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 [6]. Then we shall identify the best candidates for the PDF that characterizes the subject variable. The following, Table 1, shows the basic statistic of the variable democracy scores (DIS) of 167 countries of the world.

From Table 1, we see that the average (mean) democracy index score for all the countries of the world is 5.548 and the standard deviation is approximately 2.18. It should be noted that the data is slightly left skewed with skewness value of −0.08153. The mean is the sample mean as defined the statistical

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 [7], where they suggested to find the sample estimates of skewness by the quantity defined as,

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 Table 1.

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 Table 4, we see that, this subset of the overall data is slightly right skewed with a value of 0.77913 and it has a mean of 8.4292. The histogram of the Full Democratic Countries is given in Figure 5. From this histogram, the implication is that we need to fit some sort of mixed probability distribution.

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 Table 5 given below confirms that the best pdf for the full democratic data is the Mixed Gaussian PDF.

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 Table 6.

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 Figure 6(a).

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 Figure 6(b).

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 Figure 7(a).

The plotting of Figure 7 is very useful in the case if anyone wants to estimate the probability of any country selected at random from this subset of the population and curious about the probability of that country will have a score more than 8.5 but less than 9.5 i.e. P ( 8.5 ≤ X ≤ 9.5 ) = 1 − P ( X ≤ 8.5 ) − [ 1 − P ( X ≤ 9.5 ) ] , then that probability is 0.312 as shown in Figure 7(b).

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 Table 7, we see that, this subset of the overall data has a mean 6.67. The histogram of the subject dataset is given in Figure 8. From this histogram, the implication is that we need to fit some sort of mixed probability distribution for this data subset as well.

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

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 Table 8 given below confirms that the best pdf for the Flawed democratic data is Mixed of 3-Gaussian PDF.

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 ) 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 Table 9.

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 Figure 9.

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 Table 7. Moreover, if a country is selected at random from this cluster we expect it’s DIS will be approximately 6.67. Also, the

probability that a country will have a DIS between 6.4 and 7 would be approximately 0.4 as shown in Figure 9(b). The cdf of the Flawed Democratic DIS is given by-

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 Figure 10(a).

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 Figure 10(b).

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 Table 10, describes the basic descriptive statistics of the data subset of Hybrid democratic countries of the world. The sample mean of this subset is 4.9621. Now the histogram of the Hybrid democratic countries are given in Figure 11.

From Figure 11, it is an implication of the best fitted PDF is bell shaped Normal PDF. The justification of this selection is confirmed by the three methods of goodness-of-fit that we used in Table 11. Given below confirms that the best pdf for the full democratic data is Gaussian PDF.

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

From Table 12, we see that the estimated value of the population mean μ ^ = 4.966 , which is also the expected value of the PDF of hybrid regime is very

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 Figure 12(a).

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 Table 7 and the probability of DIS of any country greater than 5 is 0.481 as per Figure 12(b).

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 Figure 13(b).

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.

Table 13 shows the basic descriptive statistics of the Authoritarian regime of 49 countries of the world. The sample mean is approximately 2.85. The histogram of the subject data subset of overall DIS data is given below.

Figure 14, indicates that the population distribution of this data subset might follow some left skewed probability density functions. This justification is also confirmed by the three methods of Goodness-of-Fit tests. We have found that the population PDF of Authoritarian Regime countries of the world follows the Weibull Distribution and this fact is confirmed by Table 14.

From Table 14, it is clear that the population PDF of Authoritarian Regime comes from Weibull Distribution. The approximate MLEs for the given PDF of Equation (3.15) given in Table 15.

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

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 Figure 15(b).

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 Figure 16(b).