correction maximum likelihood estimation method

ppt

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2. COMPARISONS OF MAXIMUM LIKELIHOOD AND BIASED ESTIMATION METHODS USING GENERALIZED LINER MODELS M.Phil . Scholar: Syeda Salma Kazmi Supervised by: Dr. Atif Abbasi Department of Statistics King Abdullah Campus Chatter Kalas The University of Azad Jammu and Kashmir       9/25/2024 2

3. OUTLINES INTODUCTION MATERIAL AND METHODS SUMMARY AND CONCLUSIONS REFERENCES 9/25/2024 3

4. INTRODUCTION The normal linear models are based on certain assumptions such as; the mean of Y (the explained variable) should be a linear combination of the explanatory variables X’s. Additionally, with constant variance the distribution of Y is presumed to be normal. However, there are numerous experimental conditions in which the linearity and or the normality of the explained Y may not be completely applicable. For instance, a discrete random variable is a binary with two possible outcomes, success, or failure GLMs offer a method of combining several different statistical models, such as linear regression, binary regression, Poisson regression, gamma, and negative binomial regressions, etc. GLMs can handle both discrete and continuous responses, and the standard assumptions of normality and homoscedasticity are not always made on the concerned explained variable. 9/25/2024 4

5. CONTINUE Generalized linear model" is the phrase GLM is a larger class of models that McCullagh and Nelder (1982, 2nd edition 1989) proposed. In this model, the response variable is anticipated to adhere to an exponential family distribution, where the mean is represented by This distribution is made up of specific (often nonlinear) functions of , which some would refer to as "nonlinear." Although the covariates have a history of being nonlinear, McCullagh and Nelder reflect the covariates as linear because the variables only have an effect on the distribution of yi through the linear combination . 9/25/2024 5

6. COMPONENTS OF GLMs 9/25/2024 6 Random component The random component represents the conditional distribution of the dependent variable (for the i- th of n independently sampled observations), given the values of the independent variables in the model. The distribution of Yi, according to Nelder and Wedderburn's original formulation, is a member of an exponential family, such as the binomial, normal, Gamma, Poisson, or Inverse-Gaussian distributions etc. The probability density function of exponential family distributions is commonly represented as Systematic component The linear predictor is constructed as a linear combination of explanatory factors. which relates, η to a set of independent variables, X1 ,X2 ,…X k  

7. Link Function A flat and invertible linearizing link function g(. ), which converts the expected value of response variable's µi = E(Yi) to the linear predictor (John, 2015): where β is a q=p×1 vector of unidentified parameters and Xi = (x1 ,x2 ,... xk ) ′ is the vector of explanatory variables.The association between the linear predictor and the expected value of the random component is described by link function as   9/25/2024 7

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9. OBJECTIVES The undertaking research is being conducted to Estimate the parameters of the GLMs by considering binomial and Poisson regression models. Compare the maximum likelihood estimator with biased estimators like Ridge and Principal component regression estimators. Explore the effects of multicollinearity on the performance of the estimators. 9/25/2024 9

10. MATERIALS AND METHODS Various method are described which are used in GLMs for estimating the parameters . Maximum Likelihood Estimation Newton Raphson method Method of Fisher’s Scoring BIASED ESTIMATION METHODS Ridge Regression Generalizing Principal Component Regression Mean squares error 9/25/2024 10

11. Non-Linear Regression Models. Non-Linear Regression Models. we describe non-linear regression models such as Poisson and binomial regression models. Binomial Regression It is assumed that p(x) is solely dependent on the regressors' x values via a linear combination of β′x, where β is an unknown . Pdf of binomial distribution is n is number of trails, p is the probability of success. Poisson Regression The rate parameter is thought to depend on the regression with the linear predictors β′(x) through the link function when the data are to be treated as Poisson counts. Pdf of Poisson distribution is   9/25/2024 11

12. Method for detecting multicolinearity Multicolinearity Frisch (1934) was first introduced the term multicollinearity . The full rank condition in a multiple linear regression model denotes the independence of the regressors . In the event that this assumption is incorrect, multicollinearity becomes a problem and the least squares estimation breaks down. Multicollinearity is the high degree of correlation between many independent variable s. There are many criteria for the detection of multicollinearity in GLMs two of those are mentioned in the following section. Condition number In mathematical analysis, a function's condition number (CN) indicates how much a little change in the input argument can change the function's output significance. The high CN value indicated that multicollinearity is an issue. CN has the following mathematical definition: If k˃30 then there exists a multicollinearity problem. Variance inflation factor For the detection of multicollinearity one method is to calculate variance inflation factor (VIF) for each explanatory variable, value of VIF greater than 1.5 shows multicollinearity. The range of VIF signifies level of multicollinearity. The value of VIF ‘1’ is non-collinear, which is considering being negligible. Values of VIF ranging between 1 and 5 are moderate. It shows the medium level of collinearity. Values of VIF greater than 5 are highly collinear.   9/25/2024 12

13. Example:Apple juice data 4.1 . Numerical Example 1: Binomial Regression (Apple Juice data) In this section we explain the use of the maximum likelihood (ML), principal component regression (PCR) and ridge estimators to a real-life data set. To fit a binomial regression model we considered an apple juice data which has been used by Pena et al. (2011) and Özkale (2016). There are four explanatory variables in this data set which are pH (x 1 ), nisin concentration (IU/ml) (x 2 ), incubation temperature (C) (x 3 ), and soluble solids concentration (Brix) (x 4 ). The response variable is the amount of Alicyclobacillus acidoterrestris growing in apple juice, where 1 indicates growth and 0 indicates no growth. Prior to calculating the results, we standardized the independent variables and subsequently incorporated the intercept term into the model. Then, the logistic regression model is where denotes the i- th observation for the j- th explanatory variable and The eigenvalues of the matrix are obtained as λ 1 = 4.2143, λ 2 = 0.1774, λ 3 = 0.1145, λ 4 = 0.0718 and λ 5 = 0.0303       9/25/2024 13

14. . The condition number (CN) is computed as . As, the value of condition number is very large which indicates that there is a multicollinearity issue within this dataset. First, we obtain the ML estimator by an iterative procedure. For iterative ML algorithms, we generally choose (sufficiently close to zero) as the convergence criterion. The iteration ends if the norm of the difference in the parameter estimates between iterations is less than , such as where m represents the iteration step. The ordinary least square (OLS) estimator is considered as an initial estimate having the values; the initial working response variable is defined as and is the initial weight matrix computed as The ridge regression parameter is computed by following Abbasi and Özkale (2021) such as which gives 8.0874 other value of k is also selected randomly such as . For choosing the amount of principal components (PCs) we used the percentage of total variation (PTV) method. The PTV method is defined as follows: where r denotes the amount of PCs retained in the model. In this example the amount of PCs retain in the model is r =2. The results are shown in the following Table 4.1 shows the results of iteratively obtain estimators along with the SMSE values. It is seen that ridge estimators acquire the smallest SMSE as compared to ML and PCR estimators for k 1 . Then, PCR has smaller SMSE value. While the SMSE value of ML estimator is largest, which shows that multicollinearity affects the performance of ML estimator. The results show that the presentation of the ridge estimator is best amongst all other estimators to encounter the multicollinearity problem for k 1 . However, for k 2 the PCR estimator has smaller SMSE value as compared to ridge estimator. Thus, the ridge estimator performs better for large value of k whereas for small k value PCR performs better than ridge estimator. Table 4.1 shows the results only for two values of k while figure 4.1 is given to assess the performance of the estimators for remaining values of k. From figure 4.1 it is clear when the k values fall below approximately 0.14 then PCR estimator performs better than ridge estimator. However, when k is greater than approximately 0.14 then ridge estimator outperforms its counterparts.     9/25/2024 14

15. Table 4.1 Iteratively obtain estimators and their SMSE values for k1 and k2 Binomial response (Apple Juice data) ML Ridge PCR Ridge Coefficients =8.0874 =8.0874 -1.3159 -0.3187 38.5127 -1.2817 8.9941 0.1184 -7.1565 8.7376 -10.7939 -0.1598 -9.0721 -10.4668 6.1903 0.0671 -3.1447 5.9583 -5.8053 -0.0911 -6.2584 -5.6525 SMSE 61.4822 4.2419 10.0057 58.8241 ML Ridge PCR Ridge Coefficients -1.3159 -0.3187 38.5127 -1.2817 8.9941 0.1184 -7.1565 8.7376 -10.7939 -0.1598 -9.0721 -10.4668 6.1903 0.0671 -3.1447 5.9583 -5.8053 -0.0911 -6.2584 -5.6525 SMSE 61.4822 4.2419 10.0057 58.8241 9/25/2024 15

16. Figure 4.1: SMSE values of the estimators for different k values 9/25/2024 16

17. Table 4.2 Iteratively obtain estimators and their SMSE values for k1 and k2. ML Ridge PCR Ridge Coefficients =0.07776 =0.255 0.1262 0.1450 0.92500 0.1811 1.5576 1.5226 0.1673 1.4448 2.6709 2.5805 0.3033 2.4004 -1.4157 -1.3522 -0.1210 -1.2281 3.8847 3.0314 25.8305 2.5819 SMSE 0.1262 0.1450 0.92500 0.1811 ML Ridge PCR Ridge Coefficients 0.1262 0.1450 0.92500 0.1811 1.5576 1.5226 0.1673 1.4448 2.6709 2.5805 0.3033 2.4004 -1.4157 -1.3522 -0.1210 -1.2281 3.8847 3.0314 25.8305 2.5819 SMSE 0.1262 0.1450 0.92500 0.1811 9/25/2024 17

18. Figure 4.2: SMSE values of the estimators for different k values 9/25/2024 18

19. T able 4.3 Estimated MSE values of the estimators when p=4 and binomial response N ML Ridge PCR 50 0.75 0.85 0.95 0.99 41.4072 67.7412 154.9522 902.0276 0.0018 0.0013 0.0018 0.0010 16.5545 30.9442 11.0888 0.5428 100 0.75 0.85 0.95 0.99 31.3085 43.6502 146.4066 668.7058 0.0011 0.0021 0.0019 0.0020 10.2321 12.7278 9.4802 0.4604 200 0.75 0.85 0.95 0.99 32.2094 47.1508 124.7506 701.2450 0.0005 0.0019 0.0015 0.0015 9.1016 13.2858 8.9455 0.4862 400 0.75 0.85 0.95 0.99 33.3879 48.1748 104.3254 687.4707 0.0017 0.0003 0.0020 0.0020 9.0938 12.8648 18.3423 0.5186 600 0.75 0.85 0.95 0.99 28.1283 49.0983 120.9762 647.6524 0.0013 0.0023 0.0016 0.0020 7.4680 12.7745 10.1470 0.4944 800 0.75 0.85 0.95 0.99 30.5335 46.5441 119.2554 655.6240 0.0024 0.0016 0.0013 0.0020 8.0997 12.1224 9.8435 0.5136 50 0.75 0.85 0.95 0.99 41.4072 67.7412 154.9522 902.0276 0.0018 0.0013 0.0018 0.0010 16.5545 30.9442 11.0888 0.5428 100 0.75 0.85 0.95 0.99 31.3085 43.6502 146.4066 668.7058 0.0011 0.0021 0.0019 0.0020 10.2321 12.7278 9.4802 0.4604 200 0.75 0.85 0.95 0.99 32.2094 47.1508 124.7506 701.2450 0.0005 0.0019 0.0015 0.0015 9.1016 13.2858 8.9455 0.4862 400 0.75 0.85 0.95 0.99 33.3879 48.1748 104.3254 687.4707 0.0017 0.0003 0.0020 0.0020 9.0938 12.8648 18.3423 0.5186 600 0.75 0.85 0.95 0.99 28.1283 49.0983 120.9762 647.6524 0.0013 0.0023 0.0016 0.0020 7.4680 12.7745 10.1470 0.4944 800 0.75 0.85 0.95 0.99 30.5335 46.5441 119.2554 655.6240 0.0024 0.0016 0.0013 0.0020 8.0997 12.1224 9.8435 0.5136 9/25/2024 19

20. Table 4.4 Estimated MSE values of the estimators when p=6 and binomial response. N ML Ridge PCR 50 0.75 0.85 0.95 0.99 76.1947 150.1024 217.1285 1764.5085 0.0008 0.0049 0.0022 0.0009 22.6668 40.9033 37.0063 0.5074 100 0.75 0.85 0.95 0.99 52.7569 130.6881 197.5604 1394.5707 0.0005 0.0012 0.0040 0.0012 17.7369 26.7614 28.7083 0.4354 200 0.75 0.85 0.95 0.99 48.4059 82.4005 228.0517 1136.2293 0.0016 0.0017 0.0011 0.0001 13.9079 17.0723 21.2104 0.4485 400 0.75 0.85 0.95 0.99 45.0608 74.5023 187.0555 976.9752 0.0019 0.0018 0.0034 0.0020 12.0324 15.2926 26.5472 0.4178 600 0.75 0.85 0.95 0.99 50.2852 76.4025 234.3184 1080.1857 0.0004 0.0022 0.0023 0.0021 13.2633 15.1403 20.9875 0.4149 800 0.75 0.85 0.95 0.99 50.2052 73.5587 215.4897 1009.1168 0.0014 0.0004 0.0004 0.0001 13.2322 15.1288 19.7981 0.4222 50 0.75 0.85 0.95 0.99 76.1947 150.1024 217.1285 1764.5085 0.0008 0.0049 0.0022 0.0009 22.6668 40.9033 37.0063 0.5074 100 0.75 0.85 0.95 0.99 52.7569 130.6881 197.5604 1394.5707 0.0005 0.0012 0.0040 0.0012 17.7369 26.7614 28.7083 0.4354 200 0.75 0.85 0.95 0.99 48.4059 82.4005 228.0517 1136.2293 0.0016 0.0017 0.0011 0.0001 13.9079 17.0723 21.2104 0.4485 400 0.75 0.85 0.95 0.99 45.0608 74.5023 187.0555 976.9752 0.0019 0.0018 0.0034 0.0020 12.0324 15.2926 26.5472 0.4178 600 0.75 0.85 0.95 0.99 50.2852 76.4025 234.3184 1080.1857 0.0004 0.0022 0.0023 0.0021 13.2633 15.1403 20.9875 0.4149 800 0.75 0.85 0.95 0.99 50.2052 73.5587 215.4897 1009.1168 0.0014 0.0004 0.0004 0.0001 13.2322 15.1288 19.7981 0.4222 9/25/2024 20

21. Table 4.5 Estimated MSE values of the estimators when p=8 and Binomial response N ML Ridge PCR 50 0.75 0.85 0.95 0.99 161.0264 251.4752 522.0259 4608.8392 0.0027 0.0006 0.0005 0.0010 67.6362 124.4677 40.6199 0.5098 100 0.75 0.85 0.95 0.99 90.7808 147.3133 324.2545 2035.3495 0.0029 0.0009 0.0025 0.0015 24.6248 36.8634 22.2733 0.4041 200 0.75 0.85 0.95 0.99 76.7998 115.6691 322.8567 1719.4271 0.0008 0.0024 0.0025 0.0006 17.1233 26.6892 28.7476 0.3783 400 0.75 0.85 0.95 0.99 69.5682 108.1238 313.2324 1640.6706 0.0010 0.0003 0.0002 0.0017 15.9373 23.8345 29.1138 0.3532 600 0.75 0.85 0.95 0.99 67.5956 94.2755 304.6276 1570.9421 0.0006 0.0017 0.0120 0.0019 14.8812 20.9240 27.5979 0.3532 800 0.75 0.85 0.95 0.99 70.6118 106.0120 323.7606 1526.1289 0.0013 0.0011 0.0003 0.0012 15.4955 23.3277 30.8585 0.3985 50 0.75 0.85 0.95 0.99 161.0264 251.4752 522.0259 4608.8392 0.0027 0.0006 0.0005 0.0010 67.6362 124.4677 40.6199 0.5098 100 0.75 0.85 0.95 0.99 90.7808 147.3133 324.2545 2035.3495 0.0029 0.0009 0.0025 0.0015 24.6248 36.8634 22.2733 0.4041 200 0.75 0.85 0.95 0.99 76.7998 115.6691 322.8567 1719.4271 0.0008 0.0024 0.0025 0.0006 17.1233 26.6892 28.7476 0.3783 400 0.75 0.85 0.95 0.99 69.5682 108.1238 313.2324 1640.6706 0.0010 0.0003 0.0002 0.0017 15.9373 23.8345 29.1138 0.3532 600 0.75 0.85 0.95 0.99 67.5956 94.2755 304.6276 1570.9421 0.0006 0.0017 0.0120 0.0019 14.8812 20.9240 27.5979 0.3532 800 0.75 0.85 0.95 0.99 70.6118 106.0120 323.7606 1526.1289 0.0013 0.0011 0.0003 0.0012 15.4955 23.3277 30.8585 0.3985 9/25/2024 21

22. Table4. 6 Estimated MSE values of the estimators when p=4 and Poisson response N ML Ridge PCR 50 0.75 0.85 0.95 0.99 5.8703 8.9103 27.5446 160.6970 0.0027 0.0219 0.0016 0.0006 4.2111 6.1693 7.2321 1.2465 100 0.75 0.85 0.95 0.99 6.9027 9.5563 21.5211 203.2916 0.0185 0.0112 0.0037 0.0011 5.9813 6.1911 5.4875 0.4825 200 0.75 0.85 0.95 0.99 7.8466 10.4445 26.3674 153.5938 0.0045 0.0122 0.0024 0.0003 6.4162 7.6284 11.3546 1.2875 400 0.75 0.85 0.95 0.99 7.4201 10.3342 28.8553 140.3262 0.0008 0.0108 0.0003 0.0014 5.7976 7.7587 13.7033 0.2718 600 0.75 0.85 0.95 0.99 7.2841 10.3102 29.3240 154.0565 0.0024 0.0017 0.0014 0.0002 6.2151 7.9933 11.2555 8.8921 800 0.75 0.85 0.95 0.99 7.2601 10.7023 31.2335 152.1678 0.0046 0.0034 0.0005 0.0016 5.9231 7.8649 10.2621 1.8070 50 0.75 0.85 0.95 0.99 5.8703 8.9103 27.5446 160.6970 0.0027 0.0219 0.0016 0.0006 4.2111 6.1693 7.2321 1.2465 100 0.75 0.85 0.95 0.99 6.9027 9.5563 21.5211 203.2916 0.0185 0.0112 0.0037 0.0011 5.9813 6.1911 5.4875 0.4825 200 0.75 0.85 0.95 0.99 7.8466 10.4445 26.3674 153.5938 0.0045 0.0122 0.0024 0.0003 6.4162 7.6284 11.3546 1.2875 400 0.75 0.85 0.95 0.99 7.4201 10.3342 28.8553 140.3262 0.0008 0.0108 0.0003 0.0014 5.7976 7.7587 13.7033 0.2718 600 0.75 0.85 0.95 0.99 7.2841 10.3102 29.3240 154.0565 0.0024 0.0017 0.0014 0.0002 6.2151 7.9933 11.2555 8.8921 800 0.75 0.85 0.95 0.99 7.2601 10.7023 31.2335 152.1678 0.0046 0.0034 0.0005 0.0016 5.9231 7.8649 10.2621 1.8070 9/25/2024 22

23. Table 4.7 Estimated MSE values of the estimators when p=6 and Poisson response N ML Ridge PCR 50 0.75 0.85 0.95 0.99 12.4254 18.5709 47.5762 265.5990 0.0007 0.0005 0.0143 0.0001 6.11885 7.6368 8.0257 0.1356 100 0.75 0.85 0.95 0.99 10.4484 14.6152 59.7272 284.5081 0.0074 0.0023 0.0005 0.0006 8.2106 9.8361 6.1725 0.1699 200 0.75 0.85 0.95 0.99 10.6856 16.5700 46.2691 302.0588 0.0049 0.0003 0.0017 0.0005 8.1323 10.8708 15.5362 0.2583 400 0.75 0.85 0.95 0.99 11.9427 20.6798 53.5402 252.3058 0.0258 0.0018 0.0129 0.0095 10.5445 14.8662 20.0278 0.1644 600 0.75 0.85 0.95 0.99 12.1469 17.9758 48.2305 259.9044 0.0124 0.0391 0.0119 0.0006 10.4575 13.1749 21.6331 0.5071 800 0.75 0.85 0.95 0.99 12.2739 17.4516 47.1687 262.7534 0.0180 0.0056 0.0466 0.0004 10.9758 12.6008 19.5004 0.3366 50 0.75 0.85 0.95 0.99 12.4254 18.5709 47.5762 265.5990 0.0007 0.0005 0.0143 0.0001 6.11885 7.6368 8.0257 0.1356 100 0.75 0.85 0.95 0.99 10.4484 14.6152 59.7272 284.5081 0.0074 0.0023 0.0005 0.0006 8.2106 9.8361 6.1725 0.1699 200 0.75 0.85 0.95 0.99 10.6856 16.5700 46.2691 302.0588 0.0049 0.0003 0.0017 0.0005 8.1323 10.8708 15.5362 0.2583 400 0.75 0.85 0.95 0.99 11.9427 20.6798 53.5402 252.3058 0.0258 0.0018 0.0129 0.0095 10.5445 14.8662 20.0278 0.1644 600 0.75 0.85 0.95 0.99 12.1469 17.9758 48.2305 259.9044 0.0124 0.0391 0.0119 0.0006 10.4575 13.1749 21.6331 0.5071 800 0.75 0.85 0.95 0.99 12.2739 17.4516 47.1687 262.7534 0.0180 0.0056 0.0466 0.0004 10.9758 12.6008 19.5004 0.3366 9/25/2024 23

24. Table 4.8 Estimated MSE values of the estimators when p=8 and Poisson response N ML Ridge PCR 50 0.75 0.85 0.95 0.99 17.1718 26.1310 72.3064 348.9901 0.0093 0.0042 0.0011 0.0016 10.0858 13.0456 20.6005 0.7934 100 0.75 0.85 0.95 0.99 14.3551 27.4195 70.7129 342.7475 0.0058 0.0552 0.0087 0.0004 10.5345 13.1810 17.4303 0.2653 200 0.75 0.85 0.95 0.99 14.8707 28.5718 86.1708 355.3073 0.0187 0.0056 0.0026 0.0001 11.5925 16.1257 19.8846 1.3481 400 0.75 0.85 0.95 0.99 17.0676 24.1639 71.4218 353.4743 0.0101 0.0089 0.0118 0.0006 13.3476 15.6384 24.7161 0.1139 600 0.75 0.85 0.95 0.99 15.0044 26.6676 58.5547 380.5351 0.0097 0.0031 0.0691 0.0027 12.3790 19.6386 24.2047 0.6782 800 0.75 0.85 0.95 0.99 15.9130 24.0228 72.4139 369.5784 0.0168 0.0090 0.1495 0.0015 13.4027 16.7514 24.5364 0.1305 50 0.75 0.85 0.95 0.99 17.1718 26.1310 72.3064 348.9901 0.0093 0.0042 0.0011 0.0016 10.0858 13.0456 20.6005 0.7934 100 0.75 0.85 0.95 0.99 14.3551 27.4195 70.7129 342.7475 0.0058 0.0552 0.0087 0.0004 10.5345 13.1810 17.4303 0.2653 200 0.75 0.85 0.95 0.99 14.8707 28.5718 86.1708 355.3073 0.0187 0.0056 0.0026 0.0001 11.5925 16.1257 19.8846 1.3481 400 0.75 0.85 0.95 0.99 17.0676 24.1639 71.4218 353.4743 0.0101 0.0089 0.0118 0.0006 13.3476 15.6384 24.7161 0.1139 600 0.75 0.85 0.95 0.99 15.0044 26.6676 58.5547 380.5351 0.0097 0.0031 0.0691 0.0027 12.3790 19.6386 24.2047 0.6782 800 0.75 0.85 0.95 0.99 15.9130 24.0228 72.4139 369.5784 0.0168 0.0090 0.1495 0.0015 13.4027 16.7514 24.5364 0.1305 9/25/2024 24

25. Summary and Conclusions This study is aiming to give a smooth approach what to do when facing the problem of multicollinearity. Specifically in generalized linear models where the response variable may not be normally distributed. For the detection of this problem two methods were discussed in this study that are condition number (CN) and variance inflation factor (VIF) which suggest the level of multicollinearity. This problem can overcome by using biased estimation methods. These methods are ridge, PCR and ML estimation. The study includes two non-linear regression models for the estimation of parameters that are Poisson and binomial regression models. For ML estimation Iterative reweighted least square technique is used. Two iterative procedures Newton-Raphson and Fisher’s scoring are used for estimation. 9/25/2024 25

26. Continue Monte Carlo Simulation experiment also used in the study for binomial and Poisson response for different sizes and different independent variables.The performance evaluation criteria of this study are the expected mean square error (EMSE). The results show that the ridge estimator obtains the smallest SMSE as compared to ML and PCR estimators for numerical examples as well as for the simulation studies. It is conclude that Ridge estimator is the best amongst three for large value of k, while for smaller value of k PCR performs better. 9/25/2024 26

27. REFERENCES Abbasi, A., & Özkale, R. (2021). The r-k class estimator in generalized linear models applicable with simulation and empirical study using a Poisson and Gamma responses. Hacettepe Journal of Mathematics and Statistics, 50(2), 594-611.. Abdulkabir, M., Edem, U., Tunde, R., & Kemi, B. (2015). An empirical study of generalized linear model for count data. Journal of Applied and Computational Mathematics, 4, 253. Agresti, A. (2015). Foundations of linear and generalized linear models. John Wiley & Sons. Akay, K. U., & Ertan, E. (2022). A new improved Liu-type estimator for Poisson regression models. Hacettepe Journal of Mathematics and Statistics, 1-20. Cessie, S. L., & Houwelingen, J. V. (1992). Ridge estimators in logistic regression. Journal of the Royal Statistical Society Series C: Applied Statistics, 41(1), 191-201. Ertan, E., & Akay, K. U. (2022). A new Liu-type estimator in binary logistic regression models. Communications in Statistics-Theory and Methods, 51(13), 4370-4394. 9/25/2024 27

28. Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications. Hoerl, A. E., & Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1), 55- 67. Hubert, M. H., & Wijekoon, P. (2006). Improvement of Liu estimator in linear regression model . Statistical papers, 47(3), 471-479. Hussein, S. M., & Yousaf, H. M. (2015). A comparisons among some biased estimators in generalized linear regression model in present of multicolinearity. Al-Qadisiyah Journal For Administrative and Economic sciences, 17(2). Kurtoglu, F., & Özkale, M. R. (2016). Liu estimation in generalized linear models : application on gamma distributed response variable. Statistical papers, 57(4), 911-928. 9/25/2024 28

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30. Thank you 9/25/2024 30