An iterative method for estimating the parameters of a three-parameter Weibull distribution based on small samples.

Abstract

This invention discloses an iterative method for estimating the parameters of a three-parameter Weibull distribution based on small samples, comprising: Step 1: setting the shape parameter of the product lifespan to be estimated to be between 1.5 and 2.5; Step 2: deriving the model between the location parameter and the minimum sample value and sample size, obtaining a first theoretical model with a 95% confidence level to obtain a conservative estimate; Step 3: obtaining a second theoretical model to obtain a point estimate; Step 4: calculating the initial estimate of the scale parameter based on the set shape parameter, initial location parameter, and random sample using the maximum likelihood method, and substituting the initial estimate of the scale parameter into the theoretical model of Step 2 or 3 for multiple iterative calculations to obtain stable scale and location parameters. This method establishes a theoretical model describing the relationship between the location parameter, the minimum sample value, and the sample size, and proposes an iterative method for estimating the location and scale parameters.

Description

Technical Field

[0001] This invention belongs to the technical field of product life assessment methods, and relates to an iterative method for estimating the parameters of a three-parameter Weibull distribution based on a small sample. Background Technology

[0002] The Weibull distribution is one of the most popular statistical models for describing the probabilistic behavior of real-world phenomena and plays an important role in reliability analysis. For the same type of product operating in similar environments, its shape parameter is approximately constant, making it particularly useful for small sample sizes. The two-parameter Weibull distribution is widely used due to its greater simplicity.

[0003] Typically, parameter estimation requires a large number of samples. However, for large and complex equipment, economic and time constraints often limit the availability of data to small samples, making parameter estimation difficult. Currently, the problem of estimating the parameters of the three-parameter Weibull distribution has attracted considerable attention from scholars both domestically and internationally. Commonly used methods for this type of problem include the maximum likelihood method and the correlation coefficient method.

[0004] The maximum likelihood method mainly aims to find the maximum value of the likelihood function, that is, to make the partial derivative of each parameter equal to 0, i.e., to solve the following three equations:

[0005]

[0006]

[0007]

[0008] In the formula, γ, β, and η represent the location, shape, and scale parameters of the Weibull distribution. Solving these three equations is quite complex. If the shape parameter is not fixed, the maximum likelihood method may not provide a solution for estimating the parameters of the three-parameter Weibull distribution in the case of a small sample.

[0009] The main idea of ​​the correlation coefficient method is to find the location parameter when the correlation coefficient of the samples is maximized, and then use the least squares method to find the shape parameter and scale parameter respectively. However, when using this method, it is possible to find that the location parameter is 0, causing the three-parameter Weibull distribution to degenerate into a two-parameter Weibull distribution, which is overly conservative. Summary of the Invention

[0010] To address the aforementioned technical problems, this invention starts with the minimum sample value of the three-parameter Weibull distribution, establishes a relationship model between the location parameter, the minimum sample value, and the sample size, and proposes a new method for parameter estimation of the three-parameter Weibull distribution.

[0011] An iterative method for estimating the parameters of a three-parameter Weibull distribution based on small samples, according to the present invention, includes:

[0012] Step 1: Determine the shape parameters. Set the shape parameter range for the product life to be estimated to be between 1.5 and 2.5.

[0013] Step 2: Derivation of the model between location parameters, minimum sample value, and sample size. If a conservative estimate is to be obtained, a first theoretical model with a confidence level of 95% is obtained.

[0014] Step 3: Derivation of the model between location parameters, minimum sample value, and sample size. In order to obtain point estimates, the second theoretical model is obtained.

[0015] Step 4: Based on the set shape parameters, initial position parameters, and random samples, calculate the initial estimate of the scale parameters using the maximum likelihood method. Depending on the estimation requirements, substitute the initial estimate of the scale parameters into the theoretical model of Step 2 or 3, and use the maximum likelihood method and the corresponding theoretical model to perform multiple iterative calculations to obtain stable scale parameters and position parameters, which are the estimated values ​​of the position parameters and scale parameters.

[0016] In the iterative method for estimating the parameters of the three-parameter Weibull distribution based on small samples in this invention, the derivation process of the first theoretical model with a 95% confidence level in step 2 is as follows:

[0017] (1) The probability density function of the three-parameter Weibull distribution is:

[0018]

[0019] In the formula, γ, β, and η are the location, shape, and scale parameters of the Weibull distribution, satisfying γ > 0, β > 0, η > 0, and t ≥ γ; t is the sample.

[0020] (2) The cumulative distribution function of the three-parameter Weibull distribution is:

[0021]

[0022] (3) According to f(t) (1,n) )=n[1-F(t)] n-1 f(t), the minimum sample value t of the three-parameter Weibull distribution (1,n) The probability density function is:

[0023]

[0024] Minimum sample value t of a three-parameter Weibull distribution (1,n) The cumulative distribution function is:

[0025]

[0026] To ensure a conservative estimate of the location parameters, the confidence level is set to 95%, i.e., F(t) (1,n) With the value set to 95%, the first theoretical model relating the location parameters to the minimum sample value and sample size is as follows:

[0027]

[0028] Where n is the sample size.

[0029] In the iterative method for estimating the three-parameter Weibull distribution parameters based on small samples in this invention, the second theoretical model is derived in step 3 to obtain the point estimate as follows:

[0030] To obtain the point estimate, according to formula (4), F(t) (1,n) A second theoretical model can be obtained by using 50% to obtain the relationship between the location parameters, the minimum sample value, and the sample size:

[0031]

[0032] Where n is the sample size.

[0033] In the iterative method for estimating the parameters of the three-parameter Weibull distribution based on small samples in this invention, step 4 specifically includes:

[0034] Step 4.1: When the shape and position parameters are determined, the scale parameter is estimated using the maximum likelihood method as follows:

[0035]

[0036] Step 4.2: Use Monte Carlo to randomly draw a three-parameter Weibull distribution (2, 1000, 1000~4000), that is, shape parameter β=2, scale parameter η=1000, and location parameter γ=1000~4000, to obtain 5 samples, namely t1, t2, …, t5;

[0037] Step 4.3: Set the number of iterations N to 1000, and the absolute value of the difference between the results of two iterations to 10e-5;

[0038] Step 4.4: Set the initial estimate of the location parameters to 0, that is:

[0039] Step 4.5: Substitute the initial estimates of the shape parameter, sample value, and position parameter into formula (7) to calculate the initial estimate of the scale parameter.

[0040]

[0041] Step 4.6: Based on different estimation needs, the initial estimates of the scaling parameters are... Substituting these values ​​into the model of formula (5) or (6), the estimated values ​​of the new location parameters can be calculated.

[0042] Step 4.7: Estimate the new location parameters Substituting into formula (7), the estimated value of the new scale parameter can be calculated.

[0043] Step 4.8: Repeat steps 4.6-4.7 to calculate the estimated values ​​of the position parameter and scale parameter in this iteration. Compare these values ​​with the estimated values ​​of the position parameter and scale parameter calculated in the previous iteration. If the absolute value of the difference between the two iteration results of the two parameters is less than 10e-5, stop the calculation. The result in this iteration is the final estimate. Otherwise, continue iterating for 1000 times to obtain the final result.

[0044] In the iterative method for parameter estimation of a three-parameter Weibull distribution based on small samples in this invention, the sample extracted in step 4.2 follows a three-parameter Weibull distribution with shape parameter β = 2, scale parameter η = 1000, and position parameter γ = 2000.

[0045] In the iterative method for estimating the three-parameter Weibull distribution parameters based on small samples in this invention, the shape parameter is taken as an empirical value of 2.0 in the estimation of the rolling bearing life distribution parameters.

[0046] The iterative method for estimating the parameters of a three-parameter Weibull distribution based on small samples, as proposed in this invention, has at least the following beneficial effects:

[0047] 1) The method of the present invention has a simple and convergent calculation process. When using empirical values ​​for shape parameters, stable position and scale parameters can be obtained.

[0048] 2) Compared with traditional methods such as maximum likelihood estimation and correlation coefficient estimation, the parameter estimation results obtained by this invention are more stable and accurate. Attached Figure Description

[0049] Figure 1 This is a flowchart of an iterative method for estimating the parameters of a three-parameter Weibull distribution based on small samples, according to the present invention.

[0050] Figure 2 This is the iterative flowchart used in step 4 of the present invention;

[0051] Figures 3a-3d This is an iterative graph of point estimation using four randomly selected samples in this invention; wherein, Figure 3a This is a diagram illustrating the iterative process of the first set of samples. Figure 3b This is a diagram illustrating the iterative process of the second set of samples. Figure 3c This is a diagram illustrating the iterative process of the third set of samples. Figure 3d This is a diagram illustrating the iterative process of the fourth set of samples. Detailed Implementation

[0052] like Figure 1 As shown, an iterative method for estimating the parameters of a three-parameter Weibull distribution based on small samples according to the present invention includes:

[0053] Step 1: Determine the shape parameters. Set the shape parameter range for the product life to be estimated to be between 1.5 and 2.5.

[0054] Step 2: Derivation of the model relating location parameters, minimum sample value, and sample size. To obtain a conservative estimate, a first theoretical model with 95% confidence is derived as follows:

[0055] (1) The probability density function of the three-parameter Weibull distribution is:

[0056]

[0057] In the formula, γ, β, and η are the location, shape, and scale parameters of the Weibull distribution, satisfying γ > 0, β > 0, η > 0, and t ≥ γ; t is the sample.

[0058] (2) The cumulative distribution function of the three-parameter Weibull distribution is:

[0059]

[0060] (3) According to f(t) (1,n) )=n[1-F(t)] n-1 f(t), the minimum sample value t of the three-parameter Weibull distribution (1,n) The probability density function is:

[0061]

[0062] Minimum sample value t of a three-parameter Weibull distribution (1,n) The cumulative distribution function is:

[0063]

[0064] To ensure a conservative estimate of the location parameters, the confidence level is set to 95%, i.e., F(t) (1,n) With the value set to 95%, the first theoretical model relating the location parameters to the minimum sample value and sample size is as follows:

[0065]

[0066] Where n is the sample size.

[0067] Step 3: Derivation of the model between location parameters, minimum sample value, and sample size. In order to obtain point estimates, the second theoretical model is obtained.

[0068] To obtain the point estimate, according to formula (4), F(t) (1,n) A second theoretical model can be obtained by using 50% to obtain the relationship between the location parameters, the minimum sample value, and the sample size:

[0069]

[0070] Where n is the sample size.

[0071] Step 4: Based on the set shape parameters, initial position parameters, and random samples, calculate the initial estimate of the scale parameters using the maximum likelihood method. Depending on the estimation requirements, substitute this initial estimate into the theoretical model from Step 2 or 3, and perform multiple iterative calculations using the maximum likelihood method and the corresponding theoretical model to obtain stable scale and position parameters, which are the estimated values ​​of the position and scale parameters. Figure 2 As shown, step 4 specifically involves:

[0072] Step 4.1: When the shape and position parameters are determined, the scale parameter is estimated using the maximum likelihood method as follows:

[0073]

[0074] Step 4.2: Use Monte Carlo to randomly draw a three-parameter Weibull distribution (2, 1000, 1000~4000), that is, shape parameter β=2, scale parameter η=1000, and location parameter γ=1000~4000, to obtain 5 samples, namely t1, t2, …, t5;

[0075] In practice, the extracted samples follow a three-parameter Weibull distribution with shape parameter β = 2, scale parameter η = 1000, and location parameter γ = 2000.

[0076] Step 4.3: Set the number of iterations N to 1000, and the absolute value of the difference between the results of two iterations to 10e-5;

[0077] Step 4.4: Set the initial estimate of the location parameters to 0, that is:

[0078] Step 4.5: Substitute the initial estimates of the shape parameter, sample value, and position parameter into formula (7) to calculate the initial estimate of the scale parameter.

[0079]

[0080] Step 4.6: Based on different estimation needs, the initial estimates of the scaling parameters are... Substituting these values ​​into the model of formula (5) or (6), the estimated values ​​of the new location parameters can be calculated.

[0081] Step 4.7: Estimate the new location parameters Substituting into formula (7), the estimated value of the new scale parameter can be calculated.

[0082] Step 4.8: Repeat steps 4.6-4.7 to calculate the estimated values ​​of the position parameter and scale parameter in this iteration. Compare these values ​​with the estimated values ​​of the position parameter and scale parameter calculated in the previous iteration. If the absolute value of the difference between the two iteration results of the two parameters is less than 10e-5, stop the calculation. The result in this iteration is the final estimate. Otherwise, continue iterating for 1000 times to obtain the final result.

[0083] In practical implementation, the method of the present invention can be used in the estimation of rolling bearing life distribution parameters, in which case the shape parameter is taken as an empirical value of 2.0.

[0084] In practice, four sets of samples t1, t2, ..., t5 were randomly selected, and the estimated iterative graph is as follows. Figures 3a-3d As shown, the estimation results are shown in Table 1 below.

[0085] Table 1 Estimation Results

[0086]

[0087] like Figures 3a-3d As shown, the method was verified by randomly selecting 4 groups of samples. When the sample size was 5, the convergence speed of the iterative method of the present invention was relatively fast. It could achieve convergence in about 10 iterations and obtain stable scale parameter η and position parameter γ. The estimated scale parameter was around 1000 and the estimated position parameter was around 2000, indicating that the iterative method is simple, effective and convergent.

Claims

1. An iterative method for estimating the parameters of a three-parameter Weibull distribution based on small samples, characterized in that, include: Step 1: Determine the shape parameters. Set the shape parameter range for the product life to be estimated to be between 1.5 and 2.

5. Step 2: Derivation of the model between location parameters, minimum sample value, and sample size. In order to obtain a conservative estimate, a first theoretical model with a confidence level of 95% is obtained. Step 3: Derivation of the model between location parameters, minimum sample value, and sample size. In order to obtain point estimates, the second theoretical model is obtained. Step 4: Based on the set shape parameters, initial position parameters, and random samples, calculate the initial estimate of the scale parameters using the maximum likelihood method. Substitute the initial estimate of the scale parameters into the theoretical model of Step 2 or 3 according to different estimation requirements. Use the maximum likelihood method and the corresponding theoretical model to perform multiple iterations to obtain stable scale parameters and position parameters, which are the estimated values ​​of the position parameters and scale parameters. The derivation process of the first theoretical model with a 95% confidence level in step 2 is as follows: (1) The probability density function of the three-parameter Weibull distribution is: In the formula, γ, β, and η are the location, shape, and scale parameters of the Weibull distribution, satisfying γ > 0, β > 0, η > 0, and t ≥ γ; t is the sample. (2) The cumulative distribution function of the three-parameter Weibull distribution is: (3) According to f(t) (1,n) )=n[1-F(t)] n-1 f(t), the minimum sample value t of the three-parameter Weibull distribution (1,n) The probability density function is: Minimum sample value t of a three-parameter Weibull distribution (1,n) The cumulative distribution function is: To ensure a conservative estimate of the location parameters, the confidence level is set to 95%, i.e., F(t) (1,n) With the value set to 95%, the first theoretical model relating the location parameters to the minimum sample value and sample size is as follows: Where n is the sample size; In step 3, to obtain the point estimate, the second theoretical model is derived as follows: To obtain the point estimate, according to formula (4), F(t) (1,n) Using 50%, the second theoretical model obtained is as follows: (This model relates to the location parameters, the minimum sample value, and the sample size.) Where n is the sample size; Step 4 specifically involves: Step 4.1: When the shape and position parameters are determined, the scale parameter is estimated using the maximum likelihood method as follows: Step 4.2: Use Monte Carlo to randomly draw a three-parameter Weibull distribution (2, 1000, 1000~4000), that is, shape parameter β=2, scale parameter η=1000, and location parameter γ=1000~4000, to obtain 5 samples, namely t1, t2, …, t5; Step 4.3: Set the number of iterations N to 1000, and the absolute value of the difference between the results of two iterations to 10e-5; Step 4.4: Set the initial estimate of the location parameters to 0, that is: Step 4.5: Substitute the initial estimates of the shape parameter, sample value, and position parameter into formula (7) to calculate the initial estimate of the scale parameter. Step 4.6: Based on different estimation needs, the initial estimates of the scaling parameters are... Substituting these values ​​into the model of formula (5) or (6), the estimated values ​​of the new location parameters can be calculated. Step 4.7: Estimate the new location parameters Substituting into formula (7), the estimated value of the new scale parameter can be calculated. Step 4.8: Repeat steps 4.6-4.7 to calculate the estimated values ​​of the position parameter and scale parameter in this iteration. Compare these values ​​with the estimated values ​​of the position parameter and scale parameter calculated in the previous iteration. If the absolute value of the difference between the two iteration results of the two parameters is less than 10e-5, stop the calculation. The result in this iteration is the final estimate. Otherwise, continue iterating for 1000 times to obtain the final result.

2. The iterative method for estimating the parameters of a three-parameter Weibull distribution based on small samples as described in claim 1, characterized in that, The samples extracted in step 4.2 follow a three-parameter Weibull distribution with shape parameter β = 2, scale parameter η = 1000, and position parameter γ = 2000.

3. The iterative method for estimating the parameters of a three-parameter Weibull distribution based on small samples as described in claim 1, characterized in that, In the estimation of rolling bearing life distribution parameters, the shape parameter is taken as an empirical value of 2.0.

Images