Оцінювання параметрів узагальнених лінійних моделей в аналізі актуарних ризиків
Methods of estimating the parameters of generalized linear models for the case of paying insurance premiums to clients are considered. The iterative-recursive weighted least squares method, the Adam optimization algorithm, and the Monte Carlo method for Markov chains were implemented. Insurance indi...
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| author | Panibratov, Roman Bidyuk, Petro |
| author_facet | Panibratov, Roman Bidyuk, Petro |
| author_institution_txt_mv | [
{
"author": "Roman Panibratov",
"institution": "Educational and Research Institute for Applied System Analysis of the National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Petro Bidyuk",
"institution": "Educational and Research Institute for Applied System Analysis of the National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
}
] |
| author_sort | Panibratov, Roman |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2023-08-07T15:49:29Z |
| description | Methods of estimating the parameters of generalized linear models for the case of paying insurance premiums to clients are considered. The iterative-recursive weighted least squares method, the Adam optimization algorithm, and the Monte Carlo method for Markov chains were implemented. Insurance indicators and the target variable were randomly generated due to the problem of public access to insurance data. For the latter, the normal and exponential law of distribution and the Pareto distribution with the corresponding link functions were used. Based on the quality metrics of model learning, conclusions were made regarding their construction quality. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2023.2.10 |
| first_indexed | 2025-07-17T10:28:19Z |
| format | Article |
| fulltext |
R.S. Panibratov, P.I. Bidyuk, 2023
Системні дослідження та інформаційні технології, 2023, № 2 139
TIДC
МАТЕМАТИЧНІ МЕТОДИ, МОДЕЛІ,
ПРОБЛЕМИ І ТЕХНОЛОГІЇ ДОСЛІДЖЕННЯ
СКЛАДНИХ СИСТЕМ
UDC 004.852
DOI: 10.20535/SRIT.2308-8893.2023.2.10
ESTIMATION OF THE PARAMETERS OF GENERALIZED
LINEAR MODELS IN THE ANALYSIS OF ACTUARIAL RISKS
R.S. PANIBRATOV, P.I. BIDYUK
Abstract. Methods of estimating the parameters of generalized linear models for the
case of paying insurance premiums to clients are considered. The iterative-recursive
weighted least squares method, the Adam optimization algorithm, and the Monte
Carlo method for Markov chains were implemented. Insurance indicators and the
target variable were randomly generated due to the problem of public access to in-
surance data. For the latter, the normal and exponential law of distribution and the
Pareto distribution with the corresponding link functions were used. Based on the
quality metrics of model learning, conclusions were made regarding their construc-
tion quality.
Keywords: actuarial risk, generalized linear models, simulation modeling, exponen-
tial family of distributions, iterative-recursive weighted least squares method, Adam
method, Monte Carlo method for Markov chains.
INTRODUCTION
Actuarial risk is classified as the risk that the assumptions implemented by
actuaries in the model for estimating the prices of insurance policies may be
inaccurate or incorrect. This term is also identified as “insurance risk”. The level
of actuarial risk is directly proportional to the reliability of the assumptions
implemented in the pricing models used by insurance companies to set premiums.
Probability estimates are used to set the price of insurance policies, allowing
insurers to implement payments subject to normal business operations. If the
proposed hypotheses are wrong, the events that were not considered become the
reasons for increasing the frequency of payments, which, in turn, leads to serious
financial consequences for the insurer.
Monograph [1] provides useful methodology for system analysis of complex
processes. The methodology was applied by the authors to analysis of actuarial
processes. In studies [2; 3], mathematical modeling issues related to complicated
non-stationary processes and systems are discussed.
Generalized Linear Models (GLM) allow making explicit assumptions about
the nature of the insurance data and their relationship with the predicted variables.
Furthermore, GLM provides statistical diagnostics that helps in selecting only
significant variables and testing model assumptions. This approach is widely
R.S. Panibratov, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 140
recognized as a standard way of pricing for different types of insurance in
different markets of countries.
The GLM consists of a wide variety of models, including the linear
regression model as a special case. Assumptions for the latter, usually including
normal distribution, constant variance, and additivity of effects, are rejected. For
example, the target variable can be taken from an exponential family of
distributions [4].
The exponential family of distributions has the following general form:
),(
)(
)(
exp ),,( i
i
iii
iii yc
a
by
yf ,
where )(ia , )(b , and ),( iyc are functions, that are defined at the beginning;
i is a parameter related to the mean value; is a scale parameter related to variance.
The variance can vary together with the mean of the distribution. The
influence of explanatory variables is assumed to be additive on an another scale.
The following assumptions are made for GLM:
Stochastic component: each component of Y is independent and is taken
from the one distribution of exponential family.
Systematic component: р covariates (explanatory variables) form linear
predictor :
X .
Link function: the relationship between the random and systematic compo-
nents is established through a link function that is differentiable and monotonic:
)( ][ 1 gYE .
PROBLEM STATEMENT
The purpose of the study is to implement methods for estimating GLM parame-
ters for the analysis of actuarial risks, using different distribution laws and link
functions for the predicted variable.
METHODS OF ESTIMATING PARAMETERS OF GENERALIZED LINEAR
MODELS
GLM parameter estimation is a rather important issue and deserves appropriate
attention. The following algorithms were used to evaluate parameters for the
purpose of comparing methods: iterative-recursive weighted least squares
method; Adam optimization algorithm, Monte Carlo method for Markov chains.
Iterative-recursive weighted least squares method
The heteroskedastic model can be adjusted using the weighted least squares
method (WLS),
WyXWXX TT 1) (
~ ,
Estimation of the parameters of generalized linear models in the analysis of actuarial risks
Системні дослідження та інформаційні технології, 2023, № 2 141
where y is a target variable;
12
)(
i
i
iyVarDiagW is a diagonal matrix of
weights; X is a matrix of covariates.
We will use Fisher’s scoring [5]:
))( () ( )(1)1( yAXWXWXX TtTTt ,
where
1
i
i
iyVarDiagA .
The matrices A and W are related in a next way
i
iDiagWWA .
Then equation for estimating parameters of GLM can be rewritten as
follows:
) () ( )(1)1( WzXWXWXX TtTTt ,
where ) (
yz .
At each step of algorithm:
1. The current estimate of is used to calculate a new working variable z
and a set of weights W .
2. Update (regress) z using X and W to obtain new values.
Iteratively reweighted least squares with random effects (IRWLSR), which
was proposed in [6], successfully overcomes the challenge posed by high dimen-
sional intractable integrals in fitting GLMMs for non-normal data. IRWLSR is
used for Maximum Likelihood estimations of generalized linear mixed effects
models (GLMMs). The benefit is that a working linear mixed effects model
(WWLMM) for normal data performs the bulk of the calculation and prediction.
Only the working responses and weights in the WWLMM are updated using the
GLMM distribution and link function. The complete algorithm can be applied
with ease even if high-dimensional intractable integrals are present in the likeli-
hood function because it does not require any numerical evaluations of intractable
integrals.
Adam (Adaptive moment estimation)
The adaptive movement estimation algorithm or Adam is an extension of the
gradient optimization algorithm. It was created to speed up the optimization
process to improve the ability of the optimization algorithm. This is achieved by
increasing the step size for each input parameter being optimized. Each step size
is automatically adapted through a search process based on partial derivatives or
gradients for each input variable. This involves computing the first and second
moments of the gradient as an exponentially decreasing first moment and second
moment for the input variables.
R.S. Panibratov, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 142
First, the moment vector and exponentially weighted norm at infinity are
adjusted for each optimization parameter, which is denoted as m and v. Their
initial values are equal to zero.
The algorithm consists of the following steps [7]:
The gradient of the target function for the current step is calculated:
))1(( )( kxfkg .
1. Next, the first moment is updated using the gradient and the hyperparame-
ter 1 :
)() 1()1( )( 11 kgkmkm .
2. The second moment is then updated using the gradient square and the
hyperparameter 2 :
)(* ) 1 ()1( * )( 2
12 kgkvkv .
The last two values are biased because their initial values are equal to zero.
3. The first and second moments are adjusted according to the following
formulas:
k
km
km
1 1
)(
)(~
;
k
kv
kv
2 1
)(
)(~
.
4. The optimum point is calculated by the expression:
)(~
)(~
* )1( )(
kv
km
kxkx ,
where is the hyperparameter of step size; is a small value that ensures there
will not be zero division error.
According to [8], the Adam optimization algorithm combining adaptive co-
efficients and composite gradients based on randomized block coordinate descent
is proposed, which improves the algorithm’s performance. Starting with the im-
provement of the Adam algorithm to accelerate the convergence speed, accelerate
the search for the global optimal solution, and enhance the high-dimensional data
processing ability. The suggested approach improves the Adam optimization al-
gorithm’s performance to some degree, but it ignores the effects of second-order
momentum and various learning rates on the performance of the original algo-
rithm.
Quasi-Hyperbolic Momentum (QHM) and Quasi-Hyperbolic Adam (QHA-
dam) are computationally affordable, easily understood, and straightforward to
use. They were implemented in [9] by the authors. In many situations, they can be
great substitutes for momentum and the Adam algorithm. They specifically make
use of high exponential discount factors possible by employing rapid discounting.
As objective functions, the log-likelihood functions of the distributions of
the predicted variables were used with their link functions.
Bayesian approach for parameter estimation
The flexibility of the Bayesian technique to adapt, or the capacity to estimate
model parameters iteratively in the situation where new measurements are con-
Estimation of the parameters of generalized linear models in the analysis of actuarial risks
Системні дослідження та інформаційні технології, 2023, № 2 143
tinually coming in, is one of its benefits. The Monte Carlo technique for Markov
chains (MCMC) may be used to estimate the model’s parameters to the data at a
particular step i, and in general, the process can be expressed as follows:
)(
~
~~
1 iii ,
where i
~
is an estimation of some parameter of the model at step i; is a
weight factor; )(
~
i is the increment in the value of the estimate to be evaluated.
The Markov condition for representing stochastic processes using Markov
chains is satisfied since the estimate of the parameter at the subsequent stage sole-
ly depends on the estimate at the preceding phase. The Monte Carlo approach is
used to generate data from the suitable distribution of the parameter, which is re-
quired in order to estimate the rise in the estimate. Therefore, the Monte Carlo
approach for Markov chains gets its broad name.
This architecture has concerns with re-estimating the parameter distribution
in response to fresh observations and producing data from the proper distribution
for estimating unknown parameters. The MCMC method may be used to answer
these queries.
Typically, the Bayesian method for creating a regression model looks like
follows:
), ... ( ~ 11 knnkk xaxaNy , mk ,...,1 ,
where N is the Normal distribution with appropriate parameters.
In general, the distribution type can be chosen based on how the data are ac-
tually distributed. It can be rewritten in the matrix form:
),( ~ ), ,( ~ 0,0 KNIXNy e ,
where 0 is a vector of mean values at zero step; 0,K is the corresponding co-
variance matrix.
The zero step refers to the initial estimation that was made and will be im-
proved as new data and slow adaptation become available.
Ordinary Least Squares (OLS) can be used when the measured factors and
the dependent variable are both normally distributed random variables. The con-
venience of MCMC, however, comes from the ability to update the values of a
few distribution parameters without performing a full recalculation of all matrix
operations. This is made possible by the so-called property of conjugate prior dis-
tributions from the family of normal, gamma, and their inverse distributions.
The regression model has the following form:
bxay .
Primary estimation of vector 0 can be written in the next form:
0
0
0
b
a
and covariance matrix:
2
0,
2
0,
0,
0
0
b
aK .
R.S. Panibratov, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 144
Applying the aforementioned property to each new pair of observations, it is
possible to arrive at a formula for the updated values ),( 11 st , ),( 22 st of the distri-
bution parameters:
)( 1
0
1
0,1,1 yyKKK
;
111
0,1, ) (
yKKK ;
T
y tt
ss
tt
stst
21
21
21
1221
,
;
2
2
21
2
2
21
2
2
21
1
1
20
0
e
e
y
tt
tt
t
tt
t
K .
However, in the majority cases, when working with non-stationary proc-
esses, there is a case of probabilistic uncertainty, or uncertainty about the distribu-
tion, which means it is impossible to guarantee an unchanged normal distribution
of variables and parameters, making it impossible to always use the above prop-
erty. It is also theoretically incorrect to continue using OLS because the condi-
tions required for its use are broken. Even in situations when the kind of distribu-
tion sought is unknown, MCMC offers a method for approximating the unknown
distribution by producing data for the posterior distribution.
Let’s say that an unknown parameter’s distribution has to be calculated. To
do this, it is recommended that a large number of points be produced from this
distribution, from which the required distributional parameters may then be esti-
mated using the law of large numbers. Nevertheless, as the posterior distribution
of the parameters is unknown, a tool for such creation is required. The Metropo-
lis-Hastings method is a potent tool for resolving the given issue. The algorithm’s
primary steps may be summarized as follows:
1. The method begins by working with an initial value 0 , which can ini-
tially be any random integer.
2. The next step is to choose a new value for * that is suggested as being
produced from the chosen distribution. Some propositional function ),( 0
* g
serves as the foundation for the production of * . The function g is often se-
lected to be Gaussian and must be symmetric. The Bayes theorem is used to ac-
count for the effect of the newly acquired observations in this process.
3. Then calculations are made to determine the resulting value’s acceptance
rate β :
)( ) |(
)( ) |(
00
**
gyg
gyg
.
4. If u , or the newly created point * , is accepted and becomes 1 , a
random number 10 u is chosen; if not, the value is rejected, and the procedure
stays at the previous point.
Estimation of the parameters of generalized linear models in the analysis of actuarial risks
Системні дослідження та інформаційні технології, 2023, № 2 145
5. Until there are sufficient observations, steps 2, 3, 4 are repeated.
The Metropolis-Hastings method asserts that the remaining points originate
from the posterior distribution ),( Xyf after certain initial values are discarded
since the values created in the first stages are not yet stable.
The computation of new posterior distributions is repeated using the consid-
ered technique when new data is introduced since the prior posterior distributions,
according to the Bayes scheme, become a priori for the following iteration.
In [10], authors analyzed insurance claim data and represented the total
claims during a specific time frame as a random sum of individual claims’ indi-
vidual positive random variables. According to the data, the variances of both the
random total and the random claim size are as big as cubic powers of their respec-
tive means. They used natural exponential family modeling to match distributions
with cubic variance functions to the insurance data. The authors took into account
three discrete counting factors for the random sum and three positive continuous
distributions for the claim size, both of which were derived from natural exponen-
tial families. In order to create sampling algorithms, they provided a comprehen-
sive study of the nontrivial discrete counting variables. They ran Monte Carlo
simulations to compute tail probabilities, particularly for big losses, using samples
from the overall claim distribution. Two methods increased these models’ effec-
tiveness. The first is that convolutions belong to the same family as the individual
distribution because the claim size distributions fulfill the reproducibility condi-
tion. The use of importance sampling is the second enhancement.
In [11], authors presented a variety of machine learning regression ap-
proaches ranging from multivariate adaptive regression splines and kernel regres-
sion to ordinary and generalized least-squares regression variants over GLM and
generalized additive model (GAM) approaches for high-dimensional variable se-
lection applications, such as the calibration step in the least-squares Monte Carlo
(LSMC) framework. Regression algorithms proved to be suitable machine learn-
ing methods for proxy modeling of life insurance companies in their slightly dis-
guised real-world example and given LSMC setting, with potential for both
performance and computational efficiency gains by fine-tuning model hyper-
parameters and implementation designs. After all, none of the author’s tried and
true methods could entirely remove the bias seen in the validation figures and
produce findings that were consistent throughout the validation sets. The range of
suggested regression techniques can be further reduced by looking at whether
these findings are systematic for the approaches, the consequence of the Monte
Carlo error, or a mix of the two. Although though such assessments would be
quite expensive computationally, they would be extremely helpful in revealing
new ways to improve the quality of approximations.
NUMERICAL EXPERIMENT
Insurance data is not always publicly available, so it was decided to generate
insurance indicators and target variables randomly. The data consisted of the
following variables:
age (range from 18 to 64 years);
sex;
body mass index (normal distribution was used);
R.S. Panibratov, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 146
number of children (range from 0 to 5);
smoker status;
region (generated from sample view [‘north’, ‘south’, ‘east’, ‘west’, ‘center’]);
charges.
The last variable is the target and the following distribution laws with the
corresponding link functions were used for it:
normal distribution with a known variance and a logarithmic link function;
exponential distribution with the identity link function;
pareto distribution with a known scale parameter mx and a link function
of the form xxf – 1 )( .
Gaussian noise with variable variance, which is a linear function, was added
to the predicted variable.
RESULTS OF THE EXPERIMENTS
The following metrics were used to estimate the quality of the models:
Mean squared error:
2
)
1
(
1
ii
n
i
YY
n
MSE
.
Root mean squared error:
2
)
1
(
1
ii
n
i
YY
n
RMSE
.
Mean absolute error:
ii
n
i
YY
n
MAE ˆ
1
1
.
The results of the estimation of GLM parameters using three methods are
presented in Tables 1, 2, 3.
T a b l e 1 . Results of GLM construction for a target variable with a Gaussian
distribution with known variance and a logarithmic link function
Metric MCMC ADAM IRWLS
MSE 4410.78 686.65 2458.21
RMSE 66.41 26.20 49.58
MAE 51.88 25.81 49.58
T a b l e 2 . Results of GLM construction for the target variable with a Pareto
distribution with a known scale parameter and a negative linear link function
Metric MCMC ADAM IRWLS
MSE 52725.56 82188.07 638121.57
RMSE 229.62 286.69 798.83
MAE 154.02 205.31 773.33
Estimation of the parameters of generalized linear models in the analysis of actuarial risks
Системні дослідження та інформаційні технології, 2023, № 2 147
T a b l e 3 . Results of GLM construction for the target variable with an
exponential distribution and an identity link function
Metric MCMC ADAM IRWLS
MSE 213.52 148.95 288.53
RMSE 14.61 12.20 16.98
MAE 11.17 3.64 15.66
According to the results of GLM construction for three cases, it can be seen
that in most cases, the Adam method showed quite good results. The MCMC
method also showed good results for the case of the Pareto distribution. The final
choice of parameter estimation method is made after practical application of mod-
els to solve the problem of predicting possible losses.
CONCLUSIONS
The methods of estimating GLM parameters for the analysis of actuarial risks in
case of payment of insurance premiums to clients are considered. The following
three methods are implemented: the iterative-recursive weighted least squares
method, the Adam optimization algorithm, and the Monte Carlo method for Mar-
kov chains. Since insurance data are often not publicly available, the insurance
indicators and the target variable were randomly generated: age, gender, body
mass index, number of children, smoking status, region, and charges. The latter
was generated using: a normal distribution with a known variance and a logarith-
mic link function; exponential distribution with the identity link function; of the
Pareto distribution with a known scale parameter and a link function of the form
of a negative linear function. According to the results of the experiments, it can be
concluded that the Adam optimization algorithm demonstrated good results in
most cases. The Monte Carlo method for Markov chains also showed good results
for the case of the Pareto distribution.
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Hall, 1989, 532 p.
6. T. Zhang, “Iteratively reweighted least squares with random effects for maximum
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tation and Simulation, vol. 91, no. 16, pp. 3404–3425, 2021. doi:
10.1080/00949655.2021.1928127.
R.S. Panibratov, P.I. Bidyuk
ISSN 1681–6048 System Research & Information Technologies, 2023, № 2 148
7. D. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv preprint
arXiv:1412.6980, 2014. doi: 10.48550/arXiv.1412.6980.
8. M. Liu et al., “An Improved Adam Optimization Algorithm Combining Adaptive
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arXiv preprint arXiv:1810.06801, 2018. doi: 10.48550/arXiv.1810.06801.
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Received 26.01.2023
INFORMATION ON THE ARTICLE
Roman S. Panibratov, ORCID: 0000-0002-8604-4420, Educational and Research Insti-
tute for Applied System Analysis of the National Technical University of Ukraine “Igor
Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: roman.panibratov@gmail.com
Petro I. Bidyuk, ORCID: 0000-0002-7421-3565, Educational and Research Institute for
Applied System Analysis of the National Technical University of Ukraine “Igor Sikorsky
Kyiv Polytechnic Institute”, Ukraine, e-mail: pbidyuke_00@ukr.net
ОЦІНЮВАННЯ ПАРАМЕТРІВ УЗАГАЛЬНЕНИХ ЛІНІЙНИХ МОДЕЛЕЙ
В АНАЛІЗІ АКТУАРНИХ РИЗИКІВ / П.І. Бідюк, Р.С. Панібратов
Анотація. Розглянуто методи оцінювання параметрів узагальнених лінійних
моделей для аналізу актуарних ризиків у випадку виплат страхових премій
клієнтам. Було реалізовано ітеративно-рекурентно зважуваний метод най-
менших квадратів, алгоритм оптимізації Adam та метод Монте-Карло для лан-
цюгів Маркова. Страхові показники та цільова змінна генерувалися випадко-
вим чином у зв’язку з проблемою публічного доступу страхових даних. Для
останньої використовувався нормальний та експоненціальний закон розподілу
і розподіл Парето з відповідними функціями зв’язку. На основі метрик якості
навчання моделей були зроблені висновки щодо їх якості побудови.
Ключові слова: актуарний ризик, узагальнені лінійні моделі, імітаційне моде-
лювання, експоненційна множина розподілів, ітеративно-рекруентно зважува-
ний метод найменших квадратів, метод Adam,метод Монте-Карло для марків-
ських ланцюгів.
|
| id | journaliasakpiua-article-285518 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:19Z |
| publishDate | 2023 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
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| resource_txt_mv | journaliasakpiua/9f/9e46967362f1ada7b6dfa5403533519f.pdf |
| spelling | journaliasakpiua-article-2855182023-08-07T15:49:29Z Estimation of the parameters of generalized linear models in the analysis of actuarial risks Оцінювання параметрів узагальнених лінійних моделей в аналізі актуарних ризиків Panibratov, Roman Bidyuk, Petro актуарний ризик узагальнені лінійні моделі імітаційне моделювання експоненційна множина розподілів ітеративно-рекруентно зважуваний метод найменших квадратів метод Adam метод Монте-Карло для марківських ланцюгів actuarial risk generalized linear models simulation modeling exponential family of distributions iterative-recursive weighted least squares method Adam method Monte Carlo method for Markov chains Methods of estimating the parameters of generalized linear models for the case of paying insurance premiums to clients are considered. The iterative-recursive weighted least squares method, the Adam optimization algorithm, and the Monte Carlo method for Markov chains were implemented. Insurance indicators and the target variable were randomly generated due to the problem of public access to insurance data. For the latter, the normal and exponential law of distribution and the Pareto distribution with the corresponding link functions were used. Based on the quality metrics of model learning, conclusions were made regarding their construction quality. Розглянуто методи оцінювання параметрів узагальнених лінійних моделей для аналізу актуарних ризиків у випадку виплат страхових премій клієнтам. Було реалізовано ітеративно-рекурентно зважуваний метод найменших квадратів, алгоритм оптимізації Adam та метод Монте-Карло для ланцюгів Маркова. Страхові показники та цільова змінна генерувалися випадковим чином у зв’язку з проблемою публічного доступу страхових даних. Для останньої використовувався нормальний та експоненціальний закон розподілу і розподіл Парето з відповідними функціями зв’язку. На основі метрик якості навчання моделей були зроблені висновки щодо їх якості побудови. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2023-06-30 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/285518 10.20535/SRIT.2308-8893.2023.2.10 System research and information technologies; No. 2 (2023); 139-148 Системные исследования и информационные технологии; № 2 (2023); 139-148 Системні дослідження та інформаційні технології; № 2 (2023); 139-148 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/285518/279578 |
| spellingShingle | актуарний ризик узагальнені лінійні моделі імітаційне моделювання експоненційна множина розподілів ітеративно-рекруентно зважуваний метод найменших квадратів метод Adam метод Монте-Карло для марківських ланцюгів Panibratov, Roman Bidyuk, Petro Оцінювання параметрів узагальнених лінійних моделей в аналізі актуарних ризиків |
| title | Оцінювання параметрів узагальнених лінійних моделей в аналізі актуарних ризиків |
| title_alt | Estimation of the parameters of generalized linear models in the analysis of actuarial risks |
| title_full | Оцінювання параметрів узагальнених лінійних моделей в аналізі актуарних ризиків |
| title_fullStr | Оцінювання параметрів узагальнених лінійних моделей в аналізі актуарних ризиків |
| title_full_unstemmed | Оцінювання параметрів узагальнених лінійних моделей в аналізі актуарних ризиків |
| title_short | Оцінювання параметрів узагальнених лінійних моделей в аналізі актуарних ризиків |
| title_sort | оцінювання параметрів узагальнених лінійних моделей в аналізі актуарних ризиків |
| topic | актуарний ризик узагальнені лінійні моделі імітаційне моделювання експоненційна множина розподілів ітеративно-рекруентно зважуваний метод найменших квадратів метод Adam метод Монте-Карло для марківських ланцюгів |
| topic_facet | актуарний ризик узагальнені лінійні моделі імітаційне моделювання експоненційна множина розподілів ітеративно-рекруентно зважуваний метод найменших квадратів метод Adam метод Монте-Карло для марківських ланцюгів actuarial risk generalized linear models simulation modeling exponential family of distributions iterative-recursive weighted least squares method Adam method Monte Carlo method for Markov chains |
| url | https://journal.iasa.kpi.ua/article/view/285518 |
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