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1 | (3) |
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4 | (25) |
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Solutions to some problems |
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4 | (4) |
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Monotonic increasing and concave regression |
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8 | (11) |
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10 | (6) |
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16 | (3) |
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General adjustments in the estimation method when changing the concave/convex and the up/down restrictions |
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19 | (6) |
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Changes in the estimation procedure |
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20 | (2) |
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The function systems in the inclusion-part |
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22 | (2) |
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The changing of pj in the inclusion-part |
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24 | (1) |
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The changing of order relations between βj-1 and βj |
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24 | (1) |
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24 | (1) |
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25 | (4) |
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29 | (6) |
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Some Properties for the L.S. Estimation Method |
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35 | (18) |
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Convergence of the algorithm |
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35 | (1) |
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36 | (1) |
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Some properties of the weights of the bending points |
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36 | (10) |
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46 | (7) |
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53 | (36) |
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Bias obtained from the estimation method |
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53 | (3) |
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Bias of Y(xk) for a related problem |
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53 | (3) |
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Bias of Y(xk) for isotonic concave-up regression functions |
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56 | (5) |
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Bias of Y(xk) for isotonic-concave regression functions with constant curvature |
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57 | (2) |
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Bias of Y(xk) for isotonic-concave regression functions with linear functions |
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59 | (2) |
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Bias of Ymax for concave-unimodal regression functions |
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61 | (18) |
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The influence on bias of Ymax by the curvature of the regression function |
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61 | (1) |
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Bias of Ymax for concave-unimodal regression functions with constant Curvature |
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62 | (5) |
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Bias of Ymax for sysmmetric regression functions with piecewise linear functions |
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67 | (2) |
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The influence on bias of Ymax by the Skewness of the regression function |
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69 | (1) |
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The influence by the skewness on bias of Ymax when the regression function is a third-degree polynomial |
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69 | (1) |
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The influence by the skewness on bias of Ymax for piecewise linear regression functions |
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70 | (1) |
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Illustration of the nature of bias of Ymax by theoretical calculations and by simulations |
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71 | (1) |
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Comparison between E(Ymax) and the ordered E(Yi) |
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72 | (2) |
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Distinction between bias caused by the curvature and bias caused by the estimation method |
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74 | (5) |
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Bias of Xmax for concave-unimodal regression functions |
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79 | (5) |
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Bias of Xmax for symmetric regression functions |
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79 | (2) |
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The influence on bias of Xmax by the skewness of the regression function |
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81 | (1) |
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The influence on bias of Xmax by the skewness when the regression function is a third-degree polynomial |
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81 | (1) |
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The influence on bias of Xmax by the skewness for regression functions with piecewise linear functions |
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82 | (2) |
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Bias of the inflection point in sigmoid regression |
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84 | (5) |
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Bias of the estimate of the inflection point of symmetric regression functions |
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84 | (2) |
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Bias of the estimate of the inflection point of non-symmetric regression functions |
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86 | (1) |
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Comparison between bias of Xinf in sigmoid regression and bias of Xmax using the corresponding slopes between successive observations |
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87 | (2) |
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89 | (49) |
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The variance of the constructed points for fixed bending points |
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89 | (3) |
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The influence on the variance of Yi by the curvature of the increasing and concave regression function |
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92 | (5) |
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Variance of Y(xk) for increasing and concave regression functions with constant curvature |
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93 | (2) |
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Variance of Y(xk) for increasing and concave regression functions consisting of linear functions |
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95 | (2) |
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Some variance properties of the concave-unimodal regression function for stochastic Xi |
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97 | (9) |
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The effect on the variance of Ymax by the curvature of the concave-unimodal regression function |
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97 | (1) |
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Variance of Ymax for concave-unimodal regression functions with constant curvature |
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98 | (1) |
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Variance of Ymax for concave-unimodal regression functions with piecewise linear functions |
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99 | (2) |
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The influence on the variance of Xmax by the curvature of the concave-unimodal regression function |
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101 | (1) |
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Variance of Xmax for concave-unimodal regression functions with constant curvature |
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102 | (1) |
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Variance of Xmax for concave-unimodal regression functions with piecewise linear functions |
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103 | (3) |
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The influence on the variance of some estimators by the skewness of the regression function |
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106 | (1) |
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Some variance properties of the inflection point of the sigmoid regression function for stochastic Xi |
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106 | (6) |
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The variance of the estimated y-value of the inflection point |
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107 | (2) |
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The variance of the estimated x-value of the inflection point |
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109 | (2) |
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Comparison between the variance of Xinf in sigmoid and the variance of Xmax using the corresponding slopes between successive observations |
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111 | (1) |
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112 | (26) |
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Improvement of the proposed variance estimation method for regression functions with big curvature |
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119 | (3) |
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Properties of the proposed variance estimation method for a regression function with constant curvature |
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122 | (7) |
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Properties of the proposed variance estimation method for regression functions with piecewise linear functions |
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129 | (3) |
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Properties of the proposed variance estimation method for a skew regression function |
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132 | (3) |
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Properties of the proposed variance estimation method for sigmoid regression functions |
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135 | (3) |
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The Distribution of Some Estimators |
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138 | (26) |
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139 | (10) |
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The distribution of Y|X for increasing and concave regression functions with constant curvature |
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139 | (8) |
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The distribution of Y|X for increasing and concave regression functions with linear functions |
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147 | (2) |
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The distribution of the maximum point in concave-unimodal regression |
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149 | (10) |
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The influence on the distribution of Ymax by the curvature of the regression function |
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149 | (1) |
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The distribution of Ymax for regression functions with constant curvature |
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150 | (1) |
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The distribution of Ymax for symmetric regression functions with piecewise linear functions |
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151 | (1) |
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Comparison between the cumulative distribution of Ymax and the extreme-value function |
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152 | (3) |
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The influence on the distribution of Xmax by the curvature of the regression function |
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155 | (1) |
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The distribution of Xmax for regression functions with constant curvature |
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155 | (1) |
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The distribution of Xmax for regression functions with piecewise linear functions |
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156 | (2) |
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The influence on the distributions of some estimators by the skewness of the regression function |
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158 | (1) |
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The estimation of the distribution of the inflection point in sigmoid regression for stochastic Xi |
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159 | (5) |
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The distribution of the estimated y-value of the inflection point |
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160 | (2) |
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The distribution of the estimated x-value of the inflection point |
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162 | (2) |
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164 | (4) |
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168 | (23) |
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Simulation results in some different situations using binomially distributed variables with equal weights |
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178 | (8) |
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Comparison of bias obtained for sigmoid regression to bias obtained in some symmetric and one non-symmetric situations |
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179 | (3) |
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Comparison of the variance obtained using sigmoid regression to the variance obtained in some alternative situations |
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182 | (4) |
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Comparison between sigmoid regression and an ML estimation method proposed by Schmoyer |
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186 | (5) |
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The ML estimation method proposed by Schmoyer |
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186 | (2) |
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An example using Schmoyer's estimation method |
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188 | (2) |
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190 | (1) |
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Comparison Between the Proposed Estimation Method and Other Similar Methods |
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191 | (60) |
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Comparison between concave-up and isotonic regression |
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191 | (10) |
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The LSE-method of isotonic regression |
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192 | (1) |
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Comparison of some properties between concave-up and isotonic regression |
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193 | (1) |
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Comparison of regression functions with constant curvature |
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194 | (3) |
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Comparison of symmetric regression functions with linear functions |
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197 | (4) |
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Comparison of theoretical results |
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201 | (1) |
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201 | (1) |
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Comparison between concave and unimodal regression |
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201 | (13) |
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The LSE-method of unimodal regression |
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202 | (1) |
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Comparison of some properties of concave and unimodal regression |
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203 | (1) |
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Comparison of estimates of regression functions with constant curvature |
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204 | (4) |
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Comparison of regression functions with piecewise linear functions |
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208 | (4) |
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Comparison of theoretical results |
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212 | (1) |
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213 | (1) |
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Comparison between concave regression and and estimation method proposed by Fraser and Massam |
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214 | (12) |
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The estimation method proposed by Fraser and Massam |
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214 | (3) |
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An example using The Fraser and Massam estimation method |
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217 | (6) |
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The solution obtained from our proposed estimation method |
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223 | (3) |
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Comparison between concave regression and an estimation method proposed by Wu |
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226 | (14) |
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The estimation method proposed by Wu |
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226 | (4) |
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The approximation proposed by Wu |
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230 | (1) |
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An example using Wu's approximation method |
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231 | (1) |
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Some properties of Wu's approximation method |
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232 | (8) |
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Comparison between sigmoid regression and some parametric estimation methods |
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240 | (6) |
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Comparison when the regression function is symmetric |
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241 | (3) |
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Comparison when the regression function is non-symmetric |
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244 | (2) |
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Comparison between sigmoid regression, logistic regression and a discrete alternative to the logistic regression proposed by Nash |
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246 | (5) |
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The estimation method proposed by Nash |
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247 | (1) |
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Two examples using the discrete alternative estimation method proposed by Nash |
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248 | (3) |
| References |
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251 | |
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