View
1
Download
0
Category
Preview:
Citation preview
Adjusted scores for regression models
Euloge Clovis Kenne Paguikenne@stat.unipd.it
Department of Statistical SciencesUniversity of Padua
Joint work with Alessandra Salvan and Nicola Sartori
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 1 / 27
Outline
1 Background on MLE
2 Mean bias reduction
3 Median bias reduction
4 Applications
5 Conclusions
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 2 / 27
Basics
Parametric model with p-dimensional parameter θ and log likelihood
`(θ) based on sample of size n.
With θ = (θ1, . . . , θp), derivatives of `(θ):
. Ur= ∂`(θ)/∂θr , Urs = ∂2`(θ)/∂θr∂θs , r , s = 1, . . . , p;
. score U(θ) with components Ur ;
. observed information j(θ) with entry −Urs .
Expected values of derivatives of `(θ):
. expected information ı(θ) = Eθ{j(θ)} with entry ırs ;
. ırs entry of ı(θ)−1;
. νr ,st = Eθ(UrUst) and νr ,s,t = Eθ(UrUsUt).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 3 / 27
Basics
θ denotes the maximum likelihood estimator (MLE).
First-order properties of MLE:
. θ ∼Np(θ, ı(θ)−1)
. mean unbiasedness: Eθ(θ) = θ + O(n−1)
. (marginal) median unbiasedness: Pθ(θr < θr ) = 12 + O(n−1/2)
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 4 / 27
Mean bias correction
In regular parametric models, the bias of MLE has the form
Eθ(θ − θ) = b(θ) + O(n−2),
where b(θ) = −ı(θ)−1A∗(θ) = O(n−1), with
A∗r =1
2tr[ı(θ)−1{Pr + Qr}], r = 1, . . . , p,
where Pr = Eθ{U(θ)U(θ)TUr} and Qr = Eθ{−j(θ)Ur}
θBC = θ − b(θ) has bias of order O(n−2).
Asymptotically equivalent alternatives, such as bootstrap and
jackknife, can be obtained without b(θ).
Lack of equivariance under reparameterizations.
Available only if θ is finite.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 5 / 27
Mean bias reduction
Firth (1993), proposes a modification of the score vector of the form
U∗(θ) = U(θ) + A∗(θ) .
The modification term A∗(θ) is of order O(1).
The estimator θ∗, solution of U∗(θ) = 0 has a bias of order O(n−2)
and does not require finiteness of θ.
Notable examples: generalized linear models (glm) implemented in
the R package brglm2.
θ∗ ∼Np(θ, ı(θ)−1).
Still depends on parameterization.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 6 / 27
Median bias reduction
Kenne Pagui et al. (2017): a new modified score that
. does not require θ, as mean bias reduction;
. estimates all components of the parameter simultaneously;
. gives an estimator that is marginally third-order median unbiased for
each component;
. maintains (some) equivariance.
Development:
. Scalar parameter;
. Scalar parameter with nuisance parameters;
. Multidimensional parameter.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 7 / 27
Median bias reduction
Kenne Pagui et al. (2017): a new modified score that
. does not require θ, as mean bias reduction;
. estimates all components of the parameter simultaneously;
. gives an estimator that is marginally third-order median unbiased for
each component;
. maintains (some) equivariance.
Development:
. Scalar parameter;
. Scalar parameter with nuisance parameters;
. Multidimensional parameter.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 7 / 27
Median bias reduction: scalar parameter
For scalar θ, a median modification of the score is obtained using a
Cornish-Fisher expansion
U(θ) = U(θ) + A(θ) ,
where A(θ)=−Meθ{U(θ)}=νθ,θ,θ/{6ı(θ)} = O(1).
For continuous random variables and under weak regularity conditions
on the score, the solution of U(θ) = 0, θ, is third-order median
unbiased
Prθ(θ ≤ θ) = Prθ(U(θ) ≤ 0) =1
2+ O(n−3/2) .
θ is equivariant.
θ ∼N(θ, ı(θ)−1).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 8 / 27
Median bias reduction: scalar parameter
For scalar θ, a median modification of the score is obtained using a
Cornish-Fisher expansion
U(θ) = U(θ) + A(θ) ,
where A(θ)=−Meθ{U(θ)}=νθ,θ,θ/{6ı(θ)} = O(1).
For continuous random variables and under weak regularity conditions
on the score, the solution of U(θ) = 0, θ, is third-order median
unbiased
Prθ(θ ≤ θ) = Prθ(U(θ) ≤ 0) =1
2+ O(n−3/2) .
θ is equivariant.
θ ∼N(θ, ı(θ)−1).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 8 / 27
Toy example: normal model
y1, . . . , yn random sample from N(µ0, σ2), with µ0 known.
σ2 = σ2∗ = s(µ0)/n, with s(µ0) =∑n
i=1(yi − µ0)2.
The median modified score is
U(σ2) = −n−2/3
2σ2+
s(µ0)
2(σ2)2
which leads to σ2 = s(µ0)/(n−2/3).
σ2 is equal to the exact median unbiased estimator, s(µ0)/χ2n;0.5, plus
an error of order O(n−2).
Both σ2 and σ2 are equivariant; hence, for instance, for ω =√σ2, we
have ω = {σ2}1/2 and ω = {σ2}1/2.
On the other hand, ω∗ = {s(µ0)/(n−1/2)}1/2.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 9 / 27
Toy example: normal model
y1, . . . , yn random sample from N(µ0, σ2), with µ0 known.
σ2 = σ2∗ = s(µ0)/n, with s(µ0) =∑n
i=1(yi − µ0)2.
The median modified score is
U(σ2) = −n−2/3
2σ2+
s(µ0)
2(σ2)2
which leads to σ2 = s(µ0)/(n−2/3).
σ2 is equal to the exact median unbiased estimator, s(µ0)/χ2n;0.5, plus
an error of order O(n−2).
Both σ2 and σ2 are equivariant; hence, for instance, for ω =√σ2, we
have ω = {σ2}1/2 and ω = {σ2}1/2.
On the other hand, ω∗ = {s(µ0)/(n−1/2)}1/2.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 9 / 27
Scalar parameter with nuisance parameters
Let θ = (ψ, λ), with ψ a scalar parameter of interest.
Substitute λ with λψ and get the profile score UP(ψ) = Uψ(θψ).
Using a Cornish-Fisher expansion for the median of the standardized
profile score we obtain
UP(ψ) = UP(ψ)−κ1ψ(θψ) +1
6
κ3ψ(θψ)
κ2ψ(θψ),
where κ1ψ(·), κ2ψ(·) and κ3ψ(·) involve ırs , νr ,s,t and νr ,st , and are
the leading terms of the first three cumulants of UP(ψ).
UP(ψ) = 0 gives ψP which is
. third-order median unbiased,
. invariant with respect to interest preserving reparameterizations,
. asymptotically N(ψ, ıψψ).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 10 / 27
Scalar parameter with nuisance parameters
Let θ = (ψ, λ), with ψ a scalar parameter of interest.
Substitute λ with λψ and get the profile score UP(ψ) = Uψ(θψ).
Using a Cornish-Fisher expansion for the median of the standardized
profile score we obtain
UP(ψ) = UP(ψ)−κ1ψ(θψ) +1
6
κ3ψ(θψ)
κ2ψ(θψ),
where κ1ψ(·), κ2ψ(·) and κ3ψ(·) involve ırs , νr ,s,t and νr ,st , and are
the leading terms of the first three cumulants of UP(ψ).
UP(ψ) = 0 gives ψP which is
. third-order median unbiased,
. invariant with respect to interest preserving reparameterizations,
. asymptotically N(ψ, ıψψ).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 10 / 27
Multi-dimensional parameter
First idea: jointly solve system with “profile versions” for each
parameter, with all quantities evaluated at θ.
Second idea: profile score coincides with the efficient score
Uψ = Uψ − ıψλı−1λλUλ,
evaluated at θψ, indeed
UP(ψ) = Uψ(θψ)− Uψλ(θψ)Uλλ(θψ)−1Uλ(θψ) = Uψ(θψ) .
Idea: solve the profile versions system with efficient scores instead ofprofile scores (and all quantities evaluated at θ).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 11 / 27
Multi-dimensional parameter
First idea: jointly solve system with “profile versions” for each
parameter, with all quantities evaluated at θ.
Second idea: profile score coincides with the efficient score
Uψ = Uψ − ıψλı−1λλUλ,
evaluated at θψ, indeed
UP(ψ) = Uψ(θψ)− Uψλ(θψ)Uλλ(θψ)−1Uλ(θψ) = Uψ(θψ) .
Idea: solve the profile versions system with efficient scores instead ofprofile scores (and all quantities evaluated at θ).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 11 / 27
Multi-dimensional parameter
The vector of efficient scores has elements Ur =∑p
s=1(ırs/ırr )Us .
Define θ as the solution of
Ur = Ur + Mr = 0 (r = 1, . . . , p), where Mr = −κ1r +1
6
κ3rκ2r·
Properties:
. θr − θrP = Op(n−3/2) (r = 1, . . . , p);
. θr is third-order median unbiased, i.e. Prθ(θr ≤ θr ) = 12 + O(n−3/2);
. θ is equivariant under joint reparameterizations that transform each
component of θ separately;
. θ ∼Np(θ, ı(θ)−1).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 12 / 27
Multi-dimensional parameter
The vector of efficient scores has elements Ur =∑p
s=1(ırs/ırr )Us .
Define θ as the solution of
Ur = Ur + Mr = 0 (r = 1, . . . , p), where Mr = −κ1r +1
6
κ3rκ2r·
Properties:
. θr − θrP = Op(n−3/2) (r = 1, . . . , p);
. θr is third-order median unbiased, i.e. Prθ(θr ≤ θr ) = 12 + O(n−3/2);
. θ is equivariant under joint reparameterizations that transform each
component of θ separately;
. θ ∼Np(θ, ı(θ)−1).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 12 / 27
Computational aspects
The estimating equation can be reformulated in terms of an
adjustment to the original score:
U(θ) = U(θ) + A(θ), A(θ) = A∗(θ)− ı(θ)F (θ).
F (θ) has elements
Fr = [ı(θ)−1]Tr Fr (r = 1, . . . , p),
with Fr having components
Fr ,t = tr [hr{(1/3)Pt + (1/2)Qt}] (t = 1, . . . , p).
For both mean and median bias reduction, a quasi-Fisher scoring-type
algorithm has kth iteration
θ(k+1) = θ(k) + ı−1(θ(k))B(θ(k)) + ı−1(θ(k))U(θ(k)),
where B(θ) can be A∗(θ) or A(θ), respectively.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 13 / 27
Computational aspects
The estimating equation can be reformulated in terms of an
adjustment to the original score:
U(θ) = U(θ) + A(θ), A(θ) = A∗(θ)− ı(θ)F (θ).
F (θ) has elements
Fr = [ı(θ)−1]Tr Fr (r = 1, . . . , p),
with Fr having components
Fr ,t = tr [hr{(1/3)Pt + (1/2)Qt}] (t = 1, . . . , p).
For both mean and median bias reduction, a quasi-Fisher scoring-type
algorithm has kth iteration
θ(k+1) = θ(k) + ı−1(θ(k))B(θ(k)) + ı−1(θ(k))U(θ(k)),
where B(θ) can be A∗(θ) or A(θ), respectively.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 13 / 27
Logistic regression: endometrial cancer grade
Study designed to evaluate the relationship between the histology of
the endometrium of 79 patients and three risk factors: neovasculation
(NV), pulsatility index of arteria uterina (PI) and endometrium height
(EH) (Agresti, 2015, Section 5.7.1).
Maximum likelihood estimation leads to infinite estimate of βNV
(quasi-complete separation).
Both mean and median bias reduction provide a solution to the
problem of separation in logistic regression.
Bias reduction in generalized linear models (Kosmidis et al., 2018).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 14 / 27
Logistic regression: endometrial cancer grade (cont.)
Endometrial cancer grade: estimates (s.e.).intercept NV PI EH
β 4.305 (1.637) +∞ (+∞) -0.042 (0.044) -2.903 (0.846)
β 3.969 (1.552) 3.869 (2.298) -0.039 (0.042) -2.708 (0.803)
β∗ 3.775 (1.489) 2.929 (1.551) -0.035 (0.040) -2.604 (0.776)
Both β and β∗ are finite.
β is intermediate between β and β∗.
e βNV is third order median unbiased for eβNV while e β∗NV is not a bias
reduced estimate of eβNV .
R package: brglm2 (on CRAN).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 15 / 27
R package: brglm2 (on CRAN)
Maximun likelihood estimates> endometrial_ML <- glm(HG~NV+PI+EH,family=binomial,data=endometrial)
> endometrial_ML$coefficients
(Intercept) NV PI EH
4.3045178 18.1855558 -0.0421834 -2.9026056
Mean bias reduced estimates> endometrial_BR <- update(endometrial_ML,method="brglmFit",
type="AS_mean")
> endometrial_BR$coefficients
(Intercept) NV PI EH
3.77455971 2.92927335 -0.03475176 -2.60416392
Median bias reduced estimates> endometrial_MBR <- update(endometrial_ML,method="brglmFit",
type="AS_median")
> endometrial_MBR$coefficients
(Intercept) NV PI EH
3.96935983 3.86920663 -0.03867797 -2.70793447
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 16 / 27
Logistic regression: simulation
Simulate 10000 samples with covariates values fixed as in the original
sample.
True parameter β0 = (−1.19, 2.00,−0.39,−1.79).
About 28% of MLE of βNV are infinite, while β∗NV and βNV are
always finite.
Performance of the maximum likelihood (ML), mean bias reduction
(meanBR) and median bias reduction (medianBR) in terms of
estimated
. percentage of underestimation;
. bias;
. coverage of 95% Wald-type confidence intervals.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 17 / 27
Logistic regression: simulation
4446
4850
5254
56Percentage of underestimation
βint βNV βPI βEH
●
●
●
●
●
MLmeanBRmedianBR
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 18 / 27
Logistic regression: simulation
−0.
2−
0.1
0.0
0.1
0.2
0.3
Bias
βint βNV βPI βEH
●
●
●●
●
MLmeanBRmedianBR
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 18 / 27
Logistic regression: simulation
95.0
95.5
96.0
96.5
97.0
97.5
Coverages of 95% Wald CI
βint βNV βPI βEH
●
●
●
●
●
MLmeanBRmedianBR
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 18 / 27
Double index beta regression model
Y1, . . . ,Yn independent beta random variables with
fYi(yi ;µi , φi ) =
Γ(φi )
Γ(µiφi )Γ((1− µi )φi )yµiφi−1i (1− yi )
(1−µi )φi−1,
0 < yi < 1, 0 < µi < 1 and φi > 0.
Eθ(Yi ) = µi and Vθ(Yi ) = µi (1− µi )/(1 + φi ), so that φi is a
precision parameter.
Link functions connect the mean and precision with the linear
predictors g1(µi ) = xTi β and g2(φi ) = zTi γ, respectively.
xi = (xi1, . . . , xip)> and zi = (zi1, . . . , ziq)> are vectors of covariates.
Inference about θ = (β1, . . . , βp, γ1, . . . , γq)>.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 19 / 27
Simulation: a constant precision
Consider a logit link on the mean structure as
logµi
1− µi= β0 + β1xi1 + β2xi2, i = 1, . . . , n,
where the xi1 are n realizations of a standard normal and xi2 = logui ,
with ui generated from a uniform U(1, 2).
Simulate 10000 samples, with xi1 and xi2 held constant throughout
the replications.
Parameter values were fixed as β0 = 1.5, β1 = 0.5, β2 = 2 and
φ = 200.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 20 / 27
Simulation: a constant precision
3035
4045
5055
60Percentage of underestimation
βint β1 β2 φ
●
● ●
●
●
MLmeanBRmedianBR
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 21 / 27
Simulation: a constant precision
010
2030
4050
60
Bias
βint β1 β2 φ
● ● ● ●
●
MLmeanBRmedianBR
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 21 / 27
Simulation: a constant precision
8890
9294
96
Coverages of 95% Wald CI
βint β1 β2 φ
●
●●
●
●
MLmeanBRmedianBR
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 21 / 27
Reading skills data
Reading skills data from Smithson and Verkuilen (2006), also available
in the R package betareg.
The analysis is on the reading skills for nondyslexic and dyslexic
Australian children.
The children are aged between eight years and five months and twelve
years and three months.
The data consists of 44 observations of children of which 19
are dyslexic children.
The variables accuracy (the score on a reading accuracy test), iq (the
score on a nonverbal intelligent quotient test) and dyslexia (binary
variable on whether the child is dyslexic) were recorded.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 22 / 27
R function: mbrbetareg (on GitHub)
The mbrbetareg function produces the same summary output asbetareg.
> data("ReadingSkills", package = "betareg")
> rs_f <- accuracy ~ dyslexia * iq | dyslexia * iq
> rs_mbr<- mbrbetareg(rs_f, data = ReadingSkills, type = "medianBR")
> summary(rs_mbr)$coefficients
$mean
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.9962230 0.1507239 6.609589 3.853879e-11
dyslexia -0.6146817 0.1507239 -4.078197 4.538635e-05
iq 0.6999005 0.1338485 5.229048 1.703853e-07
dyslexia:iq -0.7776387 0.1338485 -5.809840 6.253246e-09
$precision
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.7486473 0.2566998 10.707633 9.372729e-27
dyslexia 1.6463254 0.2566998 6.413427 1.422845e-10
iq 1.2841753 0.2573632 4.989739 6.046090e-07
dyslexia:iq -0.7399326 0.2573632 -2.875052 4.039611e-03
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 23 / 27
Reading skills data
Model: log µi1−µi = β1 + β2dyslexiai + β3iqi + β4zi and
logφi = γ1 + γ2dyslexiai + γ3iqi + γ4zi , where the variable z
represents the interaction between the variables dyslexia and iq.
Reading skills data: estimates (s.e.).β1 β2 β3 β4 γ1 γ2 γ3 γ4
β 1.019 -0.638 0.690 -0.776 3.040 1.768 1.437 -0.611(0.145) (0.145) (0.127) (0.127) (0.258) (0.258) (0.257) (0.257)
β 0.996 -0.615 0.700 -0.778 2.749 1.646 1.284 -0.740(0.151) (0.151) (0.134) (0.134) (0.257) (0.257) (0.257) (0.257)
β∗ 0.985 -0.603 0.707 -0.784 2.721 1.634 1.281 -0.759(0.150) (0.150) (0.133) (0.133) (0.256) (0.256) (0.257) (0.257)
β is obtained from the R function mbrbetareg (on GitHub) while β
and β∗ are calculated from the R package betareg (on CRAN).
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 24 / 27
Reading skills data: simulation
3540
4550
55Percentage of underestimation
β1 β2 β3 β4 γ1 γ2 γ3 γ4
●
●
●
●
●●
●
●
●
MLmeanBRmedianBR
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 25 / 27
Reading skills data: simulation
−0.
050.
000.
050.
100.
150.
20
Bias
β1 β2 β3 β4 γ1 γ2 γ3 γ4
●
●●
● ●
●
●
●
●
MLmeanBRmedianBR
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 25 / 27
Reading skills data: simulation
8486
8890
9294
Coverages of 95% Wald CI
β1 β2 β3 β4 γ1 γ2 γ3 γ4
● ●●
●
●
●
●
●
●
MLmeanBRmedianBR
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 25 / 27
Closing
The proposed method is effective for median centering of components
of the estimator.
Gives a solution to boundary estimates by means of a model-based
penalization.
Wald confidence intervals have good coverage, but using
score statistic seems more promising.
Recover some global invariance (affine transformations)?
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 26 / 27
Main references
Agresti, A. (2015). Foundations of Linear and Generalized Linear
Models. Wiley,New York.
Firth, D. (1993). Bias reduction of maximum likelihood estimates.
Biometrika, 80, 27–38.
Kenne Pagui, E. C., Salvan, A. and Sartori, N. (2017). Median bias
reduction of maximum likelihood estimates. Biometrika, 104,
923–938.
Kosmidis, I., Kenne Pagui, E. C. and Sartori, N. (2018). Mean and
median bias reduction in generalized linear models. Statistics andComputing (accepted) http://arxiv.org/abs/1804.04085.
Smithson, M. and Verkuilen, J. (2006). A Better lemon squeezer?
maximum likelihood regression with beta-distributed dependent
variables. Psychological Methods, 11, 54–71.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 27 / 27
Median modification term
κ1r = −1
2
p∑s=1
p∑t=1
p∑u=1
ırsνtu(νs,tu + νs,t,u)/ırr , with νtu = ıtu − ıtr ıru/ırr ,
κ2r = 1/ırr ,
κ3r =
p∑s=1
p∑t=1
p∑u=1
ırs ırt ıruνs,t,u/(ırr )3.
Euloge Clovis Kenne (Univ. of Padua) Adjusted scores PRIN 2015 meeting, Padova 28 / 27
Recommended