# Minimum Penalized ϕ-Divergence Estimation under Model Misspecification

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Some Asymptotic Properties of MP$\varphi $Es

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Theorem**

**1.**

- (a)
- ${\widehat{\theta}}_{\varphi ,h}\stackrel{a.s.}{\u27f6}{\theta}_{0}$.
- (b)
- $\sqrt{n}\left(\begin{array}{c}{\widehat{\pi}}^{+}-{\pi}^{+}\\ {\widehat{\theta}}_{\varphi ,h}-{\theta}_{0}\end{array}\right)\stackrel{\mathcal{L}}{\u27f6}{N}_{m+s}(0,A{\mathsf{\Sigma}}_{{\pi}^{+}}{A}^{t}),$ where ${A}^{t}=({I}_{m},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{G}^{t})$ and G is defined in Equation (7). In particular,$$\sqrt{n}({\widehat{\theta}}_{\varphi ,h}-{\theta}_{0})\stackrel{\mathcal{L}}{\u27f6}{N}_{s}(0,G{\mathsf{\Sigma}}_{{\pi}^{+}}{G}^{t})$$
- (c)
- $\sqrt{n}\left(\begin{array}{c}{\widehat{\pi}}^{+}-{\pi}^{+}\\ P\left({\widehat{\theta}}_{\varphi ,h}\right)-P\left({\theta}_{0}\right)\end{array}\right)\stackrel{\mathcal{L}}{\u27f6}{N}_{2m}(0,B{\mathsf{\Sigma}}_{{\pi}^{+}}{B}^{t})$, where ${B}^{t}=({I}_{m},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{G}^{t}{D}_{1}\left(P\left({\theta}_{0}\right)\right))$, with ${D}_{1}\left(P\left(\theta \right)\right)$ defined in Equation (8).

**Remark**

**1.**

**Remark**

**2.**

**Corollary**

**1.**

- (a)
- For $\pi \in \mathcal{P}$,$$T=\frac{2n}{{\varphi}_{1}^{\u2033}\left(1\right)}\{{D}_{{\varphi}_{1},{h}_{1}}(\widehat{\pi},P({\widehat{\theta}}_{{\varphi}_{2},{h}_{2}}))-{\varphi}_{1}\left(1\right)\}\stackrel{\mathcal{L}}{\u27f6}{\chi}_{k-s-1}^{2}.$$
- (b)
- For $\pi \in {\Delta}_{k}({\varphi}_{2},\mathcal{P},{h}_{2})-\mathcal{P}$, let ${\theta}_{0}=\mathrm{arg}{\mathrm{min}}_{\theta}{D}_{{\varphi}_{2},{h}_{2}}(\pi ,P\left(\theta \right))$. Then$$W=\sqrt{n}\{{D}_{{\varphi}_{1},{h}_{1}}(\widehat{\pi},P({\widehat{\theta}}_{{\varphi}_{2},{h}_{2}}))-{D}_{{\varphi}_{1},{h}_{1}}(\pi ,P\left({\theta}_{0}\right))\}\stackrel{\mathcal{L}}{\u27f6}N(0,{\varrho}^{2})$$$${a}^{t}=\left({\varphi}_{1}^{\prime}\left(\frac{{\pi}_{1}}{{p}_{1}\left({\theta}_{0}\right)}\right),\dots ,{\varphi}_{1}^{\prime}\left(\frac{{\pi}_{m}}{{p}_{m}\left({\theta}_{0}\right)}\right),{v}_{1},\dots ,{v}_{m},\underset{k-m\mathrm{times}}{\underbrace{{h}_{1},\dots ,{h}_{1}}}\right),$$

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 3. Application to Bootstrapping Goodness-Of-Fit Tests

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Theorem**

**2.**

**Remark**

**6.**

- Calculate the observed value of the test statistic for the available data $({X}_{1},\dots ,{X}_{k})$, ${T}_{obs}$.
- Generate B bootstrap samples $({X}_{1}^{b*},\dots ,{X}_{k}^{b*})\sim {\mathcal{M}}_{k}(n;P({\widehat{\theta}}_{{\varphi}_{2},{h}_{2}}))$, $b=1,\cdots ,B$, and calculate the test statistic for each bootstrap sample obtaining ${T}^{*b}$, $b=1,\cdots ,B$.
- Approximate the p-value by means of the expression$${\widehat{p}}_{boot}=\frac{\mathrm{card}\{b:{T}_{b}^{*b}\ge {T}_{obs}\}}{B}.$$

## 4. Application to the Evaluation of the Thematic Classification in Global Land Cover Maps

## 5. Proofs

- (i)
- ${\widehat{\theta}}_{\varphi}={({g}_{1}({\widehat{\pi}}^{+}),\dots ,{g}_{s}({\widehat{\pi}}^{+}))}^{t}$, $\forall \phantom{\rule{0.277778em}{0ex}}n\ge {n}_{0}$, for some ${n}_{0}\in \mathbb{N}$;
- (ii)
- ${\theta}_{0}={({g}_{1}({\pi}^{+}),\dots ,{g}_{s}({\pi}^{+}))}^{t}$;
- (iii)
- $g={({g}_{1},\dots ,{g}_{s})}^{t}$ is continuously differentiable in U and the $s\times m$ Jacobian matrix of g at $({\pi}_{1},\dots ,{\pi}_{m})$ is given by$$G={\mathbb{D}}_{2}^{-1}{D}_{1}\left(P\left({\theta}_{0}\right)\right)Diag\left(\varpi \right)$$$${D}_{1}\left(P\left(\theta \right)\right)=\left(\frac{\partial}{\partial \theta}{p}_{1}\left(\theta \right),\dots ,\frac{\partial}{\partial \theta}{p}_{m}\left(\theta \right)\right),$$$${\varpi}_{i}=\frac{{\pi}_{i}}{{p}_{i}^{2}\left({\theta}_{0}\right)}{\varphi}^{\u2033}\left(\frac{{\pi}_{i}}{{p}_{i}\left({\theta}_{0}\right)}\right),$$

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Corollary**

**1.**

**Proof**

**of**

**Theorem**

**2.**

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MLE | maximum likelihood estimator |

M$\varphi $E | minimum $\varphi $-divergence estimator |

MP$\varphi $E | minimum penalized $\varphi $-divergence estimator |

RMSD | root mean square deviation |

B | bootstrap |

A | asymptotic |

GLC | global land cover |

EBL | evergreen broadleaf trees |

DBL | deciduous broadleaf trees |

ENL | evergreeen needleleaf trees |

U | urban/built up |

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**Table 1.**Type I error probabilities obtained using asymptotic approximation for Example 1 with $\theta =0.3333$, ${\varphi}_{1}=P{D}_{\lambda}$, $\lambda \in \{-2,1,2\}$, ${\varphi}_{2}=P{D}_{-2}$, and ${h}_{1}={h}_{2}\in \{0.5,1,2\}$.

${\mathit{\varphi}}_{1}={\mathit{PD}}_{-2}$ | ${\mathit{\varphi}}_{1}={\mathit{PD}}_{1}$ | ${\mathit{\varphi}}_{1}={\mathit{PD}}_{2}$ | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{h}}_{\mathbf{1}}={\mathit{h}}_{\mathbf{2}}$ | ${\mathit{h}}_{\mathbf{1}}={\mathit{h}}_{\mathbf{2}}$ | ${\mathit{h}}_{\mathbf{1}}={\mathit{h}}_{\mathbf{2}}$ | |||||||

n | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2 |

100 | 0.996 | 0.996 | 0.998 | 0.995 | 0.997 | 0.996 | 0.995 | 0.997 | 0.997 |

0.996 | 0.996 | 0.998 | 0.995 | 0.997 | 0.996 | 0.995 | 0.997 | 0.997 | |

150 | 0.995 | 0.995 | 0.996 | 0.994 | 0.995 | 0.996 | 0.994 | 0.994 | 0.995 |

0.995 | 0.995 | 0.996 | 0.994 | 0.995 | 0.996 | 0.994 | 0.994 | 0.995 | |

200 | 0.992 | 0.993 | 0.994 | 0.992 | 0.994 | 0.991 | 0.993 | 0.993 | 0.994 |

0.992 | 0.994 | 0.994 | 0.992 | 0.994 | 0.991 | 0.993 | 0.993 | 0.994 |

**Table 2.**Type I error probabilities obtained using asymptotic approximation for Example 1 with $n=200$, $\theta =0.3333$, ${\varphi}_{1}={\varphi}_{2}=P{D}_{-2}$, ${h}_{1}\ne {h}_{2}$, and ${h}_{1},{h}_{2}\in \{0.5,1,2\}$.

$({\mathit{h}}_{1},{\mathit{h}}_{2})$ | (0.5, 1) | (1, 0.5) | (0.5, 2) | (2, 0.5) | (1, 2) | (2, 1) |
---|---|---|---|---|---|---|

0.989 | 0.997 | 0.998 | 0.998 | 0.994 | 0.998 | |

0.999 | 0.997 | 0.998 | 0.998 | 0.994 | 0.999 |

**Table 3.**Type I error probabilities obtained using asymptotic approximation for Example 2 with $\theta =0.24$, ${\varphi}_{1}=P{D}_{\lambda}$, $\lambda \in \{-2,1,2\}$, ${\varphi}_{2}=P{D}_{-2}$, and ${h}_{1}={h}_{2}\in \{0.5,1,2\}$.

${\mathit{\varphi}}_{1}={\mathit{PD}}_{-2}$ | ${\mathit{\varphi}}_{1}={\mathit{PD}}_{1}$ | ${\mathit{\varphi}}_{1}={\mathit{PD}}_{2}$ | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{h}}_{\mathbf{1}}={\mathit{h}}_{\mathbf{2}}$ | ${\mathit{h}}_{\mathbf{1}}={\mathit{h}}_{\mathbf{2}}$ | ${\mathit{h}}_{\mathbf{1}}={\mathit{h}}_{\mathbf{2}}$ | |||||||

n | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2 |

100 | 0.016 | 0.017 | 0.017 | 0.013 | 0.013 | 0.014 | 0.013 | 0.014 | 0.015 |

0.034 | 0.036 | 0.036 | 0.031 | 0.030 | 0.031 | 0.030 | 0.033 | 0.033 | |

150 | 0.018 | 0.019 | 0.017 | 0.014 | 0.014 | 0.014 | 0.013 | 0.015 | 0.016 |

0.035 | 0.039 | 0.037 | 0.031 | 0.033 | 0.032 | 0.035 | 0.033 | 0.032 | |

200 | 0.024 | 0.022 | 0.022 | 0.014 | 0.016 | 0.016 | 0.014 | 0.015 | 0.016 |

0.043 | 0.042 | 0.040 | 0.032 | 0.034 | 0.032 | 0.032 | 0.035 | 0.033 |

**Table 4.**Type I error probabilities obtained using asymptotic approximation for Example 2 with $n=200$, $\theta =0.24$, ${\varphi}_{1}={\varphi}_{2}=P{D}_{-2}$, ${h}_{1}\ne {h}_{2}$, and ${h}_{1},{h}_{2}\in \{0.5,1,2\}$.

$({\mathit{h}}_{1},{\mathit{h}}_{2})$ | (0.5, 1) | (1, 0.5) | (0.5, 2) | (2, 0.5) | (1, 2) | (2, 1) |
---|---|---|---|---|---|---|

0.017 | 0.017 | 0.018 | 0.019 | 0.018 | 0.016 | |

0.035 | 0.033 | 0.035 | 0.040 | 0.036 | 0.034 |

**Table 5.**Type I error probabilities obtained using asymptotic approximation for Example 3 with $\theta =0.8$, ${\varphi}_{1}=P{D}_{\lambda}$, $\lambda \in \{-2,1,2\}$, ${\varphi}_{2}=P{D}_{-2}$, and ${h}_{1}={h}_{2}\in \{0.5,1,2\}$.

${\mathit{\varphi}}_{1}={\mathit{PD}}_{-2}$ | ${\mathit{\varphi}}_{1}={\mathit{PD}}_{1}$ | ${\mathit{\varphi}}_{1}={\mathit{PD}}_{2}$ | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{h}}_{\mathbf{1}}={\mathit{h}}_{\mathbf{2}}$ | ${\mathit{h}}_{\mathbf{1}}={\mathit{h}}_{\mathbf{2}}$ | ${\mathit{h}}_{\mathbf{1}}={\mathit{h}}_{\mathbf{2}}$ | |||||||

n | 0.5 | 1 | 2 | 0.5 | 1 | 2 | 0.5 | 1 | 2 |

100 | 0.063 | 0.066 | 0.074 | 0.095 | 0.107 | 0.111 | 0.122 | 0.136 | 0.131 |

0.122 | 0.120 | 0.125 | 0.157 | 0.165 | 0.161 | 0.181 | 0.190 | 0.182 | |

150 | 0.063 | 0.064 | 0.066 | 0.083 | 0.082 | 0.084 | 0.099 | 0.105 | 0.100 |

0.114 | 0.118 | 0.113 | 0.137 | 0.134 | 0.136 | 0.153 | 0.159 | 0.152 | |

200 | 0.062 | 0.061 | 0.061 | 0.075 | 0.079 | 0.074 | 0.086 | 0.091 | 0.086 |

0.111 | 0.111 | 0.115 | 0.129 | 0.137 | 0.123 | 0.145 | 0.148 | 0.144 |

**Table 6.**Type I error probabilities obtained using asymptotic approximation for Example 3 with $n=200$, $\theta =0.8$, ${\varphi}_{1}={\varphi}_{2}=P{D}_{-2}$, ${h}_{1}\ne {h}_{2}$, and ${h}_{1},{h}_{2}\in \{0.5,1,2\}$.

$({\mathit{h}}_{1},{\mathit{h}}_{2})$ | (0.5, 1) | (1, 0.5) | (0.5, 2) | (2, 0.5) | (1, 2) | (2, 1) |
---|---|---|---|---|---|---|

0.060 | 0.062 | 0.063 | 0.062 | 0.063 | 0.058 | |

0.108 | 0.114 | 0.113 | 0.112 | 0.113 | 0.109 |

**Table 7.**Asymptotic and bootstrap type I error probabilities for Example 1 with $\theta =0.3333$, ${\varphi}_{1}=P{D}_{\lambda}$, $\lambda \in \{-2,1,2\}$, ${\varphi}_{2}=P{D}_{-2}$, ${h}_{1}={h}_{2}\in \{0.5,1,2\}$.

${\mathit{h}}_{1}={\mathit{h}}_{2}$ | $0.5$ | 1 | 2 | ||||
---|---|---|---|---|---|---|---|

${\mathit{\varphi}}_{\mathbf{1}}$ | n | B | A | B | A | B | A |

$P{D}_{-2}$ | 100 | 0.051 | 0.996 | 0.048 | 0.996 | 0.048 | 0.998 |

0.110 | 0.996 | 0.103 | 0.996 | 0.109 | 0.998 | ||

150 | 0.055 | 0.995 | 0.050 | 0.995 | 0.056 | 0.996 | |

0.106 | 0.995 | 0.101 | 0.995 | 0.109 | 0.996 | ||

200 | 0.053 | 0.992 | 0.053 | 0.993 | 0.056 | 0.994 | |

0.103 | 0.992 | 0.106 | 0.994 | 0.108 | 0.994 | ||

$P{D}_{1}$ | 100 | 0.057 | 0.995 | 0.056 | 0.997 | 0.055 | 0.996 |

0.110 | 0.995 | 0.110 | 0.997 | 0.107 | 0.996 | ||

150 | 0.054 | 0.994 | 0.052 | 0.995 | 0.055 | 0.996 | |

0.110 | 0.994 | 0.104 | 0.995 | 0.114 | 0.996 | ||

200 | 0.055 | 0.992 | 0.051 | 0.994 | 0.052 | 0.991 | |

0.106 | 0.992 | 0.103 | 0.994 | 0.106 | 0.991 | ||

$P{D}_{2}$ | 100 | 0.055 | 0.995 | 0.056 | 0.997 | 0.054 | 0.997 |

0.110 | 0.995 | 0.109 | 0.997 | 0.107 | 0.997 | ||

150 | 0.054 | 0.994 | 0.055 | 0.994 | 0.056 | 0.995 | |

0.107 | 0.994 | 0.106 | 0.994 | 0.110 | 0.995 | ||

200 | 0.054 | 0.993 | 0.053 | 0.993 | 0.055 | 0.994 | |

0.107 | 0.993 | 0.105 | 0.993 | 0.108 | 0.994 |

**Table 8.**Asymptotic and bootstrap type I error probabilities for Example 1 with $n=200$, $\theta =0.3333$, ${\varphi}_{1}={\varphi}_{2}=P{D}_{-2}$, ${h}_{1}\ne {h}_{2}$, and ${h}_{1},{h}_{2}\in \{0.5,1,2\}$.

$({\mathit{h}}_{1},{\mathit{h}}_{2})$ | (0.5, 1) | (1, 0.5) | (0.5, 2) | (2, 0.5) | (1, 2) | (2, 1) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

B | A | B | A | B | A | B | A | B | A | B | A | |

0.061 | 0.989 | 0.050 | 0.997 | 0.059 | 0.996 | 0.042 | 0.998 | 0.044 | 0.994 | 0.063 | 0.998 | |

0.107 | 0.999 | 0.113 | 0.997 | 0.106 | 0.996 | 0.095 | 0.998 | 0.105 | 0.994 | 0.115 | 0.999 |

**Table 9.**Asymptotic and bootstrap type I error probabilities for Example 2 with $\theta =0.24$, ${\varphi}_{1}=P{D}_{\lambda}$, $\lambda \in \{-2,1,2\}$, ${\varphi}_{2}=P{D}_{-2}$, and ${h}_{1}={h}_{2}\in \{0.5,1,2\}$.

${\mathit{h}}_{1}={\mathit{h}}_{2}$ | $0.5$ | 1 | 2 | ||||
---|---|---|---|---|---|---|---|

${\mathit{\varphi}}_{\mathbf{1}}$ | n | B | A | B | A | B | A |

$P{D}_{-2}$ | 100 | 0.057 | 0.016 | 0.055 | 0.017 | 0.051 | 0.017 |

0.111 | 0.034 | 0.110 | 0.036 | 0.102 | 0.036 | ||

150 | 0.049 | 0.018 | 0.048 | 0.019 | 0.051 | 0.017 | |

0.097 | 0.035 | 0.103 | 0.039 | 0.101 | 0.036 | ||

200 | 0.051 | 0.024 | 0.055 | 0.022 | 0.051 | 0.022 | |

0.099 | 0.043 | 0.102 | 0.042 | 0.099 | 0.040 | ||

$P{D}_{1}$ | 100 | 0.058 | 0.013 | 0.054 | 0.013 | 0.051 | 0.014 |

0.114 | 0.031 | 0.113 | 0.030 | 0.106 | 0.031 | ||

150 | 0.050 | 0.014 | 0.051 | 0.014 | 0.052 | 0.014 | |

0.098 | 0.031 | 0.103 | 0.031 | 0.100 | 0.032 | ||

200 | 0.049 | 0.014 | 0.054 | 0.016 | 0.052 | 0.016 | |

0.099 | 0.032 | 0.104 | 0.034 | 0.099 | 0.032 | ||

$P{D}_{2}$ | 100 | 0.055 | 0.013 | 0.053 | 0.014 | 0.050 | 0.015 |

0.110 | 0.030 | 0.108 | 0.033 | 0.104 | 0.033 | ||

150 | 0.050 | 0.013 | 0.052 | 0.015 | 0.051 | 0.016 | |

0.097 | 0.032 | 0.103 | 0.033 | 0.098 | 0.032 | ||

200 | 0.049 | 0.014 | 0.051 | 0.015 | 0.051 | 0.016 | |

0.100 | 0.032 | 0.102 | 0.035 | 0.098 | 0.033 |

**Table 10.**Asymptotic and bootstrap type I error probabilities for Example 2 with $n=200$, $\theta =0.24$, ${\varphi}_{1}={\varphi}_{2}=P{D}_{-2}$, ${h}_{1}\ne {h}_{2}$, and ${h}_{1},{h}_{2}\in \{0.5,1,2\}$.

$({\mathit{h}}_{1},{\mathit{h}}_{2})$ | (0.5, 1) | (1, 0.5) | (0.5, 2) | (2, 0.5) | (1, 2) | (2, 1) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

B | A | B | A | B | A | B | A | B | A | B | A | |

0.048 | 0.017 | 0.051 | 0.017 | 0.052 | 0.018 | 0.053 | 0.019 | 0.050 | 0.018 | 0.049 | 0.016 | |

0.101 | 0.035 | 0.099 | 0.033 | 0.100 | 0.035 | 0.105 | 0.040 | 0.103 | 0.036 | 0.101 | 0.034 |

**Table 11.**Asymptotic and bootstrap type I error probabilities for Example 3 with $\theta =0.8$, ${\varphi}_{1}=P{D}_{\lambda}$, $\lambda \in \{-2,1,2\}$, ${\varphi}_{2}=P{D}_{-2}$, and ${h}_{1}={h}_{2}\in \{0.5,1,2\}$.

${\mathit{h}}_{1}={\mathit{h}}_{2}$ | $0.5$ | 1 | 2 | ||||
---|---|---|---|---|---|---|---|

${\mathit{\varphi}}_{\mathbf{1}}$ | n | B | A | B | A | B | A |

$P{D}_{-2}$ | 100 | 0.066 | 0.063 | 0.058 | 0.066 | 0.044 | 0.074 |

0.119 | 0.122 | 0.101 | 0.120 | 0.086 | 0.125 | ||

150 | 0.053 | 0.063 | 0.050 | 0.064 | 0.045 | 0.066 | |

0.098 | 0.114 | 0.095 | 0.118 | 0.093 | 0.113 | ||

200 | 0.051 | 0.062 | 0.047 | 0.061 | 0.046 | 0.061 | |

0.099 | 0.111 | 0.096 | 0.111 | 0.100 | 0.115 | ||

$P{D}_{1}$ | 100 | 0.049 | 0.095 | 0.049 | 0.107 | 0.041 | 0.111 |

0.103 | 0.157 | 0.098 | 0.065 | 0.084 | 0.161 | ||

150 | 0.050 | 0.083 | 0.040 | 0.082 | 0.040 | 0.084 | |

0.098 | 0.137 | 0.090 | 0.134 | 0.087 | 0.136 | ||

200 | 0.046 | 0.075 | 0.048 | 0.079 | 0.044 | 0.074 | |

0.095 | 0.129 | 0.102 | 0.137 | 0.092 | 0.123 | ||

$P{D}_{2}$ | 100 | 0.043 | 0.122 | 0.045 | 0.136 | 0.037 | 0.131 |

0.099 | 0.181 | 0.046 | 0.190 | 0.077 | 0.182 | ||

150 | 0.040 | 0.099 | 0.047 | 0.105 | 0.035 | 0.100 | |

0.041 | 0.153 | 0.093 | 0.159 | 0.081 | 0.152 | ||

200 | 0.043 | 0.086 | 0.048 | 0.091 | 0.043 | 0.086 | |

0.092 | 0.145 | 0.097 | 0.148 | 0.090 | 0.144 |

**Table 12.**Asymptotic and bootstrap type I error probabilities for Example 3 with $n=200$, $\theta =0.8$, ${\varphi}_{1}={\varphi}_{2}=P{D}_{-2}$, ${h}_{1}\ne {h}_{2}$, and ${h}_{1},{h}_{2}\in \{0.5,1,2\}$.

$({\mathit{h}}_{1},{\mathit{h}}_{2})$ | (0.5, 1) | (1, 0.5) | (0.5, 2) | (2, 0.5) | (1, 2) | (2, 1) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

B | A | B | A | B | A | B | A | B | A | B | A | |

0.047 | 0.060 | 0.048 | 0.062 | 0.051 | 0.063 | 0.049 | 0.062 | 0.048 | 0.063 | 0.044 | 0.058 | |

0.095 | 0.108 | 0.099 | 0.114 | 0.099 | 0.113 | 0.097 | 0.112 | 0.099 | 0.113 | 0.092 | 0.109 |

Globcover Map | LC-CCI Map | ||
---|---|---|---|

Classified Data | EBL | 165 | 172 |

DBL | 13 | 5 | |

ENL | 7 | 5 | |

U | 0 | 0 |

**Table 14.**Results of the goodness-of-fit test applied to the thematic classification of the EBL class.

Globcover Map | LC-CCI Map | ||||||
---|---|---|---|---|---|---|---|

${\widehat{\mathbf{\theta}}}_{-\mathbf{2},\mathbf{0.5}}=\mathbf{0.9490}$ | ${\widehat{\mathbf{\theta}}}_{-\mathbf{2},\mathbf{0.5}}=\mathbf{0.9721}$ | ||||||

${\varphi}_{1}$ | $P{D}_{-2}$ | $P{D}_{1}$ | $P{D}_{2}$ | $P{D}_{-2}$ | $P{D}_{1}$ | $P{D}_{2}$ | |

${T}_{obs}$ | 2.3015 | 2.7618 | 3.0111 | 0.1432 | 0.1432 | 0.1433 | |

${\widehat{p}}_{boot}$ | 0.1700 | 0.2253 | 0.2926 | 0.9283 | 0.9200 | 0.9148 | |

${\widehat{\mathbf{\theta}}}_{-\mathbf{2},\mathbf{1}}=\mathbf{0.9503}$ | ${\widehat{\mathbf{\theta}}}_{-\mathbf{2},\mathbf{1}}=\mathbf{0.9725}$ | ||||||

${T}_{obs}$ | 2.7686 | 3.3752 | 3.6962 | 0.2821 | 0.2823 | 0.2826 | |

${\widehat{p}}_{boot}$ | 0.1801 | 0.2325 | 0.2671 | 0.8431 | 0.9162 | 0.9182 | |

${\widehat{\mathbf{\theta}}}_{-\mathbf{2},\mathbf{2}}=\mathbf{0.9527}$ | ${\widehat{\mathbf{\theta}}}_{-\mathbf{2},\mathbf{2}}=\mathbf{0.9732}$ | ||||||

${T}_{obs}$ | 3.6352 | 4.5400 | 5.0219 | 0.5492 | 0.5508 | 0.5514 | |

${\widehat{p}}_{boot}$ | 0.1300 | 0.2492 | 0.2584 | 0.7526 | 0.8144 | 0.8291 |

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**MDPI and ACS Style**

Alba-Fernández, M.V.; Jiménez-Gamero, M.D.; Ariza-López, F.J.
Minimum Penalized *ϕ*-Divergence Estimation under Model Misspecification. *Entropy* **2018**, *20*, 329.
https://doi.org/10.3390/e20050329

**AMA Style**

Alba-Fernández MV, Jiménez-Gamero MD, Ariza-López FJ.
Minimum Penalized *ϕ*-Divergence Estimation under Model Misspecification. *Entropy*. 2018; 20(5):329.
https://doi.org/10.3390/e20050329

**Chicago/Turabian Style**

Alba-Fernández, M. Virtudes, M. Dolores Jiménez-Gamero, and F. Javier Ariza-López.
2018. "Minimum Penalized *ϕ*-Divergence Estimation under Model Misspecification" *Entropy* 20, no. 5: 329.
https://doi.org/10.3390/e20050329