The importance of a chosen criterion in creating a sound statistical choice cannot be overstated [1]. The undefined behavior of random parameters is the most challenging to investigate however, an appropriately established criterion, the ambiguity produced ought to be minimized [2]. The “Entropy” concept is extensively used in the mathematics and recently started to use in social sciences. Therefore, entropy concept is used for the building models in the planning process [3]. Entropy can be used to gauge the degree of disorder in a given system, resulting in a measurement of the degree of data uncertainty [4]. The strategy's entropy, or level of uncertainty, can be reduced by information collection [5]. The less unknown the system's condition is, the more knowledge there is about entropy and information are frequently used as indicators of uncertainty in probability distributions [6].

Despite being expressed in a complex mathematical language, the idea of probability represents the characteristics of probability that are frequently seen in daily life. For instance, each set of spots produced by the simple die toss corresponds to an actual random event whose probability is represented by the positive real number. When a simple die is thrown, the probability of two (or more) digits of spots is equal to the product of their probabilities (relation). All conceivable numbers of spots' cumulative probability are normed to one link.

The theory of probability and mathematical statistics serve as the mathematical foundation for both applied statistics and statistical analytic techniques. Entropy is a measure of uncertainty and probability distribution in mathematics statistics. Information theory is quantitatively defined in mathematics and is sometimes referred to as informational or statistical entropy. Statistical and informational science has long debated the functional link between entropy and the corresponding probability distribution.

Numerous connections have been made based on the characteristics of entropy. Characteristics, such as additivity, extensivity in the Shannon information theory, are posited in the conventional information theory and some of its extensions. It is often referred to as Shannon's entropy in mathematics. Here, we took a mathematical statistics approach to the widely studied decision subject. Our framework's starting point is a normative definition of uncertainty that connects a physical system's uncertainty measure to evidence via a probability distribution. The paper is structured as follows: Section 2 analyses various existing methods reviews employed so far. Section 3 is a detailed explanation of the proposed methodology. The performance analysis of the proposed method is estimated, and the outcomes are projected in Section 4. At last, the conclusion of the work is made in Section 5.

The following list includes several entropies that have been proposed based on supported entropy features. The most well-known of these entropies is the Shannon informational entropy, or Boltzmann-Gibbs entropy (S = -∑ipiln pi) [7]. which is virtually and usually engaged in non-equilibrium dynamics and equilibrium thermodynamics. The scientific community is divided about whether Shannon entropy is a unique and valuable indicator of statistical uncertainty or information [8]. The maximum entropy density is obtained by maximizing Shannon's [9] entropy measure.

Jaynes's maximization of entropy (maxent) principle asserted that the Shannon entropy is the only reliable indicator of uncertainty maximized in maxent [10]. One naturally wonders what would happen if some of these features changed because of this particular information property from the Shannon postulates [11].

Some of the entropies are discovered via mathematical reasoning that modifies Shannon's logic [12]. Non-extensive statistics (NES) were recently proposed using some entropy for stochastic dynamics and thermodynamics of particular non-extensive systems [13, 14]. NES has sparked a lot of publications with very different perspectives on equilibrium and non-equilibrium systems, leading to a lot of discussion [15] among statistical physicists. In the discussion, some critical issues include whether Boltzmann Gibbs-Shannon entropy should be swapped out for another in a different physical scenario. What may be maximized to have the maximum probability distribution?

The entropy forms used in maxent applications must be either explicitly posited or derived from the entropy's claimed properties [16]. The reliability of the calculated probability distributions serves as evidence for the soundness of these entropies. Ahmad et al. [17], the amount of information is measured by decreasing the entropy of such a system. The amount of information acquired in the complete clarification of the state of a certain physical system is equal to the entropy of this system as shown in Equation (1)

The average (or complete) information obtained from all possible individual observations can be rewritten in the form of the mathematical expectation of the logarithm of the state probability with the opposite sign

For continuous random variables, expression (2) is written in the form

where, f(x) – distribution density of a random variable x.

Therefore, it is necessary that the statistics of the criterion ensures the receipt of the maximum amount of information from the available statistical material about the system. Let us consider the limiting case when information about the system is represented by a sample of independent random variables X₁and X₂ of minimal volume n = 2. In the absence of other data, the principle of maximum uncertainty postulates the use of a uniform distribution on the interval [a; b] [18] where, a = min{X₁, X₂} and b = max{X₁, X₂}. For definiteness, let us consider a = 0 and b = 1.

To compare two independent random variables X₁ and X₂, evenly distributed in the interval [0; 1], we use two main metrics δ1=X1X2 and δ₂ = X₂ − X₁, X₁ ≤ X₂. Note that besides δ₁ and δ₂ other metrics are possible, which, in essence, are functions δ₁ and δ₂. However, for any transformation of the original random variable δ the loss of information is inevitable; the total conditional entropy of the system does not exceed its unconditional entropy [19]

To compare the information content of metrics δ₁ and δ₂ [their entropies (3)] it is necessary to determine the distribution density of f(δ₁) and g(δ₂). The density of the joint distribution of ordered random variables, uniformly distributed in the interval [0; 1], will be written in the form [3]

where, f_x(x) = 1 and f_y(y) = 1 – distribution density of independent random variables x and y.

Let us consider transformation variables

or

The Jacobian of the transformation has the form [20]

Then the joint distribution density [21]

where,

That is, the metric δ₁ obeys the uniform distribution law on the interval [0; 1]. Figure 1 shows a histogram of the metric distribution δ₁.

It can be seen that when the hypothesis of a uniform distribution is rejected, an error is made with probability (attainable level of significance) p = 0.67. Therefore, the reasons to reject the hypothesis that the metric is uniformly distributed with density

are absent.

Difference density δ₂ = X₂ − X₁ of random variables X₁ ≤ X₂ [1].

Figure 2 shows the result of checking the uniformity of a random variable υ = G(δ₂), where G(δ₂) – metric distribution function δ₂.

As the p-level is p = 0.15, then we can say that the results of the experiment indicate an error when rejecting the hypothesis of a uniform distribution of the random variable υ = G(δ₂) with probability of 0.15. This is more than a level of significance α = 0.1. Therefore, the sufficient grounds for rejecting the hypothesis of the uniform distribution of the random variable υ = G(δ₂) aren't present.

Thus, density (7) describes the law of distribution of the modulus of the difference of independent random variables uniformly distributed over the interval [0; 1].

Figure 3 shows a histogram of the metric distribution δ₂, from the nature of the density of which it can be seen that it belongs to the class of beta distributions [22]

with parameters a = 1 and b = 2, after substitution of which we obtain expression (7). Thus, in one interval [0; 1] we have distributions of two random metrics with different laws.

It was shown above that the informativeness of the criteria is determined by the Shannon entropy (3), which for these metrics will take the following values:

It is not hard to see that H_δ1 > H_δ2, meaning that according to entropy the metric δ₁ dominates the metric δ₂.

The ratio of two independent ordered random variables, uniformly distributed on the interval [0; 1], is more informative than their difference. In practice, this conclusion means that to construct a criterion based on a sample of independent random variables uniformly distributed in the interval [0; 1], in the absence of any additional conditions, preference should be given to their ratio.

The uniform law can serve as the basis for the criteria for making statistical decisions, while being a very simple distribution to implement and tabulate. Therefore, its identification (testing the hypothesis of a uniform distribution law) is a topical research topic in order to determine the most powerful goodness-of-fit criteria. Much attention has been paid to this issue recently, and the result of a comprehensive analysis was work [23], in which the authors investigated the power of the known criteria for samples of size n ≥ 10.

Samples of a smaller size are considered to be small, the theory of making statistical decisions on which, under the conditions of non-asymptotic formulation of problems, currently still needs to be scientifically substantiated and developed. The complexity of the formulation and solution of the problems of constructing the best estimates for a given volume of statistical material is due to the fact that the desired solution often strongly depends on the specific type of distribution the sample size and cannot be an object of a sufficiently general mathematical theory [24].

The probabilistic model provides for the summation of independent random variables, then the sum is naturally described by the normal distribution. In our work, we consider the limiting case when information about the system is represented by a sample of minimum size. The principle of maximum entropy stated that typical distributions of probabilities of states of an uncertain situation are those it increases the selected measure of uncertainty for specified information in relation to the “behavior” of the situation. In the absence of other data, the principle of maximum uncertainty postulates the use of a uniform distribution on the interval [a; b]. Since it is customary to use entropy as a measure of the uncertainty of a certain physical system. A stochastic multi-criteria preference model (SMCPM) method integrated with optimization approach will be developed for addressing stochasticity of input information.

Figure 8 shows the results of testing for the uniformity of the hypothesis H0:F(x)=x,0    x∈[0;1] and alternatives to the uniform law in the form of a hypothesis H₁:F(x) = B_I(1.5, 1.5, 1, 0) about the beta distribution of the first kind.

Similar results for samples up to without figures are summarized in Table 1.

The table shows that the proposed criterion has a high efficiency of distinguishing between hypotheses. H₀ И H₁ in the specified range n. In the traditional assessment of the power of the goodness-of-fit test, Table 2 shows its values for the right-sided critical region and for a sample of n ≤ 10 with the number of realizations 5,000 for each volume.

It can be seen from the table that the cardinality of the convolution of the VIC test is higher than the cardinality of the test Z_A Zhang at n = 10, which is at the top of preference among the criteria. Even with minimal sample sizes, it is higher than that of the criterion Z_A Zhang at n = 10, which gives a tangible advantage in distinguishing between such close hypotheses. Thus, the uniformity check procedure, which is simple to implement, can serve as a worthy tool in the study of small-volume samples.

Sample volume n > 10 should be broken down into k = 5...7 intervals as for the Kolmogorov criterion or χ². Calculate the theoretical value for each interval F(x) and empirical F_∋(x). Then you build k of private VIC criteria

These criteria are ranked υ₁ ≤ υ₂ ≤ ... ≤ υ_k and the convolution is constructed as shown above.

Decision makers are often tasked with complicated problems that have multiple objectives and uncertainties. Decision analysis is an analytical framework with methods to overcome these challenges and allow decision making to be informative and effective. In conclusion, we note that the given criterion of uniformity, built on the basis of the maximum entropy of the metric, has not only high efficiency, but also simplicity of implementation: in a manual computing process, using a computer - software and hardware, in various indicators similarity, recognition, etc. The given criterion of uniformity can also be used to solve other problems of statistical analysis. An information-entropy-based stochastic multi-criteria preference model was developed to systematically quantify the uncertainties associated with the evaluation of contaminated site remediation alternatives.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Conceptualization: DG and NL. Methodology: SE and IC. Investigation: SB. Writing and editing: RK, VB, and AP. All authors have read and agreed to the published version of the manuscript.