THE DETERMINATION OF DISTRIBUTION PARAMETERS OF ONE-DIMENSIONAL CONTINUOUS RANDOM VARIABLE ACCORDING TO ITS INITIAL CHARACTERISTICS BY FINANCIAL RISKS MODELLING

THE DETERMINATION OF DISTRIBUTION PARAMETERS OF ONEDIMENSIONAL CONTINUOUS RANDOM VARIABLE ACCORDING TO ITS INITIAL CHARACTERISTICS BY FINANCIAL RISKS MODELLING Abstracts. The conception of direct and inverse problem of random variable modelling is introduced. The direct problem is a problem for getting value of continuous random variable, which is contributed according to the given distribution law, which parameters are known. The inverse problem is a problem for defining the distribution law parameters, which are necessary for modelling of continuous one-dimensional random variable, for which the distribution law, mathematical expectation and dispersion are known. For its solution by known type of distribution it is necessary to find the parameter dependence of simulated distribution on set initial characteristics – ensemble average and standard deviation. The assigned problem is solved in explicit form for the following cases: normal distribution, exponential distribution, Laplace distribution, extreme value minimum distribution, extreme value maximum distribution, double exponential distribution, logistic distribution, gamma distribution, Erlang distribution of n-th order, Rayleigh distribution, Maxwellian distribution, parabolic distribution, Simpson distribution, arc sine distribution, inverse Gaussian distribution , Cauchy distribution, one-parameter distribution of n-dimansional random value, hyperexponential distribution, beta distribution, commonbeta distribution, Birnbaum-Sanders distribution. For random variables, which are distributed according to the laws: Erlang second order, beta-distribution of second order, logarithmic normal distribution, it is described the interactive procedure to solve the modelling inverse problem, which realizes the Newton's method for solution of linear equation system. The expressions for elements of matrix solution are received. The solution procedure of assigned task for Weibull and Nakagami distribution is set, which is based on construction of regressive equations, which interpolate the table values to determine links of distribution law parameters and initial characteristics of random variable, which is distributed according to the given law.

Introduction.For many years the statistical modeling is one of the most routine methods of determination of financial risks effects.In detail its application for this goal is described in works [1…11].Statistical modeling technique in different degree of details is described in works [12…15].Its main and integral part is the procedure of reception of random quantity with designated partition law.In this research the problem will be considered, which arises through the necessity to obtain the finite set of one-dimensional pseudorandom variables, which imitate the order of random variables with designated partition.Let us accept that one-dimensional random variable X is assigned by its density , where Q -is vector of parameters.Accept that this vector length does not rank over two.For example for the random variable X which is distributed according to normal law, we get that: In situation (1) parameter m q = 1 -is location parameter and parameter s q = 2 -is scale parameter.As is known from work [16], that m = m and s = s , where m -is expectation of relative X, s is its standard deviation.To get the normal distribution of random variable X with parameters s m, - ) , ( s m x N , using the correlation, which is given at work [16], we obtain: ). 1 , 0 ( ) , ( Taking account of the convention of getting the value ) 1 , 0 ( x N , which is given at work [16] and the connection between parameters of this distribution s m, and initial characteristics, we get, that: In situation (3) it is set, that i r is i -ic realization of quasirandom, uniformly distributed quantity on the interval [0,1].Methods of its obtaining are described in detail at works [12…15].
Let us consider a density of model distribution like: ) From work [16] follows that for this distribution m / 1 = = l q .Modeling correlation used for obtaining quasirandom quantity i x , distributed according to demonstrative law with parameter l will assume the form: The given examples show, that for a start with the modeling process of continuous random quantity it is necessary to choose its distribution law and its parameters, at the same time, usually, at the research beginning only desired values of average (mods, medians) and standard deviations are known.Problem definition.Within the framework of this work two-parameter distribution will be considered, in other cases it will be mentioned.It is supposed that such dependencies are established: It is required to get such dependencies: In convention (7) value of a quantity s m, are known before the modeling beginning.At the beginning we examine distributions of random continuous quantity, which admit problem (7) explicit solution.
Publications analysis.In available to authors of this report publications the similar problem statement wasn't discovered.At work [17] this task solution is got by carrying out actuarial calculations.
Received results.At tab. 1 are showed the results of solution of the given problem for distributions, which allow to receive result in explicit form.By compiling this table for the first, second and third columns we used data given at work [16].
Let us consider the solving of the given problem for one-parameter distribution module of ndimensional random quantity, which view and numerical characteristics are given in work [16].Density of this distribution has form: In convention ( 8) is accepted that quantity n -is positive, integer and known before the beginning of modeling, parameter а>0.Mathematical expectation of this distribution is: Let us consider the solving of the given problem for modeling of continuous random quantity, which is distributed in concordance with beta -distribution, which view and numerical characteristics are given in work [16].

( ) ( )
. (12) Numerical characteristics of this distribution are connected with its parameters form requirements [16]: ) Solving ( 13) and ( 14) relative to u and v we will receive, that: The cited above distributions can be applied to more or less famous distributions.Then let us consider less common Birnbaum-Sanders distribution, which is used by modeling insurance risks, connected with life assurance.Work [18] contains information about it.Density of this distribution has the form: so it is density of standard normal distribution.At the system definition STATGRAPHICS v. XV.I [19] is given modeling ratio to get random variables, which are distributed according to Birnbaum-Sanders law with parameters Numerical characteristics of Birnbaum-Sanders distribution, given at system STATGRAPHICS v. XV.I definition are: Сonsidering conventions ( 20) and (21) as a system of equations concerning variables a and b and accepting, that: Then let us consider distribution of random continuous quantities, for which problem solution under desired conditions (7) in explicit form is impossible.In this situation for solution of given problem Newton method for system solution of two nonlinear equations is used, in the view, which is described at work [20].Let us represent the system (7) like: Jacobian for system (31) assumes the form: Successive values of roots of this system are adduced: Hereinafter, where it will not rise misunderstanding index number of iteration n is skipped.Let us consider the solving of the given problem as applied to general Erlang distribution of the second order, which features are described at work [16].Density of this distribution has such form: Jacobian (28) for system (31) assumes the form: Let us note form determinant (29) for system (36), taking into account the convention (37) like: ( ) Let us note form determinant (30) for system (36), taking into account the condition (37) like: ( ) Let us take that: ( ) ) Then iterative procedure for inverse solution for general Erlang distribution of the second order will have the form: ; Initial value 0 0 , m l , using the references given at work [16] and accepted, that :   Let us consider the solving of the given problem as applied to beta-disrtibution of the second kind, which features are described at work [16].Density of this distribution has the form: Mathematical expectation of the considered distribution has the form: Dispersion of beta-disrtibution of the second kind has the form: Conditions (27) will have in this case the form: Jacobian (28) for system (27) will assume the form: -- Jacobian value is received under condition: Let us note form determinant (29) for system (27), taking into account the condition (28)-(30) like: Let us note form determinant (30) for system (27), taking into account the condition (28 -30), like: Taking into account the condition (28), we will get, that: Then iterative procedure of search of inverse solution for beta-distribution of the second kind will have the form: ; Initial value 0 0 , v u , using the references given at work [16] we will get as conventions : Let us consider the solving of the given problem for normal logarithmic distribution, which characteristics are described at work [16].Density of this distribution has the form: Mathematical expectation of the considered distribution has the form: Variance of logarithmically normal distribution has the form: 27) in this case assume the form: Jacobian (28) for system (65) assumes the form: . 2 Let us note form determinant (29) for system (65), taking into account the condition (66)-(68) like: Let us note form determinant (30) for system (65), taking into account the condition (66)-(68) like: Taking into account the condition (31), we will get, that: where: ( ) Then iterative procedure of search of inverse solution for logarithmically normal distribution will have the form: ; Initial value 0 0 , v u , using the references given at work [16] and accepted, that coefficient of variation: we will get in the form of conditions: ( ).
The considered examples are representative because of combined equations ( 27), which solve the given problem, contain forms, which components are elementary functions.In many cases for dependence expression of initial characteristics of random variable on its distribution parameters are used the expressions, which contain special functions.In this case solving of the given problem for each type of distribution turns into independent task.The problem solving is greatly simplified by using methods based on construction of interpolational dependences.Let us consider the possible methods of its solution in terms of Weibull and Nakagami distributions.Solution of the given problem for Weibull distribution, which features are described at work [16].Density of this distribution has the form: Mathematical expectation m of random variable, which is distributed by Weibull law, has the form: For problem solving of parametrization of Weibull distribution, which correspondent to predetermined numerical, the next procedure is used.
1. Initial data were mathematical expectation m and standard deviation s as in previous problems.
It should be especially considered the procedure of function evaluation To solve the problem of parametrization of Nakagami distribution, which correspondents to predetermined numeric, we used the next procedure.Calculation of value of form function (83) is easy using almost all basic mathematical packets.The problem will become complicated if the user tries to solve it by using such distributed pack EXCEL.Calculation of gamma-function as wired subprogram is provided by version of EXCEL-2013, this procedure is absent at previous versions.In such case for calculation of convention numerical value (56), it should be used the expression given at work [21, p.151 To check expression 5 (57) applicability for function calculation, which is specified by condition (57), are made calculations, which results are given at tab.3. ) (v g 0,967 0,948 0,930 0,915 0,903 0,895 0,889 0,886 0,886 0,886 0,890 ) ( 1 v g 0,966 0,948 0,931 0,916 0,905 0,896 0,889 0,886 0,885 0,885 0,889 From the given results follows, that the convention (84) at the basic range of changing operation of variation coefficient J gives good value approximation, which is indicated at convention (83).
Let us consider the solving of the given problem for Nakagami distribution, which characteristics are described at work [16].Density of this distribution has the form: The procedure by solving the given problem is following.
1. Initial data were mathematical expectation m and standard deviation s as in previous problems.
In convention (88) upper character ( ) * stands for numerical value of α parameter is defined according to convention (87).

Conclusions.
1.The explicit solution of inverse problem of continuous one-dimensional random variable modeling is defined.For its solution by known type of distribution it is necessary to find the parameter dependence of simulated distribution on set initial characteristics -ensemble average and standard deviation.2. The assigned problem is solved in explicit form for the following cases: normal distribution, exponential distribution, Laplace distribution, extreme value minimum distribution, extreme value maximum distribution, double exponential distribution, logistic distribution, gamma distribution, Erlang distribution of n-th order, Rayleigh distribution, Maxwellian distribution, parabolic distribution, Simpson distribution, arc sine distribution, inverse Gaussian distribution, Cauchy distribution, one-parameter distribution of n-dimansional random value, hyperexponential distribution, beta distribution, common-beta distribution, Birnbaum-Sanders distribution.3. The solution procedure of modeling inverse problem of random variables, which are distributed according to the laws: Erlang second order, beta-distribution of second order, logarithmic normal distribution, is described.4. The solution procedure of assigned task for Weibull and Nakagami distribution is suggested.5. Received results may be used by numerical simulation of random variable estimation of financial risk effects.
mathematical expectation m , dispersion D and standard deviation s equalities: conventions (35) instead convention (34) simplifies problem solving and does not increase difficulties by choosing initial data for the solving of the given problem.Convention (27) will have in such case the form:

2 .
Using convention (50) we defined a quantity of expected coefficient of variationJ .3. According to data from tab. 3.1, given at [6, p.151], we got interpolation equations )

Table 1 .
The results of solution of inverse problem of statistical modeling for distributions, which allow to receive result in explicit form.

Table 3 Comparison
2. Using convention (50) we defined a quantity of expected coefficient of variationJ .3. According to data from tab. 3.3, given at [16, p.179], we got interpolation equations