Pareto distribution parameter. See pages where the term pareto distribution is mentioned

9. The Pareto Rule Building a business as a whole, as well as its separate direction, requires attention to various details. How to figure out where to start? Business activity requires taking into account various factors, which are sometimes many. To guarantee

PARETO RULE

PARETO RULE In general, this rule states that 80% of the results of the system are provided by 20% of the costs. This law was introduced by the Italian economist Vilfredo Pareto in the 19th century. Studying the distribution of wealth, he drew attention to the fact that most of the funds

Pareto principle

The Pareto principle The well-known Italian economist and sociologist Vilfredo Pareto (1848–1923) in 1897 formulated the principle that was later named after him. This principle is also called the 80/20 rule or the principle of least effort. Pareto established that people in society are divided into

Pareto's Law: 80/20

Pareto's Law: 80/20 According to the Pareto's Law, 80% of the consequences come from 20% of the causes, and 80% of the results come from 20% of the effort and time spent. Try to organize your life according to this law, and to do this, determine: what 20% of sources create 80% of your problems and

Theoretical analysis based on special probability times, as well as a series of computer-based computational experiments, showed that only a narrow family of probability distributions, the simplest of which is the Pareto distribution, at O ​​1, reliably provides a concentration of > 75% of industrial oil reserves of less than in 10% of deposits. It is these figures and patterns that are typical for the vast majority of oil and gas basins and the world [l]

Rice. 10.5. Pareto Distribution Example (Values ​​from Table 10.1)

The Pareto distribution is a left-truncated distribution whose probability density and distribution function are expressed as x

The Pareto distribution can be modified in such a way that it can be used to describe symmetric probability distributions. Introducing a new variable t = X - B, we get

The distribution of price changes generally refers to Pareto distributions (see Appendix B). The distribution of trading P L can be considered a transformation of the price distribution. This transformation is the result of trading methods, when traders try to reduce their losses and increase their profits, therefore, the distribution of trading P L can be attributed to Pareto distributions. However, the distribution we will be studying is not a Pareto distribution. The Pareto distribution, like all other distribution functions, models a certain probabilistic phenomenon. It models the distribution of sums of independent, identically distributed random variables. The distribution function , which we will study, does not model a specific probabilistic phenomenon. It models many unimodal distribution functions. Therefore, it can replicate the shape and probability density of the Pareto distribution, as well as any other unimodal distribution. Now we will create this function. First, consider the following equation

We believe that the distribution of these 150 billion unaccounted for in the absence of the fiscal and redistributive impact of the state obeys the Pareto distribution law 20% of the richest receive 80% of all income (120 out of 150 additional billion).

After studying a fairly extensive statistical material, Pareto came to the conclusion that the parameters of this distribution are approximately the same and do not differ fundamentally in different countries and in different countries. different time. The income distribution curve is remarkably stable, it changes slightly, although the circumstances of the time and place in which it is observed change greatly, Pareto wrote in Socialist Systems. The shape of this curve depends on the biologically given distribution of the psychological characteristics of people. Pareto's law gave rise to an extensive economic literature, both critical and interpretive of the Pareto distribution in relation to a wide variety of applications - economic, social, biological, demographic, etc.

In the previous chapter, we saw a possible replacement for the normal distribution as a probability function for describing market returns. This replacement has been called, alternately, Leah's stable distributions, Pareto stable distributions, or Pareto-Levy distributions. Now we can add fractal distributions, a name that better describes them. Since the traditional names are named after the mathematicians who created them, we will use all these names interchangeably.

The remainder of this chapter is devoted to an analysis of the various probability distributions applicable in estimating the behavior of return on assets, subject to appropriate assumptions. Let's start with two continuous distributions - normal and lognormal. Then consider two discrete distributions - binomial and Poisson. Let's finish the group's consideration of other continuous distributions, including the Pareto-Levy distribution. Let us explain the most desirable characteristics of distributions from the point of view of a financial analyst.

Such a family of distributions are stable distributions, so called because when adding distributions (multiplying linear combinations of functions characterizing them) of this family, another distribution is obtained that belongs to the same family. Stable distributions in turn consist of other underlying distributions. Distributions built on the basis of the Pareto distribution (whose probability density function DA) = a/A +1 for X> 1) have the required characteristics (symmetry, high peak and fat tails) for specific values ​​of the four defining parameters. These four options

This will produce a Pareto distribution (see Figure 27) and will be able to identify several critical types of failures, which typically account for about 70% of all failures. When the information is distributed in descending order of importance, you can focus on those areas, the study of which will give the greatest effect.

Rice. 27 borrowed from a report on failures found in cars in Sweden during the mandatory annual inspection. It shows a typical picture of the Pareto distribution.

The Pareto distribution is graphically presented in fig. 12.5.

Income Fig. 12.5. Pareto distribution

The x-axis shows income, and the /(l) axis shows the number of households or individuals with income equal to or greater than a certain limit (x0). The Pareto distribution is used in practice when approximating a series of income recipients ranked by income level within the interval, i.e., it is used to describe the level of income from the number of recipients whose incomes are above or below given levels.

In connection with relation (1), it is appropriate to recall that in mathematical statistics a distribution with a power-law density decrease is well known - this is a Pareto distribution with a density (a > O, b > 0)

Consider the Pareto distribution with density

More recently, traditional portfolio models have come under heavy criticism, as price changes are thought to be best described by a Pareto distribution with infinite (or indeterminate) variance. However, many studies show that markets have become closer to a normal distribution in recent years (i.e., to limited variance and independence of results), on which the criticized portfolio models are based. Portfolio models use a distribution of returns, rather than a distribution of price changes. Although the distribution of profits is a transformed distribution of price changes (as a result of closing losing trades and holding winning positions for as long as possible), these distributions are usually different. The distribution of profits is not necessarily a class of Pareto distributions, so in Chapter 4 we modeled the distribution of P L with a managed distribution. Moreover, there are derivatives, such as options, which have limited semi-dispersion or variance. For example, a vertical debit option spread guarantees a limited dispersion of profits. I am not trying to challenge the reasonable criticism of current portfolio models. Models should be used provided that we are aware of their shortcomings. Of course, more perfect models portfolios. I am not claiming that current models are adequate, but only that the inputs to portfolio models, current or future, should be based on trading one unit at the optimal level - or at the level we believe will be optimal. For example, if we apply the E-V theory (Markowitz model), the inputs are the expected return, the variance of returns, and the correlations of returns between market systems. The input data should be determined based on the trading of one unit for each market system at the level of the optimal Model

The third physical distribution, characteristic mainly for natural risks, is the Pareto distribution (or self-similar distribution). The probability density function of the damage distribution in this case decreases according to the power law

In the previous section, we assumed that the government is the arbiter in an externality situation, setting a fee for the right to an externality that will make the distribution Pareto efficient. But suppose the state is unable or unwilling to intervene. Will the participants in this situation be able to figure it out without his participation and what will be the outcome of this trial

In the case of EMH, the theory has been developed to justify the use of statistical tools that require independence or, at best, very short-term memory. Theory often came into conflict with observed behavior. For example, according to the EMH, the frequency of price changes should be well represented by a normal distribution. We saw in Chapter 2 that this is not the case. There are too many large up and down changes in all frequencies to fit this normal curve to these distributions. However, such large changes were labeled as special events or "anomalies" and were not included in the frequency distribution. The result of eliminating large changes and renormalizing is a normal distribution. Price changes were labeled as "approximately normal". Alternatives to the normal distribution, such as the Pareto stable distribution, have been rejected, even though they match the observed costs without modification. Why Standard Statistical Analysis could not be applied using such distributions. income distribution. The latter was found to fit well with the lognormal distribution, except for about three percent of the highest individual incomes. At this point, income begins to follow an inverse power law, which gives a thickening of the tail. Roughly speaking, the probability that one person is ten times richer than another follows a normal distribution, but the probability of a hundredfold excess of wealth turns out to be much greater than what is predicted normal distribution. Pareto suggested that this thickened tail probably arises because the rich can multiply their wealth more efficiently than the average individual in order to achieve higher wealth and higher incomes. A similar inverse power law was found by Zipf (G. K. Zipf, 1948) for frequencies using stable distributions behave in the same way as Pareto distributions. In this sense, the "tail" part of the stable distributions is of the Pareto type.

Note that often, especially in the financial literature, distributions of the Pareto type and even simply Pareto distributions are called probability distributions , whose density decreases at infinity (as in a-stable laws with 0

Pareto distribution in probability theory, a two-parameter family of absolutely continuous distributions that are power-law. It is called by the name of Wilfredo Pareto. It occurs in the study of various phenomena, in particular, social, economic, physical and others. Outside the field of economics, it is sometimes also called the Bradford distribution.

Definition

Let the random value X (\displaystyle X) is such that its distribution is given by the equality: