Small sample Weibull parameter estimation method based on empirical cumulative distribution function

Abstract

The invention discloses a small sample Weibull fatigue life distribution iteration parameter estimation method based on an empirical cumulative distribution function, and relates to the technical field of product life evaluation, and the method comprises the steps: determining an empirical cumulative distribution parameter formula and the value range of empirical cumulative distribution parameters; setting an initial value of the empirical cumulative distribution parameter formula in iterative calculation, and setting the initial value as 0.3 according to an approximate median rank formula; calculating initial values of shape parameters and scale parameters based on a least square regression method according to the empirical cumulative distribution parameter formula and fatigue test life sample data arranged in an ascending order, and iterating the empirical cumulative distribution parameter formula based on a Monte Carlo method and a minimum mean square relative error; according to the method, an optimal empirical cumulative distribution parameter formula and convergent shape parameters and scale parameters are obtained, so that visual and accurate Weibull parameter estimation can be realized under the condition of small sample data capacity, a better estimation effect is obtained, and estimation errors are reduced.

Description

Technical Field

[0001] The invention relates to the technical field of product life assessment, in particular to an iterative parameter estimation method for small sample Weibull fatigue life distribution based on an empirical cumulative distribution function. Background Art

[0002] The Weibull probability distribution model is a widely used statistical distribution model in the study of fatigue life distribution of mechanical structures or products. It has a strong adaptability to various fatigue test data and plays an important role in structural reliability design. In fatigue statistical 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 not too large and basically belongs to the category of small samples, which makes the Weibull parameter estimation more difficult.

[0003] The Weibull model is relatively complex, and the reasonable estimation of its model parameters has become a key issue in practical applications. The existing parameter estimation methods involve a wide range of theories, including (weighted) least squares regression, moment estimation, maximum likelihood estimation, linear moment estimation, and Weibull parameter optimization estimation methods based on various intelligent algorithms and objective functions. Generally speaking, these parameter estimation methods can obtain good estimation results when the sample data volume is large, and they are basically consistent with the increase in the number of samples. However, in the case of small samples, the parameter estimation accuracy of these methods decreases to varying degrees. In engineering practice, compared with the maximum likelihood method, for complete data with small sample sizes, the least squares regression method can give more reasonable solution parameters.

[0004] When using the least squares regression method to estimate the parameters of the Weibull distribution, the sample data must be arranged in ascending order, and then the empirical cumulative distribution function is used to calculate the cumulative probability value of each sample. There are many empirical cumulative distribution functions in the prior art. Under large sample sizes, the empirical cumulative probability curves of different empirical cumulative distribution functions basically overlap, and the parameter estimation results are basically consistent. However, under small sample sizes, these functions have large differences in the beginning and end areas of the empirical cumulative probability curves, and their parameter estimation results have large errors.

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

[0006] In order to solve the above problems, the present invention proposes an iterative parameter estimation method for a small sample Weibull fatigue life distribution based on an empirical cumulative distribution function, which can determine the empirical cumulative distribution parameter formula and the value range of 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; calculate the initial values ​​of the shape parameter and scale parameter of the Weibull distribution model based on the least squares regression method according to the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, iterate the empirical cumulative distribution parameter formula multiple times based on the Monte Carlo method and the minimum mean square relative error criterion, and obtain the optimal empirical cumulative distribution parameter formula and the converged shape parameter and scale parameter. The converged shape parameter and scale parameter are the parameter estimation values ​​of the Weibull distribution model, so that for fatigue life test data with a small sample size, intuitive and accurate Weibull parameter estimation can be achieved, and better estimation effect can be achieved. The method is simple in thought, converges quickly, and has strong versatility. It can be extended to parameter estimation of other similar probability distribution models, and has wide engineering application value.

[0007] In order to achieve the above object, the present invention is implemented by the following technical solutions:

[0008] The present invention is an iterative parameter estimation method for small sample Weibull fatigue life distribution based on empirical cumulative distribution function, comprising:

[0009] Determine the empirical cumulative distribution parameter formula and the value range of the empirical cumulative distribution parameter;

[0010] Setting an 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;

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

[0012] A further improvement of the present invention is that the formula for determining the empirical cumulative distribution parameter and the value range of the empirical cumulative distribution parameter specifically include:

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

[0014]

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

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

[0017] Add a constraint condition, that is, b=1-2a, when n=1, i=1, the empirical cumulative probability is 0.5, and the formula for determining the empirical cumulative distribution parameter is:

[0018]

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

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

[0021] The following calculation is performed based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order:

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

[0023] X i =ln(x i )

[0024] In the formula, x i represents the fatigue test life sample data arranged in ascending order; F(x i) represents the function for calculating the empirical cumulative probability; 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, Y i and X i Represents the discrete data points after logarithmic transformation of fatigue test life sample data;

[0025] Calculate the initial value of the shape parameter of the Weibull distribution model according to the least squares regression method and the initial value of the scale parameter

[0026]

[0027] In the formula, Represents the initial value of shape parameters; represents the initial value of the scale parameter; i=1,2,...,n represents the ascending order of the fatigue test life sample data; Σ represents the summation symbol; exp() represents the exponential function with the natural constant e as the base; Y i and X i represents the 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 is to carry out a Monte Carlo numerical simulation experiment and set the initial value of the shape parameter of the Weibull distribution model and the initial value of the scale parameter 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 consistent with the fatigue test life sample data capacity, set the number of Monte Carlo numerical simulation experiments to N, and obtain N groups of Weibull random sample data.

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

[0030] A further improvement of the present invention is that: under the condition of the empirical cumulative distribution parameter formula with different empirical cumulative distribution parameters a, the mean square relative error between the parameter estimation value and the parameter true value of the Weibull distribution model is calculated, and the mean square relative error calculation formula is as follows:

[0031]

[0032] Where MSRE is the mean square relative error between the parameter estimate and the true value of the Weibull distribution model; N is 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 scale parameter of the Weibull distribution model; represents the shape parameter estimate of the Weibull distribution model; represents the estimated value of the scale parameter of the Weibull distribution model; m represents the number of Monte Carlo numerical simulation experiments.

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

[0034] A further improvement of the present invention is that: the above steps are repeated until the parameter estimation value of the Weibull distribution model converges stably to obtain the parameter estimation value of the optimal Weibull distribution model, and the convergence condition is set to that the sum of the mean square relative errors of the parameter estimation values ​​of the Weibull distribution model in the two previous and subsequent iterations is less than the error threshold θ th ,Right now

[0035]

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

[0037] The beneficial effects of the present invention are as follows: the present invention proposes an iterative parameter estimation method for small sample Weibull fatigue life distribution based on the empirical cumulative distribution function. For fatigue life test data with a small sample capacity, intuitive and accurate Weibull parameter estimation can be achieved, and better estimation effect can be achieved. The method is concise in thought, 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. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1A specific technical roadmap for an iterative parameter estimation method for small sample Weibull fatigue life distribution based on an empirical cumulative distribution function provided in an embodiment of the present invention;

[0039] Figure 2 An iterative calculation flow chart of an iterative parameter estimation method for a small sample Weibull fatigue life distribution based on an empirical cumulative distribution function provided in an embodiment of the present invention;

[0040] Figure 3 A probability density curve iteration process diagram of a Weibull distribution model of an iterative parameter estimation method for a small sample Weibull fatigue life distribution based on an empirical cumulative distribution function provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0041] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present 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, a small sample Weibull fatigue life distribution iterative parameter estimation method based on an empirical cumulative distribution function in this embodiment includes:

[0043] Determine the empirical cumulative distribution parameter formula and the value range of the empirical cumulative distribution parameter;

[0044] Setting an 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;

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

[0046] When Pratt & Whitney conducted fatigue tests on titanium alloy fan discs using a pit rotating fatigue testing machine, it obtained a set of low cycle fatigue test lives when the disc surface crack expanded to 0.8 mm. The life data were 29525, 36400, 26704, 22387, 14583, and 25995, respectively. These data are all typical small sample fatigue life data. The following example analysis of the present invention is based on the above small sample fatigue test life data.

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

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

[0049]

[0050] In the formula, x i represents the fatigue test life sample data in ascending order, i.e., 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 order of the fatigue test life sample data; n is the capacity of the fatigue test life sample data, in this example n = 7; a, b represent the empirical cumulative distribution parameters;

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

[0052] Add a constraint condition, that is, b=1-2a, when n=1, i=1, the empirical cumulative probability is 0.5, and the formula for determining the empirical cumulative distribution parameter is:

[0053]

[0054] In the formula, x i represents the fatigue test life sample data arranged in ascending order; F(x i ) represents the function for calculating the empirical cumulative probability; i = 1, 2, ..., n represents the ascending order of the fatigue test life sample data; n is the capacity of the fatigue test life sample data; a represents the empirical cumulative distribution parameter, and the value range of the 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 based on the approximate median rank formula.

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

[0057] The following calculation is performed based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order:

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

[0059] X i =ln(x i )

[0060] In the formula, x i represents the fatigue test life sample data arranged in ascending order; F(x i ) represents the function for calculating the empirical cumulative probability; 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, Y i and X i Represents the discrete data points after logarithmic transformation of fatigue test life sample data;

[0061] Calculate the initial value of the shape parameter of the Weibull distribution model according to the least squares regression method and the initial value of the scale parameter

[0062]

[0063] In the formula, Represents the initial value of shape parameters; represents the initial value of the scale parameter; i=1,2,...,n represents the ascending order of the fatigue test life sample data; Σ represents the summation symbol; exp() represents the exponential function with the natural constant e as the base; Y i and X i represents the 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 value of the shape parameter of the Weibull distribution model based on the least squares regression method and the initial value of the scale parameter

[0065] Carry out Monte Carlo numerical simulation experiments and set the initial value of the shape parameter of the Weibull distribution model and the initial value of the scale parameter 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 consistent with the fatigue test life sample data capacity, set the number of Monte Carlo numerical simulation experiments to N=5000, and obtain 5000 groups of Weibull random sample data.

[0066] Based on the empirical cumulative distribution parameter formula of different empirical cumulative distribution parameters a, parameter calculation is performed on each group of Weibull random sample data through the least squares regression method, and the empirical cumulative distribution parameter a is continuously taken at intervals of 0.001.

[0067] The mean square relative error between the parameter estimate and the true value of the parameter of the Weibull distribution model under the empirical cumulative distribution parameter formula conditions with different empirical cumulative distribution parameters a is calculated. The mean square relative error calculation formula is as follows:

[0068]

[0069] Where MSRE is the mean square relative error between the parameter estimate and the true value of the Weibull distribution model; N is 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 scale parameter of the Weibull distribution model; represents the shape parameter estimate of the Weibull distribution model; represents the estimated value of the scale parameter of the Weibull distribution model; m represents the number of Monte Carlo numerical simulation experiments.

[0070] Select the empirical cumulative distribution parameter a with the smallest mean square relative error, use the least squares regression method to perform parameter calculation on the fatigue test life sample data, and obtain the shape parameter estimate α of the Weibull distribution model in the current iteration process j and the scale parameter estimate β j .

[0071] Repeat the above steps until the parameter estimation value of the Weibull distribution model converges stably to obtain the parameter estimation value of the optimal Weibull distribution model. The convergence condition is set to the sum of the mean square relative errors of the parameter estimation values ​​of the Weibull distribution model in the two iterations before and after being less than the error threshold θ th ,Right now

[0072]

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

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

[0075] Through the introduction of the above embodiments, the present invention can determine the empirical cumulative distribution parameter formula and the value range of the empirical cumulative distribution parameter through an iterative parameter estimation method for small sample Weibull fatigue life distribution based on the empirical cumulative distribution function; 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; calculate the initial values ​​of the shape parameters and scale parameters of the Weibull distribution model based on the least squares regression method according to the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, and iterate the empirical cumulative distribution parameter formula multiple times based on the Monte Carlo method and the minimum mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape parameters and scale parameters. The converged shape parameters and scale parameters are the parameter estimation values ​​of the Weibull distribution model, so that for fatigue life test data with a small sample size, intuitive and accurate Weibull parameter estimation can be achieved, and better estimation effect can be achieved. The method is simple in thought, converges quickly, and has strong versatility. It can be extended to parameter estimation of other similar probability distribution models, and has wide engineering application value.

[0076] It will be understood by those skilled in the art that, unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as those generally understood by those skilled in the art in the art to which the present invention belongs. It should also be understood that terms such as those defined in common dictionaries should be understood to have the same meaning as in the context of the prior art, and will not be interpreted with an idealized or overly formal meaning unless defined similarly here.

[0077] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method 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 in the scope of protection of the present invention.

Claims

1. A small sample Weibull fatigue life distribution iterative parameter estimation method based on empirical cumulative distribution function, characterized in that: include: Determine the empirical cumulative distribution parameter formula and the value range of the empirical cumulative distribution parameter; Setting an 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; According to the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, the initial values ​​of the shape parameters and scale parameters of the Weibull distribution model are calculated based on the least squares regression method, and the empirical cumulative distribution parameter formula is iterated multiple times based on the Monte Carlo method and the minimum mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape parameters and scale parameters. The converged shape parameters and scale parameters are the parameter estimation values ​​of the Weibull distribution model.

2. The iterative parameter estimation method for small sample Weibull fatigue life distribution based on empirical cumulative distribution function according to claim 1 is characterized in that: The formula for determining the empirical cumulative distribution parameter and the value range of the empirical cumulative distribution parameter specifically include: The empirical cumulative distribution parameter formula is as follows: In the formula, x i represents the fatigue test life sample data arranged in ascending order; F(x i ) represents the function of calculating the empirical cumulative probability; i = 1, 2, ..., n represents the ascending order of the fatigue test life sample data; n is the capacity of the fatigue test life sample data; a, b represent the empirical cumulative distribution parameters; Among them, the a value needs to be less than 1, otherwise when i=1, the empirical cumulative probability value is negative; and the b value needs to be no greater than 1, so the value ranges of the empirical cumulative distribution parameters are set to 0≤a<1 and 0≤b≤1 respectively; Add a constraint condition, that is, b=1-2a, when n=1, i=1, the empirical cumulative probability is 0.5, and the formula for determining the empirical cumulative distribution parameter is: In the formula, x i represents the fatigue test life sample data arranged in ascending order; F(x i ) represents the function for calculating the empirical cumulative probability; i = 1, 2, ..., n represents the ascending order of the fatigue test life sample data; n is the capacity of the fatigue test life sample data; a represents the empirical cumulative distribution parameter, and the value range of the parameter a is 0≤a≤0.

5.

3. The small sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function according to claim 1 is characterized in that: The method of calculating the initial values ​​of the shape parameters and scale parameters of the Weibull distribution model based on the least squares regression method according to the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order, and iterating the empirical cumulative distribution parameter formula multiple times based on the Monte Carlo method and the minimum mean square relative error criterion to obtain the optimal empirical cumulative distribution parameter formula and the converged shape parameters and scale parameters, wherein the converged shape parameters and scale parameters are the parameter estimation values ​​of the Weibull distribution model, specifically includes: The following calculation is performed based on the empirical cumulative distribution parameter formula and the fatigue test life sample data arranged in ascending order: Y i =ln{-ln[1-F(x i )]} X i =ln(x i ) In the formula, x i represents the fatigue test life sample data arranged in ascending order; F(x i ) represents the function for calculating the empirical cumulative probability; 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, Y i and X i Represents the discrete data points after logarithmic transformation of fatigue test life sample data; Calculate the initial value of the shape parameter of the Weibull distribution model according to the least squares regression method and the initial value of the scale parameter In the formula, Represents the initial value of shape parameters; represents the initial value of the scale parameter; i=1,2,...,n represents the ascending order of the fatigue test life sample data; Σ represents the summation symbol; exp() represents the exponential function with the constant e as the base; Y i and X i represents the 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.

4. The small sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function according to claim 3 is characterized in that: Carry out Monte Carlo numerical simulation experiments and set the initial value of the shape parameter of the Weibull distribution model and the initial value of the scale parameter 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 consistent with the fatigue test life sample data capacity, set the number of Monte Carlo numerical simulation experiments to N, and obtain N groups of Weibull random sample data.

5. The iterative parameter estimation method for small sample Weibull fatigue life distribution based on empirical cumulative distribution function according to claim 4, characterized in that: Based on the empirical cumulative distribution parameter formula of different empirical cumulative distribution parameters a, parameter calculation is performed on each group of Weibull random sample data through the least squares regression method, and the empirical cumulative distribution parameter a is continuously taken at preset intervals.

6. The small sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function according to claim 5 is characterized in that: The mean square relative error between the parameter estimate and the true value of the parameter of the Weibull distribution model under the empirical cumulative distribution parameter formula conditions with different empirical cumulative distribution parameters a is calculated. The mean square relative error calculation formula is as follows: Where MSRE is the mean square relative error between the parameter estimate and the true value of the Weibull distribution model; N is the number of Monte Carlo numerical simulation experiments; α true Represents the true value of the shape parameter of the Weibull distribution model; represents the true value of the scale parameter of the Weibull distribution model; represents the shape parameter estimate of the Weibull distribution model; represents the estimated value of the scale parameter of the Weibull distribution model; m represents the number of Monte Carlo numerical simulation experiments.

7. The small sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function according to claim 6 is characterized in that: Select the empirical cumulative distribution parameter a with the smallest mean square relative error, use the least squares regression method to perform parameter calculation on the fatigue test life sample data, and obtain the shape parameter estimate α of the Weibull distribution model in the current iteration process j and the scale parameter estimate β j .

8. The small sample Weibull fatigue life distribution iterative parameter estimation method based on the empirical cumulative distribution function according to claim 7 is characterized in that: Repeat the above steps until the parameter estimation value of the Weibull distribution model converges stably to obtain the parameter estimation value of the optimal Weibull distribution model. The convergence condition is set to the sum of the mean square relative errors of the parameter estimation values ​​of the Weibull distribution model in the two iterations before and after being less than the error threshold θ th ,Right now In the formula, α j-1 Represents the estimated value of the shape parameter of the Weibull distribution model in the previous iteration; α j represents the estimated value of the shape parameter of the Weibull distribution model in the next iteration; β j-1 represents the estimated value of the scale parameter of the Weibull distribution model in the previous iteration; β j represents the estimated value of the scale parameter of the Weibull distribution model in the next iteration; θ th Represents the error threshold.

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