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l.govInferential Statistics and Likelihood Ratio Confidence Bounds Established for Weibull Distribution ParametersA. Parikh1, C. James1, S. F. Duffy1 and R.L. Bratton2

September 21, 2011

1Cleveland State University2Idaho National Laboratory

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INTRODUCTIONIn an earlier draft of the ASME nuclear code for graphite a simple procedure was proposed for accepting a graphite material based on strength as part of the acceptance criterion. We wish to present an enhanced approach that incorporates the inherent variability associated with parameter estimation. Hopefully the approach will be incorporated in a later version of the code.

Strength is a stochastic variable characterized here by a two parameter Weibull distribution. The Weibull distribution is an extreme value distribution and this facet makes it a preferred distribution to characterize the “minimum strength” of graphite.

Distribution parameters are estimated using maximum likelihood estimators. Point estimates are approximate values of the true distribution parameter values and the method of maximum likelihood is a commonly used technique because the estimating functions are efficient in that they produce the least amount of bias.

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The point estimates are dependent on the number of strength tests conducted and the question of whether the estimated values are of sufficiently high quality is directly related to the fundamental question of “how many samples must be tested?” However, the question that should be asked is “how many samples must be tested to establish a given confidence level for a stipulated component reliability?” We wish to address the latter question properly by utilizing interval estimates along with hypothesis testing.

Confidence intervals are used to indicate the quality of estimated parameters. Confidence intervals on parameter estimates represent the range of values for the distribution parameters that are both reasonable and plausible. Within the interval one can expect the true value of a distribution parameter with a quantifiable degree of confidence. When confidence intervals are combined with hypothesis testing a rejection region is established, and the notion of a likelihood ratio ring can be developed wherein the estimates should reside, and moreover, where the true distribution parameter pair can be found.

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Hypothesis testing is applied to the distribution parameters (1, 2). Achieving statistical significance is equivalent to accepting that the observed results (the point estimates) are plausible if the null hypothesis (H0) is not rejected. The null hypothesis helps in deriving a rejection region in a distribution parameter space.

A test statistic is developed that aids in our decision to reject the null hypothesis. The statistic, based on a ratio of likelihood functions, helps in defining the relative size of the rejection region.

The objective here is to define this process relative to strength testing of graphite materials that establishes a robust material acceptance criteria. The process presented here will hopefully be incorporated into the ASME nuclear design code.

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Hypothesis TestingSince the sample population is assumed to be a two parameter Weibull distribution, then a vector of distribution parameters whose components are the MLE parameter estimates is identified as

A null hypothesis is stipulated such that

that is the region 0 contains all the MLE parameter estimates, and the alternative hypothesis is stipulated as

with 0C representing all point estimates that are not the MLE parameter

estimates.

)ˆ,ˆ(),( 2121

)ˆ,ˆ(: 2100 H

cH 01 :

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In essence we are testing the null hypothesis that the true distribution parameters are equal to the MLE parameter estimates with an alternative hypothesis that the true distribution parameters are not equal to the MLE parameter estimates.

Likelihood functions are expressed as

and

A test statistic is introduced that is defined as the natural log of the ratio of the likelihood functions, i.e.,

)ˆ,ˆ|(ˆ21

10

n

iixfL

),|(ˆ21

11

n

iixfL

0

1

ˆˆ

ln2LLT

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The Neyman-Pearson lemma states that this likelihood ratio test is the most powerful test statistic available for testing the null hypothesis. We can rewrite this last expression as

where

and

LLT *2

)ˆ,ˆ|(lnˆln* 211

0

n

iixfLL

),|(lnˆln 211

1 n

iixfLL

L(θ1,θ2)θ2

θ1

θ'2

1̂

2̂

θ'1

T/2L̂

Lˆ

88

The test statistic is designed in such a way that the probability of a Type I error does not exceed a constant , a value that we control. Thus the probability of a Type I error is fixed and we search for the test statistic that maximizes (1 – ) where again is the probability of a Type II error.

In our case where inferences are being made on parameters from a two parameter Weibull distribution the degree of freedom for the 2 - distribution is one and the values of the 2 - distribution are easily calculated.

One can compute the likelihood ratio T and compare −2ln(T) to the 2 value corresponding to a desired significance level and define a rejection region. This is outlined in the next section.

The ratio of the two likelihood functions defined previously should be low in the optimal critical region – a result of minimizing and maximizing (1 – ). The ratio is high in the complementary region. A high ratio corresponds to a high probability of a correct decision under H0.

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Likelihiood Ratio Ring (Region of Acceptance)

17.0, 400.0

15.223, 393.381

375

380

385

390

395

400

405

410

5 10 15 20 25Wei

bull

Cha

ract

eris

tic S

tren

gth

(σθ)

Weibull Modulus (m)

Confidence Ringγ = 90%

Confidence Ring

True value

Estimated Value

The likelihood ratio confidence bound is based on the equation

For a two parameter Weibull distribution the previous equation is expressed as

0ˆ,ˆ, 2

21;

emLmL

21;)ˆ(

)(ln2

LL

1010

The material acceptance approach depends on the ability to compute the probability of failure of the component under design. This will be quantified using a hazard rate format, i.e, expressed as a fraction with a numerator of one. The hazard rate is simply the reciprocal of the probability of failure a quantity usually expressed as a percentage.

We assume component probability of failure is modeled using a weakest link (series) reliability model. The underlying strength is characterized by a two parameter Weibull distribution. Thus a component probability of failure curve can be generated in an m –0 graph.

Points along the curve represent parameter pairs equal to a specified probability of failure. This curve will be referred to as a component performance curve. We overlay this graph with point estimates of the Weibull distribution parameters obtained from tensile strength data that a typical material supplier might have. Point estimates from this data to the right of the component design curve represent lower probability of failures. Conversely points to the left of this curve are associated with performance curves with higher probability of failures. Thus one component design curve defines two regions of the m – 0 space, an acceptable performance region and a rejection region relative to a specified component probability of failure.

Material Performance Curves

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The figure to the right is an example of a component performance curve. An increase in the failure rate would shift the curve to the left, a decrease would shift the curve to the right.

This particular curve was developed for a thin wall pressure vessel. The geometry will be presented momentarily.

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Through Weibull size scaling the material characteristic strength 0 can be converted to a characteristic strength associated with a test specimen. For this effort the four point bend bar is chosen, and the characteristic strength is designated (). Thus the component performance curve (m – 0graph) can be converted to an equivalent test specimen performance curve plotted in an m – graph.

Once again parameter pairs to the right of the material performance curve have lower probability of failures, and parameter pairs to the left are associated with higher probability of failures.

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A simple generic component was used to illustrate a surface flaw analysis, i.e., a thin walled pressure vessel. Model geometry is as follows:

Inner Radius = 100 mm

Thickness = 5 mm

Length = 400 mm

A finite element mesh is shown but this problem can be anlaysed by hand. The component performance curve was generated using the following expression

compm

m

f AP211exp1

0

max

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The material performance curve was generated from the following expression

For sake of illustration a four point bend specimens is utilized. The test specimen was selected in accordance with the guidelines set forth in ASTM C1161 configuration B. The ASTM standard for this specimen stipulates overall dimensions of 4 mm x 3 mm x 45 mm. The outer support span (Lo ) of 40 mm (with 2.5 mm overhang on each side) and the inner support span (Li ) of 20 mm.

To obtain the expression above the following relationship between the material Weibull material characteristic strength and the Weibull characteristic strength was utilized.

A

A

AA

m

f

om

mm

A PdbL

1

max0 )1ln()(2

212

meffectiveA

1

0

Am

AA

o

i

Aeffective m

mLL

bmdLA

1

0 111

1

1515

We now marry the likelihood confidence ring and the material perfmormance curve.

Generic data sets were generated via Monte Carlo simulation. The true distribution parameters are known to us. The red curve corresponds to a data set with 10 test specimens. The blue curve is the material performance curve.

Point estimates were made using maximum likelihood estimators.

The ring was generated at the 90th confidence level and it is partitioned by the material performance curve.

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To insure that the acceptance region enclosed by the likelihood ratio ring is completely to the right of the material performance curve we can either adjust the probability of failure by increasing it (not a good approach)….

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… or we can increase the confidence level until the likelihood ratio ring is tangent to the material performance curve (see the figure to the right).

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• The two most common statistical inference procedures are confidence interval estimation and hypothesis testing. Confidence interval estimates provide a range of possible values within which the true, unknown population parameters reside. Hypothesis testing speaks to the quality of the data.

• The two concepts can be combined to produce a material acceptance criterion.

• Part of the process presented here was considered for incorporationinto the ASME nuclear design code.

• The approach presented herein is not mature. A Weibull size scaling approach was utilized and size dependence in graphite appears to be a function of density. A more robust size scaling methodology is needed that incorporates density in a coherent fashion.

Summary