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The Economist Intelligence Unit (EIU), collected information from 167 countries of the world to classify each of the countries into four categories of Democracy and they have classified those countries based on the Democracy Index Score (DIS). EIU derived the DIS from the subject data and proceeded descriptively to use the DIS score to classify each of the countries into one of the four types of democracy . In this paper, we have identified the overall probability density function (PDF) of the DIS as well as the PDF of each of the individual type of democracy defined by EIU. Knowing the PDF it can probabilistically characterize the behavior of the overall DIS data and each of the four types of democracy. It is found that the overall PDF of DIS is mixer distribution with their corresponding weights and some of the PDF of individual category follows the same probability density function.

The EIU has introduced the democracy index scores [

In this study, we have studied the data on the DIS and estimated the probability density functions to represent the behavior of the scores statistically. By doing so, one can utilize the estimated pdf to calculate the probability of the random variable falling within a particular range of values, in our case, it is from 0 to 10 as defined by the EIU. Therefore, it is practically relevant to the fact that if anyone wants to find the probability of a group of countries falling in a certain range of scores then that will give them an idea of that particular countries quality of democracy with certain type of assurance statistically.

The Economist Intelligence Unit (EIU) index of democracy is based on the view that measures of democracy that reflect the state of political freedoms and civil liberties are not thick enough. Their democracy index is based on five categories: 1) Electoral Process and Pluralism (EPP); 2) Civil Liberties (CL); 3) Functioning of Government (FG); 4) Political Participation (PP); and 5) Political Culture (PC). The present study is to find the probability density function (PDF) of the scores by the EIU. Having identified the PDF of the EIU data we can probabilistically characterize the democracy behavior that is driven by the collected data.

In addition, having the PDF of the EIU index scores we can obtain other useful information such as the probability of a given country being one of the categories of Democracy [

In this study, we are using the data from the Economist Intelligence Units (EIU) Democracy Index [

As part of our preliminary preparation of the dataset, we have checked to see that the data was randomly collected to determine if there is any biasness and it does not contain any outliers. So, after these aforementioned tests, we proceeded to find the best Probability Distribution Functions (PDF) of all the DIS scores and each of the four classification of Democracy, namely, Full, Flawed, Hybrid, and Authoritarian Regime.

In the process of finding the best fitted PDF, we have implemented the methodology of graphing the variable DIS which will give us an initial idea of what the

distribution may look like [

From

literature as, x ¯ = 1 n ∑ i = 1 n x i and the sample standard deviation is calculated by the formula, s = 1 n − 1 ∑ i = 1 n ( x i − x ¯ ) 2 . The skewness and kurtosis are estimated

by the formulas explained by Joanes and Gill [

b 1 = 1 n ∑ i = 1 n ( x i − x ¯ ) 3 [ 1 n − 1 ∑ i = 1 n ( x i − x ¯ ) 2 ] 3 / 2 and the kurtosis was estimated from the sample data by g 1 = 1 n ∑ i = 1 n ( x i − x ¯ ) 4 [ 1 n ∑ i = 1 n ( x i − x ¯ ) 2 ] 2 − 3 . A histogram of the scores also supports the

same information provided in the descriptive statistics given in

We proceeded by testing the goodness-of-fit for a number of well defined PDFs using three statistical tests, namely, Kolomogrov-Smirnov [

Descriptive Statistics of DIS Countries | ||||
---|---|---|---|---|

Mean | Median | Std. Deviation | Skewness | Kurtosis |

5.548 | 5.792 | 2.177 | −0.082 | −1.034 |

We have found that, the Mixed Gaussian PDF best fits all the DIS data as it is supported by the results given in

Thus, we proceed to discuss and fit the Mixed Gaussian PDF of the DIS of 167 countries of the world.

After passing the data through the aforementioned three goodness-of-fit tests [^{th} component has a mean of μ k and standard deviation of σ k for the univariate case. In our case K = 2 because in the estimated pdf we have mixure of two distributions as postulated in

f ( x ) = ∑ i = 1 k ϕ i N ( x | μ i , σ i 2 ) , with N ( x | μ i , σ i ) = 1 σ i 2 π exp ( − ( x − μ i ) 2 σ i 2 ) , − ∞ ≤ X ≤ ∞ (3.1)

The mean and the variance is 5.554 and 4.912, respectively, with standard deviation of 2.216. Alternative analytical form of the PDF given in Equation (3.1) has the following form of PDF:

f ( x ) = ( ϕ 1 e − ( x − μ 1 ) 2 2 σ 1 2 2 π σ 1 ( ϕ 1 + ϕ 2 ) + ϕ 2 e − ( x − μ 2 ) 2 2 σ 2 2 2 π σ 2 ( ϕ 1 + ϕ 2 ) 0, otherwise (3.2)

For our data, the approximate maximum likelihood estimates (MLE) of the parameters ( σ i , μ i , and ϕ i ) of 2 are given in

Thus, the estimated analytical form of the subject PDF is given by-

f ( x ) = ( 0.144 e − 0.53 ( x − 3.11 ) 2 + 0.18 e − 0.24 ( x − 6.88 ) 2 , 0 ≤ X ≤ 10 0, otherwise (3.3)

The graph of 3 is given by

Thus, if a country was selected at random from the 167 countries, one can identify the probability of its classification of the four categories of Democracy. By using the plots given in

α | p-value | Do Not Reject/Reject | |
---|---|---|---|

Kolmogorov-Smirnov | 0.05 | 0.9993 | Do Not Reject |

Anderson-Darling | 0.05 | 0.9840 | Do Not Reject |

Chi-Squared | 0.05 | 0.5268 | Do Not Reject |

MLEs of DIS scores | |||||
---|---|---|---|---|---|

μ ^ 1 | μ ^ 2 | σ ^ 1 | σ ^ 2 | ϕ ^ 1 | ϕ ^ 2 |

3.107 | 6.877 | 0.974 | 1.437 | 0.351 | 0.649 |

anyone calculates the probability of DIS within the range of 7.64 to 10, then the corresponding probability would be the probability of any country falling in the “Fully Democratic” category and so on. Furthermore, the moment generating function of 3 is given by

M X ( t ) = 0.351 e 3.11 t + 0.47 t 2 + 0.65 e 6.88 t + 1.033 t 2 (3.4)

The moment generating function (MGF) is given in the Equation (3.4) can be used to calculate the moments of higher order and consequently to calculated the mean and variance of the Mixed Gaussian PDF. Thus, if a country is selected at random from the population of 167 countries we will expect its DIS to be 5.554. Also, we calculate the variance, V [ X ] = 4.912 and standard deviation, S T D V [ X ] = 2.216 . Note that these estimates are close to the basic statistics given in

The Cumulative Distribution Function of the DIS is as follows:

F ( x ) = P ( X ≤ x ) = ϕ 1 erfc ( μ 1 − x 2 σ 1 ) 2 ( ϕ 1 + ϕ 2 ) + ϕ 2 erfc ( μ 2 − x 2 σ 2 ) 2 ( ϕ 1 + ϕ 2 ) (3.5)

where, θ 1 and θ 2 are 0.351 & 0.649 respectively and erfc is the “cumulative error function”

The graph of cumulative distribution function is by following

Now we will proceed to find the PDF for each of the four classified categories of Democracy in the following sections.

Here we shall proceed to find the probability distribution that characterize the probabilistic behavior of only the DIS data for Full Democracy. To do this we have implemented the same steps we have used in finding the overall PDF of DIS for all democracy classifications. For this purpose, we have started with the basic descriptive statistics of Fully Democratic countries.

From

Using the three goodness-of-fit tests to the present data of fully democratic countries we have identified that the data can be characterized probabilistically by the “Mixed distribution of 2-Gaussian PDF”. The justification of this selection is confirmed by the three methods of goodness-of-fit that we used in

Descriptive Statistics of DIS of Fully Democratic Countries | ||||
---|---|---|---|---|

Mean | Median | Std. Deviation | Skewness | Kurtosis |

8.4292 | 8.168 | 0.63265 | 0.77913 | −0.46872 |

α | p-value | Do Not Reject/Reject | |
---|---|---|---|

Kolmogorov-Smirnov | 0.05 | 0.916 | Do Not Reject |

Anderson-Darling | 0.05 | 0.986 | Do Not Reject |

Chi-Squared | 0.05 | 0.9462 | Do Not Reject |

Thus, the fitted theoretical PDF of the subject data is given by-

f ( x ) = ∑ i = 1 k ϕ i N ( x | μ i , σ i 2 ) , Here N ( x | μ i , σ i 2 ) = 1 σ i 2 π exp ( − ( x − μ i ) 2 σ i 2 ) (3.6)

The approximate MLEs of the parameters that drive the estimated Mixed Gaussian PDF are given by

Also, ∑ i = 1 2 ϕ ^ i = ϕ ^ 1 + ϕ ^ 2 = 0.53 + 0.47 = 1.00 , are the weights of 2-Gaussian PDF of the mixed distribution. Thus, the analytical structure of the estimated PDF of Fully Democratic countries of the world is given by

f ( x ) = ( 0.36 e − 1.8 ( x − 9.02 ) 2 + 1.23 e − 17.21 ( x − 7.93 ) 2 , 7.6 ≤ X ≤ 10 0 , otherwise (3.7)

The graph of the PDF of 3.7 is given in

The expected value and variance of Fully Democratic data subset is 8.4482 and 0.4453 respectively. That is, if a country is selected at random from this cluster we expect it’s DIS will be approximately 8.45. Also, the probability that a country will have a DIS of more than 9 is 0.246 as shown in

The CDF of the Fully Democratic countries of the world is given by-

F ( x ) = P ( X ≤ x ) = 1 4 erfc ( μ 1 − x 2 σ 1 ) + 1 4 erfc ( μ 2 − x 2 σ 2 ) (3.8)

The graph of F ( x ) in Equation (3.8) is given below by

The plotting of

We shall now proceed to find the probability distribution that characterize the probabilistic behavior of only the DIS data for Flawed Democracy. To do this we have implemented the same steps we have used in finding the overall PDF of DIS for all democracy classifications. For this purpose, we have started with the basic descriptive statistics of Flawed Democratic countries.

From

Using the three goodness-of-fit tests to the present data of Flawed democratic countries we have identified that the data can be characterized probabilistically

MLE | |||
---|---|---|---|

μ ^ 1 | μ ^ 2 | σ ^ 1 | σ ^ 2 |

9.024 | 7.93 | 0.525 | 0.1704 |

Descriptive Statistics of Flawed Democratic Countries | ||||
---|---|---|---|---|

Mean | Median | Std. Deviation | Skewness | Kurtosis |

6.665 | 6.672 | 0.5592 | 0.0745 | −1.0085 |

by the “Mixed distribution of 3-Gaussian PDF”. The justification of this selection is confirmed by the three methods of goodness-of-fit that we used in

Thus, the fitted theoretical PDF of the subject data is given by:

α | p-value | Do Not Reject/Reject | |
---|---|---|---|

Kolmogorov-Smirnov | 0.05 | 0.996011 | Do Not Reject |

Anderson-Darling | 0.05 | 0.999374 | Do Not Reject |

Chi-Squared | 0.05 | 0.964295 | Do not Reject |

f ( x ) = ∑ i = 1 k ϕ i N ( x | μ i , σ i ) Here N ( x | μ i , σ i ) = 1 σ i 2 π exp ( − ( x − μ i ) 2 σ i 2 ) , (3.9)

Also, k = 3 and ∑ i = 1 k ϕ i = 1 as well.

The approximate MLEs of the parameters that drive the estimated Mixed Gaussian PDF are given by

And at the same time the analytical structure of the parameterized probability density function estimated from the data is given in the Equation (3.10) as follows:

f ( x ) = 0.83 e − 41.77 ( x − 7.48 ) 2 + 0.79 e − 5.36 ( x − 6.59 ) 2 + 1.353 e − 202.8 ( x − 5.84 ) 2 , 5 .6 ≤ X ≤ 7 .8 (3.10)

Also, the weights for each of the Gaussian density estimated from the data are ϕ 1 = 0.168518 ϕ 2 = 0.602757 and ϕ 3 = 0.228725 that makes ∑ i = 1 3 ϕ i = 1 .

The graph of the PDF of 3.10 is given in

The expected value and variance of Flawed Democratic data subset is E ( x ) = 6.667 and V ( X ) = 0.328 respectively and this value closely match with the values given in

μ ^ 1 | μ ^ 2 | μ ^ 3 | σ ^ 1 | σ ^ 2 | σ ^ 3 |
---|---|---|---|---|---|

7.475 | 6.594 | 5.834 | 0.109 | 0.305 | 0.049 |

probability that a country will have a DIS between 6.4 and 7 would be approximately 0.4 as shown in

F ( x ) = P ( X ≤ x ) = ϕ 1 erfc ( μ 1 − x 2 σ 1 ) 2 ( ϕ 1 + ϕ 2 + ϕ 3 ) + ϕ 2 erfc ( μ 2 − x 2 σ 2 ) 2 ( ϕ 1 + ϕ 2 + ϕ 3 ) + ϕ 3 erfc ( μ 3 − x 2 σ 3 ) 2 ( ϕ 1 + ϕ 2 + ϕ 3 ) (3.11)

It’s graph is given by

From the figures given above, we can extract some very useful information. Such as, the probability of any country’s DIS is greater or equal to 7 would be approximately 0.283 as shown in

We shall now proceed to find the probability distribution that characterize the probabilistic behavior of only the DIS data for Hybrid Democracy. To do this we have implemented the same steps we have used in finding the overall PDF of DIS for all democracy classifications. For this purpose, we have started with the basic descriptive statistics of Hybrid Regime countries.

The

From

Descriptive Statistics of Hybrid Democratic Countries | ||||
---|---|---|---|---|

Mean | Median | Std. Deviation | Skewness | Kurtosis |

4.96 | 5.013 | 0.668 | −0.0597 | 2.113 |

α | p-value | Do Not Reject/Reject | |
---|---|---|---|

Kolmogorov-Smirnov | 0.05 | 0.977 | Do Not Reject |

Anderson-Darling | 0.05 | 0.9695 | Do Not Reject |

Chi-Squared | 0.05 | 0.6472 | Do not Reject |

The MLEs of this pdf fitted to Hybrid democratic countries data is presented in the following table:

From

MLEs of Hybrid Democratic Countries | |
---|---|

μ ^ | σ ^ |

4.966 | 0.7044 |

close to the sample mean 4.9621. The analytical structure of the PDF of Hybrid democratic countries is given in the Equation (3.12).

f ( x | μ , σ ) = ( 1 σ 2 π exp ( − ( x − μ ) 2 σ 2 ) , − ∞ ≤ x ≤ ∞ 0, otherwise (3.12)

The analytical structure of the PDF of Hybrid regime countries with the estimated parameters is given by:

f ( x ) = ( 0.566 e − 1.007 ( x − 4.966 ) 2 , 3 .5 ≤ X ≤ 6 .5 0, otherwise (3.13)

The graph of the pdf given in 13 is shown in

From the above, one can calculate the expected DIS score of any country randomly selected from this cluster of the population is E ( X ) = 4.966 and the variance V ( X ) = 0.496241 . The estimated expected value is a very close match to the sample mean of 4.9621 given in

The CDF of Hybrid Regime countries of the world is given by:

F ( x ) = P ( X ≤ x ) = 1 2 erfc ( μ − x 2 σ ) , 3.5 ≤ X ≤ 6.5 (3.14)

The graph of the CDF mentioned in Equation (3.14) is postulated as follows:

From the figures given above, we can extract some very useful information. Such as, the probability of randomly selected any country’s DIS is less than 5 (i.e. P ( X ≤ 5 ) = 1 − P ( X > 5 ) ) will be 0.52 as shown in

We have implemented the same steps we have used in finding the overall PDF of DIS for all democracy classifications. For this purpose, we have started with the basic descriptive statistics of Authoritarian Regime countries.

From

The analytical structure of the PDF of Authoritarian Regime countries of the world is given as follows:

Descriptive Statistics of Authoritarian Regime Countries | ||||
---|---|---|---|---|

Mean | Median | Std. Deviation | Skewness | Kurtosis |

2.85 | 3.012 | 0.724 | −0.644 | 2.43 |

α | p-value | Do Not Reject/Reject | |
---|---|---|---|

Kolmogorov-Smirnov | 0.05 | 0.5889 | Do Not Reject |

Anderson-Darling | 0.05 | 0.4602 | Do Not Reject |

Chi-Squared | 0.05 | 0.4996 | Do not Reject |

f ( x ) = ( α e − ( x β ) α ( x β ) α − 1 β , if x > 0 0 , otherwise (3.15)

The parameterized PDF of 3.15 is given by

f ( x ) = ( 0.0228 x 3.667 e − 0.005 x 4.667 , 0 ≤ X ≤ 3.89 0, otherwise (3.16)

The corresponding graph of 3.16 is given below.

The expected value from this PDF is 2.858, which is very close to the sample mean x ¯ = 2.851 . This indicates that our density estimation process is statistically correct. Also, it tells the fact that, if any country is randomly selected from this population, then the expected democracy index score would be approximately 2.86. Moreover, if a country is randomly selected from this cluster of the population then the probability of that country’s DIS is greater 3 will be approximately 0.44 as shown in

The CDF for the Authoritarian Regime countries is given by-

F ( X ) = 1 − P ( X ≥ x ) = 1 − e − ( x β ) α (3.17)

The plot of the above CDF given in Equation (3.17) is as follows:

Thus, if any country is randomly selected from this sub-population, then the expected democracy index score would be 2.86 and the probability of that country being selected and having score less than or equal to 3 would be approximately 0.56 as shown in

In the present study, we have found the probability distribution function, PDF, of the Democracy Index Scores, DIS, that have been documented by the Economist Intelligence Unit, EIU. Having identified the PDF of the subject data we can characterize the probabilistic behavior of different types of Democracy of

MLEs of Authoritarian Regime | |
---|---|

α ^ | β ^ |

4.6678 | 3.126 |

different countries of the world. The EIU collected information of 167 countries in the world and descriptively classified each country as 1) Full Democracy, 2) Flawed Democracy, 3) Hybrid Regime, and 4) Authoritarian Regime. We proceeded to find the PDF and it’s CPDF of

a) All the DIS scores for 167 countries to be the “Mixture of 2-Gaussian Probability Density Function”.

b) The PDF of Fully Democratic 35 countries out of 167 to be “2-Mixed Gaussian Probability Density Function”.

c) The PDF of 47 Flawed Democratic countries out of 167 countries follows “3-Mixed Gaussian Probability Density Function”.

d) The PDF of 36 Hybrid Regime countries out of 167 countries of the world follows the “Normal Probability Density Function”.

e) The PDF of 49 Authoritarian countries out of 167 countries have the “Weibull Probability Density Function”.

Thus, we can characterize the probabilistic behavior of all the DIS scores or the DIS for each of the four categories of democracy around the globe and obtain other useful information.

The authors declare no conflicts of interest regarding the publication of this paper.

Bashar, A.K.M.R. and Tsokos, C.P. (2019) Statistical Parametric Analysis on Democracy Data. Open Access Library Journal, 6: e5828. https://doi.org/10.4236/oalib.1105828