408105.9453.88457.8272.56246.6732.212105.9453.88457.8272.56241.7572.0490.8156.5215.76667.1783.15060.0212.906156.5215.76667.1783.15051.6782.608156.5215.76667.1783.15045.3092.3690.9315.66011.80279.6994.17769.0533.734315.66011.80279.6994.17757.3853.224315.66011.80279.6994.17748.9362.8370.993440.840126.65466.7204.38851.3993.4783440.840126.65466.7204.38836.8612.6053440.840126.65466.7204.38827.8252.05950.712.3252.26512.0792.21412.0312.20412.3252.26512.0792.21411.9592.19012.3252.26512.0792.21411.8892.1750.817.6323.30417.1263.19017.0283.16817.6323.30417.1263.19016.8833.13617.6323.30417.1263.19016.7403.1040.933.2186.51531.3996.04431.0585.95833.2186.51531.3996.04430.5605.83433.2186.51531.3996.04430.0785.7150.99327.37571.474206.26536.676191.84833.624327.37571.474206.26536.676173.33629.931327.37571.474206.26536.676157.74726.990100.77.1523.4807.1223.4667.1173.4647.1523.4807.1223.4667.1083.4607.1523.4807.1223.4667.0993.4550.810.6555.15710.5885.12610.5755.11910.6555.15710.5885.12610.5555.11010.6555.15710.5885.12610.5355.1010.921.14310.22220.86310.09120.80810.06521.14310.22220.86310.09120.72610.02721.14310.22220.86310.09120.6449.9890.99222.411106.264190.37691.494185.17589.089222.411106.264190.37691.494177.95885.746222.411106.264190.37691.494171.36082.6850.910.77.1523.4807.1143.4637.1073.459105.9453.88450.9092.34839.9401.987105.9453.88450.9092.34835.2931.8240.810.6555.15710.5695.11710.5525.109156.5215.76657.3862.81343.0192.281156.5215.76657.3862.81337.3082.0540.921.14310.22220.78410.05420.71410.022315.66011.80265.2863.57245.9842.698315.66011.80265.2863.57238.8022.3520.99222.411106.264183.04588.102176.97985.2923440.840126.65446.4473.18224.9661.8853440.840126.65446.4473.18218.6631.50350.712.3252.26512.0102.20011.9492.18812.3252.26512.0102.20011.8582.16912.3252.26512.0102.20011.7692.1510.817.6323.30416.9863.15916.8623.13117.6323.30416.9863.15916.6793.09117.6323.30416.9863.15916.5003.0510.933.2186.51530.9145.92230.4915.81633.2186.51530.9145.92229.8765.66533.2186.51530.9145.92229.2845.5220.99327.37571.474186.21332.474170.94829.471327.37571.474186.21332.474151.79425.902327.37571.474186.21332.474136.06123.107100.77.1523.4807.1143.4637.1073.4597.1523.4807.1143.4637.0953.4547.1523.4807.1143.4637.0843.4480.810.6555.15710.5695.11710.5525.10910.6555.15710.5695.11710.5265.09710.6555.15710.5695.11710.5015.0850.921.14310.22220.78410.05420.71410.02221.14310.22220.78410.05420.6109.97321.14310.22220.78410.05420.5069.9250.99222.411106.264183.04588.102176.97985.292222.411106.264183.04588.102168.70281.451222.411106.264183.04588.102161.27477.996Table 13
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 50 and p = 7 when there is no outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.70.3550.3790.3370.3600.3340.3560.3550.3790.3370.3600.3290.3510.3550.3790.3370.3600.3250.3460.80.5560.5930.5130.5460.5060.5380.5560.5930.5130.5460.4950.5270.5560.5930.5130.5460.4850.5160.91.1881.2720.9801.0490.9481.0151.1881.2720.9801.0490.9050.9681.1881.2720.9801.0490.8660.9270.9914.11514.9993.7783.9953.3503.54214.11514.9993.7783.9952.8653.02614.11514.9993.7783.9952.5002.63950.70.3650.3830.3640.3820.3640.3820.3650.3830.3640.3820.3640.3820.3650.3830.3640.3820.3640.3810.80.5420.5740.5400.5720.5390.5710.5420.5740.5400.5720.5390.5710.5420.5740.5400.5720.5380.5700.91.1121.1861.1031.1761.1011.1741.1121.1861.1031.1761.0981.1711.1121.1861.1031.1761.0951.1680.9913.44814.32711.94112.71811.68212.44113.44814.32711.94112.71811.31512.04913.44814.32711.94112.71810.97311.683100.70.3200.3390.3200.3390.3200.3390.3200.3390.3200.3390.3200.3390.3200.3390.3200.3390.3200.3390.80.4900.5150.4890.5150.4890.5150.4900.5150.4890.5150.4890.5150.4900.5150.4890.5150.4890.5140.91.0221.0831.0201.0811.0201.0811.0221.0831.0201.0811.0201.0801.0221.0831.0201.0811.0191.0800.9912.42413.11212.10012.77312.03812.70712.42413.11212.10012.77311.94512.61012.42413.11212.10012.77311.85412.5140.710.70.3550.3790.3180.3380.3110.3320.3550.3790.3180.3380.3030.3230.3550.3790.3180.3380.2960.3150.80.5560.5930.4680.4980.4550.4840.5560.5930.4680.4980.4370.4650.5560.5930.4680.4980.4220.4480.91.1881.2720.8070.8630.7640.8161.1881.2720.8070.8630.7090.7571.1881.2720.8070.8630.6640.7080.9914.11514.9992.0332.1441.7441.83714.11514.9992.0332.1441.4371.51114.11514.9992.0332.1441.2241.28450.70.3650.3830.3630.3810.3630.3810.3650.3830.3630.3810.3620.3800.3650.3830.3630.3810.3620.3790.80.5420.5740.5370.5690.5360.5680.5420.5740.5370.5690.5350.5670.5420.5740.5370.5690.5340.5650.91.1121.1861.0911.1631.0861.1591.1121.1861.0911.1631.0801.1521.1121.1861.0911.1631.0741.1460.9913.44814.32710.41611.0899.97810.62113.44814.32710.41611.0899.3939.99513.44814.32710.41611.0898.8809.447100.70.3200.3390.3200.3390.3200.3380.3200.3390.3200.3390.3200.3380.3200.3390.3200.3390.3200.3380.80.4900.5150.4890.5140.4890.5140.4900.5150.4890.5140.4880.5140.4900.5150.4890.5140.4880.5140.91.0221.0831.0181.0791.0171.0781.0221.0831.0181.0791.0161.0771.0221.0831.0181.0791.0151.0750.9912.42413.11211.69512.34811.56012.20612.42413.11211.69512.34811.36312.00012.42413.11211.69512.34811.17311.8010.910.70.3550.3790.3090.3290.3020.3220.3550.3790.3090.3290.2930.3110.3550.3790.3090.3290.2840.3020.80.5560.5930.4500.4780.4350.4620.5560.5930.4500.4780.4160.4410.5560.5930.4500.4780.3990.4230.91.1881.2720.7470.7980.7020.7491.1881.2720.7470.7980.6470.6891.1881.2720.7470.7980.6020.6420.9914.11514.9991.6441.7311.4021.47314.11514.9991.6441.7311.1521.20814.11514.9991.6441.7310.9861.03050.70.3650.3830.3630.3800.3620.3800.3650.3830.3630.3800.3620.3790.3650.3830.3630.3800.3610.3780.80.5420.5740.5360.5680.5350.5660.5420.5740.5360.5680.5330.5650.5420.5740.5360.5680.5320.5630.91.1121.1861.0851.1571.0791.1511.1121.1861.0851.1571.0721.1431.1121.1861.0851.1571.0641.1350.9913.44814.3279.80310.4339.3169.91313.44814.3279.80310.4338.6799.23313.44814.3279.80310.4338.1348.651100.70.3200.3390.3200.3380.3200.3380.3200.3390.3200.3380.3200.3380.3200.3390.3200.3380.3190.3380.80.4900.5150.4880.5140.4880.5140.4900.5150.4880.5140.4880.5130.4900.5150.4880.5140.4880.5130.91.0221.0831.0171.0781.0161.0771.0221.0831.0171.0781.0141.0751.0221.0831.0171.0781.0131.0730.9912.42413.11211.50312.14711.33611.97112.42413.11211.50312.14711.09411.71812.42413.11211.50312.14710.86311.476Table 14
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 50 and p = 7 when there is one outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.72.3680.3942.2660.3772.2460.3742.3680.3942.2660.3772.2180.3702.3680.3942.2660.3772.1910.3650.83.7160.6263.4350.5823.3840.5743.7160.6263.4350.5823.3110.5633.7160.6263.4350.5823.2410.5520.97.5041.3196.2081.1026.0041.0687.5041.3196.2081.1025.7251.0237.5041.3196.2081.1025.4720.9820.9988.32615.69021.5174.21418.8043.73588.32615.69021.5174.21415.7503.19188.32615.69021.5174.21413.4762.78150.70.8410.4100.8400.4100.8400.4090.8410.4100.8400.4100.8400.4090.8410.4100.8400.4100.8390.4090.81.2960.6231.2930.6211.2930.6211.2960.6231.2930.6211.2920.6201.2960.6231.2930.6211.2910.6200.92.7841.2812.7681.2712.7641.2692.7841.2812.7681.2712.7601.2672.7841.2812.7681.2712.7551.2640.9938.78815.97634.68914.22433.98013.92238.78815.97634.68914.22432.97313.49438.78815.97634.68914.22432.02913.094100.70.4280.3470.4280.3470.4280.3470.4280.3470.4280.3470.4280.3460.4280.3470.4280.3470.4280.3460.80.6690.5290.6690.5290.6690.5290.6690.5290.6690.5290.6690.5290.6690.5290.6690.5290.6680.5290.91.3991.1111.3971.1091.3971.1091.3991.1111.3971.1091.3961.1081.3991.1111.3971.1091.3951.1080.9917.30213.50016.82813.15116.73713.08417.30213.50016.82813.15116.60112.98417.30213.50016.82813.15116.46812.8860.710.72.3680.3942.1440.3582.1040.3522.3680.3942.1440.3582.0480.3442.3680.3942.1440.3581.9950.3360.83.7160.6263.1250.5353.0300.5213.7160.6263.1250.5352.9000.5033.7160.6263.1250.5352.7810.4860.97.5041.3195.0780.9184.7800.8717.5041.3195.0780.9184.3970.8107.5041.3195.0780.9184.0750.7610.9988.32615.69010.5952.2578.8261.93388.32615.69010.5952.2576.9511.58788.32615.69010.5952.2575.6541.34750.70.8410.4100.8390.4090.8380.4080.8410.4100.8390.4090.8380.4080.8410.4100.8390.4090.8370.4070.81.2960.6231.2900.6191.2880.6181.2960.6231.2900.6191.2860.6171.2960.6231.2900.6191.2840.6150.92.7841.2812.7461.2592.7391.2542.7841.2812.7461.2592.7271.2482.7841.2812.7461.2592.7161.2420.9938.78815.97630.49212.44329.27511.92938.78815.97630.49212.44327.64011.24138.78815.97630.49212.44326.19710.636100.70.4280.3470.4280.3460.4270.3460.4280.3470.4280.3460.4270.3460.4280.3470.4280.3460.4270.3460.80.6690.5290.6680.5290.6680.5280.6690.5290.6680.5290.6680.5280.6690.5290.6680.5290.6680.5280.91.3991.1111.3941.1071.3931.1061.3991.1111.3941.1071.3921.1051.3991.1111.3941.1071.3901.1030.9917.30213.50016.23612.71516.03912.56917.30213.50016.23612.71515.75112.35717.30213.50016.23612.71515.47412.1530.910.72.3680.3942.0880.3502.0400.3432.3680.3942.0880.3501.9740.3332.3680.3942.0880.3501.9110.3250.83.7160.6262.9920.5162.8820.5003.7160.6262.9920.5162.7330.4793.7160.6262.9920.5162.5990.4610.97.5041.3194.6630.8524.3480.8037.5041.3194.6630.8523.9520.7427.5041.3194.6630.8523.6250.6920.9988.32615.6908.2121.8206.7371.54888.32615.6908.2121.8205.2161.26688.32615.6908.2121.8204.1921.07850.70.8410.4100.8380.4080.8380.4080.8410.4100.8380.4080.8370.4070.8410.4100.8380.4080.8360.4060.81.2960.6231.2880.6181.2860.6171.2960.6231.2880.6181.2840.6151.2960.6231.2880.6181.2810.6130.92.7841.2812.7351.2532.7261.2472.7841.2812.7351.2532.7121.2392.7841.2812.7351.2532.6981.2310.9938.78815.97628.78611.72327.42311.15038.78815.97628.78611.72325.63010.39838.78815.97628.78611.72324.0819.753100.70.4280.3470.4270.3460.4270.3460.4280.3470.4270.3460.4270.3460.4280.3470.4270.3460.4270.3460.80.6690.5290.6680.5280.6680.5280.6690.5290.6680.5280.6670.5280.6690.5290.6680.5280.6670.5280.91.3991.1111.3931.1061.3921.1041.3991.1111.3931.1061.3901.1031.3991.1111.3931.1061.3881.1010.9917.30213.50015.95512.50815.71112.32717.30213.50015.95512.50815.35912.06717.30213.50015.95512.50815.02211.818Table 15
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 50 and p = 7 when there are two outliers.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.73.7080.4233.5460.3993.5150.3943.7080.4233.5460.3993.4700.3883.7080.4233.5460.3993.4260.3820.85.4090.6575.0510.6004.9860.5915.4090.6575.0510.6004.8910.5765.4090.6575.0510.6004.8000.5630.910.8101.3999.3291.1409.0891.10110.8101.3999.3291.1408.7541.04710.8101.3999.3291.1408.4460.9990.99113.43416.40042.5854.28937.9213.788113.43416.40042.5854.28932.3353.219113.43416.40042.5854.28927.9482.79150.71.3590.4351.3570.4341.3570.4341.3590.4351.3570.4341.3560.4341.3590.4351.3570.4341.3560.4340.82.1090.6652.1040.6632.1030.6632.1090.6652.1040.6632.1020.6622.1090.6652.1040.6632.1010.6620.94.4881.3614.4621.3514.4571.3494.4881.3614.4621.3514.4491.3464.4881.3614.4621.3514.4411.3430.9959.62317.05653.65015.21352.61314.89459.62317.05653.65015.21351.14014.44359.62317.05653.65015.21349.75814.020100.70.6300.3720.6300.3720.6300.3720.6300.3720.6300.3720.6290.3720.6300.3720.6300.3720.6290.3720.80.9750.5670.9750.5670.9750.5670.9750.5670.9750.5670.9740.5670.9750.5670.9750.5670.9740.5670.92.0161.1842.0141.1822.0131.1822.0161.1842.0141.1822.0131.1812.0161.1842.0141.1822.0121.1810.9924.54314.27923.98713.92423.88013.85624.54314.27923.98713.92423.72113.75424.54314.27923.98713.92423.56413.6540.710.73.7080.4233.3510.3713.2880.3633.7080.4233.3510.3713.1970.3513.7080.4233.3510.3713.1120.3400.85.4090.6574.6470.5414.5220.5245.4090.6574.6470.5414.3460.5005.4090.6574.6470.5414.1840.4790.910.8101.3997.9550.9267.5750.87110.8101.3997.9550.9267.0710.80310.8101.3997.9550.9266.6310.7460.99113.43416.40022.1222.24418.4111.906113.43416.40022.1222.24414.3781.547113.43416.40022.1222.24411.5341.29850.71.3590.4351.3550.4331.3540.4331.3590.4351.3550.4331.3530.4331.3590.4351.3550.4331.3520.4320.82.1090.6652.0980.6612.0960.6602.1090.6652.0980.6612.0930.6582.1090.6652.0980.6612.0900.6570.94.4881.3614.4281.3384.4161.3334.4881.3614.4281.3384.3981.3274.4881.3614.4281.3384.3811.3200.9959.62317.05647.50113.33345.71112.79059.62317.05647.50113.33343.29812.06159.62317.05647.50113.33341.16211.420100.70.6300.3720.6290.3720.6290.3720.6300.3720.6290.3720.6290.3720.6300.3720.6290.3720.6290.3710.80.9750.5670.9740.5670.9740.5660.9750.5670.9740.5670.9740.5660.9750.5670.9740.5670.9730.5660.92.0161.1842.0111.1802.0101.1792.0161.1842.0111.1802.0081.1782.0161.1842.0111.1802.0061.1760.9924.54314.27923.29113.48023.05813.33224.54314.27923.29113.48022.71813.11624.54314.27923.29113.48022.39012.9070.910.73.7080.4233.2610.3593.1850.3493.7080.4233.2610.3593.0760.3363.7080.4233.2610.3592.9750.3240.85.4090.6574.4700.5174.3220.4975.4090.6574.4700.5174.1190.4715.4090.6574.4700.5173.9330.4490.910.8101.3997.4230.8507.0040.79410.8101.3997.4230.8506.4590.72510.8101.3997.4230.8505.9910.6700.99113.43416.40017.1011.78813.9111.506113.43416.40017.1011.78810.5661.215113.43416.40017.1011.7888.2921.02250.71.3590.4351.3540.4331.3530.4321.3590.4351.3540.4331.3510.4321.3590.4351.3540.4331.3500.4310.82.1090.6652.0950.6592.0930.6582.1090.6652.0950.6592.0890.6572.1090.6652.0950.6592.0850.6550.94.4881.3614.4111.3314.3961.3264.4881.3614.4111.3314.3731.3174.4881.3614.4111.3314.3511.3090.9959.62317.05644.99012.57242.97711.96559.62317.05644.99012.57240.31911.16859.62317.05644.99012.57238.01410.483100.70.6300.3720.6290.3720.6290.3720.6300.3720.6290.3720.6290.3710.6300.3720.6290.3720.6290.3710.80.9750.5670.9740.5660.9740.5660.9750.5670.9740.5660.9730.5660.9750.5670.9740.5660.9730.5660.92.0161.1842.0091.1782.0081.1772.0161.1842.0091.1782.0061.1762.0161.1842.0091.1782.0031.1740.9924.54314.27922.96013.26922.67113.08524.54314.27922.96013.26922.25312.82024.54314.27922.96013.26921.85312.567Table 16
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 100 and p = 7 when there is no outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.70.1750.1850.1710.1820.1710.1810.1750.1850.1710.1820.1700.1800.1750.1850.1710.1820.1690.1790.80.2760.2930.2670.2830.2650.2810.2760.2930.2670.2830.2630.2780.2760.2930.2670.2830.2600.2760.90.5430.5730.5010.5290.4940.5210.5430.5730.5010.5290.4830.5100.5430.5730.5010.5290.4730.4990.996.1366.4162.9093.0552.6532.7876.1366.4162.9093.0552.3482.4686.1366.4162.9093.0552.1102.21850.70.1500.1580.1500.1580.1500.1580.1500.1580.1500.1580.1500.1580.1500.1580.1500.1580.1500.1580.80.2290.2420.2290.2410.2290.2410.2290.2420.2290.2410.2290.2410.2290.2420.2290.2410.2290.2410.90.4950.5240.4940.5230.4930.5220.4950.5240.4940.5230.4930.5220.4950.5240.4940.5230.4920.5210.995.6465.9555.3765.6695.3255.6155.6465.9555.3765.6695.2505.5375.6465.9555.3765.6695.1785.461100.70.1560.1640.1560.1640.1560.1640.1560.1640.1560.1640.1560.1640.1560.1640.1560.1640.1560.1640.80.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.90.4570.4800.4560.4800.4560.4800.4570.4800.4560.4800.4560.4800.4570.4800.4560.4800.4560.4790.995.3665.7355.3085.6725.2965.6605.3665.7355.3085.6725.2795.6415.3665.7355.3085.6725.2625.6230.710.70.1750.1850.1670.1770.1650.1750.1750.1850.1670.1770.1630.1730.1750.1850.1670.1770.1620.1710.80.2760.2930.2560.2710.2530.2680.2760.2930.2560.2710.2480.2620.2760.2930.2560.2710.2430.2570.90.5430.5730.4560.4820.4430.4670.5430.5730.4560.4820.4250.4480.5430.5730.4560.4820.4090.4310.996.1366.4161.7931.8841.5871.6676.1366.4161.7931.8841.3561.4246.1366.4161.7931.8841.1881.24650.70.1500.1580.1490.1580.1490.1580.1500.1580.1490.1580.1490.1580.1500.1580.1490.1580.1490.1580.80.2290.2420.2280.2410.2280.2410.2290.2420.2280.2410.2280.2410.2290.2420.2280.2410.2280.2400.90.4950.5240.4910.5200.4910.5190.4950.5240.4910.5200.4900.5180.4950.5240.4910.5200.4880.5170.995.6465.9555.0565.3314.9535.2235.6465.9555.0565.3314.8085.0705.6465.9555.0565.3314.6724.927100.70.1560.1640.1560.1640.1560.1640.1560.1640.1560.1640.1550.1630.1560.1640.1560.1640.1550.1630.80.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.90.4570.4800.4560.4790.4560.4790.4570.4800.4560.4790.4550.4790.4570.4800.4560.4790.4550.4790.995.3665.7355.2325.5915.2065.5635.3665.7355.2325.5915.1675.5215.3665.7355.2325.5915.1295.4810.910.70.1750.1850.1650.1750.1630.1730.1750.1850.1650.1750.1610.1700.1750.1850.1650.1750.1590.1680.80.2760.2930.2510.2660.2470.2620.2760.2930.2510.2660.2410.2550.2760.2930.2510.2660.2360.2500.90.5430.5730.4370.4620.4220.4460.5430.5730.4370.4620.4020.4240.5430.5730.4370.4620.3850.4060.996.1366.4161.5131.5891.3291.3956.1366.4161.5131.5891.1291.1836.1366.4161.5131.5890.9861.03250.70.1500.1580.1490.1580.1490.1580.1500.1580.1490.1580.1490.1580.1500.1580.1490.1580.1490.1570.80.2290.2420.2280.2410.2280.2410.2290.2420.2280.2410.2280.2400.2290.2420.2280.2410.2270.2400.90.4950.5240.4900.5190.4890.5180.4950.5240.4900.5190.4880.5170.4950.5240.4900.5190.4860.5150.995.6465.9554.9115.1784.7885.0495.6465.9554.9115.1784.6174.8685.6465.9554.9115.1784.4594.701100.70.1560.1640.1560.1640.1550.1630.1560.1640.1560.1640.1550.1630.1560.1640.1560.1640.1550.1630.80.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.2290.2430.90.4570.4800.4560.4790.4550.4790.4570.4800.4560.4790.4550.4780.4570.4800.4560.4790.4550.4780.995.3665.7355.1955.5515.1625.5165.3665.7355.1955.5515.1135.4645.3665.7355.1955.5515.0665.413Table 17
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 100 and p = 7 when there is one outlier.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.70.4140.1870.4070.1840.4060.1830.4140.1870.4070.1840.4040.1820.4140.1870.4070.1840.4020.1810.80.6060.2950.5910.2850.5880.2840.6060.2950.5910.2850.5830.2810.6060.2950.5910.2850.5790.2780.91.1550.5781.0920.5341.0810.5261.1550.5781.0920.5341.0650.5151.1550.5781.0920.5341.0490.5040.9912.0706.4777.4163.1116.9442.84212.0706.4777.4163.1116.3452.52112.0706.4777.4163.1115.8432.27050.70.2110.1600.2110.1600.2110.1600.2110.1600.2110.1600.2110.1600.2110.1600.2110.1600.2110.1600.80.3160.2460.3160.2450.3150.2450.3160.2460.3160.2450.3150.2450.3160.2460.3160.2450.3150.2450.90.6620.5310.6610.5290.6600.5290.6620.5310.6610.5290.6600.5290.6620.5310.6610.5290.6590.5280.997.1906.0186.8985.7316.8435.6787.1906.0186.8985.7316.7635.5997.1906.0186.8985.7316.6855.523100.70.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.80.2950.2490.2950.2490.2950.2490.2950.2490.2950.2490.2950.2490.2950.2490.2950.2490.2950.2490.90.6030.4990.6030.4980.6020.4980.6030.4990.6030.4980.6020.4980.6030.4990.6030.4980.6020.4980.996.7385.8706.6665.8066.6515.7946.7385.8706.6665.8066.6305.7756.7385.8706.6665.8066.6095.7570.710.70.4140.1870.3990.1790.3960.1770.4140.1870.3990.1790.3920.1750.4140.1870.3990.1790.3880.1730.80.6060.2950.5720.2740.5660.2700.6060.2950.5720.2740.5570.2650.6060.2950.5720.2740.5490.2600.91.1550.5781.0230.4871.0020.4731.1550.5781.0230.4870.9730.4531.1550.5781.0230.4870.9460.4360.9912.0706.4775.1191.9334.6121.71312.0706.4775.1191.9334.0041.46712.0706.4775.1191.9333.5281.28650.70.2110.1600.2110.1600.2110.1600.2110.1600.2110.1600.2110.1600.2110.1600.2110.1600.2100.1600.80.3160.2460.3150.2450.3150.2450.3160.2460.3150.2450.3150.2450.3160.2460.3150.2450.3140.2440.90.6620.5310.6580.5270.6570.5260.6620.5310.6580.5270.6560.5250.6620.5310.6580.5270.6550.5240.997.1906.0186.5515.3936.4395.2857.1906.0186.5515.3936.2805.1327.1906.0186.5515.3936.1314.988100.70.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.80.2950.2490.2950.2490.2950.2480.2950.2490.2950.2490.2950.2480.2950.2490.2950.2490.2950.2480.90.6030.4990.6020.4980.6020.4980.6030.4990.6020.4980.6020.4970.6030.4990.6020.4980.6010.4970.996.7385.8706.5725.7246.5395.6966.7385.8706.5725.7246.4925.6546.7385.8706.5725.7246.4455.6130.910.70.1630.1730.4140.1870.3950.1770.4140.1870.3950.1770.3870.1720.4140.1870.3950.1770.3820.1700.80.2470.2620.6060.2950.5640.2680.6060.2950.5640.2680.5460.2580.6060.2950.5640.2680.5360.2520.90.4220.4461.1550.5780.9930.4671.1550.5780.9930.4670.9350.4291.1550.5780.9930.4670.9050.4110.991.3291.39512.0706.4774.4221.63412.0706.4774.4221.6343.3541.22212.0706.4774.4221.6342.9151.06850.70.1490.1580.2110.1600.2110.1600.2110.1600.2110.1600.2100.1600.2110.1600.2110.1600.2100.1600.80.2280.2410.3160.2460.3150.2450.3160.2460.3150.2450.3140.2440.3160.2460.3150.2450.3140.2440.90.4890.5180.6620.5310.6570.5260.6620.5310.6570.5260.6540.5230.6620.5310.6570.5260.6520.5220.994.7885.0497.1906.0186.3935.2417.1906.0186.3935.2416.0694.9307.1906.0186.3935.2415.8944.762100.70.1550.1630.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.1990.1680.80.2290.2430.2950.2490.2950.2480.2950.2490.2950.2480.2940.2480.2950.2490.2950.2480.2940.2480.90.4550.4790.6030.4990.6020.4980.6030.4990.6020.4980.6010.4970.6030.4990.6020.4980.6010.4970.995.1625.5166.7385.8706.5265.6846.7385.8706.5265.6846.4255.5966.7385.8706.5265.6846.3665.544Table 18
Estimated MSEs of OLS, MRE, MRE, MRT, and RTMME atn = 100 and p = 7 when there are two outliers.
kSigmaRhod = 0.2d = 0.5d = 0.8α^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMMEα^α^Mα^kα^Mkα^MRTα^RTMME0.310.70.8350.1920.8270.1880.8250.1880.8350.1920.8270.1880.8230.1870.8350.1920.8270.1880.8200.1860.81.2200.3021.2000.2921.1960.2901.2200.3021.2000.2921.1910.2881.2200.3021.2000.2921.1850.2850.92.3040.5892.2200.5462.2040.5382.3040.5892.2200.5462.1810.5272.3040.5892.2200.5462.1580.5160.9923.2016.58015.7803.20214.8922.92923.2016.58015.7803.20213.7302.60423.2016.58015.7803.20212.7252.34850.70.2640.1630.2640.1630.2640.1630.2640.1630.2640.1630.2640.1630.2640.1630.2640.1630.2640.1630.80.4010.2510.4000.2500.4000.2500.4010.2510.4000.2500.4000.2500.4010.2510.4000.2500.3990.2500.90.8300.5400.8270.5380.8260.5380.8300.5400.8270.5380.8250.5370.8300.5400.8270.5380.8240.5370.998.9316.1028.5465.8108.4745.7558.9316.1028.5465.8108.3695.6758.9316.1028.5465.8108.2665.598100.70.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.80.4630.2650.4630.2650.4630.2650.4630.2650.4630.2650.4630.2650.4630.2650.4630.2650.4630.2650.90.9490.5370.9480.5370.9480.5370.9490.5370.9480.5370.9480.5370.9490.5370.9480.5370.9480.5370.9910.0996.19210.0006.1269.9806.11310.0996.19210.0006.1269.9516.09410.0996.19210.0006.1269.9216.0750.710.70.8350.1920.8160.1840.8130.1820.8350.1920.8160.1840.8070.1800.8350.1920.8160.1840.8020.1790.81.2200.3021.1750.2811.1670.2771.2200.3021.1750.2811.1550.2721.2200.3021.1750.2811.1430.2670.92.3040.5892.1200.4992.0880.4852.3040.5892.1200.4992.0420.4662.3040.5892.1200.4991.9980.4500.9923.2016.58011.2312.00410.1551.77923.2016.58011.2312.0048.8361.52623.2016.58011.2312.0047.7791.33950.70.2640.1630.2640.1630.2640.1620.2640.1630.2640.1630.2630.1620.2640.1630.2640.1630.2630.1620.80.4010.2510.3990.2500.3990.2500.4010.2510.3990.2500.3980.2490.4010.2510.3990.2500.3980.2490.90.8300.5400.8230.5360.8220.5350.8300.5400.8230.5360.8200.5330.8300.5400.8230.5360.8180.5320.998.9316.1028.0925.4667.9475.3568.9316.1028.0925.4667.7415.2008.9316.1028.0925.4667.5485.053100.70.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.80.4630.2650.4620.2650.4620.2650.4630.2650.4620.2650.4620.2650.4630.2650.4620.2650.4620.2650.90.9490.5370.9480.5360.9470.5360.9490.5370.9480.5360.9470.5360.9490.5370.9480.5360.9460.5360.9910.0996.1929.8706.0419.8266.01210.0996.1929.8706.0419.7605.96910.0996.1929.8706.0419.6955.9260.910.70.8350.1920.8110.1820.8070.1800.8350.1920.8110.1820.8000.1780.8350.1920.8110.1820.7940.1760.81.2200.3021.1630.2761.1530.2711.2200.3021.1630.2761.1380.2651.2200.3021.1630.2761.1230.2600.92.3040.5892.0740.4802.0350.4642.3040.5892.0740.4801.9800.4432.3040.5892.0740.4801.9290.4240.9923.2016.5809.7461.6988.6701.49623.2016.5809.7461.6987.3891.27323.2016.5809.7461.6986.3941.11350.70.2640.1630.2640.1620.2630.1620.2640.1630.2640.1620.2630.1620.2640.1630.2640.1620.2630.1620.80.4010.2510.3990.2490.3980.2490.4010.2510.3990.2490.3980.2490.4010.2510.3990.2490.3970.2490.90.8300.5400.8210.5340.8190.5330.8300.5400.8210.5340.8170.5310.8300.5400.8210.5340.8140.5300.998.9316.1027.8865.3107.7125.1788.9316.1027.8865.3107.4694.9948.9316.1027.8865.3107.2444.824100.70.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.3110.1780.80.4630.2650.4620.2650.4620.2650.4630.2650.4620.2650.4620.2650.4630.2650.4620.2650.4620.2650.90.9490.5370.9470.5360.9470.5360.9490.5370.9470.5360.9460.5360.9490.5370.9470.5360.9460.5350.9910.0996.1929.8076.0009.7505.96310.0996.1929.8076.0009.6675.90810.0996.1929.8076.0009.5855.855


## 6. Numerical Example

Hussein data were originally adopted by Hussein and Zari [18] and later by Lukman et al. [19, 20]. The data contain 31 observations and three explanatory variables. The variance inflation factor for the three explanatory variables is greater than 10 (VIF> 10), which indicates multicollinearity. Lukman et al. [20] identified the following observations as outliers in the y-direction: 12, 14, 15, 16, 30, and 31. Hence, this data set suffers both problems considered in this study. Table 19 shows that the estimated MSE value of the new estimator, RTMME, is smaller compared to the ridge, M-ridge, and MRT estimator.Table 19
Estimated MSE values of the OLSE, ME, RE, RME, MRT, and RTMME in Hussein data set.
Coefficientsαα^Mα^kα^MkHMα^MRTkHMR,dα^RTMMEk˜HMR,dβ0208.874173.331178.6184155.4684178.6184155.4684β1−2.0171−1.8846−2.0167−1.8844−2.0167−1.8844β21.515331.63401.51531.63391.51531.6339β3−1.3143−1.3527−1.3143−1.3527−1.3143−1.3527MSE20875.629751.862269.331952.602269.33629.58k=1.9136,kHM=1.2980,kHMR=0.9813,kHMR=0.6656, and d=0.95.The theoretical results in this study are validated through the Hussein data as follows:(i)
∑i=1pΩii=883.55<∑i=1pσ2λi−1=1850.482; thus, MSEα^RTMMEk,d<MSEα^MRTk,d for every k > 0. By Theorem 1, this implies that the RTMME outperforms the MRT estimator.(ii)
From Theorem2, k1i is calculated as [0.2470, 0.0520, 0.0519, 0.0002]. For k > 1.2980 (k> k1i), MSEα^RTMMEk,d<MSEα^Mk.(iii)
The necessary condition from Theorem3 is also proven from ∑i=1pΩii<∑i=1p∑i=1pk1+dαi2/2λi+k1+d=3026.59 =>MSEα^RTMMEk,d<MSEα^M.


## 7. Conclusion

We proposed a two-parameter ridge-type modified M-estimator to jointly handle the problem of multicollinearity and outliers in a linear regression model. Theoretically, the new estimator outperforms the existing estimators under certain conditions. The results of the simulation study and numerical example agree with the theoretical findings. A right choice ofk and d also produces better estimates using the proposed estimator. Thus, in the presence of multicollinearity and outliers, this estimator can effectively replace the following estimators: the ordinary least squares estimator, the M-estimator, the ridge estimator, the M-ridge estimator, and the modified ridge-type estimator.



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*Source: 3192852-2020-05-15.xml*