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Optimal Sample Size Allocation forMulti-Stress Tests using

Extreme Value RegressionH. K. Tony Ng

Department of Statistical Science

Southern Methodist University, Dallas, Texas

Co-authored with N. Balakrishnan (McMaster University)

& P. S. Chan (Chinese University of Hong Kong)

Optimal Sample Size Allocation for Multi-Stress Tests using Extreme Value Regression – p.1/51

Outline1. Introduction - Multi-stress Experiments



2. Introduction - Extreme value regression




3. Likelihood Inference





4. Optimal Sample Size Allocation for Estimation





4. Optimal Sample Size Allocation for Estimation• Two-stress levels





4. Optimal Sample Size Allocation for Estimation• Two-stress levels• k-stress levels






5. Optimal Sample Size Allocation for Prediction






5. Optimal Sample Size Allocation for Prediction• Two-stress levels






5. Optimal Sample Size Allocation for Prediction• Two-stress levels• k-stress levels







6. Numerical Illustration







6. Numerical Illustration

7. Monte Carlo Study for Small Sample Sizes


Multi-stress Experiments• In industrial experiments, the lifetime of an unit

may be affected by one or more factors/covariates(such as voltage, load, temperature, etc.)




• The purpose of a life-testing experiment is then tostudy the effect of these covariates on the failuretime distribution and predict the reliability undersome other operating conditions.




• The purpose of a life-testing experiment is then tostudy the effect of these covariates on the failuretime distribution and predict the reliability undersome other operating conditions.

• Nowadays, products that are tested are oftenextremely reliable with long life under normaloperating conditions.


Multi-stress Experiments• To overcome these problems, the experimenter

may resort to accelerated testing wherein theunits are subjected to higher stress levels thannormal.



may resort to accelerated testing wherein the unitsare subjected to higher stress levels than normal.

• The accelerated tests may be performed usingconstant stress or increasing stress levels.





• Assessing the effects of the acceleratingvariables(e.g. temperature, voltage, load,vibration etc.) on the lifetimes of the product.





• Assessing the effects of the acceleratingvariables(e.g. temperature, voltage, load,vibration etc.) on the lifetimes of the product.

• Information from tests at high-levels of one ormore accelerating variables is extrapolated,through a reasonable statistical model, to obtainestimates of life or long-term performance atlower, normal levels of the acceleratingvariable(s).


Multi-stress Experiments –Example

• Such testing provides a saving in time and costcompared with testing at normal conditions.




• Refrigerator door open/close test:




• Refrigerator door open/close test:• Consider the reliability of a refrigerator door,

which is designed for a median open/closecycles of 200,000 times, assuming a normalusage force is used.






• Increase the force of open/close test, therefrigerator door fail more rapidly than itwould have failed at the normal usage force.






• Increase the force of open/close test, therefrigerator door fail more rapidly than itwould have failed at the normal usage force.

• The open/close tests are done under higherforces, the data may then be extrapolated toestimate the mean lifetime under normalcondition.



• The insulating fluidin a power transformer performs two

major functions:




major functions:

• electrical insulation to withstand the high voltages

present inside the transformer.




major functions:



• heat transfer medium to dissipate heat generated within

the transformer windings.




major functions:





• Dielectric Breakdown Voltage - a measure of the ability of

the liquid to withstand electric stress without failure.




major functions:





• Dielectric Breakdown Voltage - a measure of the ability of

the liquid to withstand electric stress without failure.

• The general accepted minimum dielectric strength is 30kV

for transformers with a high-volatage rating≥ 287.5 kV

and 25kV for transformers with a high-voltage rating<

287.5kV.


Multi-stress Experiments

-

-

-

-

-

-

-

26 kV

28 kV

30 kV

32 kV

34 kV

36 kV

38 kV

Voltage (Stress)

s ss

s s ss s

s s s ss ss ss

s s sss ss s ss s

s s s s s s s ss s s s s s

ss ss ss s

s s s s s

Example: Life-testing experiment of insulating fluid at seven stress

levels

• Lifetime = function of factors or covariates


Extreme Value Regression• To study the effect of these covariates on the

failure time distribution, a regression model isused to incorporate the covariates in the analysis.




• If the life data are taken under differentenvironmental conditions, the Weibull (Extremevalue) model will be useful as it has moreflexibility.




• If the life data are taken under differentenvironmental conditions, the Weibull (Extremevalue) model will be useful as it has moreflexibility.

• Suppose the survival timeT has a Weibulldistribution with probability density function

fT (t; α, δ) =δ

α

(

t

α

)δ−1

exp

[

−

(

t

α

)δ]

, t > 0,

whereα > 0 is the scale parameter andδ > 0 isthe shape parameter.


Extreme Value Regression• In this discussion, we will focus on the case with

a single covariate.


Extreme Value Regression• In this discussion, we will focus on the case with

a single covariate.• Assume that the covariatex affects only the scale

parameterα; then, we have a proportional hazardmodel for the survival timeT (Kalbfleisch andPrentice, 1980).


Extreme Value Regression• Making a logarithmic transformation on the

survival timeT , we have an extreme value(Gumbel Type I) distribution forY = ln T withp.d.f.

fY (y; µ(x), σ) =1

σexp

[(

y − µ(x)

σ

)

− exp

(

y − µ(x)

σ

)]

, −∞ < y < ∞,

whereµ(x) = ln α(x) andσ = 1δ.


Extreme Value Regression• Making a logarithmic transformation on the

survival timeT , we have an extreme value(Gumbel Type I) distribution forY = ln T withp.d.f.

fY (y; µ(x), σ) =1

σexp

[(

y − µ(x)

σ

)

− exp

(

y − µ(x)

σ

)]

, −∞ < y < ∞,

whereµ(x) = ln α(x) andσ = 1δ.

• We can writeY in a location-scale form as

Y = µ(x) + σZ,

whereZ has a standard extreme valuedistribution with p.d.f.

fZ(z) = exp (z − exp z) , −∞ < z < ∞.


Extreme Value Regression• A simple functional form forµ(x) is given by

µ(x) = ν0 + ν1x,

whereν0 andν1 are the regression parameters.


Extreme Value Regression• A simple functional form forµ(x) is given by

µ(x) = ν0 + ν1x,

whereν0 andν1 are the regression parameters.• This model corresponds to the Weibull model

with power law relationship between the scaleparameterα and the stress levels, i.e. α = csq,whereµ(x) = ln α, x = ln s, ν0 = ln c andν1 = q.


Multi-stress Experiments

-

-

...

-

-

Stress

x1

x2

...

xk−1

xk

n1

n2

...

nk−1

nk

• Suppose we haveN items available for the test atk oredered

stress level, sayx1 < x2 < . . . < xk.

• We assignnl items for testing at stress levelxl (l = 1, 2, . . . , k)

withk∑

l=1

nl = N

• In planning such a life-testing experiment, we have the flexibility

in the choice of (n1, n2, . . . , nk) for a pre-fixed value ofN and

the stress levelsxl, l = 1, 2, . . . , k.


Likelihood Inference• One of the approaches to analyze extreme value (Weibull)

regression model is based on the asy. theory of MLEs.




• Let Zli = (Yli − ν0 − ν1xl) /σ for l = 1, 2, . . . , k,

i = 1, 2, . . . , nl, Zli follows a standard extreme value dist.




• Let Zli = (Yli − ν0 − ν1xl) /σ for l = 1, 2, . . . , k,

i = 1, 2, . . . , nl, Zli follows a standard extreme value dist.• The likelihood function based on thek-stress level model is

then

L(ν0, ν1, σ) =

k∏

l=1

1

σnl

nl∏

i=1

exp (zli − ezli ) ,

so that the log-likelihood function is

ln L(ν0, ν1, σ) = −N ln σ +

k∑

l=1

nl∑

i=1

zli −

k∑

l=1

nl∑

i=1

ezli ,

whereN =k∑

l=1

nl is the total number of units on life-test.


Likelihood Inference• We then have the likelihood equations as

∂ ln L(ν0, ν1, σ)

∂ν0= −

1

σ

(

N −

k∑

l=1

nl∑

i=1

ezli

)

= 0,

∂ ln L(ν0, ν1, σ)

∂ν1= −

1

σ

(

k∑

l=1

nlxl −

k∑

l=1

xl

nl∑

i=1

ezli

)

= 0,

∂ ln L(ν0, ν1, σ)

∂σ= −

1

σ

(

N +

k∑

l=1

nl∑

i=1

zli −

k∑

l=1

nl∑

i=1

zliezli

)

= 0,


Likelihood Inference• We then have the likelihood equations as

∂ ln L(ν0, ν1, σ)

∂ν0= −

1

σ

(

N −

k∑

l=1

nl∑

i=1

ezli

)

= 0,

∂ ln L(ν0, ν1, σ)

∂ν1= −

1

σ

(

k∑

l=1

nlxl −

k∑

l=1

xl

nl∑

i=1

ezli

)

= 0,

∂ ln L(ν0, ν1, σ)

∂σ= −

1

σ

(

N +

k∑

l=1

nl∑

i=1

zli −

k∑

l=1

nl∑

i=1

zliezli

)

= 0,

• The corresponding MLEs ofν0, ν1 andσ(denoted byν0, ν1 andσ, respectively) can beobtained by solving the above non-linearequations simultaneously by some numericalprocedures, such as the Newton-Raphson method.


Likelihood Inference• Let pl = nl

Nbe the proportion of units (of a total

of N units under test) to be assigned to stresslevelxl.


Likelihood Inference• Let pl = nl

Nbe the proportion of units (of a total

of N units under test) to be assigned to stresslevelxl.

• The Fisher information matrix in terms ofpl’s is

I(ν0, ν1, σ) =N

σ2I∗(ν0, ν1, σ)

=N

σ2

1k∑

l=1

xlpl (1 − γ)

k∑

l=1

xlpl

k∑

l=1

x2l pl (1 − γ)

k∑

l=1

xlpl

(1 − γ) (1 − γ)k∑

l=1

xlplπ2

6+ (1 − γ)2

.


Likelihood Inference• The asymptotic variance-covariance matrix of

(ν0, ν1, σ) is then

I−1 =

σ2

N

Q1 +6(1−γ)2

π2−Q2 −

6(1−γ)

π2

Q3 0

6π2

,

where

Q1 =

k∑

l=1

x2l pl

k∑

l=1

x2lpl −

(

k∑

l=1

xlpl

)2, Q2 =

k∑

l=1

xlpl

k∑

l=1

x2lpl −

(

k∑

l=1

xlpl

)2,

Q3 =1

k∑

l=1

x2lpl −

(

k∑

l=1

xlpl

)2.


OPTIMAL SAMPLE SIZE ALLOCATION

FOR ESTIMATION


For Estimation• We can now determine the optimal allocation

(p1, p2, . . . , pk−1) s.t. different optimality criteria.




• If we are interested in the estimation of the modelparametersν0, ν1 andσ, we may consideroptimality in terms of




• If we are interested in the estimation of the modelparametersν0, ν1 andσ, we may consideroptimality in terms of[1] D-optimality, wherein the determinant of the Fisher

information matrix is maximized, which results in

minimum volume for the Wald-type joint confidence

region for the model parameters(ν0, ν1, σ).




• If we are interested in the estimation of the modelparametersν0, ν1 andσ, we may consideroptimality in terms of[1] D-optimality, wherein the determinant of the Fisher

information matrix is maximized, which results in

minimum volume for the Wald-type joint confidence

region for the model parameters(ν0, ν1, σ).

[2] A-optimality, which minimizes the sum of the variances

of the parameter estimates and provides an overall

measure of variability from the marginal variabilities.


For Estimation[1] Maximization of the determinant of the Fisher

information matrix, viz.,

det[I∗] =π2

6

k∑

l=1

x2l pl −

(

k∑

l=1

xlpl

)2

.




det[I∗] =π2

6

k∑

l=1

x2l pl −

(

k∑

l=1

xlpl

)2

.

• This is equivalent to minimization of the function

D(p1, p2, . . . , pk−1) = −

k∑

l=1

x2l pl −

(

k∑

l=1

xlpl

)2

, with pk = 1 −

k−1∑

l=1

pl.




det[I∗] =π2

6

k∑

l=1

x2l pl −

(

k∑

l=1

xlpl

)2

.


D(p1, p2, . . . , pk−1) = −

k∑

l=1

x2l pl −

(

k∑

l=1

xlpl

)2

, with pk = 1 −

k−1∑

l=1

pl.

• This optimality criterion also minimizes var(ν1)since var(ν1) can be expressed as

var(ν1) =σ2

N

{

π2

6

1

det[I∗]

}

=σ2

N

k∑

l=1

x2l pl −

(

k∑

l=1

xlpl

)2

−1

.


For Estimation[2] Minimization of the trace of the

variance-covariance matrix (I−1) of the MLE’s,

viz.

tr[I∗−1] = 1 +6[(γ − 1)2 + 1]

π2+

1 +

(

k∑

l=1

xlpl

)2

k∑

l=1

x2lpl −

(

k∑

l=1

xlpl

)2.


For Estimation[2] Minimization of the trace of the

variance-covariance matrix (I−1) of the MLE’s,

viz.

tr[I∗−1] = 1 +6[(γ − 1)2 + 1]

π2+

1 +

(

k∑

l=1

xlpl

)2

k∑

l=1

x2lpl −

(

k∑

l=1

xlpl

)2.


T (p1, p2, . . . , pk−1) =

1 +

(

k∑

l=1

xlpl

)2

k∑

l=1x2

lpl −

(

k∑

l=1xlpl

)2, with pk = 1 −

k−1∑

l=1

pl.


Estimation – Two-stress Levels• Theorem 1.Whenk = 2, the functionD(p1) is

minimized atp∗1 = 12.




• Theorem 2.Whenk = 2, the functionT (p1) isminimized at

p∗1 =1

1 + d, whered2 =

x21 + 1

x22 + 1

.




• Theorem 2.Whenk = 2, the functionT (p1) isminimized at

p∗1 =1

1 + d, whered2 =

x21 + 1

x22 + 1

.

• Remark: Sincex1 < x2, d =√

x2

1+1

x2

2+1

< 1 and

consequentlyp∗1 = 11+d

> 12. This means that the

trace-optimal design allocates more units fortesting at the lower stress level than does thedeterminant optimal design.


Estimation – k-stress Levels• Theorem 3.The functionD(p1, . . . , pk−1) is

minimized atp∗1 = 12, p∗j = 0 for j = 2, . . . , k − 1,

andp∗k = 12.


Estimation – k-stress Levels• Theorem 3.The functionD(p1, . . . , pk−1) is

minimized atp∗1 = 12, p∗j = 0 for j = 2, . . . , k − 1,

andp∗k = 12.

• Theorem 4.The functionT (p1, p2, . . . , pk−1) isminimized at

p∗1 =1

1 + d, whered2 =

x21 + 1

x2k + 1

,

p∗j = 0 for j = 2, . . . , k − 1, andp∗k = d1+d

.


Estimation – k-stress Levels• Proof of Theorem 3: We havek − 1 equations,

of which the first equation is

−(x21 − x2

k) + 2(x1 − xk)

(

k∑

l=1

xlpl

)

= 0

=⇒

k∑

l=1

xlpl =x1 + xk

2.


Estimation – k-stress Levels• Proof of Theorem 3: We havek − 1 equations,

of which the first equation is

−(x21 − x2

k) + 2(x1 − xk)

(

k∑

l=1

xlpl

)

= 0

=⇒

k∑

l=1

xlpl =x1 + xk

2.

• Now, consider the first derivative of the functionD with respect topj (for j = 2, 3, . . . , k − 1)

∂D(p1, . . . , pk−1)

∂pj

= −(x2j − x2

k) + 2(xj − xk)

(

k∑

l=1

xlpl

)

= (xj − x1)(xk − xj) > 0.


Estimation – k-stress Levels• Hence,D(p1, . . . , pk−1) is a monotone increasing

function inpj for j = 2, . . . , k − 1.




• Since0 ≤ pj ≤ 1, D(p1, . . . , pk−1) is minimizedatp∗j = 0 for j = 2, 3, . . . , k − 1.




• Since0 ≤ pj ≤ 1, D(p1, . . . , pk−1) is minimizedatp∗j = 0 for j = 2, 3, . . . , k − 1.

• As a result, the only non-zeropj forj = 1, 2, . . . , k − 1 is p1 and from Theorem 1,p∗1 = 1

2 minimizes the functionD(p1, . . . , pk−1),and consequentlyp∗k = 1 − p∗1 = 1

2.


Estimation – k-stress Levels• In conclusion, in planning ak-stress level

experiment, we do not have to assign units at anystress level other than the lowest (x1) and thehighest (xk) levels.




• If optimality criterion [1] is considered, then oneshould allocate50% of the units to stress levelx1

and the remaining50% of the units to stress levelxk.




• If optimality criterion [1] is considered, then oneshould allocate50% of the units to stress levelx1

and the remaining50% of the units to stress levelxk.

• If optimality criterion [2] is considered, then oneshould allocate100p∗1% of the units to stress levelx1 and the remaining100(1− p∗1)% of the units to

stress levelxk, wherep∗1 = 11+d

with d2 = x2

1+1

x2

k+1

.


OPTIMAL SAMPLE SIZE ALLOCATION

FOR PREDICTION


For prediction• Our goal is to estimate a parameter

θ = ν0 + ν1x + hσ at a pre-specified stress levelx.




• Note that this general parameterθ can include thequantiles and the mean of the lifetime distributionat stress levelx.





• Specifically, ifθ is the mean, thenh = −γ; if θ istheq-th quantile, thenh = ln[− ln(1 − q)].





• Specifically, ifθ is the mean, thenh = −γ; if θ istheq-th quantile, thenh = ln[− ln(1 − q)].

• With this goal in mind, we may then be interestedin the optimal sample size allocation for theestimation ofθ (for a givenh).


For prediction• The MLE ofθ is given by

θ = ν0 + xν1 + hσ

with

var(θ) = var(ν0) + x2var(ν1) + h2var(σ)

+2x cov(ν0, ν1) + 2h cov(ν0, σ) + 2xh cov(ν1, σ).


For prediction• The MLE ofθ is given by

θ = ν0 + xν1 + hσ

with

var(θ) = var(ν0) + x2var(ν1) + h2var(σ)

+2x cov(ν0, ν1) + 2h cov(ν0, σ) + 2xh cov(ν1, σ).

• From the asymptotic variance-covariance matrix of theMLEs, we then have

N

σ2var(θ) = Q1 − 2xQ2 + x2Q3 +

6[(1 − γ) − h]2

π2

=

k∑

l=1

x2l pl − 2x

k∑

l=1

xlpl + x2

k∑

l=1

x2lpl −

(

k∑

l=1

xlpl

)2+

6[(1 − γ) − h]2

π2.


For predictionIn this case, optimality can simply be defined in termsof minimizing the variance of the MLE ofθ for apre-specified stress levelx, or equivalently,[3] Minimization of

V (p1, p2, . . . , pk−1) =

k∑

l=1

x2l pl − 2x

k∑

l=1

xlpl + x2

k∑

l=1

x2lpl −

(

k∑

l=1

xlpl

)2

with pk = 1 −k−1∑

l=1

pl.


Prediction – Two-stress Levels• Theorem 5.The functionV (p1) is minimized at

p∗1 =

x2−x(x1−x)+(x2−x) , if x < x1x2−xx2−x1

, if x1 < x < x2x−x2

(x−x1)+(x−x2), if x > x2,

andp∗2 = 1 − p∗1.



p∗1 =

x2−x(x1−x)+(x2−x) , if x < x1x2−xx2−x1

, if x1 < x < x2x−x2

(x−x1)+(x−x2), if x > x2,

andp∗2 = 1 − p∗1.• If the stress levelx is close tox1, p1 takes on

higher values (with the value of 1 atx1) while if xis close tox2, p1 takes on lower values (with thevalue of 0 atx2).



p∗1 =

x2−x(x1−x)+(x2−x) , if x < x1x2−xx2−x1

, if x1 < x < x2x−x2

(x−x1)+(x−x2), if x > x2,

andp∗2 = 1 − p∗1.• If the stress levelx is close tox1, p1 takes on

higher values (with the value of 1 atx1) while if xis close tox2, p1 takes on lower values (with thevalue of 0 atx2).

• As x tends to either−∞ or +∞, p1 tends to12 .


Prediction – k-stress Levels• Theorem 6.The functionV (p1, p2, . . . , pk−1) is

minimized at

p∗1 =

{

xk−x(x1−x)+(xk−x) , if x < x1,

x−xk

(x−x1)+(x−xk) , if x > xk.

p∗j = 0 for j = 2, . . . , k − 1, andp∗k = 1 − p∗1.


Prediction – k-stress Levels• Theorem 6.The functionV (p1, p2, . . . , pk−1) is

minimized at

p∗1 =

{

xk−x(x1−x)+(xk−x) , if x < x1,

x−xk

(x−x1)+(x−xk) , if x > xk.

p∗j = 0 for j = 2, . . . , k − 1, andp∗k = 1 − p∗1.

• For the case whenx1 ≤ x ≤ xk, any(p1, p2, . . . , pk−1) which satisfies the equationk

∑

l=1

xlpl = x will minimize V (p1, p2, . . . , pk−1).


NUMERICAL ILLUSTRATION


Numerical Illustration• Nelson and Meeker (1978) considered the

following data on insulating fluids in anaccelerated test that was conducted in order todetermine the relationship between time tobreakdown and voltage (which is the stressfactor).


Numerical Illustration• Nelson and Meeker (1978) considered the

following data on insulating fluids in anaccelerated test that was conducted in order todetermine the relationship between time tobreakdown and voltage (which is the stressfactor).

• 76 units were tested at seven stress levels, equallyspaced between 26kV and 38kV


Numerical IllustrationTime to breakdown for insulating fluids at seven stress levels

Stress ni Breakdown Times

26kV 3 5.79, 1579.52, 2323.70

28kV 5 68.85, 426.07, 110.29, 108.29, 1067.60

30kV 11 17.05, 22.66, 21.02, 175.88, 139.07, 144.12, 20.46, 43.40,194.90, 47.30,

7.74

32kV 15 0.40, 82.85, 9.88, 89.29, 215.10, 2.75, 0.79, 15.93, 3.91, 2.7,

0.69, 100.58, 27.80, 13.95, 53.24

34kV 19 0.96, 4.15, 0.19, 0.78, 8.01, 31.75, 7.35, 6.50, 8.27, 33.91,

32.52, 3.16, 4.85, 2.78, 4.67, 1.31, 12.06, 36.71, 72.89

36kV 15 1.97, 0.59, 2.58, 1.69, 2.71, 25.50, 0.35, 0.99, 3.99, 3.67,

2.07, 0.96, 5.35, 2.90, 13.77

38kV 8 0.47, 0.73, 1.40, 0.74, 0.39, 1.13, 0.09, 2.38

We havex1 = ln 26 = 3.2581, x2 = ln 28 = 3.3322, x3 = ln 30 = 3.4012, x4 = ln 32 =

3.4657, x5 = ln 34 = 3.5264, x6 = ln 36 = 3.5835 andx7 = ln 38 = 3.6376.


Numerical Illustration• For optimality criteria [1]:

At stress levelx1, n1 = 38At stress levelx7, n7 = 38


Numerical Illustration• For optimality criteria [1]:


• For optimality criterion [2], we would obtain

d =

√

(3.2581)2 + 1

(3.6376)2 + 1= 0.9034,

p∗1 =1

1 + d= 0.5254 and p∗7 = 0.4746.



Numerical Illustration• Prediction the mean lifetime at stress level at

20kV (x∗ = ln(20) = 2.99573):

p∗1 =3.63759 − 2.99573

(3.25810 − 2.99573) + (3.63759 − 2.99573)= 0.70984.



Numerical Illustration• Prediction the mean lifetime at stress level at

20kV (x∗ = ln(20) = 2.99573):

p∗1 =3.63759 − 2.99573

(3.25810 − 2.99573) + (3.63759 − 2.99573)= 0.70984.


• Prediction the mean lifetime at stress level at40kV (x∗∗ = ln(40) = 3.68888):

p∗1 =3.68888 − 3.63759

(3.68888 − 3.25810) + (3.68888 − 3.63759)= 0.10640



Numerical IllustrationComparison of the allocation scheme of Nelson and Meeker (1978) and the optimal allocations

(n1, n2, n3, n4) det[I∗](RE) (N/σ2)var(ν1) (N/σ2)tr[I∗−1](RE)

Example (3,5,11,15,19,15,8) .0154(0.259) 107.1449 1418.3173(0.253)

[1] (38,0,0,0,0,0,38) .0592(1.000) 27.7754 359.6750(0.997)

[2] (40,0,0,0,0,0,36) .0591(0.997) 27.8525 358.7541(1.000)

[3] (54,0,0,0,0,0,22) .0487(0.823) 33.7607 418.4271(0.857)

[4] (8,0,0,0,0,0,68) .0223(0.377) 73.7273 1029.6976(0.348)

(n1, n2, n3, n4) (N/σ2) var [θ|x∗] (RE) (N/σ2) var [θ|x∗∗] (RE)

Example (3,5,11,15,19,15,8) 28.4127(0.221) 5.3953(0.374)

[1] (38,0,0,0,0,0,38) 7.2853(0.863) 3.0191(0.669)

[2] (40,0,0,0,0,0,36) 7.0523(0.891) 3.1605(0.639)

[3] (54,0,0,0,0,0,22) 6.2853(1.000) 4.8826(0.414)

[4] (8,0,0,0,0,0,68) 28.3188(0.222) 2.0191(1.000)


MONTE CARLO STUDY FOR

SMALL SAMPLE SIZES


Monte Carlo Study for SmallSample Sizes

• As the results are based on asymptotic theory ofMLEs, Monte Carlo simulations are used toexamine that if the optimal allocations hold forsmall sample sizes (as small as 20 even).




• Under different settings, we simulated




• Under different settings, we simulated• The Fisher information matrix




• Under different settings, we simulated• The Fisher information matrix• The variance-covariance matrix of the MLEs

ν0, ν1 andσ.


Monte Carlo StudySetting (1):k = 2, ν0 = 0.0, ν1 = 1.0, σ = 1.0

Stress

-3.5 θ1 = E(Y |x = −3.5)

x1 = −2.5 -

-1.0 θ3 = E(Y |x = −1.0)

x2 = 0.5 -

1.5 θ2 = E(Y |x = 1.5)


Monte Carlo StudyComparison of different allocation schemes for setting (1), N = 20

(Relative Efficiency in parenthesis)

Asymptotic

(n1, n2) det[I] tr[I−1] var(θ1) var(θ2) var(θ3)

(12,8) 28424.5(0.96) .148(0.787) .192(0.880) .262(0.646) .083(0.975)

(10,10)[1][5] 29608.8(1.00) .130(0.894) .219(0.772) .219(0.772) .080(1.000)

(6,14)[2] 24871.4(0.84) .117(1.000) .335(0.506) .176(0.962) .090(0.894)

(16,4)[3] 18949.6(0.64) .246(0.474) .169(1.000) .482(0.351) .109(0.741)

(4,16)[4] 18949.6(0.64) .121(0.964) .482(0.351) .169(1.000) .109(0.741)

Simulated

(12,8) 60725.6(0.99) .156(0.773) .197(0.857) .280(0.627) .081(1.014)

(10,10)[1][5] 61543.3(1.00) .135(0.895) .234(0.721) .227(0.773) .082(1.000)

(6,14)[2] 51262.4(0.83) .121(1.000) .350(0.483) .184(0.958) .090(0.913)

(16,4)[3] 37331.6(0.61) .278(0.434) .169(1.000) .556(0.316) .120(0.685)

(4,16)[4] 39605.7(0.64) .125(0.969) .522(0.323) .176(1.000) .111(0.742)




Asymptotic


(25,15) 222066.1(0.94) .077(0.755) .094(0.903) .138(0.613) .042(0.960)

(20,20)[1][5] 236870.5(1.00) .065(0.894) .110(0.772) .110(0.772) .040(1.000)

(12,28)[2] 198971.2(0.84) .058(1.000) .167(0.506) .088(0.962) .045(0.894)

(32,8)[3] 151597.1(0.64) .123(0.474) .085(1.000) .241(0.351) .054(0.741)

(8,32)[4] 151597.1(0.64) .060(0.964) .241(0.351) .085(1.000) .054(0.741)

Simulated

(25,15) 315641.9(0.96) .080(0.739) .099(0.842) .144(0.607) .042(0.963)

(20,20)[1][5] 330256.9(1.00) .066(0.897) .113(0.737) .110(0.795) .040(1.000)

(12,28)[2] 280236.3(0.85) .059(1.000) .173(0.480) .088(0.988) .045(0.881)

(32,8)[3] 205492.5(0.62) .130(0.454) .083(1.000) .256(0.341) .056(0.715)

(8,32)[4] 214723.7(0.65) .063(0.942) .261(0.317) .087(1.000) .057(0.707)




Asymptotic


(35,25) 777231.4(0.97) .048(0.807) .065(0.863) .084(0.668) .027(0.983)

(30,30)[1][5] 799438.0(1.00) .043(0.894) .073(0.772) .073(0.772) .027(1.000)

(18,42)[2] 671527.9(0.84) .039(1.000) .112(0.506) .059(0.962) .030(0.894)

(48,12)[3] 511640.3(0.64) .082(0.474) .056(1.000) .161(0.351) .036(0.741)

(12,48)[4] 511640.3(0.64) .040(0.964) .161(0.351) .056(1.000) .036(0.741)

Simulated

(35,25) 971381.4(0.96) .049(0.820) .067(0.815) .087(0.669) .027(0.997)

(30,30)[1][5] 1010361.9(1.00) .045(0.907) .074(0.742) .075(0.776) .027(1.000)

(18,42)[2] 855098.1(0.85) .040(1.000) .114(0.480) .061(0.951) .031(0.864)

(48,12)[3] 619924.6(0.61) .088(0.460) .055(1.000) .174(0.332) .038(0.704)

(12,48)[4] 642403.0(0.64) .041(0.985) .166(0.330) .058(1.000) .036(0.737)


Monte Carlo StudySetting (2):k = 4, ν0 = 0.0, ν1 = 1.0, σ = 1.0

Stress

-2.5 θ1 = E(Y |x = −2.5)

x1 = −1.5 -

x2 = −0.5 -

x3 = 0.1 -

x4 = 0.66 -

-1.0 θ2 = E(Y |x = 1.5)



Asymptotic

(n1, n2, n3, n4) det[I] (RE) tr[I−1] (RE) var(θ1) (RE) var(θ2) (RE)

(5,5,5,5) 8426.01(0.549) 0.1714(0.772) 0.4549(0.475) 0.3362(0.561)

(7,3,3,7) 11233.50(0.732) 0.1517(0.872) 0.3501(0.617) 0.2817(0.670)

(10,0,0,10)[1] 15349.21(1.000) 0.1363(0.971) 0.2659(0.812) 0.2384(0.792)

(8,0,0,12)[2] 14735.24(0.960) 0.1323(1.000) 0.3158(0.684) 0.2101(0.899)

(15,0,0,5)[3] 11511.91(0.750) 0.1957(0.676) 0.2159(1.000) 0.4263(0.443)

(4,0,0,16)[4] 9823.49(0.640) 0.1563(0.847) 0.5789(0.373) 0.1888(1.000)

Simulated

(5,5,5,5) 17274.50(0.538) 0.1819(0.752) 0.4961(0.455) 0.3600(0.554)

(7,3,3,7) 23420.52(0.729) 0.1577(0.867) 0.3748(0.603) 0.2928(0.682)

(10,0,0,10)[1] 32111.97(1.000) 0.1449(0.943) 0.2826(0.799) 0.2538(0.786)

(8,0,0,12)[2] 31447.15(0.979) 0.1367(1.000) 0.3312(0.682) 0.2179(0.916)

(15,0,0,5)[3] 24531.99(0.764) 0.2102(0.651) 0.2259(1.000) 0.4626(0.431)

(4,0,0,16)[4] 20269.58(0.631) 0.1652(0.827) 0.6344(0.356) 0.1995(1.000)



Asymptotic


(10,10,10,10) 67408.08(0.549) 0.0857(0.772) 0.2275(0.475) 0.1681(0.561)

(15,5,5,15) 95419.34(0.777) 0.0742(0.892) 0.1659(0.651) 0.1361(0.692)

(20,0,0,20)[1] 122793.67(1.000) 0.0681(0.971) 0.1329(0.812) 0.1192(0.790)

(16,0,0,24)[2] 117881.92(0.960) 0.0662(1.000) 0.1579(0.684) 0.1050(0.897)

(30,0,0,10)[3] 92095.25(0.750) 0.0978(0.676) 0.1080(1.000) 0.2131(0.442)

(9,0,0,31)[4] 85648.58(0.698) 0.0746(0.887) 0.2599(0.415) 0.0942(1.000)

Simulated

(10,10,10,10) 96463.52(0.550) 0.0879(0.767) 0.2384(0.466) 0.1719(0.557)

(15,5,5,15) 133141.30(0.759) 0.0768(0.878) 0.1719(0.646) 0.1414(0.677)

(20,0,0,20)[1] 175329.44(1.000) 0.0690(0.977) 0.1371(0.810) 0.1200(0.798)

(16,0,0,24)[2] 166940.00(0.952) 0.0674(1.000) 0.1639(0.678) 0.1077(0.889)

(30,0,0,10)[3] 130971.65(0.747) 0.1024(0.658) 0.1111(1.000) 0.2249(0.426)

(9,0,0,31)[4] 122058.71(0.696) 0.0763(0.884) 0.2701(0.411) 0.0957(1.000)



Asymptotic


(15,15,15,15) 227502.28(0.549) 0.0571(0.772) 0.1516(0.475) 0.1121(0.560)

(20,10,10,20) 290766.44(0.702) 0.0514(0.858) 0.1212(0.594) 0.0963(0.653)

(30,0,0,30)[1] 414428.64(1.000) 0.0454(0.971) 0.0886(0.812) 0.0795(0.790)

(24,0,0,36)[2] 397851.49(0.960) 0.0441(1.000) 0.1053(0.684) 0.0700(0.897)

(46,0,0,14)[3] 296546.71(0.716) 0.0684(0.645) 0.0720(1.000) 0.1512(0.415)

(13,0,0,47)[4] 281351.00(0.679) 0.0504(0.875) 0.1793(0.401) 0.0628(1.000)

Simulated

(15,15,15,15) 288524.21(0.557) 0.0580(0.767) 0.1554(0.463) 0.1131(0.556)

(20,10,10,20) 365622.05(0.706) 0.0524(0.850) 0.1231(0.585) 0.0987(0.636)

(30,0,0,30)[1] 517988.16(1.000) 0.0461(0.966) 0.0893(0.806) 0.0819(0.768)

(24,0,0,36)[2] 502820.78(0.971) 0.0445(1.000) 0.1062(0.678) 0.0706(0.890)

(46,0,0,14)[3] 374893.37(0.724) 0.0696(0.639) 0.0720(1.000) 0.1545(0.407)

(13,0,0,47)[4] 351866.54(0.679) 0.0512(0.870) 0.1833(0.393) 0.0628(1.000)


Monte Carlo Study• The optimal allocation of units for optimality

criterion [1] actually increases the determinant ofthe Fisher information matrix and reduces thevariance ofν1 compared to all other allocation ofunits for sample sizes as small as 20.




• The optimal allocation of units for optimalitycriterion [2] similarly reduces the trace of thevariance-covariance matrix.




• The optimal allocation of units for optimalitycriterion [2] similarly reduces the trace of thevariance-covariance matrix.

• For largeN andk (k at least as large as 3 or 4),the difference between optimality criterion [1]and [2] is negligible (which means that either ofthese criteria can be used in practice, since theywill give nearly full efficiency).


Monte Carlo Study• The allocations based on optimality criterion [3]

can change dramatically depending on what levelof stress we are interested in predicting at, andthe corresponding relative efficiencies will alsocritically depend on those allocations.


Monte Carlo Study• The allocations based on optimality criterion [3]

can change dramatically depending on what levelof stress we are interested in predicting at, andthe corresponding relative efficiencies will alsocritically depend on those allocations.

• We have shown by means of Monte Carlosimulations that the optimal allocations hold notonly for large sample sizes but also for smallsample sizes (as small as 20 even).


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