CN108415877A - A kind of equal tails Estimating Confidence Interval method of Weibull distributed constants - Google Patents

A kind of equal tails Estimating Confidence Interval method of Weibull distributed constants Download PDF

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CN108415877A
CN108415877A CN201810228387.4A CN201810228387A CN108415877A CN 108415877 A CN108415877 A CN 108415877A CN 201810228387 A CN201810228387 A CN 201810228387A CN 108415877 A CN108415877 A CN 108415877A
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parameter
scale parameter
confidence interval
weibull
scale
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黄洪钟
曾国桓
黄土地
李彦锋
郭骏宇
余奥迪
曾颖
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University of Electronic Science and Technology of China
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Abstract

The present invention discloses a kind of equal tails Estimating Confidence Interval method of Weibull distributed constants, is applied to reliability field.For the problems of estimation Weibull distributed constant confidence intervals under small sample, the present invention establishes the indeterminate equation of the form parameter and scale parameter of Weibull distributions using likelihood ratio, by solving the indeterminate equation, the form parameter and the respective maximum of scale parameter and minimum of Weibull distributions are obtained, to obtain the form parameter and the respective confidence interval of scale parameter of Weibull distributions.The method of the present invention is compared with Fisher matrix methods, and section is wider under identical confidence level, engineer application of being more convenient for.

Description

A kind of equal tails Estimating Confidence Interval method of Weibull distributed constants
Technical field
The invention belongs to reliability field, more particularly to a kind of equal tails Estimating Confidence Interval technology.
Background technology
In life of product analysis, the complete service life obtained using reliability test or truncation service life carry out parameter Estimation, The concrete form of service life distribution is obtained, and then it is indispensable in reliability Work to obtain the value of ASSOCIATE STATISTICS characteristic quantity and section Few link.Frequently with the method for Fisher information matrix in common interval estimation.But this method is only suitable for sample The case where this amount is more than 10, and it is more optimistic to the estimation in section.
Invention content
In order to solve the above technical problems, the present invention proposes that a kind of of Weibull distributed constants waits tails Estimating Confidence Interval side Method is divided using section two on the basis of likelihood ratio theory and gives Weibull distributed constants m and σ with binomial fitting0's Equal tails confidence interval estimate.
The technical solution adopted by the present invention is：A kind of equal tails Estimating Confidence Interval method of Weibull distributed constants, uses The indeterminate equation that likelihood ratio establishes the form parameter and scale parameter of Weibull distributions obtains shape by solving the indeterminate equation Shape parameter and the respective maximum of scale parameter and minimum, to obtain form parameter and the respective confidence area of scale parameter Between.
Further, following steps are specifically included：
S1, point estimation is carried out to distributed constant using Maximum Likelihood Estimation, respectively obtains the shape of Weibull distributions The point estimate of parameter and scale parameter；
S2, likelihood ratio statistics are established, determines the distribution of likelihood ratio statistics；
S3, the likelihood ratio determined according to the point estimate and step S2 of the step S1 form parameters determined and scale parameter The distribution of statistic builds the indeterminate equation of form parameter and scale parameter；
S4, iterative formula of the indeterminate equation about form parameter and scale parameter is established；
S5, form parameter and the maximum and minimum of scale parameter are respectively obtained by calculating, to obtain parameter Confidence interval.
Further, the point estimate for the form parameter and scale parameter that Weibull described in step S1 is distributed is according to the following formula It is calculated：
Wherein, m is form parameter, σ0For scale parameter, n indicates that product quantity, i indicate that i-th of product, r indicate failure Product quantity, j indicate j-th of failure product, tυjIndicate complete lifetime data, tiIndicate to include all complete lifetime datas and institute There is some element in the set of truncation lifetime data,Indicate tiM powers,It is scale parameter σ0M powers.
Further, it is distributed as the chi square distribution that degree of freedom is 1, confidence level is α described in step S2
Further, the indeterminate equation expression formula described in step S3 is：
Wherein,For the estimated value of form parameter,For the estimated value of scale parameter.
Further, the step S5 is specially：It is expressed by the iteration of form parameter and scale parameter to step S4 Formula is solved, and form parameter and the disaggregation of scale parameter are respectively obtained；In form parameter and scale parameter, respectively disaggregation is formed Convex curves and notching curve respectively obtain form parameter and the respective maximum of scale parameter and minimum with dichotomy, into And obtain form parameter and the respective confidence interval of scale parameter.
Beneficial effects of the present invention：The equal tails Estimating Confidence Interval method of the Weibull distributed constants of the present invention, using seemingly The method of right ratio establishes the indeterminate equation in relation to form parameter and scale parameter, is solved by Newton iteration, obtains equation phase The disaggregation answered；The convex curves and notching curve that are formed in disaggregation respectively obtain the pole of form parameter and scale parameter with dichotomy Big value and minimum, and then obtain the confidence interval of parameter；And the distributed area of parameter is calculated by data；The present invention The parameter section that method obtains can cover the section obtained using Fisher methods, illustrate that the present processes are feasible effective , and the method for the present invention, compared with Fisher matrix methods, section is wider under identical confidence level, and engineering of being more convenient for is answered With.
Description of the drawings
Fig. 1 is the solution of the present invention flow chart；
Fig. 2 is the interval estimation profile diagram of form parameter provided in an embodiment of the present invention；
Fig. 3 is the interval estimation profile diagram of scale parameter provided in an embodiment of the present invention.
Specific implementation mode
For ease of those skilled in the art understand that the present invention technology contents, below in conjunction with the accompanying drawings to the content of present invention into one Step is illustrated.
The equal tails Estimating Confidence Interval method of the Weibull distributed constants of the application has using the method foundation of likelihood ratio Close m and σ0Indeterminate equation, which is solved by Newton iteration, obtains the corresponding disaggregation of the indeterminate equation.In disaggregation The convex curves and notching curve of formation respectively obtain m and σ with dichotomy0Maximum and minimum, and then obtain parameter m and σ0Confidence interval.
It is the protocol procedures figure of the application as shown in Figure 1, the technical solution of the application is：Weibull distributed constants etc. Tail Estimating Confidence Interval method, including：
S1, point estimation is carried out to distributed constant using Maximum Likelihood Estimation, respectively obtains the shape of Weibull distributions The point estimate of parameter and scale parameter；
S2, likelihood ratio statistics are established, determines the distribution of likelihood ratio statistics；
S3, the likelihood ratio determined according to the point estimate and step S2 of the step S1 form parameters determined and scale parameter The distribution of statistic builds the indeterminate equation of form parameter and scale parameter；
S4, iterative formula of the indeterminate equation about form parameter and scale parameter is established；
S5, form parameter and the maximum and minimum of scale parameter are respectively obtained by calculating, to obtain parameter Confidence interval.
If the service life of product is T, T obeys distribution function and is
Wherein, m is form parameter, m ＞ 0；σ0For scale parameter, η ＞ 0.
Under fixed time test environment, if shared n product (containing the product for continuing to participate in experiment after repair) participates in examination It tests, test period t.In test, observation is that：There are r product failure, service life to be followed successively by within the period [0, T] tυ(1)≤tυ(2)≤…≤tυ(r)；N-r product does not fail in test, and truncation lifetime data is tτ(1)≤tτ(2)≤…≤ tτ(n-r)。
Step S1 is specially：
When sample lifetime data contains right censored data tτWhen, it is not yet lost in off-test because there are portioned products Effect, the likelihood function of this part are product Π [the 1-F ((t of Reliability Functionτk))], so total likelihood function is：
Under Weibull distributions, a product is in [ti,ti+dti] probability that fails in the time is f (ti)dti, i=1, 2,...,r.Therefore the out-of-service time data that can be will be observed that are described as follows by likelihood function：
The do not fail likelihood functions of products of remaining n-r are：
Because of the existing complete lifetime data t in fixed time test dataυ, and have right truncation lifetime data tτ, according to pole The principle of maximum-likelihood method, the parameter m and σ in fixed time test0Likelihood function be：
To simplify operation, the present invention will include all complete lifetime data tυj(j≤r) and all truncation lifetime data tτk Element in the set of (k≤n-r) is denoted as ti(i≤n) obtains formula (5) abbreviation：
By L (m, σ0) take logarithm
Afterwards again to m and σ0Local derviation is sought, is had：
By maximum likelihood equationsTwo-parameter weibull distribution parameter is obtained after collated abbreviation to estimate The maximum likelihood equations of meter：
First equation includes only unknown parameter m in formula (9) equation group.It is solved by Newton iteration method in the present invention (can also be that other methods solve), substitutes into second equation and can be obtained
Step S2 is specially：
If L (β, η) is the likelihood function of sample,WithIt is the estimates of parameters of the sample.So likelihood ratio of the sample Function is：
By formula (10) equal sign both sides with logarithm and multiplication by constants -2 is taken, have：
Think that formula (11) is progressive in the chi square distribution that degree of freedom is 1, confidence level is αI.e.：
By distributed constant β and η estimated valueWithExpression, has：
Similarly, Weibull distributed constants m and σ0Estimated value expression formula be：
Step S3 is specially：
The sample likelihood function of Follow Weibull Distribution can be written as under fixed time test：
Convolution (14) and formula (15), have：
Since Weibull distributed constants have acquired its estimated value by step S1WithChi square distributionValue It can be by tabling look-up to obtain, therefore formula (16) right part is a constant C at given confidence level α, i.e.,：
It brings formula (17) into formula (16) and obtains a related m and σ0Indeterminate equation：
Step S4 is specially：
In order to facilitate solution, indeterminate equation (18) the right and left is taken into logarithm, is had：
According to the thought of Newton iteration method, enable：
Respectively with m and σ0For variable, to f (m, σ0) local derviation is sought, have：
Then there is iterative formula：
Step S5 is specially：
σ0According to given initial value and step length searching, until finding m with σ0In the section (σ of convex trend0a,σ0b), by m The section two that maximum value occurs is divided, and next round iteration is carried out, until meeting precision, then in equal tails confidences of the m at confidence level α Boundary mU=mmax；Ask the section lower bound and σ of m0Upper (lower) boundary it is similar with the above method.
The present processes are illustrated below by way of specific example：
If Fatigue Life Follow Weibull Distribution (the σ of certain product0=400, m=2), Censoring time T is 1500 small When, obtain complete lifetime data collection tυWith truncation lifetime data collection tτAs shown in table 2.1.Ask m and σ0Under confidence level 0.95 Two-sided confidence interval.
1 fixed time censoring with replacement test data (σ of table0=400, m=2)
Likelihood equation is solved, can be obtainedIn turn
σ is first set below0For definite value, the interval range of m is sought.Known to tabling look-up：
Enable σ0=400, obtain two root m of mL=2.1062 and mU=5.0632.In order to improve iteration speed, first with 50 It is solved to both sides spreading range for iteration step length, until the m of generationLThere is recessed trend, mUThere is convex trend, i.e., with σ0Increase, mLFirst reduce and increases afterwards, mUFirst increases and then decreases.In this example, in σ0Section be on [350,500] can be observed Convex and recessed trend.The region of search two is divided near upper and lower two points of inflexion on a curve, seeks the m corresponding to its midpointLAnd mU.It is right Its value determines the region of search of next step again after being compared.Iterative process is shown in Table 2.
The iterative process of 2 form parameter m of table
It can be seen that when iterating to 15 step, iteration difference error has been less than 0.01.Its profile diagram is as shown in Figure 2.
According to table 2 and Fig. 2, the value of the convex curves peak of m is near 6.5100, the value of notching curve minimum point Near 2.0982, therefore confidence intervals of the m under confidence level 95% is [2.0982,6.5100].
Likewise, to scale parameter σ0It is iterated.Since the value of m is smaller, therefore iteration step length also will accordingly reduce.Enable m =3, there is convex and recessed trend on section [2.5,3] in initial iteration step 0.1, discovery.Next it is carried out in this section Iteration, part iterative process are shown in Table 3.
3 scale parameter σ of table0Iterative process
Iteration result is observed, after a few wheel iteration, numerical value change is little, and reaction is that curve becomes on Fig. 3 Change very gentle.In order to reduce calculation amount, therefore it is not necessarily to be further continued for being calculated.
According to table 3 and Fig. 3, σ0Convex curves peak near 534, notching curve minimum point near 346, Therefore σ0Confidence interval under confidence level 95% is [346,534].So under confidence level 95%, m that this method obtains Confidence interval be [2.0982,6.5100], σ0Confidence interval be [346,534].
The confidence interval that Fisher matrix methods obtain m is [2.2150,5.7050], σ0Confidence interval be [367.2713, 504.5887].It can be seen that the parameter interval range that Fisher matrix methods acquire is narrower than likelihood ratio method.
The application establishes related m and σ in the case where Weibull is distributed occasion using the method for likelihood ratio0Indeterminate equation, lead to It crosses the Newton iteration solution equation and obtains corresponding disaggregation.The convex curves and notching curve dichotomy formed in disaggregation are distinguished Obtain m and σ0Maximum and minimum, and then obtain the confidence interval of form parameter and scale parameter at confidence level α.
The application method is compared with Fisher matrix methods, and section is wider at identical confidence level α.And the application method It calculates simply, is convenient for engineer application.
Those of ordinary skill in the art will understand that the embodiments described herein, which is to help reader, understands this hair Bright principle, it should be understood that protection scope of the present invention is not limited to such specific embodiments and embodiments.For ability For the technical staff in domain, the invention may be variously modified and varied.All within the spirits and principles of the present invention, made by Any modification, equivalent substitution, improvement and etc. should be included within scope of the presently claimed invention.

Claims (6)

  1. A kind of equal tails Estimating Confidence Interval method of 1.Weibull distributed constants, which is characterized in that established using likelihood ratio The indeterminate equation of the form parameter and scale parameter of Weibull distributions obtains form parameter and ruler by solving the indeterminate equation The respective maximum of parameter and minimum are spent, to obtain form parameter and the respective confidence interval of scale parameter.
  2. 2. a kind of equal tails Estimating Confidence Interval method of Weibull distributed constants according to claim 1, feature exist In specifically including following steps：
    S1, point estimation is carried out to distributed constant using Maximum Likelihood Estimation, respectively obtains the form parameter of Weibull distributions With the point estimate of scale parameter；
    S2, likelihood ratio statistics are established, determines the distribution of likelihood ratio statistics；
    S3, the likelihood ratio determined according to the point estimate and step S2 of the step S1 form parameters determined and scale parameter count The distribution of amount builds the indeterminate equation of form parameter and scale parameter；
    S4, iterative formula of the indeterminate equation about form parameter and scale parameter is established；
    S5, form parameter and the maximum and minimum of scale parameter are respectively obtained by calculating, to obtain the confidence of parameter Section.
  3. 3. a kind of equal tails Estimating Confidence Interval method of Weibull distributed constants according to claim 2, feature exist In form parameter and the point estimate of scale parameter are calculated according to the following formula described in step S1：
    Wherein, m is form parameter, σ0For scale parameter, n indicates that product quantity, i indicate that i-th of product, r indicate failure product number Amount, j indicate j-th of failure product, tυjIndicate complete lifetime data, tiIndicate to include all complete lifetime datas and all truncation Some element in the set of lifetime data,Indicate tiM powers,It is scale parameter σ0M powers.
  4. 4. a kind of equal tails Estimating Confidence Interval method of Weibull distributed constants according to claim 3, feature exist In being distributed as described in step S2：The chi square distribution that degree of freedom is 1, confidence level is α
  5. 5. a kind of equal tails Estimating Confidence Interval method of Weibull distributed constants according to claim 4, feature exist In the indeterminate equation expression formula described in step S3 is：
    Wherein,For the estimated value of form parameter,For the estimated value of scale parameter.
  6. 6. a kind of equal tails Estimating Confidence Interval method of Weibull distributed constants according to claim 5, feature exist In the step S5 is specially：It is solved by the iteration expression formula of form parameter and scale parameter to step S4, respectively Obtain form parameter and the disaggregation of scale parameter；In form parameter and the scale parameter convex curves and recessed that respectively disaggregation is formed Curve respectively obtains form parameter and the respective maximum of scale parameter and minimum with dichotomy, so obtain form parameter and The respective confidence interval of scale parameter.
CN201810228387.4A 2018-03-20 2018-03-20 A kind of equal tails Estimating Confidence Interval method of Weibull distributed constants Pending CN108415877A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109145502A (en) * 2018-09-17 2019-01-04 中国人民解放军海军工程大学 A kind of Weibull type cell life estimation of distribution parameters method
CN110458436A (en) * 2019-08-02 2019-11-15 北京理工大学 A kind of small sample verification method of priming system reliability index

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109145502A (en) * 2018-09-17 2019-01-04 中国人民解放军海军工程大学 A kind of Weibull type cell life estimation of distribution parameters method
CN110458436A (en) * 2019-08-02 2019-11-15 北京理工大学 A kind of small sample verification method of priming system reliability index
CN110458436B (en) * 2019-08-02 2022-02-01 北京理工大学 Small sample verification method for initiating explosive device reliability index

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Application publication date: 20180817