A Small-Sample Weibull Parameter Estimation Method Based on Empirical Cumulative Distribution Function

Abstract

This invention discloses a small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function, relating to the field of product life assessment technology. The method includes: determining the empirical cumulative distribution parameter formula and the range of values ​​for the empirical cumulative distribution parameter; setting the initial value of the empirical cumulative distribution parameter formula in the iterative calculation, and setting the initial value to 0.3 according to the approximate median rank formula; calculating the initial values ​​of shape and scale parameters based on the empirical cumulative distribution parameter formula and fatigue test life sample data arranged in ascending order using the least squares regression method; iterating the empirical cumulative distribution parameter formula based on the Monte Carlo method and the minimum mean square relative error to obtain the optimal empirical cumulative distribution parameter formula and the convergent shape and scale parameters. This allows for intuitive and accurate Weibull parameter estimation with a small sample data capacity, achieving better estimation results and reducing estimation errors.

Description

Technical Field

[0001] This invention relates to the field of product life assessment technology, specifically a small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function. Background Technology

[0002] The Weibull probability distribution model is a widely used statistical distribution model in the study of fatigue life distribution in mechanical structures or products. It exhibits strong adaptability to various fatigue test data and plays a crucial role in structural reliability design. In fatigue statistics applications, the model parameters of the Weibull distribution are calculated from the corresponding fatigue test data. However, due to time and cost constraints, the sample size of fatigue tests is generally not large, falling into the category of small samples, which makes Weibull parameter estimation quite difficult.

[0003] The Weibull model is quite complex, and the reasonable estimation of its model parameters is a key issue in practical applications. Existing parameter estimation methods cover a wide range of theories, including (weighted) least squares regression, moment estimation, maximum likelihood estimation, and linear moment estimation, as well as Weibull parameter optimization estimation methods based on various intelligent algorithms and objective functions. Generally, these parameter estimation methods achieve good estimation results with large sample sizes, and their accuracy tends to converge with increasing sample size. However, with small sample sizes, the accuracy of these methods decreases to varying degrees. In engineering practice, for small samples of complete data, least squares regression can provide more reasonable parameter estimates than maximum likelihood estimation.

[0004] When estimating the parameters of the Weibull distribution using least squares regression, the sample data must be sorted in ascending order, and then the cumulative probability value of each sample can be calculated using the empirical cumulative distribution function. Currently, there are many empirical cumulative distribution functions available. With large sample sizes, the empirical cumulative probability curves of different functions largely overlap, and the parameter estimation results are generally consistent. However, with small sample sizes, these functions show significant differences at the beginning and end of the empirical cumulative probability curves, resulting in substantial errors in the parameter estimation results.

[0005] Therefore, how to achieve intuitive and accurate Weibull parameter estimation with a small sample size, obtain better estimation results, and reduce estimation errors has become an urgent problem to be solved in the field of product life assessment technology. Summary of the Invention

[0006] To address the aforementioned issues, this invention proposes a small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function. This method determines the empirical cumulative distribution parameter formula and its value range; sets initial values ​​for the empirical cumulative distribution parameter formula in the iterative calculation, and sets these initial values ​​to 0.3 according to the approximate median rank formula; based on the empirical cumulative distribution parameter formula and fatigue test life sample data arranged in ascending order, the initial values ​​of the shape and scale parameters of the Weibull distribution model are calculated using the least squares regression method; the empirical cumulative distribution parameter formula is iterated multiple times using the Monte Carlo method and the least mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape and scale parameters. The converged shape and scale parameters are the parameter estimates of the Weibull distribution model. Therefore, for fatigue life test data with a small sample size, this method can achieve intuitive and accurate Weibull parameter estimation with good estimation results. Furthermore, the method is simple, converges quickly, and has strong versatility, and can be extended to parameter estimation of other similar probability distribution models, thus possessing broad engineering application value.

[0007] To achieve the above objectives, the present invention is implemented through the following technical solution:

[0008] This invention is a small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function, comprising:

[0009] Determine the formula for the empirical cumulative distribution parameter and the range of values ​​for the empirical cumulative distribution parameter;

[0010] Set the initial value of the empirical cumulative distribution parameter formula in the iterative calculation, and set the initial value to 0.3 according to the approximate median rank formula;

[0011] Based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, the initial values ​​of the shape and scale parameters of the Weibull distribution model are calculated using the least squares regression method. The empirical cumulative distribution parameter formula is iterated multiple times using the Monte Carlo method and the least mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape and scale parameters. The converged shape and scale parameters are the parameter estimates of the Weibull distribution model.

[0012] A further improvement of the present invention is that: the determination of the formula for the empirical cumulative distribution parameter and the range of values ​​for the empirical cumulative distribution parameter specifically includes:

[0013] The formula for the empirical cumulative distribution parameter is as follows:

[0014]

[0015] In the formula, x iThis represents the fatigue test life sample data sorted in ascending order; F(x) i ) represents the function for calculating the empirical cumulative probability; i = 1, 2, ..., n represents the ascending sequence number of the fatigue test life sample data; n is the size of the fatigue test life sample data; a, b represent the empirical cumulative distribution parameters;

[0016] Among them, the value of a needs to be less than 1, otherwise when i=1, the empirical cumulative probability value is negative; while the value of b needs to be no greater than 1. Therefore, the range of the empirical cumulative distribution parameter is set to 0≤a<1 and 0≤b≤1 respectively.

[0017] Adding the constraint b = 1 - 2a, when n = 1, i = 1, the empirical cumulative probability is 0.5. The formula for determining the empirical cumulative distribution parameter is:

[0018]

[0019] In the formula, x i This represents the fatigue test life sample data sorted in ascending order; F(x) i ) represents the function for calculating the empirical cumulative probability; i = 1, 2, ..., n represents the ascending sequence number of the fatigue test life sample data; n is the size of the fatigue test life sample data; a represents the empirical cumulative distribution parameter, and the value of parameter a is 0 ≤ a ≤ 0.5.

[0020] A further improvement of the present invention is that: based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, the initial values ​​of the shape and scale parameters of the Weibull distribution model are calculated using the least squares regression method; the empirical cumulative distribution parameter formula is iterated multiple times using the Monte Carlo method and the least mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape and scale parameters, wherein the converged shape and scale parameters are the parameter estimates of the Weibull distribution model, specifically including:

[0021] Based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, the following calculations are performed:

[0022] Y i =ln{-ln[1-F(x)} i )]}

[0023] X i =ln(x i )

[0024] In the formula, x i This represents the fatigue test life sample data sorted in ascending order; F(x) iY represents the function for calculating the cumulative probability of experience; i = 1, 2, ..., n represents the ascending order of the fatigue test life sample data; ln() represents the logarithmic function with the constant e as the base; i and X i This represents the discrete data points after logarithmic transformation of fatigue test life sample data;

[0025] The initial values ​​of the shape parameters of the Weibull distribution model are calculated using the least squares regression method. and initial values ​​of scale parameters

[0026]

[0027] In the formula, Indicates the initial values ​​of the shape parameters; The initial values ​​of the scaling parameters are represented by Σ; i = 1, 2, ..., n represent the ascending order of the fatigue test life sample data; Σ represents the summation sign; exp() represents an exponential function with the natural constant e as its base; Y i and X i The numbers represent discrete data points after logarithmic transformation of fatigue test life sample data; k and m represent the slope and intercept of the linear least squares regression line, respectively.

[0028] A further improvement of the present invention lies in: conducting Monte Carlo numerical simulation experiments, and using the initial values ​​of the shape parameters of the Weibull distribution model. and initial values ​​of scale parameters The true value α of the shape parameter of the Weibull distribution model true and the true value of the scale parameter β true ,Right now A Weibull random sample data with the same capacity as the fatigue test life sample data is randomly generated, and the number of Monte Carlo numerical simulation experiments is set to N, resulting in N sets of the Weibull random sample data.

[0029] A further improvement of the present invention is that: based on the formula of the empirical cumulative distribution parameter a for different empirical cumulative distribution parameters, the least squares regression method is used to calculate the parameters of each group of Weibull random sample data, and the empirical cumulative distribution parameter a is continuously taken at a preset interval.

[0030] A further improvement of the present invention lies in: calculating the mean square relative error between the parameter estimates and the true values ​​of the Weibull distribution model under the condition of calculating different empirical cumulative distribution parameter a using the empirical cumulative distribution parameter formula, wherein the mean square relative error is calculated as follows:

[0031]

[0032] In the formula, MSRE represents the mean square relative error between the parameter estimates and the true parameters of the Weibull distribution model; N represents the number of Monte Carlo numerical simulation experiments; α true Represents the true value of the shape parameter of the Weibull distribution model; β true Represents the true value of the scaling parameter of the Weibull distribution model; This represents the estimated shape parameter values ​​of the Weibull distribution model; represents the estimated scale parameter value of the Weibull distribution model; m represents the sequence number of the Monte Carlo numerical simulation experiment.

[0033] A further improvement of the present invention is that: the empirical cumulative distribution parameter α that minimizes the mean square relative error is selected, and the least squares regression method is used to calculate the parameters of the fatigue test life sample data to obtain the shape parameter estimate α of the Weibull distribution model in the current iteration process. j and the estimated scale parameter β j .

[0034] A further improvement of the present invention is that: the above steps are repeated until the parameter estimates of the Weibull distribution model converge stably, thus obtaining the parameter estimates of the optimal Weibull distribution model. The convergence condition is set to the sum of the mean square relative errors of the parameter estimates of the Weibull distribution model in the two iterations being less than the error threshold θ. th ,Right now

[0035]

[0036] In the formula, α j-1 α represents the estimated shape parameter value of the Weibull distribution model during the previous iteration; j β represents the estimated shape parameter of the Weibull distribution model in the next iteration. j-1 This represents the estimated scale parameter of the Weibull distribution model during the previous iteration; β j θ represents the estimated scaling parameter of the Weibull distribution model in the next iteration. th This indicates the error threshold.

[0037] The beneficial effects of this invention are as follows: This invention proposes a small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function. For fatigue life test data with a small sample size, it can achieve intuitive and accurate Weibull parameter estimation and obtain good estimation results. Moreover, the method is simple in concept, converges quickly, and has strong versatility. It can be extended to parameter estimation of other similar probability distribution models and has broad engineering application value. Attached Figure Description

[0038] Figure 1A detailed technical roadmap for a small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function, provided in this embodiment of the invention;

[0039] Figure 2 A flowchart illustrating the iterative calculation process of a small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function, provided in this embodiment of the invention.

[0040] Figure 3 The diagram shows the iterative process of the probability density curve of a Weibull distribution model for a small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function, as provided in this embodiment of the invention. Detailed Implementation

[0041] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0042] like Figure 1 As shown in this embodiment, a small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function includes:

[0043] Determine the formula for the empirical cumulative distribution parameter and the range of values ​​for the empirical cumulative distribution parameter;

[0044] Set the initial value of the empirical cumulative distribution parameter formula in the iterative calculation, and set the initial value to 0.3 according to the approximate median rank formula;

[0045] Based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, the initial values ​​of the shape and scale parameters of the Weibull distribution model are calculated using the least squares regression method. The empirical cumulative distribution parameter formula is iterated multiple times using the Monte Carlo method and the least mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape and scale parameters. The converged shape and scale parameters are the parameter estimates of the Weibull distribution model.

[0046] Pratt & Whitney used a pit-rotating fatigue testing machine to conduct fatigue tests on titanium alloy fan discs and obtained a set of low-cycle fatigue test lives with surface cracks extending to 0.8 mm. The life data were 29525, 36400, 26704, 22387, 14583, and 25995. These data are typical small sample fatigue life data. The following is an example analysis of the present invention based on the above small sample fatigue test life data.

[0047] The formula for determining the empirical cumulative distribution parameter and the range of values ​​for the empirical cumulative distribution parameter specifically include:

[0048] The formula for the empirical cumulative distribution parameter is as follows:

[0049]

[0050] In the formula, x i This represents the fatigue test life sample data sorted in ascending order, namely 14583, 22387, 25995, 26704, 29525, 36400; F(x i ) represents the function for calculating the empirical cumulative probability; i = 1, 2, ..., n represents the ascending sequence number of the fatigue test life sample data; n is the size of the fatigue test life sample data, in this example n = 7; a, b represent the empirical cumulative distribution parameters;

[0051] Among them, the value of a needs to be less than 1, otherwise when i=1, the empirical cumulative probability value is negative; while the value of b needs to be no greater than 1. Therefore, the range of the empirical cumulative distribution parameter is set to 0≤a<1 and 0≤b≤1 respectively.

[0052] Adding the constraint b = 1 - 2a, when n = 1, i = 1, the empirical cumulative probability is 0.5. The formula for determining the empirical cumulative distribution parameter is:

[0053]

[0054] In the formula, x i This represents the fatigue test life sample data sorted in ascending order; F(x) i ) represents the function for calculating the empirical cumulative probability; i = 1, 2, ..., n represents the ascending sequence number of the fatigue test life sample data; n is the size of the fatigue test life sample data; a represents the empirical cumulative distribution parameter, and the value of parameter a is 0 ≤ a ≤ 0.5.

[0055] Set the initial value of the empirical cumulative distribution parameter formula in the iterative solution calculation to 0.3 according to the approximate median rank formula.

[0056] like Figure 2As shown, based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, the initial values ​​of the shape and scale parameters of the Weibull distribution model are calculated using the least squares regression method. The empirical cumulative distribution parameter formula is then iterated multiple times using the Monte Carlo method and the least mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape and scale parameters. These converged shape and scale parameters are the parameter estimates of the Weibull distribution model, specifically including:

[0057] Based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, the following calculations are performed:

[0058] Y i =ln{-ln[1-F(x)} i )]}

[0059] X i =ln(x i )

[0060] In the formula, x i This represents the fatigue test life sample data sorted in ascending order; F(x) i Y represents the function for calculating the cumulative probability of experience; i = 1, 2, ..., n represents the ascending order of the fatigue test life sample data; ln() represents the logarithmic function with the natural constant e as the base; i and X i This represents the discrete data points after logarithmic transformation of fatigue test life sample data;

[0061] The initial values ​​of the shape parameters of the Weibull distribution model are calculated using the least squares regression method. and initial values ​​of scale parameters

[0062]

[0063] In the formula, Indicates the initial values ​​of the shape parameters; The initial values ​​of the scaling parameters are represented by Σ; i = 1, 2, ..., n represent the ascending order of the fatigue test life sample data; Σ represents the summation sign; exp() represents an exponential function with the natural constant e as its base; Y i and X i The numbers represent discrete data points after logarithmic transformation of fatigue test life sample data; k and m represent the slope and intercept of the linear least squares regression line, respectively.

[0064] Calculate the initial values ​​of the shape parameters of the Weibull distribution model using the least squares regression method. and initial values ​​of scale parameters

[0065] Monte Carlo numerical simulation experiments were conducted to initialize the shape parameters of the Weibull distribution model. and initial values ​​of scale parameters The true value α of the shape parameter of the Weibull distribution model true and the true value of the scale parameter β true ,Right now A Weibull random sample data with the same capacity as the fatigue test life sample data is randomly generated, and the number of Monte Carlo numerical simulation experiments is set to N=5000, resulting in 5000 sets of the Weibull random sample data.

[0066] Based on the formula for the empirical cumulative distribution parameter a with different empirical cumulative distribution parameters a, the parameters of each group of Weibull random sample data are calculated by the least squares regression method, and the empirical cumulative distribution parameter a takes continuous values ​​at intervals of 0.001.

[0067] Under the condition of different empirical cumulative distribution parameter 'a', the mean square relative error between the parameter estimates and the true values ​​of the Weibull distribution model is calculated. The formula for calculating the mean square relative error is as follows:

[0068]

[0069] In the formula, MSRE represents the mean square relative error between the parameter estimates and the true parameters of the Weibull distribution model; N represents the number of Monte Carlo numerical simulation experiments; α true Represents the true value of the shape parameter of the Weibull distribution model; β true Represents the true value of the scaling parameter of the Weibull distribution model; This represents the estimated shape parameter values ​​of the Weibull distribution model; represents the estimated scale parameter value of the Weibull distribution model; m represents the sequence number of the Monte Carlo numerical simulation experiment.

[0070] The empirical cumulative distribution parameter α that minimizes the mean square relative error is selected, and the least squares regression method is used to calculate the parameters of the fatigue test life sample data to obtain the shape parameter estimate α of the Weibull distribution model in the current iteration process. j and the estimated scale parameter β j .

[0071] Repeat the above steps until the parameter estimates of the Weibull distribution model converge stably, obtaining the parameter estimates of the optimal Weibull distribution model. The convergence condition is set as the sum of the mean square relative errors of the parameter estimates of the Weibull distribution model in two consecutive iterations being less than the error threshold θ. th ,Right now

[0072]

[0073] In the formula, α j-1 α represents the estimated shape parameter value of the Weibull distribution model during the previous iteration; j β represents the estimated shape parameter of the Weibull distribution model in the next iteration. j-1 This represents the estimated scale parameter of the Weibull distribution model during the previous iteration; β j θ represents the estimated scaling parameter of the Weibull distribution model in the next iteration. th This indicates the error threshold.

[0074] The optimal Weibull parameter is obtained as: α best =2.8927, β best =29389, the Weibull probability density curve during the iteration process is as follows Figure 3 As shown.

[0075] Through the above embodiments, this invention, using a small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function, can determine the empirical cumulative distribution parameter formula and the range of values ​​for the empirical cumulative distribution parameter. It sets the initial value of the empirical cumulative distribution parameter formula in the iterative calculation, and sets the initial value to 0.3 according to the approximate median rank formula. Based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, it calculates the initial values ​​of the shape and scale parameters of the Weibull distribution model using the least squares regression method. It then iterates the empirical cumulative distribution parameter formula multiple times using the Monte Carlo method and the least mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape and scale parameters. The converged shape and scale parameters are the parameter estimates of the Weibull distribution model. Therefore, for fatigue life test data with a small sample size, it can achieve intuitive and accurate Weibull parameter estimation, achieving good estimation results. Furthermore, the method is simple, converges quickly, and has strong versatility, and can be extended to parameter estimation of other similar probability distribution models, possessing broad engineering application value.

[0076] It will be understood by those skilled in the art that, unless otherwise defined, all terms used herein (including technical and scientific terms) have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. It should also be understood that terms such as those defined in general dictionaries should be understood to have the same meaning as in the context of the prior art, and should not be interpreted in an idealized or overly formal sense unless defined as herein.

[0077] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function, characterized in that, include: Determine the formula for the empirical cumulative distribution parameter and the range of values ​​for the empirical cumulative distribution parameter; Set the initial value of the empirical cumulative distribution parameter formula in the iterative calculation, and set the initial value to 0.3 according to the approximate median rank formula; Based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, the initial values ​​of the shape and scale parameters of the Weibull distribution model are calculated using the least squares regression method. The empirical cumulative distribution parameter formula is iterated multiple times using the Monte Carlo method and the least mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape and scale parameters. The converged shape and scale parameters are the parameter estimates of the Weibull distribution model. The formula for determining the empirical cumulative distribution parameter and the range of values ​​for the empirical cumulative distribution parameter specifically include: The formula for the empirical cumulative distribution parameter is as follows: In the formula, x i This represents the fatigue test life sample data sorted in ascending order; F(x) i ) represents the function for calculating the empirical cumulative probability; i = 1, 2, ..., n represents the ascending sequence number of the fatigue test life sample data; n is the size of the fatigue test life sample data; a, b represent the empirical cumulative distribution parameters; Among them, the value of a needs to be less than 1, otherwise when i=1, the empirical cumulative probability value will be negative; while the value of b needs to be no greater than 1. Therefore, the range of the empirical cumulative distribution parameter is set to 0≤a<1 and 0≤b≤1 respectively. Adding the constraint b = 1 - 2a, when n = 1 and i = 1, the empirical cumulative probability is 0.

5. The formula for determining the empirical cumulative distribution parameter is: In the formula, x i This represents the fatigue test life sample data sorted in ascending order; F(x) i ) represents the function for calculating the empirical cumulative probability; i = 1, 2, ..., n represents the ascending sequence number of the fatigue test life sample data; n is the size of the fatigue test life sample data; a represents the empirical cumulative distribution parameter, and the value range of parameter a is 0 ≤ a ≤ 0.5; The process involves calculating initial values ​​of the shape and scale parameters of the Weibull distribution model based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, using the least squares regression method. Then, the empirical cumulative distribution parameter formula is iterated multiple times using the Monte Carlo method and the least mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape and scale parameters. These converged shape and scale parameters are the parameter estimates of the Weibull distribution model, specifically including: Based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, the following calculations are performed: Y i =ln{-ln[1-F(x i )]} X i =ln(x i ) In the formula, x i This represents the fatigue test life sample data sorted in ascending order; F(x) i Y represents the function for calculating the cumulative probability of experience; i = 1, 2, ..., n represents the ascending order of the fatigue test life sample data; ln() represents the logarithmic function with the natural constant e as the base; i and X i This represents the discrete data points after logarithmic transformation of fatigue test life sample data; The initial values ​​of the shape parameters of the Weibull distribution model are calculated using the least squares regression method. and initial values ​​of scale parameters In the formula, Indicates the initial values ​​of the shape parameters; The initial values ​​of the scaling parameters are represented by ∑; i = 1, 2, ..., n represent the ascending order of the fatigue test life sample data; ∑ represents the summation sign; exp() represents an exponential function with base e; Y i and X i The discrete data points represent the fatigue test life sample data after logarithmic transformation; k and m represent the slope and intercept of the linear least squares regression line, respectively. Monte Carlo numerical simulation experiments were conducted to initialize the shape parameters of the Weibull distribution model. and initial values ​​of scale parameters The true value α of the shape parameter of the Weibull distribution model true and the true value of the scale parameter β true ,Right now And randomly generate Weibull random sample data with the same capacity as the fatigue test life sample data, set the number of Monte Carlo numerical simulation experiments to N, and obtain N sets of Weibull random sample data; Based on the formula for the empirical cumulative distribution parameter a with different empirical cumulative distribution parameters a, the least squares regression method is used to calculate the parameters for each group of Weibull random sample data, and the empirical cumulative distribution parameter a is continuously taken at a preset interval; Under the condition of calculating the empirical cumulative distribution parameter formula for different empirical cumulative distribution parameters a, the mean square relative error between the parameter estimate and the true parameter value of the Weibull distribution model is calculated; The empirical cumulative distribution parameter α that minimizes the mean square relative error is selected, and the fatigue test life sample data is calculated using the least squares regression method to obtain the shape parameter estimate α of the Weibull distribution model in the current iteration process. j and the estimated scale parameter β j ; Repeat the above steps until the parameter estimates of the Weibull distribution model converge stably, obtaining the parameter estimates of the optimal Weibull distribution model. The convergence condition is set as the sum of the mean square relative errors of the parameter estimates of the Weibull distribution model in two consecutive iterations being less than the error threshold θ. th .

2. The method for iterative parameter estimation of small-sample Weibull fatigue life distribution based on empirical cumulative distribution function according to claim 1, characterized in that, The formula for calculating the mean square relative error is as follows: In the formula, MSRE represents the mean square relative error between the parameter estimates and the true parameters of the Weibull distribution model; N represents the number of Monte Carlo numerical simulation experiments; α true Represents the true value of the shape parameter of the Weibull distribution model; β true Represents the true value of the scaling parameter of the Weibull distribution model; This represents the estimated shape parameter values ​​of the Weibull distribution model; represents the estimated scale parameter value of the Weibull distribution model; m represents the sequence number of the Monte Carlo numerical simulation experiment.

3. The small-sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function according to claim 1, characterized in that, The convergence condition is set to the sum of the mean square relative errors of the parameter estimates of the Weibull distribution model in two consecutive iterations being less than the error threshold θ. th ,Right now In the formula, α j-1 α represents the estimated shape parameter value of the Weibull distribution model during the previous iteration; j β represents the estimated shape parameter of the Weibull distribution model in the next iteration. j-1 This represents the estimated scale parameter of the Weibull distribution model during the previous iteration; β j θ represents the estimated scaling parameter of the Weibull distribution model in the next iteration. th This indicates the error threshold.

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