Transcript Section 4 g - Academic Csuohio

Section 4: Parameter Estimation
ESTIMATION THEORY - INTRODUCTION
The parameters associated with a distribution must be estimated on the basis of
samples obtained from the population the distribution is modeling. The role of
sampling as it relates to the statistical inference and parameter estimation is outlined
in the figure in the next overhead.
The point is to construct a mathematical model that captures the population under
study. This requires
• inferring the type of distribution that best characterizes the population; and
• estimating parameters once the distribution has been established.
Thus sampling a population will yield information in order to establish values of the
parameters associated with the chosen distribution.
Section 4: Parameter Estimation
Tensile Strength
Random Variable X
Realizations of random variable X:
0 < x < +
Assume random variable is characterized
by the distribution fX(x)
Experimental Observations MOR bars or Tensile Specimens
{ x1, x2, ... , xn }
Construct histogram to simulate fX(x)
f (x)
X
x
Familiar Statistical Estimators
x = ( S xi )
Inferences on
fX(x)
n
{ (xi - x) }
s = S
n-1
2
2
Section 4: Parameter Estimation
“Choosing” a distribution can be somewhat of a qualitative and subjective process.
We stress that the physics that underlie a problem should indicate an appropriate
choice. However, most times the engineer is left with somehow establishing a
rational choice, and too often histograms and their shapes are relied on. However
there are quantitative tools that can aid the engineer in his/her selection. These tools
are known as goodness-of-fit tests, e.g,
• Anderson-Darling Goodness-of-Fit Test
Usually these types of tests will only indicate when the engineer chooses badly. For
ceramics, the industry has focused on the two parameter Weibull distribution. This is
a Type III minimum extreme value statistic. Thus physics and mathematics drive this
selection.
Once the type of distribution has been made, the next step involves parameter
estimation. There are two types of parameter estimation
• Point Estimates
• Interval Estimates
Section 4: Parameter Estimation
Point estimation is concerned with the calculation of a single number, from a sample of
observations, that “best” represents the parameters associated with a chosen distribution.
Interval estimation goes further and establishes a statement on the confidence in the
estimated quantity. The result is the determination of an interval indicating the range
wherein the true population parameter is located. This range is associated with a level of
confidence.
For a given number of samples an increasing interval range increases the level of
confidence. Alternatively, increasing the sample size will tend to decrease the interval
range for a given level of confidence.
Best possible combination is a large confidence and small interval size.
The endpoints of the interval range define what is known as “confidence bounds.”
Section 4: Parameter Estimation
POINT ESTIMATION - PRELIMINARIES
In general, the objective of parameter estimation is the derivation of functions, i.e.,
estimators, that are dependent on failure data, and that yield in some sense optimum
estimates of the underlying population parameters.
Various performance criteria can be applied to ensure that optimized estimates are
obtained consistently. Two important criteria are:
• Estimate Bias
• Estimate Invariance
Bias is a measure of the deviation of the estimated parameter value from the
expected value of the population parameter. The values of point estimates computed
from a number of samples will vary from sample to sample. In this context, the word
sample implies a group of values, specifically a group of failure strengths. If enough
samples are taken one can generate statistical distributions for the point estimates, as
a function of sample size. If the mean of a distribution for a parameter estimate is
equal to the expected value of the parameter, the associated estimator is said to be
unbiased.
Section 4: Parameter Estimation
If an estimator yields biased results, the value of an individual estimate can easily be
corrected if the estimator is invariant. An estimator is invariant if the bias associated with
estimated parameter value is not functionally dependent on the true distribution
parameters that characterize the underlying population. An example of an estimator that
is not invariant is the linear regression estimators for the three-parameter Weibull
distribution. The maximum likelihood estimator (MLE) for a two-parameter Weibull
distribution is invariant.
There are three typical methods utilized in obtaining point estimates of distribution
functions:
• Method of moments
• Linear regression techniques
• Likelihood techniques
Section 4: Parameter Estimation
METHOD OF MOMENTS
Section 4: Parameter Estimation
MINIMIZING RESIDUALS
No matter how refined our physical measurement techniques become, we can never
ascertain the “true value” of anything. Thus we take repeated measurements of a
quantity (say the distance between two corners of a property) and each time a
measurement is conducted the values vary. Thus we are confronted with the dilemma
of what value best represents the quantity measured. Several options include
• Mean
• Median
• Mode
Faced with options, one should question which approach yields the “best possible”
value. To answer this question a systematic approach is needed such that one can say
“This is the best possible value since this quantity is minimized”
or
“This is the best possible answer since that quantity is maximized”
Section 4: Parameter Estimation
Thus we begin by focusing on the distance measuring example cited earlier and
identify
~
D  Best possible value for the distance between two corners (unknown)
If many observations are made of this distance, then it is quite possible that none of
the observations within a sample will coincide with the “best possible” value
(whatever that is). If we define the difference between an observation and the “best
possible” value as a residual
~
 i  D  di
where
 i  i th Residual
di
 i th Observation
A residual is similar to, but not equal to an error. We need the true value to compute
errors.
Section 4: Parameter Estimation
We now search for a method to establish a “best possible” value. A systematic approach
that yields the “best possible” value surely must minimize the residual associated with
each observation (unless the observation is aberrant for some reason, i.e., the observation
is an outlier). If we identify
S

n

i 1

i
~

D

n
i 1
 di

~
Then if the quantity S is minimized, the “best possible” value ( D ) used in the
computation above would have a quantifiable “goodness” associated with it, i.e., that the
sum of the residuals has been minimized.
Section 4: Parameter Estimation
To minimize the sum of the residuals, take the derivative of the expression above
~
~
with respect to D , set the derivative equal to zero and solve for D
n
S
~
D


~
 D  di

i 1
~
D
 (1  0) + (1  0) +  + (1  0)
 n
Setting this last expression equal to zero definitely minimizes the residuals, for if
no measurements are taken, all the residuals are zero. There is obviously a logic
fault here.
Section 4: Parameter Estimation
If minimizing the sum of the residuals is initially appealing (but the results do not
help) then minimizing the sum of the squares of the residuals should be no less
appealing. Here
2
n
~
2
S    D  di


i 1
then
n
S 2
~
D



~
D  di
i 1

2
~
D
~
 2nD  2(d1 + d 2 +  + d n )
Setting this last expression equal to zero yields
~
D 
1
n
n
d
i 1
i
Thus if we wish to minimize the sum of the squares of the residuals, then the sample
mean should be utilized as the “best possible” value.
Section 4: Parameter Estimation
Note that we developed this argument in terms of deriving a best possible value for a
series of measurements. This concept can be easily extended to estimating values for
distribution parameters, where instead of making a “measurement,” we take a sample
from the underlying population.
Minimizing the sum of the squares of the residuals is not the only systematic
approach in producing the “best possible” estimates of distribution parameters. The
maximum likelihood technique is another systematic approach where a “likelihood”
is maximized. This technique is described in a later section. In some instances the
estimators from various methods coincide, most times they do not.
In situations where different approaches produce different estimators (and estimates),
then one must choose between the different techniques. The amount of bias produced
by an estimator is one measure of assessing efficacy. Additional statistical tools are
available.
Section 4: Parameter Estimation
PROBABILISTIC REGRESSION ANALYSIS
We now wish to extend the
concepts associated with
regression analysis to parameter
estimation.
Consider an experiment where the
tensile strength data has been
collected for a given material.
The tensile strength data is
identified as the dependent
variable (since the individual
conducting the test can not control
the value of this parameter – the
material does). We need to adopt
an independent random variable.
Consider the ranked probability of
failure associated with each
tensile strength value depicted to
the right.
Experimental Data
Strength (yi)
Probability of
Failure Pi (= xi)
y1
x1
y2
x2
y3
x3
…
…
…
…
yn
xn
Note: Although the table above shows y
dependent on x, in typical Weibull plots Pi
is the vertical axis and strength is the
horizontal axis. This causes confusion,
especially using EXCEL.
Section 4: Parameter Estimation
Here
yi = ith ranked tensile strength
x i = Pi
= Associated ranked probability of failure
The data is ranked in the following fashion
y1 < y2 < y3 < ... < yn
Thus it seems reasonable to expect
P1 < P2 < P3 < ... < Pn
x1 < x2 < x3 < ... < xn
Note carefully that the individual conducting the experiment controls the value of n.
This is important, as we will see later.
Section 4: Parameter Estimation
The ranked data is in ascending order. But what are the probability values associated
with each ranked data value? Consider the following observations:
• x1 corresponds to the lowest probability of failure
P1i  0
• xn corresponds to the highest probability of failure
Pn
 1.0
• Assuming n is an odd integer
Pn +1
2
 0.5
Section 4: Parameter Estimation
To possibly account for these three observations, consider the following
expression:
Pi

i
n +1
For large n values, P1 trends to zero and Pn approaches 1.
If we adopt this expression it is quite clear that the individual conducting the
experiment influences Pi (or xi) through the choice of n prior to testing. Thus Pi
(or xi) should be considered the independent variable in the experiment.
With data collected from the experiment the individual analyzing the data now
assumes an underlying probability density function
Pi
 FY  yi ,q1 ,q 2 
If this expression can be linearized we can apply linear regression techniques to
find the parameters q1 and q2.
Section 4: Parameter Estimation
LINEAR REGRESSION –
TWO PARAMETER WEIBULL DISTRIBUTION
If we assume that the probability of failure in our experiment is governed by a twoparameter Weibull distribution, i.e.
 m
f ( s ) = 
sq
 s
 
 sq
(m-1)



 s
exp - 
  s q



m



where s is the applied stress at failure, then this expression can be linearized as follows:
ln s
=
1   1 
  ln ln 

m
1

P
   


 + ln s q 

Section 4: Parameter Estimation
If we take
yi
 ln s i 
  1 

xi = ln ln 
  1  Pi 
b  ln s q 
a 
1
m
Then
yi =
axi
+ b



Section 4: Parameter Estimation
We can now make use of the traditional linear regression expressions for a
and b
n
n xi yi

i 1
a =
n
i 1
n
 x   y
b =
i 1
i
i 1
n
n  xi 
i 1
i
i 1
 n 
   xi 
 i 1 
n  xi 
2
n
x y
i 1
2
n
n
i
2

i
2
n
n
x x y
i 1
i
i 1
2
i
i
 n 
   xi 
 i 1 
~ and s~ (where the
Once a and b are determined the Weibull parameters m
q
tilde designates the value as an estimate) can be extracted from the
expressions on the previous page.
Section 4: Parameter Estimation
PROBABILITY OF FAILURE – RANKING SCHEMES
A number of ranking schemes for Pi have been proposed in the literature. A mean
ranking scheme was introduced in the previous section. In this section a median
ranking scheme is discussed.
As Johnson (1951) points out, the usual method of statistical inference involves
constructing a histogram, from which a smoothed probability density function is
derived. However, small sample sizes present difficulties since histograms vary
greatly with changes in class intervals.
As an alternative to this, ordered statistics were developed whereby ranked failure data
is utilized. Consider a sample with five observations, where the observations are
arranged in an increasing numerical order. It would seem reasonable to assume that
the first observation (lowest value) would represent a value where 20% of the entire
population would fall below this value. Thus an estimate for Pi of 20% is assigned
this ranked value. Similarly a value of 40%, 60%, 80% and 100% would be assigned
to the other ranked observations.
Section 4: Parameter Estimation
If we concentrate on the first observation, assuming that 20% of the entire population
falls below this value is a fairly far-reaching assumption. This statement can be made
for the other four estimate Pi values Thus we will appeal to a statistical estimate of the
population fraction that lies below this 20% value.
To illustrate the concept, consider a sample of five observations taken from a
population whose probability density function and attending distribution parameters
are known. This sample of five is repeated four times, and for each sample the data is
arranged in ascending order. If the cumulative distribution function for the first value
is computed, then
F(x1) = percentage of the population below the value of x1
Section 4: Parameter Estimation
This is illustrated in the following figure taken from Lipson and Sheth (1979)
Thus in sample #1 (darkened circle) the first failure may have occurred at A, where
15% of the population has a value less than value at A. For sample #2 (open circle),
the first value occurs at B, which represents 9% of the population.
Section 4: Parameter Estimation
When this procedure is repeated many times the data generates a series of
percentage values that are randomly distributed. The median value of this
distribution is given by the expression
F  x1  
1  0 .3
n + 0 .4
which equals
F x1   .1296
when the total number of observations in the sample is n=5. From the
second observation
F  x2  
2  0.3
n + 0.4
which equals
F x2  .3148
for 5 observations.
Section 4: Parameter Estimation
Thus in general
F xi   Pi

i  0.3
n + 0.4
Another ranking scheme proposed by Nelson (1982) had found wide acceptance.
Here
F xi   Pi

i  0.5
n
This estimator yields less bias then the median rank estimator, or the mean rank
estimator. It is also the estimator accepted for use in ASTM 1239, and ISO
Designation FDIS 20501. Due to client requests over the years WeibPar has
included a host of other estimates for the probability of failure.
Section 4: Parameter Estimation
METHOD OF MAXIMUM LIKELIHOOD
The method of maximum likelihood is the most commonly used estimation
technique for brittle material strength because the estimators derived by this
approach maintain some very attractive features.
Let (X1, X2, X3, …, Xn) be a random sample of size n drawn from an arbitrary
probability density function with one distribution parameter, i.e.,
f X x,q 
Here q is an unknown distribution parameter. The likelihood function of this random
sample is defined as the joint density of the n random variables
L  Likelihood Function

n
 f  x ,q 
X
i 1

i
f X x1 ,q  f X x2 ,q  f X xn ,q 
Section 4: Parameter Estimation
Often times it is much easier to manipulate the logarithm of the likelihood function,
i.e.,
L
 ln L
The maximum likelihood estimator (MLE) of q identified as qˆ, is the root of the
expression obtained by equating the derivative of L to zero
L
q
 0
If there is more than one parameter associated with a distribution, then derivatives of
the log likelihood function are taken with respect to each unknown parameter, and
each derivative is set equal to zero, i.e.,
L
q1
 0 ,
L
q 2
 0 ,

,
L
q k
 0
Section 4: Parameter Estimation
where
L
 n

 ln  f X xi ,q1 ,q 2 ,,q k 
 i 1

And k represents the number of parameters associated with a particular distribution.
When more than one parameter must be estimated often times the system of equations
obtained by taking the derivative of the log likelihood function must be solved in an
iterative fashion, e.g., as is done with the two parameter Weibull distribution inside
WeibPar.
Section 4: Parameter Estimation
The next two graphs illustrates how a first guess, and then a subsequent iteration
affects the likelihood function. When
f( x, q1 , q 2 )

f (s , m, s q )

m

 sq
 s
 
  sq
(m-1)



 s
exp - 
  s q



m



then the first graph below
f (s , m, s q )
Fracture Stress (MPa )
plots this probability density function given above based on the first guess of m (=q1
and sq (=q2 , which are admittedly pore choices.
Section 4: Parameter Estimation
In both graphs the number of observations in the sample is n = 9. Note that all nine
observed strength values fall to the right of the peak of the function. If the “sampling”
procedure was truly random, the observed strength values would be more evenly spaced
along the probability density function. This is why the first graph represents such a poor
choice in the parameter estimates.
Moreover, the likelihood function aids in quantifying whether or not the data is
dispersed along the probability density function. To help visualize that the magnitude of
the likelihood function does this, an arrow has been attached to the associated value of
the probability function for each of the nine strength values. The value of the likelihood
function would be the product of these nine values.
Section 4: Parameter Estimation
Next consider the following graph which represents an iteration on the estimated
distribution parameter values. Note the vertical scale has been maintained from the
previous graph
f (s , m, s q )
The shape and position of the probability density function appear to be a much better fit to
the nine data points. Again, we base judgment on the assumption that our data values
represent a random sample and they should therefore span the range. Note for small
sample sizes this assumption can easily break down.
Section 4: Parameter Estimation
Again, nine arrows point to the associated values of the joint probability density function
for each of the nine failure strengths. The product of these nine values represents the
value of the likelihood function for this choice of distribution parameters. A simple
inspection is sufficient to conclude that the likelihood from the latter iteration is greater
than the likelihood from the former. If the latter choice of parameters is considered more
acceptable, then this would indicate that obtaining a “best” set of distribution parameters
involves maximizing the likelihood function.
Two important properties of maximum likelihood estimators
1. Maximum likelihood estimators yield unique solutions
2. Estimates asymptotically converge to the true parameters as the sample size
increases
Section 4: Parameter Estimation
MLE – TWO PARAMETER WEIBULL DISTRIBUTION
Let s1, s2, ... , sN represent realizations of the ultimate tensile strength (a random
variable) in a given sample, where it is assumed that the ultimate tensile strength is
characterized by the two-parameter Weibull distribution. As noted in the previous
section the likelihood function associated with this sample is the joint probability
density evaluated at each of the N sample values. Thus this function is dependent on
the two unknown Weibull distribution parameters (m, sq ). The likelihood function
for an uncensored sample under these assumptions is given by the expression
 m̂
L =  
ˆq
i=1  s
N
m̂- 1
 s i 
  
  sˆ q 
  s i m̂ 
exp   
  sˆ q  
Here mˆ  q1  and sˆq  q 2  designate estimates of the true distribution parameters m
and sq .
Section 4: Parameter Estimation
The system of equations obtained by differentiating the log likelihood function for
a censored sample is given by
N
m
(
ln ( s i )
)
s
i
S
ˆ
i=1
N
m
(
)
s
i
S
ˆ
1
N
N
S ln ( s i ) i=1
1
=0
mˆ
i=1
And

mˆ  1 
=
(
)

sˆ q  S s i  
N
  i=1
N
1/ mˆ
The top equation is solved first for m̂. Subsequentlyŝ q is computed from the
second expression. Obtaining a closed form solution for the first equation is not
possible. This expression must be solved numerically. WeibPar solves these
equations given a sample of failure data.
Section 4: Parameter Estimation
EXAMPLE - SINGLE FLAW DISTRIBUTION
WeibPar – Uncen.dat
Section 4: Parameter Estimation
PARAMETER ESTIMATE BIAS
A certain amount of intrinsic variability occurs due to sampling error. This error is
associated with the fact that only a limited number of observations are taken from an
infinite (assumed) population.
Bias - Deviation of the expected value of the estimated parameter from the true
parameter. This is the tendency of a particular estimator to give consistently high or
consistently low estimates relative to the true value.
Once it is recognized that the estimates of the Weibull modulus and characteristic strength
will vary from sample to sample, these estimates can be treated as random variables (i.e.,
statistics).
Section 4: Parameter Estimation
Given the definition of bias on the previous overhead, the amount of bias associated with a
parameter estimate can be quantified through the following general expression

E qˆ
qˆ fq qˆ dqˆ
+

ˆ

 q true population parameter being estimated 

The function fqˆ qˆ is an unknown probability density function for the parameter estimate qˆ .
This reinforces the concept that the parameter estimate is a random variable. If we focus on
the Weibull modulus, and realize the expectation can be estimated by the average, or
arithmetic mean of a sample, then
E mˆ  
+
 mˆ f mˆ  dmˆ
mˆ

1 n

mˆ i

n i 1
 m true Weibull modulus 
Section 4: Parameter Estimation
A similar expression exists for the estimate of the characteristic strength sˆq , i.e.,
E sˆq  
+
 sˆq
fsˆq sˆq  dsˆq


1 n
sˆq i

n i 1
If we focus attention on the parameter estimate of the Weibull modulus, the following
procedure, which relies heavily on Monte Carlo simulation can quantify the amount of
bias
1. Specify a two-parameter Weibull distribution where m and sq are known.
2. Fix the sample size (say n = 10) and sample from the population characterized by the
two-parameter Weibull distribution specified in the previous step.
3. Estimate the Weibull distribution parameters from the sample using maximum
likelihood estimators
4. Divide the maximum likelihood estimates of the Weibull modulus ( m̂ ) by the true
value (m) specified in the first step.
Section 4: Parameter Estimation
5. Repeat steps 2 through 4 say 10,000 times. Thus 10,000 samples are generated, along
with 10,000 parameter estimates, and 10,000 ratios.
6. The 10,000 parameter estimates of m̂ represents a sample from the distribution f mˆ mˆ  .
Compute the sample mean, or alternatively
 mˆ
1 10, 000 mˆ i


10,000 i 1 m
And plot this as a function of the number of specimens per sample (n).
7. Increment the number of specimens per sample and repeat steps 3 through 6.
Section 4: Parameter Estimation
The information from this seven step
procedure can be plotted as follows
The dashed line represents  m̂ for
each sample size. If the estimator for
the Weibull modulus (an MLE was
used in this example) was unbiased,
then
E mˆ i 
m
 1.0
 mˆ 
We see that this is only true in the
limit as the number of specimens
increases. While this fact is
reassuring it does not help the design
engineer with a finite test budget.
The ability to quantify bias allows us to remove it using the value  m̂ , as long as the
estimator used is invariant, i.e., if the procedure just outlined works for any set of true
distribution parameters chosen in Step #1. This is not always possible.
Section 4: Parameter Estimation
If one uses maximum likelihood estimators to compute the Weibull characteristic
strength, then the estimate of the parameter is essentially unbiased. This can be seen in
the following graph
The dashed line is essentially
equal to one for nearly all
values of the number of
specimens per sample.
However, WeibPar will
compute the small amount of
bias present in this type of
estimate and provide an
unbiased estimate of the
Weibull characteristic strength.
Section 4: Parameter Estimation
On a final note regarding estimate bias, the first theorem derived by Thoman, Bain &
Antle (1969) established that the ratio
^
m
m
=
Estimated modulus
True modulus
was a random variable with an exponential distribution. They also formally
proved that this random variable is independent of the true Weibull modulus and
characteristic strength of the underlying population. This allows the numerical
estimation of unbiasing factors through the use of Monte Carlo simulation just
outlined.
Section 4: Parameter Estimation
CONFIDENCE BOUNDS
Confidence bounds quantify the uncertainty associated with a point estimate
of a population parameter.
l
The values used to construct confidence bounds are based on percentile
distributions obtained from Monte Carlo simulation. For example, given a fixed
sample size (n) and a simulation that contains 10,000 estimates, then after
ranking the estimates in numerical order
l
1,000th Estimate - 10th Percentile
2,000th Estimate - 20th Percentile
9,000th Estimate - 90th Percentile
The 10th and 90th percentile define a range of values where the user is 80%
confident that the true value of the parameter estimate lies between these two
values. Confidence bounds are closely associated with the concept of hypothesis
testing.
l
Section 4: Parameter Estimation
The second theorem derived by Thoman, Bain & Antle (1969) established
that the parameter t defined as
t
=
^ ln ( s^q sq )
m
is a statistic that is functionally dependent on two random variables with
exponential distributions. The "t" random variable is independent of the true
Weibull modulus and characteristic strength of the underlying population. Once
again this allows the numerical estimation of confidence bounds through the use
of Monte Carlo simulation.
Section 4: Parameter Estimation
EXAMPLE - SINGLE FLAW DISTRIBUTION
WeibPar – Uncen.dat (focus on unbiased values and confidence bounds)
Section 4: Parameter Estimation
MULTIPLE FLAW DISTRIBUTIONS (C.A. Johnson)
The previous discussion assumes that single distribution of flaws is present
throughout the specimens being tested. Real ceramic materials contain two or more
types of defects, each with its own characteristic flaw size distribution. This can be
easily verified using careful fractographic techniques.
The other indication of
multiple flaw distributions is
a distinctive “knee” in the
data. This is depicted in the
figure to the right. This
figure appears in ASTM C
1239. Here the probability
of failure for the data points
is estimated using
Pi

i  0.5
n
Section 4: Parameter Estimation
The knee in the curve can appear as it does in
the previous graph, or the curvature can be
reversed as shown to the right. This figure
appears in a GE report (1979) by C.A. Johnson.
One can use the manner in which a knee is
generated, i.e., how the data deviates from
linearity to deduce information concerning the
nature of the flaw distributions present.
Consider a group of test specimens that when failed, have two distinctly different types
of fracture origins. Some specimens fail from flaw type “A” while the remainder fail
from flaw type “B.”
Section 4: Parameter Estimation
There are at least three ways in which these two flaw populations are present in the test
specimens, i.e., the sample.
•Both flaw distributions are present in every test specimen. This is known as
“Concurrent Flaw Populations.”
•Within a group of specimens any given specimen may contain flaws from
distribution A, or from distribution B, but not from both. This is known as
“Mutually Exclusive Flaw Populations.”
•Within a group of specimens, flaw distribution A may be present in all specimens,
while distribution B may be present in only some test specimens. This is known
as “Partially Concurrent Flaw Populations.”
A larger number of cases must be considered than three above if more than two flaw
distributions are active.
Section 4: Parameter Estimation
Examples of the three types of flaw distributions are as follows.
Concurrent Flaw Distributions – A group of test specimens machined from a
ceramic billet which contains defects distributed throughout the volume. Each
specimen then contains both machining flaws on the surface and volume defects
interior to the test specimen.
Exclusive Flaw Distributions – A group of test specimens comprised of two
subgroups purchased from two different manufacturer.
Partially Concurrent Distributions - A group of pressed and sintered test
specimens which were all sintered in the same furnace, but could not be sintered
in a single furnace cycle. The specimens sintered at the longer cycle would
contain elongated grains (considered a deleterious flaws) and all the specimens
would have strength degrading inclusions present from the powder preparation.
The parameter estimation methods presented here assumes that if multiple flaw
populations are present, then they are concurrent flaw distributions. As indicated
above, partially concurrent distributions point to processing methods that have not , or
are not mature. In addition, it is quite typical for a manufacturer of ceramic
components to identify one material supplier, and this would minimize the presence of
exclusive flaw distributions.
Section 4: Parameter Estimation
MLEs FOR MULTIPLE FLAW DISTRIBUTIONS
The likelihood function for the two-parameter Weibull distribution where concurrent
flaw distributions are present (known as a censored sample) is defined by the
expression

 r  m̂
L =  
ˆ
i=1 s

  q
 s i 
  
  sˆ q 
m̂- 1
  s i m̂  
  s i m̂ 
 N
exp      exp   
j= r +1
  sˆ q   
  sˆ q  

This expression is applied to a sample where two or more active concurrent flaw
distributions have been identified from fractographic inspection. For the purpose of
the discussion here, the different distributions will be identified as flaw types A, B, C,
etc. When the expression above is used to estimate the parameters associated with the
A flaw distribution, then r is the number of specimens where type-A flaws were found
at the fracture origin, and i is the associated index in the first summation. The second
summation is carried out for all other specimens not failing from type-A flaws (i.e.,
type-B flaws, type-C flaws, etc.). Therefore the sum is carried out from
(j = r
+ 1) to N (the total number of specimens) where j is the index in the second
summation. Accordingly, σi and σj are the maximum stress in the ith and jth test
specimen at failure.
Section 4: Parameter Estimation
The system of equations obtained by differentiating the log likelihood function for a
censored sample is given by
N
( s i )m̂ ln ( s i )
S
1
i=1
N
(s i )
S
i=1
m̂
-
r
r
S
i=1
ln ( s i ) -
1
=0
m̂
and
 N
m̂  1 
sˆ q =   S ( s i )  
r
  i=1
1/ m̂
where:
r = the number of failed specimens from a particular group of a censored sample
Section 4: Parameter Estimation
EXAMPLE - MULTIPLE FLAW DISTRIBUTIONS
Section 4: Parameter Estimation
NON-LINEAR REGRESSION ANALYSIS:
THREE PARAMETER WEIBULL DISTRIBUTION
When strength data indicates the existence of a threshold stress, a three-parameter
Weibull distribution can be employed in the stochastic failure analysis of structural
components. However, care must be taken such that careful fractography does not
indicate the presence of a partially concurrent flaw distribution, which has the
appearance of a population characterized by a three parameter Weibull distribution.
The resulting expression for the probability of failure of a component fabricated from a
material characterized by a three parameter Weibull distribution and subjected to a
single applied stress s is
Pf



s  
 dV 
= 1  exp   
 0 


if the defect population is spatially distributed throughout the volume. A similar
expression exists for failures due to area defects.
Section 4: Parameter Estimation
Here the distribution g has the physical interpretation of a threshold stress. Regression
analysis postulates a relationship between two variables. In an experiment typically one
variable can be controlled (the independent variable) while the response variable (or
dependent variable) is not. In simple failure experiments the material dictates the
strength at failure, indicating that the failure stress is the response variable. The ranked
probability of failure (Pi) can be controlled by the experimentalist, since it is
functionally dependent on the sample size (N). After arranging the observed failure
stresses (s1, s2, s3,  , sN) in ascending order, and as before specifying
Pi

i  0.5
N
then clearly the ranked probability of failure for a given stress level can be influenced by
increasing or decreasing the sample size. The procedure outlined here adopts this
philosophy. The assumption is that the specimen failure stress is the dependent variable,
and the associated ranked probability of failure becomes the independent variable.
Section 4: Parameter Estimation
Using the three-parameter version for the probability of failure, an expression can be
obtained relating the ranked probability of failure (Pi) to an estimate of the failure
strength (s~i ). Assuming uniaxial stress conditions in a test specimen with a unit
volume, then
1/ ~
~ + ~ ln 1  
~
=


si
0  
1

Pi  

~
~
~ ,  and  are estimates of the shape parameter (), the scale parameter ( ),
where 
0
0
threshold parameter (), respectively. Defining the residual as
 i = s~i - s i
where si is the ith ranked failure stress obtained from actual test data, then the sum of
the squared residuals is expressed as
N
N

~
1/ ~
~
 (  i ) =   + 0 Wi  - s i
i=1
2
i=1

2
Section 4: Parameter Estimation
Here the following notation has been adopted where
 1 
W i = ln

 1 - Pi 
Note that the forms of s~i and Wi change with specimen geometry. It should be apparent
that the objective of this method is to obtain parameter estimates that minimize the sum of
the squared residuals.
the partial derivatives of the sum of the squared residuals
~ Setting
~
~
with respect to  ,  0 and  equal to zero yields the following three expressions
 N
 N

1/  
N   s i (W i )  -   s i 
~
 i=1

 i=1

0 =
N
 N

2/ 
N  (W i )
-   (W i )1/  
i=1
 i=1

~ =



N

N


N
i=1

i=1


i=1
 N
1/  
)
(W

i


 i=1

 N
1/  
  (W i ) 
 i=1


N

i=1

 s i    (W i )2 /   -   s i (W i )1/     (W i )1/  
N



N

N

N

i=1

i=1

i=1

 (W i )2/  -   (W i )1/     (W i )1/  
Section 4: Parameter Estimation
and
 s i Wi 
N
i=1
1/ ~
 
ln Wi
N
 
 ~  s i Wi
i=1
1/ ~
 
ln Wi
~
N
 
 0  s i Wi
i=1
2 / ~
 
ln Wi
  conv
in terms of the parameter estimates. The solution of this system of equations is
iterative, where the third expression is used to check convergence at iteration.
Section 4: Parameter Estimation
EXAMPLE – THREE PARAMETER WEIBULL DISTRIBUTION
Section 4: Parameter Estimation
Section 4: Parameter Estimation