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PRACTICAL PROBLEMS ANO SOLUTIONS TO THE USE OF THEORETICAL LAWS IN THE SCIENCES OF THE 2IST CENTURY
TASHKENT, e-s MAY 2024
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ESTIMATION OF UNKNOWN PARAMETER OF WEIBULL DISTRIBUTION IN INCOMPLETE MODELS OF STATISTICS
Berdimuradov Mirkamol
PhD student of National University of Uzbekistan, University Street 4, Tashkent 100149, Uzbekistan mirkamol1403@gmail.com, ORCID ID: 0009-0006-5004-5938 https://doi.org/10.5281/zenodo.13358608 Abstract: This paper will discuss the estimation results of Weibull distribution with type 1 right-censored data using numerical methods. These methods involve simulations employing the Maximum Likelihood Estimation technique, utilizing both the Quasi-Newton rule and the Nelder-Mead simplex algorithm. The simulation includes generating random sample data from distribution with sample n sizes of 500 and 1000. The parameters used for the initial guess are obtained from example data of patients with lung cancer, specifically k = 2, X = 3. Based on the simulation results of the two estimation methods, it is evident that parameter estimation using the Quasi-Newton rule outperforms the Nelder-Mead simplex algorithm when in an uncensored state. However, the estimated results of the Nelder-Mead method show better estimated values compared to the Quasi-Newton rule after a fixed censoring time. [see, graphs and tables below].
Keywords: Weibull distribution, Quasi-Newton, Nelder-Mead algorithm, MLE, Right censoring..
1 INTRODUCTION
Usually, when estimating unknown parameters using the MLE method, it is necessary to calculate the integral and derivatives of the cumulative distribution function F(x). However, in some cases, due to the complexity of integration and derivation by the analytical method, numerical calculation in computer programs becomes necessary. In this thesis, the unknown parameters of the continuous Weibull distribution are estimated using numerical methods with random samples.The Weibull distribution is widely used in various fields, including modeling the distribution phenomena of fatigue and the lifespan of many devices, such as bearings, shafts, and motors. Luís Andrade Ferreira FEUP - Faculdade de Engenharia da Universidade do Porto Department of Mechanical Engineering Rua Dr. Roberto Frias, 4200465, Porto, Portugal and José Luís Silva ESTV - Escola Superior Tecnologia de Viseu Department of Mechanical Engineering and Industrial Management Campus Politécnico, 3504-510, Viseu, Portugal are created an article which is based on Expectation-Maximization algorithm to estimate unknown parameters of Weibull distribution with MLE method. In this article, we compare the methods of estimating unknown parameters of the Weibull distribution using the Nelder-Mead and Quasi-Newton rules with numerical solutions in Matlab, as opposed to the EM algorithm. [1,2,4].
2 RESEARCH METHODOLGY
The estimation process utilizes various statistical techniques, methods, and procedures to analyze data concerning a variable of interest, such as the time
elapsed from a clearly defined starting point (e.g., equipment installation) to a specific event (e.g., equipment or component failure). This process aims to estimate the distribution parameters for modeling the system under examination. These parameters are the distribution characteristics that reflect the behavior of a particular population and are therefore fixed for a specific system. Hence, the estimation of system parameters is derived from the data collected from the population. Parametric analysis assumes that the data conforms to a specific distribution, such as the Weibull distribution.
3 ANALYSIS OF LITERATURE ON THE SUBJECT
In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It models a broad range of random variables, largely in the nature of a time to failure or time between events. The probability density function of a Weibull random variable is
f ( k ) =
— - e
x x < 0
,-( x!X)k
x > 0
parameter and X> 0 is the distribution.
where k > 0 is the shape the scale parameter of Its complementary cumulative distribution function is a stretched exponential function. The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution ( k = 1 ).
k-i
PPSUTLSC-2024
PRACTICAL PROBLEMS AND SOLUTIONS TO THE USE OF THEORETICAL LAWS IN THE SCIENCES OF THE 2IST CENTURY
tashkent, o-8 MAv 2004 www.in~academy.uz
The cumulative distribution function for the Weibull distribution is
F (x; k ,A) = 1 - x/Af for x > 0, and F(x; k,A) = 0 for x < 0. 4 ANALYSIS AND RESULTS As we mentioned above, it is necessary to calculate the integral and derivatives of the cumulative distribution function F(x). However, in some cases, due to the complexity of integration and derivation by the analytical method, numerical calculation in computer programs becomes necessary. To present some results obtained through numerical optimization using MATLAB for the Weibull distribution, below graphs and table 1 display the findings.
[Here, we can compare the results of both methods at different censoring levels.]
Selected Values and Fitted Weibull Distributions (MLE)
Nelder-Mead Method, n=1000, Censoring = 0%
Selected Values and Fitted Weibull Distributions (MLE)
Quasi-Newton Method, n=1000, Censoring = 10%
§
5 „
a 0 £
O Success (delta=1) O Failure (delta=0)
True distribution (k=2, A=3) - - - Fitted distribution (k=1.9764, A=3.0723) .....T(i) constant_
Success (delta=1): 1000 Failure (delta=0): 0
Selected Values and Fitted Weibull Distributions (MLE)
Quasi-Newton Method, n=1000, Censoring = 0%
O Success (delta=1) O Failure (delta=0)
-True distribution (k=2, A=3)
- - -Fitted distribution (k=1.9898, A=3.0358) T(i) constant_
Success (delta=1): 1000 Falure (delta=0): 0
Selected Values and Fitted Weibull Distributions (MLE)
Nelder-Mead Method, n=1000, Censoring = 10%
Success (delta=1) Failure (delta=0) —True distribution (k=2, A=3) - Fitted distribution (k=2.0646, A=2.6207) •• T(i) constant_
Success (delta=1): 901 Failure (delta=0): 99
O Success (delta=1) O Failure (delta=0)
-True distribution (k=2, A=3)
---Fitted distribution (k=2.0816, A=2.6675)
.....T(i) constant
Success (delta=1): 902 Failure (delta=0): 98
Selected Values and Fitted Weibull Distributions (MLE)
Nelder-Mead Method, n=1000, Censoring =30%
u.o !..
O Success (delta=1) O Failure (delta=0)
True distribution (k=2, A=3) - - -Fitted distribution (k=2.1957, A=2.3091) T(i) constant
Success (delta=1): 703 Failure (delta=0): 297
Selected Values and Fitted Weibull Distributions (MLE)
Quasi-Newton Method, n=1000, Censoring = 30%
O Success (delta=1) O Failure (delta=0) — True distribution (k=2, A=3)
---Fitted distribution (k=2.2259, A=2.2236)
......T(i) constant_
Success (delta=1): 703 Failure (delta=0): 297
5 CONCLUSIONS AND SUGGESTIONS
Based on the simulation results of the two estimation methods, it is evident that parameter estimation using the Quasi-Newton rule outperforms the Nelder-Mead simplex algorithm when in an uncensored state. However, the estimated results of the Nelder-Mead method show better estimated values compared to the Quasi-Newton rule after a fixed censoring time. Estimated of unknown parameters of Weibull distribution with two rules in MLE Table 1
Interval [0;10]
T p% Weibull Distribution
Initial Guess k =2; 1=3
PPSUTLSC-2024
PRACTICAL PROBLEMS AND SOLUTIONS TO THE USE OF THEORETICAL LAWS IN THE SCIENCES OF THE 2IST CENTURY
tashkent, o-8 may 2004 www.in~academy.uz
T e [0;10] fixed constant
MLE by QuasiNewton rule MLE by Nelder-Mead rule
n=500
10 0% k =1.9815; 1 =3.0215; k =1.9118; 1 =3.0412;
2.7 10% k =2.0913; 1 =2.705; k =2.0742; 1 =2.809;
2.2 30% k =2.323; 1 =2.2342; k =2.1821; 1 =2.401;
n=1000
10 0% k =1.9898; 1 =3.0358; k =1.9764; 1 =3.0723;
2.7 10% k =2.0816; 1 =2.6103; k =2.0646; 1 =2.6207;
2.2 30% k =2.2259; 1 =2.2236; k =2.1957; 1 =2.3091;
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