# The Data-Constrained Generalized Maximum Entropy Estimator of the GLM: Asymptotic Theory and Inference

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Data-Constrained GME Formulation

## 3. Consistency and Asymptotic Normality of the GME Estimator

- R1
- The ${\epsilon}_{i}$′s are iid with ${c}_{1}+\delta \le {\epsilon}_{i}\le {c}_{J}-\delta $ for some δ > 0 and large enough finite positive ${c}_{J}=-{c}_{1}$.
- R2
- The pdf of ${\epsilon}_{i},f({\epsilon}_{i})$, is symmetric around 0 with variance ${\sigma}^{2}$.
- R3
- ${\beta}_{k}\in ({\beta}_{kL},{\beta}_{kH})$, for finite ${\beta}_{kL}\text{and}{\beta}_{kH},\forall k=1,\dots ,K$.
- R4
- X has full column rank.
- R5
- ${\scriptscriptstyle \frac{1}{N}}({X}^{\prime}X)$ is O(1) and the smallest eigenvalue of ${\scriptscriptstyle \frac{1}{N}}({X}^{\prime}X)>\epsilon $ for some ε > 0, and $\forall N>{N}^{*}$, where N
^{**}is some positive integer. - R6
- ${\scriptscriptstyle \frac{1}{N}}({X}^{\prime}X)\to \mathrm{\Omega}$, a finite positive definite symmetric matrix.

- C1
- $\sum _{\ell =1}^{{J}_{k}}}{z}_{k\ell}{p}_{k\ell}={b}_{k},{\beta}_{kL}={z}_{k1}\le {z}_{k2}\le \cdots \le {z}_{k{J}_{k}}={\beta}_{kH},\text{}k=1,\dots ,K$
- C2
- $\sum _{\ell =1}^{J}}{v}_{\ell}{w}_{i\ell}={e}_{i}={y}_{i}-{X}_{i\u2022}b={e}_{i}(b),\text{}i=1,\dots ,N$
- C3
- ${c}_{1}={v}_{1}\le {v}_{2}\le \cdots \le {v}_{J}={c}_{J}$
- C4
- $-{v}_{\ell}={v}_{J+1-\ell}(\text{thusfor}J\text{odd}{v}_{{\scriptscriptstyle \frac{J+1}{2}}}\equiv 0)$
- C5
- $\sum _{\ell =1}^{{J}_{k}}}{p}_{k\ell}=1,k=1,\dots ,K$
- C6
- $\sum _{\ell =1}^{J}}{w}_{i\ell}=1,i=1,\dots ,N$

**Theorem.**Under regularity conditions R1-R5, the GME estimator $\widehat{\beta}=Z\widehat{p}$ is a consistent estimator of β. With the addition of regularity condition R6, the GME estimator is asymptotically normally distributed as

**Proof.**Define the maximized entropy function, conditional on $b=\tau $, as:

_{i}’s around zero that:

#### 3.1. Consistency

#### 3.2. Asymptotic Normality

_{1}, c

_{J}). As $\delta \to 0$, the asymptotic variance of $\sqrt{N}(\widehat{\beta}-\beta )$ may tend to zero, but cannot grow without bound. For example, if at $\delta =0,\exists \epsilon >0$ such that $P(\epsilon \le k)\ge \epsilon (k-{c}_{1})$, all $k\in ({c}_{1},{c}_{J})$ $(\Rightarrow P(\epsilon \ge k)\ge \epsilon ({c}_{J}-k)$ all $k\in ({c}_{1},{c}_{J}))$, then $\underset{\delta \to 0}{\mathrm{lim}}{\scriptscriptstyle \frac{{\sigma}_{\gamma}^{2}}{{\xi}^{2}}}=0$.

#### 3.3. Cross-Entropy Extensions

## 4. Statistical Tests

#### 4.1. Asymptotically Normal Tests

_{z}can be used to test hypotheses about the values of the ${\beta}_{k}$’s.

#### 4.2. Wald Tests

_{0}. Similarly, for nonlinear restrictions $g(\beta )=[0]$, where $g(\beta )$ is a continuously differentiable L-dimensional vector function with $q={\scriptscriptstyle \frac{\partial g(b)}{\partial b}}$ and rank $(q(\beta ))=L\le K$, it follows that:

#### 4.3. Likelihood Ratio Tests

#### Lagrange Multiplier Tests

## 5. Monte Carlo Simulations

_{I}were chosen to be most favorable to the GME estimator, where the elements of the true β-vector are located in the center of their respective supports and the widths of the supports are relatively narrow. The supports represented by Z

_{II}are tilted to the left of β

_{1}and β

_{2}and to the right of β

_{3}and β

_{4}by 1 unit, with the widths of the supports being the same as their counterparts in Z

_{I}. The last set of supports represented by Z

_{III}are wider and effectively define an upper bound of 10 on the absolute values of each of the elements of β.

_{Z}test, and the hypothesis ${H}_{0}:{\beta}_{2}=c,{\beta}_{3}=d$ was tested using the Wald, pseudo-likelihood, and Lagrange Multiplier tests, with c and d set equal to the true values of β

_{2}and β

_{3}, i.e., c = 1 and d = −1. Critical values of the tests were based on their respective asymptotic distributions and a 0.05 level of significance. An observation on the power of the respective tests was obtained by performing a test of significance whereby c = d = 0 in the preceding hypotheses. All scenarios were analyzed using 10,000 Monte Carlo repetitions, and sample sizes of n = 25, 100, 400, and 1,600 were examined. In the course of calculating values of the test statistics, both unrestricted and restricted (by β

_{2}= c and/or β

_{3}= d) GME estimators needed to be calculated. Therefore, bias and mean square error measures relating to these and the least squares estimators were calculated as well. Monte Carlo results for the test statistics and for the unrestricted GME and OLS estimators are presented in Table 1 and Table 2, respectively, while results relating to the restricted GME and OLS estimators are presented in Table 3. Because the choice of which asymptotic covariance matrix to use in calculating the T

_{Z}and Wald tests was inconsequential, only the results for the second suggested covariance matrix representation are presented here.

_{0}is consistent with the behavior expected from the respective asymptotic distributions when n is large (sample size of 1600), their sizes being approximately .05 regardless of the choice of support for β. The sizes of the tests remain within 0.01 of their asymptotic size when n decreases to 400, except for the Lagrange Multiplier test under support Z

_{II}, which has a slightly larger size. Across all support choices and ranging over all sample sizes from small to large, the sizes of the T

_{Z}and Wald tests remain in the 0−0.10 range; for Z

_{I}supports and small sample sizes, the sizes of the tests are substantially less than 0.05. Results were similar for the pseudo-likelihood and Lagrange Multiplier tests, except for the cases of Z

_{II}support and n ≤ 100, where the size of the test increased as high as 0.36 for the pseudo-likelihood test and 0.73 for the Lagrange multiplier test when n = 25.

**Table 1.**Rejection Probabilities for True $({\beta}_{2}=1,{\beta}_{3}=-1)$ and False $({\beta}_{2}={\beta}_{3}=0)$ Hypotheses.

Supports | T_{z} | WALD | Pseudo-Likelihood | Lagrange Multiplier | ||||
---|---|---|---|---|---|---|---|---|

H_{0} | H_{0} | H_{0} | H_{0} | |||||

Z_{I} | β_{2} = 1 | β_{2} = 0 | β_{2} = 1 | β_{2} = 0 | β_{2} = 1 | β_{2} = 0 | ||

β_{2} = 1 | β_{2} = 0 | β_{3} = −1 | β_{3} = 0 | β_{3} = −1 | β_{3} = 0 | β_{3} = −1 | β_{3} = 0 | |

n = 25 | 0.000 | 0.825 | 0.004 | 0.998 | 0.021 | 1.000 | 0.059 | 1.000 |

n = 100 | 0.017 | 0.999 | 0.022 | 1.000 | 0.038 | 1.000 | 0.056 | 1.000 |

n = 400 | 0.041 | 1.000 | 0.042 | 1.000 | 0.048 | 1.000 | 0.053 | 1.000 |

n = 1600 | 0.047 | 1.000 | 0.046 | 1.000 | 0.049 | 1.000 | 0.050 | 1.000 |

Z_{II} | ||||||||

n = 25 | 0.101 | 0.047 | 0.080 | 0.894 | 0.357 | 0.980 | 0.734 | 0.995 |

n = 100 | 0.085 | 0.996 | 0.067 | 1.000 | 0.114 | 1.000 | 0.172 | 1.000 |

n = 400 | 0.053 | 1.000 | 0.048 | 1.000 | 0.058 | 1.000 | 0.066 | 1.000 |

n = 1600 | 0.052 | 1.000 | 0.052 | 1.000 | 0.055 | 1.000 | 0.057 | 1.000 |

Z_{III} | ||||||||

n = 25 | 0.038 | 0.670 | 0.070 | 0.967 | 0.097 | 0.980 | 0.088 | 0.972 |

n = 100 | 0.045 | 0.999 | 0.050 | 1.000 | 0.057 | 1.000 | 0.052 | 1.000 |

n = 400 | 0.045 | 1.000 | 0.050 | 1.000 | 0.051 | 1.000 | 0.050 | 1.000 |

n = 1600 | 0.051 | 1.000 | 0.051 | 1.000 | 0.052 | 1.000 | 0.051 | 1.000 |

_{Z}test in the case of Z

_{II}support and the smallest sample size, the latter result being indicative of a notably biased test. Overall, the choice of support did impact the power of tests for rejecting the errant hypotheses, although the effect was small for all but the T

_{Z}test.

_{I}), the GME estimator dominated the OLS estimator in terms of MSE, and GME superiority was substantial for sample sizes of n ≤ 100 (Table 2). The GME-Z

_{I}estimator and, of course, the OLS estimator, were unbiased, with the GME-Z

_{I}estimator exhibiting substantially smaller variances for smaller n. The choice of support has a significant effect on the bias and MSE of the GME estimator for small sample sizes. Neither the GME-Z

_{II}or GME-Z

_{III}estimator dominates the OLS estimator, although the GME-Z

_{III}estimator is generally the better estimator across the various sample sizes. When n = 25, the GME-Z

_{II}estimator offers notable improvement over OLS for estimating three of the four elements of β, but is significantly worse for estimating β

_{2}. For larger sample sizes, the GME-Z

_{II}estimator is generally inferior to the OLS estimator. Although the centers of the Z

_{III}support are on average further from the true β’s than are the centers of the Z

_{II}support, the wider widths of the former result in a superior GME estimator.

Estimator | β_{1} = 2 | β_{2} = 1 | β_{3} = −1 | β_{4} = 3 | ||||
---|---|---|---|---|---|---|---|---|

$E({\widehat{\beta}}_{1})$ | MSE | $E({\widehat{\beta}}_{2})$ | MSE | $E({\widehat{\beta}}_{3})$ | MSE | $E({\widehat{\beta}}_{4})$ | MSE | |

GME-Z_{I} | ||||||||

n = 25 | 2.000 | 0.015 | 1.001 | 0.038 | −1.001 | 0.028 | 3.000 | 0.006 |

n = 100 | 2.003 | 0.034 | 1.003 | 0.026 | −1.000 | 0.011 | 2.999 | 0.004 |

n = 400 | 2.000 | 0.032 | 1.001 | 0.009 | −1.000 | 0.003 | 3.000 | 0.002 |

n = 1600 | 2.000 | 0.014 | 1.000 | 0.002 | −1.000 | 0.001 | 3.000 | 0.001 |

GME-Z_{II} | ||||||||

n = 25 | 1.022 | 0.977 | 0.484 | 0.309 | −0.840 | 0.058 | 3.182 | 0.040 |

n = 100 | 1.306 | 0.519 | 0.826 | 0.056 | −0.966 | 0.013 | 3.139 | 0.023 |

n = 400 | 1.672 | 0.141 | 0.960 | 0.010 | −0.996 | 0.003 | 3.066 | 0.006 |

n = 1600 | 1.892 | 0.026 | 0.991 | 0.002 | −1.000 | 0.001 | 3.022 | 0.001 |

GME-Z_{III} | ||||||||

n = 25 | 1.278 | 0.757 | 0.946 | 0.131 | −0.881 | 0.069 | 3.092 | 0.028 |

n = 100 | 1.709 | 0.252 | 0.995 | 0.037 | −0.978 | 0.014 | 3.046 | 0.011 |

n = 400 | 1.914 | 0.068 | 0.999 | 0.010 | −0.996 | 0.003 | 3.015 | 0.003 |

n = 1600 | 1.978 | 0.017 | 0.999 | 0.002 | −0.999 | 0.001 | 3.004 | 0.001 |

OLS | ||||||||

n = 25 | 1.997 | 1.342 | 1.002 | 0.181 | −1.002 | 0.066 | 3.001 | 0.065 |

n = 100 | 2.009 | 0.283 | 1.003 | 0.041 | −1.000 | 0.014 | 2.998 | 0.014 |

n = 400 | 2.001 | 0.068 | 1.001 | 0.010 | −1.000 | 0.003 | 3.000 | 0.003 |

n = 1600 | 2.000 | 0.017 | 1.000 | 0.003 | −1.000 | 0.001 | 3.000 | 0.001 |

**Table 3.**$E({\widehat{\beta}}_{i})$ and Mean Square Error Measures – Restricted Estimators Under the Errant Restriction ${\beta}_{2}={\beta}_{3}=0$.

Estimator | β_{1} = 2 | β_{4} = 3 | ||
---|---|---|---|---|

$E({\widehat{\beta}}_{1})$ | MSE | $E({\widehat{\beta}}_{4})$ | MSE | |

GME-Z_{I} | ||||

n = 25 | 2.078 | 0.041 | 2.681 | 0.011 |

n = 100 | 2.340 | 0.191 | 2.630 | 0.142 |

n = 400 | 2.689 | 0.537 | 2.600 | 0.196 |

n = 1600 | 2.898 | 0.832 | 2.520 | 0.232 |

GME-Z_{II} | ||||

n = 25 | 1.064 | 0.915 | 2.885 | 0.018 |

n = 100 | 1.603 | 0.234 | 2.772 | 0.056 |

n = 400 | 2.330 | 0.169 | 2.630 | 0.140 |

n = 1600 | 2.776 | 0.628 | 2.543 | 0.210 |

GME-Z_{III} | ||||

n = 25 | 1.686 | 0.589 | 2.750 | 0.084 |

n = 100 | 2.468 | 0.542 | 2.601 | 0.172 |

n = 400 | 2.842 | 0.823 | 2.530 | 0.225 |

n = 1600 | 2.958 | 0.948 | 2.508 | 0.243 |

OLS | ||||

n = 25 | 3.011 | 3.342 | 2.497 | 0.342 |

n = 100 | 3.013 | 1.575 | 2.497 | 0.274 |

n = 400 | 3.005 | 1.138 | 2.499 | 0.256 |

n = 1600 | 2.999 | 1.030 | 2.500 | 0.251 |

#### Asymmetric Error Supports

_{II}is a simple translation of V

_{I}by five positive units in magnitude and retaining symmetry centered about 5. The asymmetric support V

_{III}translates the truncation points by five positive units in magnitude, but retains the center support point 0. The true error distribution is generated in two ways: a symmetric distribution specified as a N(0,1) distribution truncated at (−3,3) and an asymmetric distribution specified as a Beta(3,2) translated and scaled from support (0,1) to (−3,3) with mean 0.6. Supports on the parameter coefficients terms are retained as Z

_{I}, providing symmetric support points about the true coefficient values.

**Table 4.**Mean and MSE of 1,000 Monte Carlo Simulations with True Distribution Symmetric. Symmetric and Asymmetric Error Supports and Coefficient Support Z

_{I}.

Estimator | β_{1} = 2 | β_{2} = 1 | β_{3} = −1 | β_{4} = 3 | ||||
---|---|---|---|---|---|---|---|---|

E(β_{1}) | MSE | E(β_{2}) | MSE | E(β_{3}) | MSE | E(β_{4}) | MSE | |

GME-Z_{I},V_{I} | ||||||||

25 | 2.002 | 0.016 | 1.003 | 0.042 | −1.000 | 0.030 | 2.997 | 0.007 |

100 | 2.000 | 0.033 | 1.001 | 0.026 | −1.002 | 0.011 | 3.002 | 0.004 |

400 | 2.000 | 0.035 | 1.001 | 0.010 | −0.998 | 0.003 | 2.999 | 0.002 |

GME-Z_{I},V_{II} | ||||||||

25 | 1.259 | 0.585 | 0.815 | 0.101 | −1.009 | 0.048 | 2.209 | 0.636 |

100 | 0.208 | 3.258 | 0.804 | 0.071 | −0.944 | 0.020 | 2.381 | 0.389 |

400 | −1.144 | 9.903 | 0.868 | 0.028 | −0.959 | 0.005 | 2.640 | 0.132 |

GME-Z_{I},V_{III} | ||||||||

25 | 1.506 | 0.271 | 0.875 | 0.069 | −1.005 | 0.038 | 2.476 | 0.282 |

100 | 0.752 | 1.598 | 0.875 | 0.045 | −0.961 | 0.015 | 2.602 | 0.163 |

400 | −0.235 | 5.024 | 0.925 | 0.015 | −0.977 | 0.004 | 2.794 | 0.044 |

OLS | ||||||||

25 | 2.014 | 1.321 | 1.007 | 0.204 | −0.998 | 0.069 | 2.993 | 0.065 |

100 | 1.999 | 0.280 | 1.001 | 0.042 | −1.002 | 0.014 | 3.002 | 0.014 |

400 | 2.001 | 0.075 | 1.001 | 0.011 | −0.997 | 0.003 | 2.999 | 0.003 |

**Table 5.**Mean and MSE of 1000 Monte Carlo Simulations with True Distribution Asymmetric. Symmetric and Asymmetric Error Supports and Coefficient Support Z

_{I}.

β_{1}=2 | β_{2}=1 | β_{3}=−1 | β_{4}=3 | |||||
---|---|---|---|---|---|---|---|---|

Estimator | E(β_{1}) | MSE | E(β_{2}) | MSE | E(β_{3}) | MSE | E(β_{4}) | MSE |

GME-Z_{I},V_{I} | ||||||||

25 | 2.089 | 0.031 | 1.038 | 0.060 | −1.005 | 0.041 | 3.094 | 0.018 |

100 | 2.233 | 0.108 | 1.023 | 0.033 | −1.006 | 0.016 | 3.071 | 0.010 |

400 | 2.427 | 0.229 | 1.015 | 0.012 | −1.004 | 0.005 | 3.033 | 0.004 |

GME-Z_{I},V_{II} | ||||||||

25 | 1.358 | 0.449 | 0.843 | 0.103 | −1.021 | 0.057 | 2.305 | 0.496 |

100 | 0.410 | 2.583 | 0.826 | 0.073 | −0.966 | 0.019 | 2.463 | 0.294 |

400 | −0.860 | 8.209 | 0.890 | 0.025 | −0.966 | 0.006 | 2.700 | 0.092 |

GME-Z_{I},V_{III} | ||||||||

25 | 1.597 | 0.190 | 0.905 | 0.075 | −1.019 | 0.049 | 2.574 | 0.193 |

100 | 0.964 | 1.129 | 0.889 | 0.055 | −0.967 | 0.020 | 2.674 | 0.112 |

400 | 0.126 | 3.553 | 0.946 | 0.016 | −0.981 | 0.005 | 2.835 | 0.030 |

OLS | ||||||||

25 | 2.600 | 2.324 | 1.041 | 0.261 | −1.009 | 0.097 | 2.998 | 0.099 |

100 | 2.616 | 0.813 | 1.001 | 0.052 | −0.999 | 0.020 | 2.997 | 0.021 |

400 | 2.610 | 0.471 | 1.003 | 0.013 | −1.000 | 0.005 | 2.997 | 0.005 |

## 6. Further Results

## 7. Conclusions

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**MDPI and ACS Style**

Mittelhammer, R.; Cardell, N.S.; Marsh, T.L.
The Data-Constrained Generalized Maximum Entropy Estimator of the GLM: Asymptotic Theory and Inference. *Entropy* **2013**, *15*, 1756-1775.
https://doi.org/10.3390/e15051756

**AMA Style**

Mittelhammer R, Cardell NS, Marsh TL.
The Data-Constrained Generalized Maximum Entropy Estimator of the GLM: Asymptotic Theory and Inference. *Entropy*. 2013; 15(5):1756-1775.
https://doi.org/10.3390/e15051756

**Chicago/Turabian Style**

Mittelhammer, Ron, Nicholas Scott Cardell, and Thomas L. Marsh.
2013. "The Data-Constrained Generalized Maximum Entropy Estimator of the GLM: Asymptotic Theory and Inference" *Entropy* 15, no. 5: 1756-1775.
https://doi.org/10.3390/e15051756