a. h. el-shaarawi national water research institute and mcmaster university
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A. H. El-Shaarawi National Water Research Institute and McMaster University Southern Ontario Statistics, Graduate Student Seminar Days, 2006 McMaster University May 12, 2006. Outline. What is statistical science?. - PowerPoint PPT PresentationTRANSCRIPT
A. H. El-ShaarawiNational Water Research Institute and McMaster University
Southern Ontario Statistics, Graduate Student Seminar Days, 2006
McMaster University
May 12, 2006
Environmental Control and Statistical Science
Outline
1. What is Statistical Science?
2. What are the Sources of its foundations?
3. What are its Tools? 4. How do you become a successful statistician? 5. Some statistical issues related to environmental protection and control
What is statistical science?
A coherent system of knowledge that has its own methods and areas of applications.
The success of the methods is measured by their universal acceptability and by the breadth of the scope of their applications.
Statistics has broad applications (almost to all human activities including science and technology).
Environmental problems are complex and subject to many sources of uncertainty and thus statistics will have greater role to play in furthering the understanding of environmental problems.
The word “ENVIRONMETRICS” refers in part to Environmental Statistics
What are the Sources of the foundations? Concepts and abstraction. Schematization == Models Models and reality (deficiency in theory leads
to revision of models)
What are the Tools?
Philosophy “different schools of statistical inference”.
Mathematics. Science and technology.
How to become a successful statistician? Continue to upgrade your statistical knowledge. Improve your ability to perform statistical
computation. Be knowledgeable in your area of application. Understand the objectives and scope of the problem
in which you are involved. Read about the problem and discuss with experts in
relevant fields. Learn the art of oral and written communication. The
massage of communication is dependent on the interest of to whom the message is intended.
Environmental Problem
ionsCommunicatnsApplicatiosearch Re
Tools for:•Data Acquisition•Analysis & Interpretation•Modeling•Model Assessment
•Trend Analysis•Regulations•Improving Sampling Network•Estimation of Loading•Spatial & Temporal Change
•E Canada•H Canada•DFO•INAC•Provincial•EPA•International
Hazards Exposure Control
Data Acquisition
Data Analysis Empirical Models Process Models
Information
Prior Information
Modeling
Data
Time Space
Seasonal Trend Input-output Net-work
Error +Covariates
Measurements
Input System Output
Desirable Qualities of Measurements Effects Related Easy and Inexpensive Rapid Responsive and more Informative (high statistical power)
Burlington Beach
Applications:
1. Microbiological Regulations (Human health) Current U.S. Environmental Protection Agency (USEPA) guidelines for:
a) designated beaches specify a 30-day geometric mean and a single-sample sample maximum corresponding to the 75th percentile based on that 30-day mean [USEPA, 1986].
b) drinking water specify the arithmetic mean coliform density of all standard samples examined per month shall not exceed one per 100ml.
EPA recent workshop to establish Recreational Water Quality Criteria, Chapel Hill, North Carolina last February: Objective was not only to determine compliance but also to relate waterborne illness to bacteriological indicator’s density
2. Estimation of Chemical Concentrations and Loadings (Ecosystem Health)
Designing Sampling Program for Recreational Water (EC, EPA)
Sampling Grid for bathing beach water quality
Setting the regulatory limits: Select the indicators;Determine indicators illness association; Select indicators levelsThat corresponds to acceptable risk level.
Sampling Problems
Sampling Designs
Model based Design based Examples of sampling designs
1. Simple random sampling
2. Composite sampling
3. Ranked set sampling
Composite Sampling
Individual samples
Composite sample
Individual samples
Composite sample
Efficiency of Composite Sampling
Table 1 Density of frequently used distributions in the analysis of bacteriological data
Distribution
Normal
Log-normal
Poisson
Negative binomial
Density
)(xf
2
2
2
2
)(
x
e
22
2
))(ln(
2
2
2
x
ex
!x
e x
)()()()(
)(x
x
x
x
Mean
2/2e
Variance2
2
)1(22 e
+ 2/
Skewness3
3 /
0
1)2(22
ee
2/1 )(
2
Kurtosis4
4 /
3
332222 234 eee
13
2163
Efficiency for estimating the mean and variance of the distribution
Moments for the sample mean
344
233
2
)/()33()(;)/()(
/)(;)(
mkmkxmkx
mkxVarxE
.
Number of Composite samples = mNumber of sub-samples in a single C sample = k
Properties of the estimator of Variance:1. It is an unbiased estimator of regardless of the values taken by k and . The variance of this estimator is given by
This expression shows that for: , composite sampling improves the efficiency of as an estimator of regardless of the value of k and in this case the maximum efficiency is obtained for k =1 which corresponds to discrete sampling. , the efficiency of composite sampling depends only on m and is completely independent of k., the composite sampling results in higher variance and for fixed m the variance is maximized when k =1. It should be noted that the frequently used models to represent bacterial counts belong to case c above. This implies that the efficiency declines by composite sampling and maximum efficiency occurs when k = 1. Case b corresponds to the normal distribution where the efficiency is completely independent of the number of the discrete samples included in
the composite sample.
1
23
)1(
)}332()1{()( 4
42
mmkkmm
mkmmsVar
.
Health Survey
Summer of Mean Indicator Density –Swimming—Association Gastroenteritis Rate From Trails of All U.S. Studies
Location Beach1 Year E.coli
Density Enterococcus
Density Number
Swimmers Number Illnesses
Number Nonswimmers
Number Illnesses
Gastroenteritis Rate nor-1000
Lake Erie A 1979 23 5.2 3020 17.2 2349 14.9 2.3 B 47 13 2056 19.5 2349 14.9 4.6
Keystone Lake E 138 38.8 3059 20.6 970 15.5 5.1 W 19 6.8 2440 20 970 15.5 0.5
Lake Erie A 1980 137 25 2907 16.5 2944 11.7 4.8 B 236 71 2427 26.4 2944 11.7 14.7*
Keystone Lake E 52 23 5121 13.5 1211 8.1 5.2 W 71 20 3562 11.2 1211 8.1 3.0
Lake Erie B 1982 146 20 4374 24.9 1650 13.9 11.0*
1A=Beach 7 , B=Beach 11, E=Washington Irving Cove Beach, W=Salt Creek Cove—Keystone Ramp Beaches * Indicate swimmer-norswimmer illness rate difference significant at p=0.05 level
The effects of exposure to contaminated water
Surface water quality criteria (CFU/100mL) proposed by EPA for primary contact recreational use
Water Indicator Geometric Mean
Single Sample Maximum
Marine Enterococci 35 104
Fresh Enterococci
E. coli
33
126
61
235
Water Indicator Geometric Mean
Single Sample Maximum
Marine Enterococci 35 104
Fresh Enterococci
E. coli
33
126
61
235
Based on not less than 5 samples equally spaced over a 30-day
period.The selection of :IndicatorsSummary statistics, number of samples and the reporting periodControl limits
Approximate expression for probability of compliance with the regulations
Let
b and
a
)(
)(
g
where b is the geometric mean ; a is single sample maximum )( is the pdf of standard normal distribution
)( is the CDF of standard normal distribution
)())((1Pr ngnob
)()1
)((2Pr
2
n
gg
ganob
Sample size n=5 and 10 # of simulations =10000
Ratio of single sample rejection probability to that of the mean rule (n = 5,10 and 20)
nagprob
bXprob nn
1
)(1log
)(1
)(1log
)(
Modeling the Accumulation of Contaminants in Aquatic Environment
The fish (trout) contamination data:
1. Lake Ontario (n = 171); Lake Superior (61)2. Measurements (total PCBs in whole fish, age, weight, length, %fat)
– fish collected from several locations (representative of the population in the lake because the fish moves allover the lake)
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
0.5
L.ONT
PCB
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
L.Sup
PCB
2 4 6
PCBs
0.0
0.2
0.4
0.6
0.8
1.0
Cd
f
L.OntL.SupCrit ical Level
Let x(t) be a random variable representing the contaminant level in a fish at age t. The expected value of x(t) is frequently represented by the expression
where b is the asymptotic accumulation level and λ is the growth parameter. Note that 1 – exp(-λt) is cdf of E(λ ) and so an immediate generalization of this is
The expected instantaneous accumulation rate is f(t; λ)/F(t; λ).
One possible extension is to use the Weibull cdf
)exp(1)( tbt
,)( tbFt
)exp(1)( mtbt
1 2 3 4 5
02
04
06
08
01
00
PCB Concentration
14 years
7 years
3 years
Modeling: Consider a continuous time systems with a stochastic perturbations
)()( xxbdt
dx
with initial condition x(0) = x0, b(x) is a given function of x and t
σ(x) is the amplitude of the perturbation ξ = dw/dt is a white noise assumed to be the time derivative of a Wiener process
Examples for σ(x)=0 : 1. b(x) = - λx μ(x) = μ(0) exp(- λt ) (pure decay) 2. b(x) = λ{μ(0) - μ(x)} μ(x) = μ(0) {1- exp(- λt )} Bertalanffy equation
When σ(x) > 0, a complete description of the process requires finding the pdf f(t,x) and its moments given f(0,x).
d
ji
d
i
d
i
i
ji
ij Rxtx
fb
xx
fa
t
f
1, 1
2
,0,)()(
2
1
The density f (t,x) satisfies the Fokker-Planck equation or Kolmogorov forward equation
Where
d
kjkikij xxxa
1
)()()( . When d = 1 this equation simplifies to
x
bf
x
f
dt
df
)()(5.0
2
22
Multiplying by xn and integrating we obtain the moments equation
})({})({)1(5.0 122 nnn xxbnExxEnndt
dm
Clearly dm0/dt = 0 and dm1 /dt = E(b)
tt exVaretxVar
txtx
222
))0(()1(2
))((
)exp())0(())((
In the first example with b(x) = -λx and σ(x) = σ, we have
In the second example with b(x) = λ{B - μ(x)} and σ(x) = σ, we have
)()1(2
)(
},)1({2
}1{)(
222
2
2222
tetm
meBdt
dm
eBt
t
t
t
The Quasi Likelihood Equations
0)(1'
yV
and the variance of
11^^
)()(
VVaris
Upstream-Downstream Water Quality Monitoring Human and Ecosystem Health
Regulations and Control
S0 S1 S2 . . . Sk-1 Sk
Niagara River Overview of U-D M: Purpose, Design and
Examples
Univariate Series and Ratio (Trend & Seasonality)
Bivariate Series
Fraser River (BC)
b) Modeling: Semi Parametric Mixed Model Spline regression with AR error + random effects models.
Yijt = Pijm(t) + Sit + it + Zit i + it
where
Pjm= j0 + j1t + …+jmtm for tj-1 t tj j = 0, 1, . . ., J
~AR(p), ~ MVN(0, V) and it ~ N(0, i2)
The EM is used to obtain the ML estimates
0 100 200 300
0.0
00
.05
0.1
00
.15
day
Estim
ate
d C
on
ce
ntr
atio
n o
f T
P(m
g/L
)
Hope
Red PassHansardMargueriteHope
Hansard/Red Pass
0 100 200 300
51
01
52
02
53
03
5
day
Ra
tio
of th
e T
P C
on
ce
ntr
atio
n
0 100 200 300
1.0
1.5
2.0
2.5
3.0
day
Ra
tio o
f th
e T
P C
on
cen
tra
tion
Features of the Data: Period : 2 April 86 - Weekly (March 96) then biweekly Missing values Nondetects (contaminats) Explanatory variables (Flow & Solid concentration) Models a) Niagara River Individual Station
Y ~ Log-normal (x x , 2)
(Y-1)/ ~ N (x x , 2)
Two Stations
Ratio R = Y2/Y1
R ~ Log-normal (x x , 2) (R-1)/ ~ N (x x , 2)
Censoring Patterns for the Ratio of the Concentrations
Censoring Status at FE
Censoring Status at NOTL
Censoring Status for the Ratio
A1 : Y1>d1
A1 : Y1>d1
B1 : Y1d1
B1 : Y1d1
A2 : Y2>d2
B2 : Y2d2 A2 : Y2>d2
B2 : Y2d2
Y2/Y1 observed Y2/Y1d2/y1
Y2/Y1y2/d1
0 <Y2/Y1 <
Inference and the Profile Likelihood of The profile likelihood of is
PL() = L(, *(), *())
The relative profile likelihood of is
RPL() = PL()/PL(**)
Ratio of GEV Distributions
Example is Canadian Ecological Effects Monitoring (EEM) Program for Pulp Mills
Risk Identification Risk Assessment Risk Management
Objectives of Environmental Effects Monitoring Program: Does effluent cause an effect in the environment? Is effect persistent over time? Does effect warrant correction? What are the causative stressors? From 1992, all new effluent regulations require sites to do EEM. Pulp and Paper Pilot program
Modelling the Toxicity of Canadian Pulp and Paper Effluent on the Reproduction and Survival of Ceridaphnia:
Environmental Effects Monitoring
Environmental Effects Monitoring: Canadian Pulp and Paper Industry
Structure Data and Objective
survivalgrowthalAsurvivalgrowthLarvalonreproductiSurvival
SelanastumnowFatheaddubiaiaCeriodaphn
MillMillMillMill
Tests
IndustryPaperandPulp
Cycles
Ii
Ii
lg
.3min.2.1
321
321
21
21
Example of data (daphnia survival and reproduction)
No. of neonates produced per replicates and total female adult mortality
100000000000100
9000101000050
12642222510
525
210
13
610
13
10
611
10
012.5
016
12
11
15
915
12
14
10
10
6.25
023
14
16
22
10
22
11
13
15
28
3.13
1020
19
16
13
16
17
14
14
28
1.56
045
44
40
41
38
46
49
35
39
50
0
Mortality
10
987654321%Effluent
Example of reproduction data (one cycle)
0 20 40 60 80 100
Dilut ion Factor
01
02
03
04
05
0
Co
un
ts
Exp1Exp2Exp3All
Accounting for over dispersion
First Source
)(~| PoisN
)exp(z Frequently
)(~ kgamma leads to negative binomial distribution. Less frequently
),0(log~ normal , gives
0
2
)1()1(
}2/)1)((exp{)1()(
n
nnnNp
n
Some observations: No closed form for the likelihood is available so simulation is the most
convenient approach for the maximization of the likelihood function.
Given ork , quasi likelihood is available for estimating the regression
parameters and this could be followed by the method of moments for estimating the dispersion parameters.
Given the equality of the first two moments of it is easy to show the third and fourth central moments under the gamma assumption are less than those under the lognormal model.
Second Source
)1(Iprob
!/)))((exp))(()1()(
var
)0,|()1()1,|()|(
)(
0
1
smmm
sSprob
NSiablesmofsumConsider
IxnNgIxnNgxnNg
ds
s
ds
mm
m
ii
)()(
)())(1()(
)1()(2
SESVar
SEmSVar
mSE
ds
ds
Animal
Alive Dead
Young No Young Young No Young
Sample mean vs geometric mean
),(~ 2NX , then )exp(XY ~ log-normal distribution The th moment of Y is 2/)(exp{)}{exp()( 2 XEYE
The ratio of MSE for the arithmetic mean to the geometric mean is
)exp(2/)1(exp2)/2exp(
)exp()2exp(222
22
nnnn
R
0.0 0.5 1.0 1.5 2.0 2.5 3.0
tau^2
02
46
810
R
The Maximum Likelihood Estimator
Consider the model )exp( iii aY
The MLE of
)exp()exp(),( 210
21010 a
is
),( 10
^
0 )exp(^
21
^
0
))/21log(exp()),(( 212
2202
1010
^
ncE f , (1)
where pnf and aAAac 1)( .
This expression immediately shows that ),( 10
^
is: 1. Asymptotically unbiased and consistent. 2. Positively biased that is overestimates ),( 10 for finite n.
3. Defined only for .2 21n
The last property particularly shows the serious limitation of the ML estimator since its expectation does not exist when does not satisfy (3) and so the ML estimator in this case is meaningless. For finite samples MLE > largest observation
Some simulation results (MLE)
22 5 10 20 30 50
0.10 0.0006 0.0004 0.0002 0.0001 0.0001 0.0000
0.50 0.0190 0.0108 0.0058 0.0030 0.0020 0.0012
1.00 0.1014 0.0474 0.0244 0.0124 0.0083 0.0050
2.00 - 0.2481 0.1098 0.0522 0.0343 0.0203
3.00 - 0.8825 0.2905 0.1263 0.0808 0.0470
Table 1:Relative Bias in the ML estimator for different n and 2
n
n 2
2 5 8 12 20 30 50
0.01 0.030 0.012 0.008 0.005 0.003 0.002 0.001 0.10 0.311 0.129 0.081 0.054 0.033 0.022 0.013 0.50 2.936 0.909 0.560 0.373 0.224 0.149 0.090 1.00 NA 3.798 1.752 1.072 0.617 0.406 0.242 2.00 NA NA 29.373 5.364 2.189 1.316 0.749 3.00 NA NA NA 109.194 6.910 3.136 1.588
Table 2 Skewness
MLE has a heavy right tail distribution (skewed to the right)
n 2
2 5 8 12 20 30 50
0.01 -1.959 -1.984 -1.990 -1.993 -1.996 -1.997 -1.998 0.10 -1.458 -1.807 -1.883 -1.924 -1.955 -1.970 -1.982 0.50 NA 0.642 -0.764 -1.297 -1.630 -1.770 -1.869 1.00 NA 83.616 7.089 1.481 -0.576 -1.212 -1.594 2.00 NA NA NA 154.729 11.447 2.850 -0.118 3.00 NA NA NA NA 274.759 26.896 4.703
Table 3 Kurtosis
MLE has heavy tails and sharp central part for kurtosis>0 while tails are lighter and the central part is flatter for kurtosis<0
UMVU Estimator
UMVU for is
0
^22^~
!2
1)exp(
j jj
jj
jb
ca
where aAAac 1)( and )2/)((/)2/)2(()(2 pnjpnpnb jjj .
j
i iji
j
j jj
jjj
bb
bij
jb
ccaVar
0
2
0
222
~
!2
1))(22exp()(
The mean square error is used to compare those estimators, that is
2^^
)()( BVarMSE and )()(~~
VarMSE .
UMVU : Closed form expression for n=2m-1
Theorem: )(^
nng the MLE based on df of )exp()( 2 h satisfies the recurrence relation
2
^
2
^
2 )(2
)(
n
nnn
nn gd
dng
where 2/2nn ns and 2
ns is the MLE of 2 based on n df.
It is easy to show that )2cosh()( 11
^
1 g and
12|)2cosh()(
1
1
12
^
12
mm
m
mmd
dg
For n =1, 3, 5 and 7 are
25.2
2/3
/)2cosh(3/)25.1)(2sinh(158/)2sinh(34/)2cosh(3
2/)2sinh()2cosh(
For sample of size 2 MLE reduces to the sample mean
UMVU: n even
For n =2m we have to start the recurrence relation
)()!(
)2/()( 0
^
022 I
ig
i
i
Where I0 (z) is the modified Bessel function of order 0 which can be written in the form
0
0 )cos(exp1
)( dzzI .
Approximation of this function will be needed to start the recurrence relation.
Modified Estimator
Modified Estimator for the Sample Mean We consider an estimator of the form 5.0,exp 2 sxm Select that minimizes MSE. This leads to
2
)1(
)3(611
6exp s
nn
nnxm
Note that as n ,^
m
0.0 0.1 0.2 0.3 0.4 0.5
c
0.2
0.4
0.6
0.8
1.0
RM
SE
n = 5
s= .01s= .10
s= .50
s= 1.00
s= 1.50
0.0 0.1 0.2 0.3 0.4 0.5
c
0.0
0.2
0.4
0.6
0.8
1.0
RM
SE
n = 10
s= .01s= .10
s= .50
s= 1.00
s= 1.50
s= 2.00s= 2.50s= 3.00
0.0 0.1 0.2 0.3 0.4 0.5
c
0.5
1.0
1.5
RM
SE
n = 20
s= .01s= .10s= .50
s= 1.00s= 1.50s= 2.00
s= 2.50s= 3.00
0.0 0.1 0.2 0.3 0.4 0.5
c
12
3
RM
SE
n = 40
s= .01
s= .1
s= .5s= 1.0
1.5
s= 2.5
s= 2
s= 3
Sample mean vs geometric mean
),(~ 2NX , then )exp(XY ~ log-normal distribution The th moment of Y is 2/)(exp{)}{exp()( 2 XEYE
The ratio of MSE for the arithmetic mean to the geometric mean is
)exp(2/)1(exp2)/2exp(
)exp()2exp(222
22
nnnn
R
0.0 0.5 1.0 1.5 2.0 2.5 3.0
tau^2
02
46
810
R
The Maximum Likelihood Estimator
Consider the model )exp( iii aY
The MLE of
)exp()exp(),( 210
21010 a
is
),( 10
^
0 )exp(^
21
^
0
))/21log(exp()),(( 212
2202
1010
^
ncE f , (1)
where pnf and aAAac 1)( .
This expression immediately shows that ),( 10
^
is: 1. Asymptotically unbiased and consistent. 2. Positively biased that is overestimates ),( 10 for finite n.
3. Defined only for .2 21n
The last property particularly shows the serious limitation of the ML estimator since its expectation does not exist when does not satisfy (3) and so the ML estimator in this case is meaningless. For finite samples MLE > largest observation
Some simulation results (MLE)
22 5 10 20 30 50
0.10 0.0006 0.0004 0.0002 0.0001 0.0001 0.0000
0.50 0.0190 0.0108 0.0058 0.0030 0.0020 0.0012
1.00 0.1014 0.0474 0.0244 0.0124 0.0083 0.0050
2.00 - 0.2481 0.1098 0.0522 0.0343 0.0203
3.00 - 0.8825 0.2905 0.1263 0.0808 0.0470
Table 1:Relative Bias in the ML estimator for different n and 2
n
n 2
2 5 8 12 20 30 50
0.01 0.030 0.012 0.008 0.005 0.003 0.002 0.001 0.10 0.311 0.129 0.081 0.054 0.033 0.022 0.013 0.50 2.936 0.909 0.560 0.373 0.224 0.149 0.090 1.00 NA 3.798 1.752 1.072 0.617 0.406 0.242 2.00 NA NA 29.373 5.364 2.189 1.316 0.749 3.00 NA NA NA 109.194 6.910 3.136 1.588
Table 2 Skewness
MLE has a heavy right tail distribution (skewed to the right)
n 2
2 5 8 12 20 30 50
0.01 -1.959 -1.984 -1.990 -1.993 -1.996 -1.997 -1.998 0.10 -1.458 -1.807 -1.883 -1.924 -1.955 -1.970 -1.982 0.50 NA 0.642 -0.764 -1.297 -1.630 -1.770 -1.869 1.00 NA 83.616 7.089 1.481 -0.576 -1.212 -1.594 2.00 NA NA NA 154.729 11.447 2.850 -0.118 3.00 NA NA NA NA 274.759 26.896 4.703
Table 3 Kurtosis
MLE has heavy tails and sharp central part for kurtosis>0 while tails are lighter and the central part is flatter for kurtosis<0
UMVU Estimator
UMVU for is
0
^22^~
!2
1)exp(
j jj
jj
jb
ca
where aAAac 1)( and )2/)((/)2/)2(()(2 pnjpnpnb jjj .
j
i iji
j
j jj
jjj
bb
bij
jb
ccaVar
0
2
0
222
~
!2
1))(22exp()(
The mean square error is used to compare those estimators, that is
2^^
)()( BVarMSE and )()(~~
VarMSE .
UMVU : Closed form expression for n=2m-1
Theorem: )(^
nng the MLE based on df of )exp()( 2 h satisfies the recurrence relation
2
^
2
^
2 )(2
)(
n
nnn
nn gd
dng
where 2/2nn ns and 2
ns is the MLE of 2 based on n df.
It is easy to show that )2cosh()( 11
^
1 g and
12|)2cosh()(
1
1
12
^
12
mm
m
mmd
dg
For n =1, 3, 5 and 7 are
25.2
2/3
/)2cosh(3/)25.1)(2sinh(158/)2sinh(34/)2cosh(3
2/)2sinh()2cosh(
For sample of size 2 MLE reduces to the sample mean
UMVU: n even
For n =2m we have to start the recurrence relation
)()!(
)2/()( 0
^
022 I
ig
i
i
Where I0 (z) is the modified Bessel function of order 0 which can be written in the form
0
0 )cos(exp1
)( dzzI .
Approximation of this function will be needed to start the recurrence relation.
Modified Estimator
Modified Estimator for the Sample Mean We consider an estimator of the form 5.0,exp 2 sxm Select that minimizes MSE. This leads to
2
)1(
)3(611
6exp s
nn
nnxm
Note that as n ,^
m
0.0 0.1 0.2 0.3 0.4 0.5
c
0.2
0.4
0.6
0.8
1.0
RM
SE
n = 5
s= .01s= .10
s= .50
s= 1.00
s= 1.50
0.0 0.1 0.2 0.3 0.4 0.5
c
0.0
0.2
0.4
0.6
0.8
1.0
RM
SE
n = 10
s= .01s= .10
s= .50
s= 1.00
s= 1.50
s= 2.00s= 2.50s= 3.00
0.0 0.1 0.2 0.3 0.4 0.5
c
0.5
1.0
1.5
RM
SE
n = 20
s= .01s= .10s= .50
s= 1.00s= 1.50s= 2.00
s= 2.50s= 3.00
0.0 0.1 0.2 0.3 0.4 0.5
c
12
3
RM
SE
n = 40
s= .01
s= .1
s= .5s= 1.0
1.5
s= 2.5
s= 2
s= 3