Whether Economic Freedom Is Significantly Related to Death of COVID-19

COVID-19 has caused a huge mayhem globally. Different economic freedom leads to different performances of a country's reaction to the pandemic. We study 164 countries and apply mathematical and statistical approaches to tackle the problem: whether economic freedom has a significant impact on the death of COVID-19. We devise a metric, some norms, and some orderings to construct an absolute reference and the actual relation via binary sequences. Then, we use the theoretical binary sequences to construct a probability distribution which linearises the strength of relation between economic freedom and death of COVID-19. Then, the actual relation from the data analysis provides an evidence to the hypothetical testing. Our analysis and model show that there is no significant relation between economic freedom and death of COVID-19.


Introduction
.1ptDue to COVID-19 pandemic, there are many fatalities across the world. Many countries are baffled by whether to open the market or impose lockdown [1][2][3]. It creates a huge chaos in either economic or social stability [4,5]. is motivates us to study the relation between economic freedom and the death rate or tolls of COVID-19. We locate 164 countries from some datasets [6,7]-because some of the countries lack statistics of either the economic freedom or the death information regarding COVID-19. en, we use a series of mathematical and statistical approaches to reach a conclusion. For the mathematical part, we define a new concept of metric d which could measure the difference between the scoring structures-this is hardly the case if one adopts the usual Euclidean metric. For reference purpose, one fixes the referential structure e → (or scoring system) first.
en, one could compute the distances between all the (sampled) multivalued data N points v → i : 1 ≤ i ≤ N and I, i.e., d( e → , v → i ): 1 ≤ i ≤ N . Based on these distances, we could then create an ordering for v → i : 1 ≤ i ≤ N with respect to the referential structure e → .

Notations and Symbols.
For a vector w → , we use | w → | to denote its length; for any set H, we use |H| to denote its size (cardinality). Moreover, we use w → (j) to denote the j-th element in w → . Let b → denote a binary vector, i.e., each element in b → is either 0 or 1. Let B k denote the set of all the binary vectors with total length k. Let C � C 1 , C 2 , . . . , C m be a set of countries. Let A ef � A 1 , A 2 , . . . , A n be a set of attributes of economic freedom (regarded as independent variables). Let B j be a set of result (regarded as dependent variables). Each time we fix one B j to study the relation between the attributes and B j . In this article, we restrict our attribute values to be numerical numbers. e theoretical table is shown in Table 1, and for the actual forms, one could refer to Sections 3.2.1 and 3.2.2.

Binary Subvectors and Norm
Definition 1 (subvectors). Suppose b → is a binary vector. We use Sub( b → ) to denote all its truncated subvectors consisting of only 1.
Indeed Sub( b → ) reveals the structure of an independentdependent variable relation.
One could, according to real situations, adopt other numbers (for example, replace 2 with other numbers) or other forms other than the one provided here.
Proof. It follows immediately from the definition.
Remark 1. A binary norm indeed serves an important technique in revealing the relation between dependent and independent variables. Given two pairs (x 1 , y 1 ) and (x 2 , y 2 ) of numerical data with x 1 ≠ x 2 , if (x 2 − x 1 ) · (y 2 − y 1 ) > 0 (i.e., they act proportionally), we associate it with a value 1 to indicate such relation and 0, otherwise (i.e., they act inversely). Such mechanism gives a way to look into the fundamental relation between X and Y variables. is kind of analysis is in particular useful when the precision of the data is questionable or when the actual numbers are unknown or more suitable to be interpreted via ranks.
Definition 4 (relational vector). Suppose that b → is a sign vector; we associate it with a relational vector Rel( b → ) whose i-th element is assigned 1 iff b → (i) � b → (i + 1) and 0, otherwise.
to denote the equivalence class whose elements' norms are all p.

Probability Distribution
{ } k is the sampling population. Define a statistic BN on B k by its binary norm. e range for BN is B # k . Define a counting ρ: could define the probability distribution for BN by prob: . (1) One observes that is probability distribution reveals the relation between the independent variables and the dependent variables. is would serve our theoretical distribution for our statistical testing H 0 : the economic freedom and the death of COVID-19 has no significant relation, i.e., the economic freedom has no great impact on the death of COVID-19. For a concrete construction of such probability distribution, one could refer to Section 6.1.

Metric.
A metric or a distance function is a non-negative function d on X × X satisfying identity, symmetry, and triangle properties. In this article, it suffices to define a metric on a closed interval of real number. Fix I � [a, b]⊆R, where a, b ∈ R and a < b. Let v → be a finite vector whose first element is a, last element is b, and all the other elements are incrementally increased and lie between a and b. Let Definition 7 (metric). Define d: Proof. is can be shown by the definitions and some techniques.
□ is metric will be used in Section 3.3.
is metric basically measures the differences between the structures of the attributes in the scoring system. e more similar the structures are, the lower the distances are. Unlike the static Euclidean distance, this metric takes the interval structures into consideration.

Procedures.
Let us summarise the whole procedure of our modelling for the sake of data analysis. Let e → � (100, 100, 100, 100, . . . , 100, 100). Let Death(C i ) denote the death rate (or tolls, depending on the context) for the country i. (2) Rank C via the sorted distances with a rank function

Data Analysis
Following the procedures in Section 2.5, we start to collect, analyse, and produce a report via data analysis. Since the data are huge and hard to handle by the one-off approach, we resort to the sampling technique and reach a conclusion via statistical testing.

3.1.
Sampling. e raw data consist of 164 countries (we use 1 to 164 to name the countries) up to 2020, June 27 th . Since the size is too huge, we apply the Monte Carlo approach to sample the 164 countries. We do 20 times (or 20 batches: S1 to S20) sampling with 25 countries over the 164 countries per sampling. e sampled batches are listed in Tables 2 and 3.

Economic Freedom.
Corresponding to the form listed in Table 1, we associate C with S1 and define Freedom. e attribute values are based on a 100-point scoring system [6]. Due to the limitation of space, we list only the first sampling (or S1) regarding its attributes of economic freedom in Table 4. We omit the other 19 similar tables of this form. A ef serves as the set of our independent variables.

COVID-19
. Now we start to introduce the dependent variables. Indeed we tackle an individual dependent variable each time. Since the data are huge, we only extract the data [7] for sampling one (or S1) as shown in Table 5.
Corresponding to the form listed in Table 1, we associate C with S1 and define B 1 ≡ Total Confirmed COVID-19 Cases, B 2 ≡ Death Toll of COVID-19, B 3 ≡ Total Recovered COVID-19 Cases, and B 4 ≡ Population of the Countries. Due to the limitation of space, we list only the first sampling (or S1) regarding its dependent variables. We omit the other 19 similar tables of this form. Moreover, in the later analysis, we only take and fix B 2 as our dependent variable. If the readers are interested in other dependent variables (or B 1 , B 3 , or other mixed forms), they could simply follow the same approach provided in this article.

Metric.
Since we have defined an interval metric in Section 2.4, we could apply it over here. Here we measure the distance between every sampled data and the fixed reference vector e → � (100, 200, . . . , 1100, 1200). We construct the distances for the 164 countries based on economic freedom (for example, the data of sample one could be referred from  , 1100, 1200). Each country C sampled in S1 will be transformed into C → , for example, �→ � (0, 64.8, 145.7, 247.5, 379.9, . . . , 1170, 1200) are the converted data for the first country sampled in the first sampling or country 68. e economic freedom vector for each sampled country is converted by the same way. e converted data are not tabulated. en, we apply d in Section 2.4 on the converted data and repeat the whole processes for other samplings. e complete results regarding the distance for the 20 sampled countries are presented in Tables 6 and 7. e (i, j) cell in the tables means the value d( e → , C → ij ), where C ij denotes the i-th country sampled in j-th sampling and C → ij denotes the converted data for C ij .

Absolute Reference
By the derived distances presented in Tables 6 and 7, we could construct absolute references. e absolute references would server as the benchmarks for other internal structures.
Let us use C s to denote the set of sampled countries in s-th sampling. Let C → si , C → sj ∈ C s be arbitrary.
Definition 8 (ordering of the sampled countries).
Based on this ordering, we could generate the absolute references (Tables 8 and 9). Let us take S1 for example: C 68 > C 112 > C 92 > · · · > C 41 > C 85 > C 14 . From these absolute references (or ordering for the samplings), we could view the structure (or interval) difference between the ideal scoring (or e → ) and real scoring results. Indeed, an absolute reference is a reference acting like ordering without specific scales. Such reference is useful when the precise values are unknown or when the precision of the data is questionable. In Table 2: 20 sampled batches-S1 to S10.

Ordering for COVID-19 Fatalities
Based on Table 5 and other omitted tables, we start to construct the ordering (or ranking) based on the fatalities of COVID-19.
where Death(C is ) is the death toll for i-th country sampled in s-th sampling.
Based on this ordering, we have the results presented in Tables 10 and 11. Let us take the cells in S1 for example:

Norm and Probability
In this experiment, we only consider N � 23 and construct its distribution accordingly. Hence, the domain is 0, 1 { } 23 and the range lies between 0 and 2 23 − 1 � 8388607 (indeed some of the values' probability is 0). is section generalises Example 3. e higher the value is, the higher the impact of independent variables on dependent variable is.

Probability Distribution.
We have already constructed the theoretical setting of probability distribution for our testing in Section 2.3. Based on that framework and the data given, we could create the theoretical probability distribution prob in Figure 1. e (one-tailed) critical values for 5 and 10 percentages are 138 and 78 (via numerical computation), respectively; that is, if the sampled value is larger than the critical values, we should reject H o : there is no significant relation between the economic freedom and death of COVID-19.

Real Results.
In comparison with the absolute reference, we could generate the binary sign vectors for the real data from each sampling S j (or simply j) in Table 12-for the formula and explanation of sign vectors, one could refer to Section 2.2. However, in these 0 and 1 representations, it separates the proportional and inversely proportional relation between the economic freedom and death of COVID-19. To take all the factors into consideration, one further analyses the alternative behaviour of 0 and 1. If there are too many alternations between 0 and 1, it would indicate that there is a less relation between those two. On the other hand, if the alternative times are few, then it leads to the longer length of subvector consisting of pure 1.
e alternative results are shown in Table 13.

Conclusion and Future Work
e contribution of death in COVID-19 is very complicated. We use economic freedom to capture a potential factor in such contribution. To verify the truth of great impact from economic freedom, we devise a metric, two norms, absolute ordering, binary ordering, and probability distribution for the statistical testing population. Based on our research, we find out that the economic freedom has no significant relation to the death of COVID-19. is might provide some reference for the decision makers of the countries. In the future research, one could further study the relation between economic freedom and other ratios related to COVID-19. One could also use other nonparametric approaches to enrich the statistical testing. ere is another related paper on the same topic [8]. In that paper, the authors use two-step estimators: negative binomial regression and nonlinear least squares, and find out there is a close relation between economic freedom and fatalities of COVID-19. In essence, their approach focuses more on statistical techniques, while ours focuses more on mathematical approaches. For the future researcher, he could compare or combine these methods to yield a comprehensive or generalised theory that could accommodate and single out the factors that cause the discrepancies.
Data Availability e data supporting the findings of this study are included within the article.

Conflicts of Interest
e author declares that there are no conflicts of interest.