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Section 4: Parameter Estimation – Fast Fracture
ESTIMATION THEORY - INTRODUCTION
The parameters associated with a distribution must be estimated on the basis of a
sample distributions obtained from a population. 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 – Fast Fracture
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 = ( Sxi )
Inferences on
fX(x)
n
{ S(xi - x) }
s =
n-1
2
2
Section 4: Parameter Estimation – Fast Fracture
“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 drives
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 – Fast Fracture
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 the level of confidence increases with an increasing
interval range. 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 range define the “confidence bounds.”
Section 4: Parameter Estimation – Fast Fracture
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. 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 – Fast Fracture
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.
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 – Fast Fracture
METHOD OF MOMENTS
Section 4: Parameter Estimation – Fast Fracture
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 – Fast Fracture
Thus we begin by focusing on the distance measuring example cited earlier and
identify
~
D  Best possible value for the distance between two corners
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. If we
define the difference
~
 i  D  di
where
 i  i th Residual
di
 i th Observation
Section 4: Parameter Estimation – Fast Fracture
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
Si

n

i 1

i
~

D

n
i 1
 di

~
Then if this quantity is minimized, the resulting “best possible” value ( D ) would have a
quantifiable “goodness” associated with it, i.e., that the sum of the residuals has been
minimized.
Section 4: Parameter Estimation – Fast Fracture
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
Si
~
D


~
 D  di
i 1

~
D
 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 – Fast Fracture
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
Si 
2


n
~
D  di
i 1

2
then
n
Si
~
D
2



i 1
1
n

~
D  di

2
~
D
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 – Fast Fracture
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. 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. There are other statistical tools
available.
Section 4: Parameter Estimation – Fast Fracture
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 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
Section 4: Parameter Estimation – Fast Fracture
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.
Section 4: Parameter Estimation – Fast Fracture
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 even integer
Pn / 2
 0.5
Section 4: Parameter Estimation – Fast Fracture
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 – Fast Fracture
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 load at failure, then this expression can be linearized as follows:
ln s
=
1   1 
  ln ln 
  + ln s q 
m
1

P
   
 
Section 4: Parameter Estimation – Fast Fracture
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 – Fast Fracture
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 
Once a and b are determined the Weibull parameters m and sq can be
extracted from the expressions on the previous page.
Section 4: Parameter Estimation – Fast Fracture
PROBABILITY OF FAILURE – RANKING SCHEMES
A number of ranking schemes for Pi have been proposed. 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 observation, 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 a 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 – Fast Fracture
If we concentrate on the first observation, assuming that 20% of the entire population
falls below this value is a fairly far-reaching assumption. Thus we will appeal to a
statistical estimate of the population fraction that lies below this 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 – Fast Fracture
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 – Fast Fracture
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
Where n is the number of observations within a sample. Fro the second
observation
F  x2  
2  0.3
n + 0.4
Section 4: Parameter Estimation – Fast Fracture
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.
Section 4: Parameter Estimation – Fast Fracture
METHOD OF MAXIMUM LIKELIHOOD
The method of maximum likelihood is the most commonly used estimation
technique 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  LikelihoodFunction

n
 f  x ,q 
X
i 1

i
f X x1 ,q  f X x2 ,q  f X xn ,q 
Section 4: Parameter Estimation – Fast Fracture
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 are more than one parameter associated with a distribution, then derivative of
the log likelihood function is 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 – Fast Fracture
where
L

n
 f  x ,q ,q
X
i
1
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 the
WeibPar algorithm. The next two graphs illustrates how a first guess and then a
subsequent iteration affects the likelihood function.
Section 4: Parameter Estimation – Fast Fracture
The parameters associated with this first iteration is a rather poor choice. Here
the sample size is n = 9. Note that all nine observed strength values fall to the
right of the peak of the function. Keep in mind that the likelihood function is a
joint probability density function. If the “sampling” procedure was truly random,
the observed strength values would be more evenly spaced along the joint
probability density function.
Section 4: Parameter Estimation – Fast Fracture
The likelihood function aids in quantifying whether or not the data is dispersed along
the joint 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
joint probability function for each of the nine strength values. The value of the
likelihood function would be the product of these nine values.
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.
Section 4: Parameter Estimation – Fast Fracture
The shape and position of the joint 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.
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 – Fast Fracture
MLE – TWO PARAMETER WEIBULL DISTRIBUTION
Section 4: Parameter Estimation – Fast Fracture
Section 4: Parameter Estimation – Fast Fracture
MULTIPLE FLAW DISTRIBUTIONS
Section 4: Parameter Estimation – Fast Fracture