CN114428940A - Weibull distribution three-parameter minimum difference estimation method - Google Patents

Weibull distribution three-parameter minimum difference estimation method Download PDF

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CN114428940A
CN114428940A CN202210097281.1A CN202210097281A CN114428940A CN 114428940 A CN114428940 A CN 114428940A CN 202210097281 A CN202210097281 A CN 202210097281A CN 114428940 A CN114428940 A CN 114428940A
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谢里阳
吴宁祥
杨小玉
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Northeastern University China
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Abstract

The invention discloses a Weibull distribution three-parameter minimum difference estimation method, which comprises the following steps of 1: determining a relational expression of the scale parameter, the cumulative distribution function, the shape parameter and the position parameter; step 2: estimating a cumulative distribution function through a median rank, substituting the cumulative distribution function into the relational expression in the step 1 to obtain a three-parameter relational expression; and step 3: randomly extracting n samples, setting a series of shape parameters and position parameters for inspection, respectively using the n sample values, estimating according to a three-parameter relational expression, and taking a group of shape parameters and position parameters with the minimum standard deviation of the n scale parameter values as final estimation values; and 4, step 4: and (4) substituting the final estimation values of the shape parameters and the position parameters obtained in the step (3) into a three-parameter relational expression to obtain n scale parameter estimation values, and averaging the n scale parameter estimation values to obtain the final estimation value of the scale parameters. The estimation method of the invention is more stable, more conservative, more accurate and simpler to carry out the parameter estimation of the three-parameter Weibull distribution.

Description

Weibull distribution three-parameter minimum difference estimation method
Technical Field
The invention belongs to the technical field of parameter estimation methods of three-parameter Weibull distribution, and relates to a Weibull distribution three-parameter minimum difference estimation method.
Background
The weibull distribution is widely used to describe product life because of its applicability. However, the two parameter weibull distribution is most commonly used. The three parameter weibull distribution has further advantages, especially for long life, high reliability products. Research on fatigue life of metal specimens shows that a three-parameter weibull distribution can well describe life dispersion, whereas a two-parameter weibull distribution cannot.
However, the more parameters in the probability distribution function, the more complicated the parameter estimation. Unless the sample size is large enough, the error of conventional parameter estimation methods (such as maximum likelihood, graphical or median rank regression curve fitting) is very large and unstable.
Parametric estimation of three-parameter weibull distributions has long been the focus of methodology and practical applications. In particular, the maximum likelihood method draws wide attention, and the maximum likelihood mainly establishes a likelihood function, makes partial derivatives of each parameter equal to 0, and solves an equation set. The maximum likelihood method is more complex to carry out three-parameter Weibull distribution parameter estimation and calculation, and no solution can occur in some cases. When the sample size is small, classical statistical analysis may lead to limitations in the estimation results. Therefore, for the three-parameter Weibull distribution, the invention of a more accurate, more robust and more effective parameter estimation method has important significance.
Disclosure of Invention
Aiming at the limitation of a three-parameter Weibull parameter estimation result possibly caused by the classical statistical analysis when the sample size is small, a more accurate, more stable and more effective Weibull distribution three-parameter minimum difference estimation method is provided.
The invention provides a Weibull distribution three-parameter minimum difference estimation method, which comprises the following steps:
step 1: determining a relational expression of the scale parameter, the cumulative distribution function, the shape parameter and the position parameter;
step 2: substituting the accumulated distribution function estimated by the median rank into the relational expression in the step 1 to obtain a three-parameter relational expression;
and 3, step 3: randomly extracting n samples, setting a series of shape parameters and position parameters for inspection, respectively using the n sample values, estimating according to a three-parameter relational expression, and taking a group of shape parameters and position parameters with the minimum standard deviation of the n scale parameter values as final estimation values;
and 4, step 4: and (4) substituting the final estimation values of the shape parameters and the position parameters obtained in the step (3) into a three-parameter relational expression to obtain n scale parameter estimation values, and averaging the n scale parameter estimation values to obtain the final estimation value of the scale parameters.
In the Weibull distribution three-parameter minimum difference estimation method of the present invention, the derivation process of the relationship between the scale parameter and the cumulative distribution function, the shape parameter and the position parameter in the step 1 is as follows:
(1) the cumulative distribution function of the three-parameter weibull distribution is:
Figure BDA0003491351900000021
wherein gamma, beta and eta are respectively a position parameter, a shape parameter and a scale parameter of Weibull distribution, and the conditions that gamma is more than 0, beta is more than 0, eta is more than 0 and t is more than or equal to gamma are met;
(2) according to the formula (1), the relationship between the derived scale parameter and the cumulative distribution probability, the derived shape parameter and the derived position parameter is as follows:
Figure BDA0003491351900000022
in the formula, t is a sample.
In the weibull distribution three-parameter minimum difference estimation method of the present invention, the cumulative distribution function estimated by the median rank in step 2 is specifically:
step 2.1: at n sample values tiIf (i ═ 1,2, …, n) is available, let t be1,n,t2,n,…,tn,n(t1,n≤t2,n≤,…,≤tn,n) Representing ordered sample values, the cumulative distribution function is estimated by the median rank:
Figure BDA0003491351900000031
step 2.2: substituting equation (3) into equation (2) yields the three-parameter relationship as follows:
Figure BDA0003491351900000032
in the minimum difference estimation method for three parameters of Weibull distribution, the step 3 is specifically as follows:
step 3.1: using Monte Carlo to randomly draw three-parameter Weibull distribution Weibull (2,1000,100), i.e. shape parameter beta is 2, scale parameter eta is 1000 and position parameter gamma is 1000, obtaining n samples ti,i＝1…n；
Step 3.2: setting j shape parameters beta1,β2,...,βj；
Step 3.3: for each shape parameter value set in step 3.2, the k position parameter values γ are tested one by one1,γ2,...,γk；
Step 3.4: the calculation of t for each sample using equation (4)iJ × k scale parameters (η)i,1,ηi,2,…,ηi,j×k) And calculates the n samples t1,t2,…,tnIn taking the same (beta)j,γk) N scale parameters (eta) obtained in time1,j×k,η2,j×k,…,ηn,j×k) And finding a group corresponding to the smallest standard deviation from the j × k standard deviations
Figure BDA0003491351900000033
As final estimated values of the shape parameters and the position parameters.
In the weibull distribution three-parameter minimum difference estimation method of the present invention, the final estimation value of the scale parameter in the step 4 is calculated according to the following formula:
Figure BDA0003491351900000034
wherein, for the final estimate of the shape parameters,
Figure BDA0003491351900000035
is the final estimate of the position parameter.
In the weibull distribution three-parameter minimum difference estimation method of the present invention, the sample obtained in step 3.1 is subjected to a three-parameter weibull distribution, where the shape parameter β is 2, the scale parameter η is 1000, and the position parameter γ is 1000.
In the Weibull distribution three-parameter minimum difference estimation method, five shape parameters beta are considered for roughly estimating the scale parameters1,β2,...,β51.0, 1.5, 2.0, 3.0 and 4.0, respectively; for each shape parameter value, a series of position parameters gamma are tested one by one1,γ2,...,γ 410, 50, 100, …, 2000, respectively, with Δ γ equal to 50.
In the weibull distribution three-parameter minimum difference estimation method of the present invention, for more accurate estimation, the increment of the shape parameter may be set smaller as Δ β ═ 0.1, and the increment of the position parameter may be set smaller as Δ γ ═ 1.
The Weibull distribution three-parameter minimum difference estimation method provided by the invention at least has the following beneficial effects:
1) scale parameter η of each sample estimateiThe difference between (i ═ 1,2, …, n) depends on the sample uncertainty, the error in the position parameter γ, and the error in the shape parameter β. By minimizing the difference in the estimated scale parameters for each sample, the correct shape parameters and position parameters can be easily found. The scale parameters are then finally estimated by averaging the scale parameters estimated for each sample value, using the correct shape parameters and position parameters.
2) Compared with a maximum likelihood method and a median rank regression curve fitting method, the method is more stable, more conservative, more accurate and simpler to carry out parameter estimation of three-parameter Weibull distribution.
Drawings
FIG. 1 is a flow chart of a Weibull distribution three-parameter minimum difference estimation method of the present invention;
fig. 2 is a standard deviation of scale parameters estimated in the present invention with respect to different shape parameters and location parameters.
Detailed Description
As shown in fig. 1, the method for estimating minimum difference of three parameters in weibull distribution of the present invention comprises the following steps:
step 1: determining a relational expression of the scale parameter, the cumulative distribution function, the shape parameter and the position parameter;
in specific implementation, the derivation process of the relationship between the scale parameter and the cumulative distribution function, the shape parameter and the position parameter is as follows:
(1) the cumulative distribution function of the three-parameter weibull distribution is:
Figure BDA0003491351900000051
wherein gamma, beta and eta are respectively a position parameter, a shape parameter and a scale parameter of Weibull distribution, and the conditions that gamma is more than 0, beta is more than 0, eta is more than 0 and t is more than or equal to gamma are met;
(2) according to the formula (1), the relationship between the scale parameter and the cumulative distribution probability, the shape parameter and the position parameter is deduced as follows:
Figure BDA0003491351900000052
in the formula, t is a sample.
Step 2: substituting the median rank estimation cumulative distribution function into the relational expression in the step 1 to obtain a three-parameter relational expression, which specifically comprises the following steps:
step 2.1: at n sample values tiIf (i ═ 1,2, …, n) is available, let t be1,n,t2,n,…,tn,n(t1,n≤t2,n≤,…,≤tn,n) Representing ordered sample values, the cumulative distribution function is estimated by the median rank:
Figure BDA0003491351900000053
step 2.2: substituting equation (3) into equation (2) yields the three-parameter relationship as follows:
Figure BDA0003491351900000054
and step 3: randomly extracting n samples, setting a series of shape parameters and position parameters for inspection, estimating according to a three-parameter relational expression by using the n sample values respectively, and taking a group of shape parameters and position parameters with the minimum standard deviation of the n scale parameter values as final estimation values, wherein the step 3 specifically comprises the following steps:
step 3.1: using Monte Carlo to randomly draw three-parameter Weibull distribution Weibull (2,1000,1000), i.e. shape parameter beta is 2, scale parameter eta is 1000 and position parameter gamma is 1000, obtaining n samples ti,i＝1…n；
In specific implementation, the sample is subjected to a three-parameter weibull distribution, where the shape parameter β is 2, the scale parameter η is 1000, and the position parameter γ is 1000. 15 samples t are taken1,t2,…,t151372.9, 1464.2, 1573.9, 1615.6, 1657.6, 1687.8, 1716.2, 1899.9, 1980.6, 2097.6, 2105.0, 2170.8, 2274.7, 2446.1, 2639.3, respectively.
Step 3.2: setting j shape parameters beta1,β2,...,βj；
Step 3.3: for each shape parameter value set in step 3.2, the k position parameter values γ are tested one by one1,γ2,...,γk；
Step 3.4: the calculation of t for each sample using equation (4)iJ × k scale parameters (η)i,1,ηi,2,…,ηi,j×k) And calculating the n samples t1,t2,…,tnIn taking the same (beta)j,γk) N scale parameters (eta) obtained in time1,j×k,η2,j×k,…,ηn,j×k) And finding a group corresponding to the smallest standard deviation from the j × k standard deviations
Figure BDA0003491351900000061
As final estimated values of the shape parameters and the position parameters. FIG. 2 is a view about a different shapeThe standard deviation of the estimated scale parameter values of the parameter and the position parameter, which is shown as the smallest standard deviation of the estimated scale parameter when β is 2.0 and γ is 1200;
in the specific implementation, five shape parameters beta are considered for roughly estimating the scale parameters1,β2,…,β51.0, 1.5, 2.0, 3.0 and 4.0, respectively; for each shape parameter value, a series of position parameters gamma are tested one by one1,γ2,…,γ 410, 50, 100, …, 2000, respectively, with Δ γ equal to 50. For each sample 5 × 41 scale parameters can be calculated according to equation (4), and these 15 samples t are calculated1,t2,…,t15Taking the same group (. beta.)j,γk) 15 scale parameters (eta) obtained in time1,j×k,η2,j×k,…,η15,j×k) And finding a group corresponding to the smallest standard deviation from the j × k standard deviations
Figure BDA0003491351900000062
As final estimated values of the shape parameters and the position parameters.
In specific implementation, more shape parameters are needed to obtain higher precision, and smaller increment can be set; to achieve higher accuracy, more position parameters are required and smaller increments can be set. An increment of the shape parameter, for example, Δ β ═ 0.1; the increment Δ γ of the position parameter is 1.
And 4, step 4: and (4) substituting the final estimation values of the shape parameters and the position parameters obtained in the step (3) into a three-parameter relational expression to obtain n scale parameter estimation values, and averaging the n scale parameter estimation values to obtain the final estimation value of the scale parameters.
The final estimate of the scale parameter is calculated according to the following equation:
Figure BDA0003491351900000071
wherein, is
Figure BDA0003491351900000072
Root of common StichopusThe final estimate of the number of the bits,
Figure BDA0003491351900000073
is the final estimate of the position parameter.
In particular, the average of the 15 scale parameters estimated from the 15 sample values is 806.0, so the scale parameter is finally estimated to be 806.0.

Claims (8)

1. A Weibull distribution three-parameter minimum difference estimation method is characterized by comprising the following steps:
step 1: determining a relational expression of the scale parameter, the cumulative distribution function, the shape parameter and the position parameter;
step 2: substituting the accumulated distribution function estimated by the median rank into the relational expression in the step 1 to obtain a three-parameter relational expression;
and step 3: randomly extracting n samples, setting a series of shape parameters and position parameters for inspection, respectively using the n sample values, estimating according to a three-parameter relational expression, and taking a group of shape parameters and position parameters with the minimum standard deviation of the n scale parameter values as final estimation values;
and 4, step 4: and (4) substituting the final estimation values of the shape parameters and the position parameters obtained in the step (3) into a three-parameter relational expression to obtain n scale parameter estimation values, and averaging the n scale parameter estimation values to obtain the final estimation value of the scale parameters.
2. The weibull distribution three-parameter minimum difference estimation method according to claim 1, wherein the derivation process of the relationship between the scale parameter and the cumulative distribution function, the shape parameter, and the position parameter in step 1 is:
(1) the cumulative distribution function of the three-parameter weibull distribution is:
Figure FDA0003491351890000011
wherein gamma, beta and eta are respectively a position parameter, a shape parameter and a scale parameter of Weibull distribution, and the conditions that gamma is more than 0, beta is more than 0, eta is more than 0 and t is more than or equal to gamma are met;
(2) according to the formula (1), the relationship between the derived scale parameter and the cumulative distribution probability, the derived shape parameter and the derived position parameter is as follows:
Figure FDA0003491351890000012
in the formula, t is a sample.
3. The weibull distribution three-parameter minimum difference estimation method according to claim 2, wherein the cumulative distribution function estimated by the median rank in step 2 is specifically:
step 2.1: at n sample values tiIf (i ═ 1,2, …, n) is available, let t be1,n,t2,n,…,tn,n(t1,n≤t2,n≤,…,≤tn,n) Representing ordered sample values, the cumulative distribution function is estimated by the median rank:
Figure FDA0003491351890000021
step 2.2: substituting equation (3) into equation (2) yields the three-parameter relationship as follows:
Figure FDA0003491351890000022
4. the weibull distribution three-parameter minimum difference estimation method according to claim 3, wherein the step 3 is specifically:
step 3.1: using Monte Carlo to randomly draw three-parameter Weibull distribution Weibull (2,1000,1000), i.e. shape parameter beta is 2, scale parameter eta is 1000 and position parameter gamma is 1000, obtaining n samples ti,i＝1…n；
Step 3.2: setting j shape parameters beta1,β2,…,βj；
Step 3.3: for each shape parameter value set in step 3.2, the k position parameter values γ are tested one by one1,γ2,…,γk；
Step 3.4: the calculation of t for each sample using equation (4)iJ × k scale parameters (η)i,1,ηi,2,…,ηi,j×k) And calculating the n samples t1,t2,…,tnIn taking the same (beta)j,γk) N scale parameters (eta) obtained in time1,j×k,η2,j×k,…,ηn,j×k) And finding a group corresponding to the smallest standard deviation from the j × k standard deviations
Figure FDA0003491351890000023
As final estimated values of the shape parameters and the position parameters.
5. The weibull distribution three-parameter minimum variance estimation method as claimed in claim 4, wherein the final estimate of the scale parameter in step 4 is calculated according to the following formula:
Figure FDA0003491351890000024
wherein, is
Figure FDA0003491351890000031
The final estimate of the shape parameters is,
Figure FDA0003491351890000032
is the final estimate of the position parameter.
6. The weibull distribution three-parameter minimum variance estimation method of claim 4, wherein the samples taken in step 3.1 are subjected to a three-parameter weibull distribution with a shape parameter β of 2, a scale parameter η of 1000, and a position parameter γ of 1000.
7. The Weibull distribution three-parameter minimum difference estimation method of claim 4, wherein for rough estimation of the scale parameter, five shape parameters β are considered1,β2,…,β51.0, 1.5, 2.0, 3.0 and 4.0, respectively; for each shape parameter value, a series of position parameters gamma are tested one by one1,γ2,…,γ410, 50, 100, …, 2000, respectively, with Δ γ equal to 50.
8. The weibull distribution three-parameter minimum variance estimation method of claim 7, wherein the increment of the shape parameter is set smaller such as Δ β ═ 0.1 and the increment of the position parameter is set smaller such as Δ γ ═ 1 for more accurate estimation.
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