Metodološkizvezki,Vol. 14,No. 1,2017,49–59
PretestShrinkageEstimatorsfortheShape
ParameterofaParetoModelusingPriorPoint
KnowledgeandRecordObservations
LeilaBarmoodehandMehranNaghizadehQomi
1
Abstract
Considering a Pareto model with unknown shape and scale parameters α and
β, respectively, we are interested in Thompson shrinkage test estimation for the
shape parameter α under the Squared Log Error Loss (SLEL) function. We find
a risk-unbiased estimator for α and compute its risk under the SLEL. According to
Thompson(1986),weconstructthepretestshrinkage(PTS)estimatorsforαwiththe
help of a point guess value α
0
and record observations. We investigate the risk-bias
oftheseestimatorsandcomputetheirrisksnumerically. Acomparisonisperformed
between the PTS estimators and a risk-unbiased estimator. A numerical example is
presentedforillustrativeandcomparativepurposes. Weendthepaperbydiscussion
andconcludingremarks.
1 Introduction
In many situations, we have a point guess value regarding the parameter of interest from
pastinvestigationsoranyothersourceswhatsoever,whichisconsideredasnonsamplein-
formationoruncertainpriorinformation. Thompson(1968)proposedlinearpointshrink-
age estimators by combining sample information and nonsample information by moving
theunbiasedestimatorclosertoapointguessvalueinthehopethatitwillperformbetter
thantheunbiasedestimator.
Many researchers have considered the problem of shrinkage estimation, see Pandey
and Singh (1980) and Singh et al. (1996) among others. Pretest estimators may be con-
structedforincorporatingapretestonguessvalue,whenthepriorknowledgeisnottrust-
worthy. Pandey and Singh (1993) proposed shrinkage pretest estimators for the Weibull
shape parameter. Baklizi (2005) developed a pretest estimator for the exponential scale
parameter. Prakash and Singh (2007) and Prakash and Singh (2008) dealt with shrink-
age pretest estimation under the LINEX loss in Pareto and exponential distribution, re-
spectively. New researches are in works by Belaghi et al.(2015), Naghizadeh Qomi and
Barmoodeh(2015)andKiapourandNaghizadehQomi(2016).
ArandomvariableX issaidtohaveaParetodistribution,denotedbyX ∼ Par(α,β),
ifitscumulativedistributionfunction(cdf)is
F(x;α;β) = 1−

β
x

α
, x > β, α > 0, β > 0,
1
DepartmentofStatistics,UniversityofMazandaran,Babolsar,Iran;m.naghizadeh@umz.ac.ir

50 L.BarmoodehandM.NaghizadehQomi
andtheprobabilitydensityfunction(pdf)is
f(x;α;β) = αβ
α
x
−(α+1)
, x > β, α > 0, β > 0. (1.1)
Inthispaper,weareinterestedintheconstructionofPTSestimatorsbasedonrecords
fromParetodistributionundertheSLELintroducedbyBrown(1968)oftheform
L(α,δ) = (lnδ−lnα)
2
=

ln
δ
α

2
,
whereδ is an estimator ofα. This loss is convex for Δ =
δ
α
≤ e and concave otherwise,
but has a unique minimum at Δ = 1. Also when Δ > 1, this loss increases sublinearly,
while when 0 < Δ < 1, it rises rapidly to infinity at zero. The SLEL function is useful
insituationswhereunderestimationismoreseriousthanoverestimation; seeSanjariFar-
sipour and Zakerzadeh (2005), Kiapour and Nematollahi (2011) and Naghizadeh Qomi
andBarmoodeh(2015).
The paper is organized as follows. In section 2, we present the form of data and give
the maximum likelihood estimator (MLE) of α and β. A risk-unbiased estimator of α
undertheSLELisobtainedinsection3. ThePTSestimatorsareobtainedandtheirrisks
are computed under the SLEL in section 5. A comparsion study between PSE and RUE
is performed in section 5. An illustrated example is presented in section 6. We conclude
insection7withasummaryofourfindingsandsomeremarks.
2 Record-breakingData
Considerasequence{X
i
, i≥ 1}ofindependentandidenticallydistributed(iid)continu-
ousrandomvariableshavingacdfF andapdff. AnobservationX
j
willbecalledtobea
lowerrecordvalueifitsvalueissmallerthanallpreviousobservationsX
1
,X
2
,...,X
j−1
.
By conventionX
1
is the first lower record value. An analogous definition deals with up-
perrecords. Suchdatamayberepresentedby(R,K) := (R
1
,k
1
,...,R
m
,k
m
),whereR
i
is
thei-threcordvaluemeaningnewminimum(ormaximum)andk
i
isthenumberoftrials
following the observation of R
i
that are needed to obtain a new record value R
i+1
, see
Doostparast and Balakrishnan (2012). Chandler (1952) began studing the distributions
of lower records for iid random variables. Records and their properties have been exten-
sively studied in literature, see Arnold et al. (1998) and the references therein for more
detailsonapplicationsofrecords.
Consider a sequence of independent random variables X
1
,X
2
,X
3
,... drawn from a
pdf f(.) and items are presented sequently and sampling is terminated when the mth
minimum is observed. We assume that only successive minima are observable, so that
the observed value may be represented as (r,k) := (r
1
,k
1
,r
2
,k
2
,...,r
m
,k
m
), where r
i
is the value of the ith observed minimum, and k
i
is the number of trials required to
obtain the next new minimum. The likelihood function associated with the sequence
r
1
,k
1
,...,r
m
,k
m
isoftheform
L(r,k) =
m
Y
i=1
f(r
i
)[1−F(r
i
)]
k
i
−1
I(−∞,r
i−1
), (2.1)

PretestShrinkageEstimators... 51
wherer
0
≡∞,k
m
≡ 1andI(A)istheindicatorfunctionofthesetA. Thetotalnumber
ofitemssampledisarandomnumber,andk
m
isdefinedtobeoneforconvenience. Con-
sideringthesequenceR
1
,k
1
,...,R
m
,k
m
iscomingfromPar(α,β)in(1.1),thelikelihood
functionin(2.1)basedon (R
1
,k
1
,...,R
m
,k
m
)at (r
1
,k
1
,r
2
,k
2
,...,r
m
,k
m
)isgivenby
L(α,β|r,k) =
α
m
β
α
P
m
i=1
k
i
Q
m
i=1
r
αk
i
+1
i
, 0 < β≤ r
m
, α > 0.
Aftersomesimplealgebraiccalculations,themaximumlikelihoodofβandαare
ˆ
β = R
m
and ˆ α =
m
Tm
respectively, where T
m
=
P
m−1
i=1
k
i
log
R
i
Rm
which is distributed as Gamma
(m−1,α
−1
)orequivalently,2αT
m
∼ χ
2
2(m−1)
,seeDoostparastandBalakrishnan(2012).
3 ARisk-unbiasedEstimator
Lehmann (1951) provided the concept of risk-unbiased estimator. An estimatorδ ofα is
saidtoberiskunbiasedifitsatisfies
E[L(α,δ)]≤ E[L(α
0
,δ)], ∀α
0
6= α.
FormtheSLELsetting,wehave
E

ln
2
δ
α

−E

ln
2
δ
α
0

= (ln
2
α−ln
2
α
0
)−2(lnα−lnα
0
)E[lnδ].
IfweconsiderE[lnδ] = lnα,weconcludethat
E

ln
2
δ
α

−E

ln
2
α
α
0

=−(lnα−lnα
0
)
2
< 0.
Therefore,anestimatorδ ofαisrisk-unbiasedundertheSLELifitsatisfiesinthecondi-
tion E[lnδ] = lnα or equivalently E[ln(δ/α)] = 0. Note that, if E[ln(δ/α)] > 0(< 0),
thentheestimatorδ ofαispositively(negatively)risk-biased.
The following lemma is useful for deriving a risk-unbiased estimator of α under the
SLEL.
Lemma3.1. LetY ∼ χ
2
2a
, Γ(a)denotesthecompletegammafunctiongivenby
Γ(a) =
Z
∞
0
t
a−1
e
−t
dt,
Ψ(a) =
d
da
Γ(a) is the digamma function, and Ψ
0
(.) is the trigamma function which is
definedas Ψ
0
(a) =
d
da
Ψ(a). Thenwehave
(i) E[lnY] = ln2+Ψ(a).
(ii) E[ln
2
Y] = [ln2+Ψ(a)]
2
+Ψ
0
(a).
Proof. Foraproof,seeNaghizadehQomi(2017).
Inthefollowingtheorem,wefindarisk-unbiasedestimatorofαbasedon ˆ α.

52 L.BarmoodehandM.NaghizadehQomi
Theorem 3.2. The estimator ˆ α
u
= d
1
ˆ α, where d
1
= m
−1
exp(Ψ(m− 1)), is a risk-
unbiasedestimatorforαundertheSLELanditsriskisR(α, ˆ α
u
) = Ψ
0
(m−1).
Proof. Byassuminga = m−1andY = 2αT
m
∼ χ
2
2(m−1)
inLemma3.1.i,wehave
E[ln ˆ α
u
] = E

ln

d
1
m
T
m

= lnd
1
+lnm+E

ln

2α
Y

= Ψ(m−1)−E[ln(Y)]+ln2α
= lnα.
Alsotheriskof ˆ α
u
is
R(α, ˆ α
u
) = E

ln
2

d
1
ˆ α
α

= E

ln
2

2md
1
Y

= (ln
2
(2md
1
))+E[ln
2
Y]−2{ln(2md
1
)}E[lnY]
= Ψ
0
(m−1).
4 PretestShrinkageEstimation
4.1 ShrinkageEstimators
We consider the following Thompson shrinkage estimators for the parameter α using a
priorpointguessα
0
ofαas
ˆ α
s
= lˆ α
u
+(1−l)α
0
, (4.1)
wherel∈ [0,1]denotestheshrinkagefactor. Thevalueof(1−l)maybeassignedbythe
experimenteraccordingtoconfidenceinthepriorvalueofα
0
. Ideally,thecoefficientl is
chosen to minimize the risk of the estimator (4.1), see Ahmed (1992). The risk of (4.1)
undertheSLELisgivenby
R(α, ˆ α
s
) = E

ln
2

lˆ α
u
+(1−l)α
0
α

= E

ln
2

2mld
1
Y
+(1−l)α
?

=
Z
∞
0
ln
2

2mld
1
y
+(1−l)α
?

g(y)dy, (4.2)
whereα
?
= α
0
/αandg(y)isthepdfofY = 2αT
m
∼ χ
2
2(m−1)
.
4.2 PTSEstimatorsandtheirRisks
For checking the guess α
0
is close to α, a pretest H
0
: α = α
0
versus H
0
: α 6= α
0
is
performed. We can construct our PTS estimators based on acceptance or rejection of the

PretestShrinkageEstimators... 53
nullH
0
. Thegeneralformoftheproposedestimatorsislˆ α
u
+(1−l)α
0
, ifH
0
: α = α
0
isacceptedor ˆ α
u
,otherwise. IfH
0
: α = α
0
isacceptedatthelevelofγ,thenwehave
Pr

q
1
≤ 2α
0
T
m
≤ q
2

= 1−γ.
whereq
1
= χ
2
γ/2,2(m−1)
andq
2
= χ
2
1−γ/2,2(m−1)
are left quantiles of the chi-square distri-
butionwith 2(m−1)degreesoffreedom. Therefore,theproposedPTSestimatorcanbe
writtenas
ˆ α
st
= (lˆ α
u
+(1−l)α
0
)I(t
1
≤ T
m
≤ t
2
)+ ˆ α
u
I(T
m
< t
1
or T
m
> t
2
) (4.3)
wheret
1
= q
1
/2α
0
andt
2
= q
2
/2α
0
. Therisk-biasofthePTSestimatorundertheSLEL
isgivenby
E

ln

ˆ α
st
α

= E

ln

lˆ α
u
+(1−l)α
0
α

I(t
1
≤ T
m
≤ t
2
)

+E

ln

ˆ α
u
α

I(T
m
< t
1
or T
m
> t
2
)

= E

ln

(1−l)α
?
+
2mld
1
Y

I(y
1
≤ Y ≤ y
2
)

+E

ln

2md
1
Y

−E

ln

2md
1
Y

I(y
1
≤ Y ≤ y
2
)

=
Z
y
2
y
1

ln

(1−l)α
?
+
2mld
1
y

−ln

2md
1
y

g(y)dy, (4.4)
where y
1
= q
1
/α
?
and y
2
= q
2
/α
?
. Figure 1, shows the plot of (4.4) for selected values
of m = 2(1)5 and γ = 0.01 with respect to α
?
, which is computed numerically using
the statistical packageR version 3.1.2. It is observed that the risk-bias may be negative,
zero or positive, then we can state that the estimator ˆ α
st
may be negatively risk-biased,
risk-unbiasedorpositivelyrisk-biased.
Using a derivation similar to the above, the risk of the PTS estimator given in (4.3)
undertheLSELfunctionis
R(α, ˆ α
st
) = E

ln
2

(1−l)α
?
+
2mld
1
Y

I(y
1
≤ Y ≤ y
2
)

+E

ln
2

2md
1
Y

−E

ln
2

2md
1
Y

I(y
1
≤ Y ≤ y
2
)

=
Z
y
2
y
1

ln
2

(1−l)α
?
+
2mld
1
y

−ln
2

2md
1
y

g(y)dy +Ψ
0
(m−1).

54 L.BarmoodehandM.NaghizadehQomi
0 1 2 3 4 5
−1.0 0.0 0.5
m=2,γ=0.01
α
*
l=0.2
l=0.4
l=0.6
l=0.8
0 1 2 3 4 5
−0.8 −0.2 0.2
m=3,γ=0.01
α
*
l=0.2
l=0.4
l=0.6
l=0.8
0 1 2 3 4 5
−0.6 −0.2 0.2
m=4,γ=0.01
α
*
l=0.2
l=0.4
l=0.6
l=0.8
0 1 2 3 4 5
−0.6 −0.2 0.2
m=5,γ=0.01
α
*
l=0.2
l=0.4
l=0.6
l=0.8
Figure1: Therisk-biasofthePTSestimator ˆ α
st
forselectedvaluesofm = 2(1)5, γ = 0.01
and l = 0.2(0.2)0.8withrespectto α
?
5 Comparison between PTS Estimator and a Risk-unbi-
asedEstimator
Inthissection,weevaluatetheperformanceoftheproposedestimators. Forcomparison,
therelativeefficiency(RE)oftheestimator ˆ α
st
withrespecttotherisk-unbiasedestimator
ˆ α
u
iscalculatedas
RE(ˆ α
st
, ˆ α
u
) =
R(α, ˆ α
u
)
R(α, ˆ α
st
)
. (5.1)
Figures 2–4 give the relative efficiency (5.1). Figure 2 shows the RE for the selected
values of m = 2(1)5, γ = 0.01 and l = 0.2(0.2)0.8 with respect to α
?
= α
0
/α. Note
that we used the notation low(step)up for presentation of values. From this figure, we
find that no PTS estimator perform uniformly better than the ˆ α
u
. We see that the PTS

PretestShrinkageEstimators... 55
0 1 2 3 4 5
0 2 4 6
m=2,γ=0.01
α
*
l=0.2
l=0.4
l=0.6
l=0.8
0 1 2 3 4 5
1 2 3 4 5
m=3,γ=0.01
α
*
l=0.2
l=0.4
l=0.6
l=0.8
0 1 2 3 4 5
1 2 3 4 5 6
m=4,γ=0.01
α
*
l=0.2
l=0.4
l=0.6
l=0.8
0 1 2 3 4 5
1 3 5 7
m=5,γ=0.01
α
*
l=0.2
l=0.4
l=0.6
l=0.8
Figure2: TheREbetweenthePTSestimatorandtherisk-unbiasedestimatorforselected
valuesof m = 2(1)5, γ = 0.01and l = 0.2(0.2)0.8withrespectto α
?
estimatorsarebetterthanthe ˆ α
u
forthevaluesofα
?
neartoone(α
0
closetoα). TheRE
betweenthePTSandtherisk-unbiasedestimatorisplottedinFigure3forselectedvalues
of m = 2(1)5 and γ = 0.01,0.05,0.1 with respect to shrinkage factor l, when α
?
= 1.
This figure show that the RE is decreasing in l, i.e., the PTS estimators with small l
perform better than other estimators when m and γ are fixed. Also, the PTS estimators
withsmallγ aregoodforfixedmandl. Finally,fromFigure4, weobservethatthePTS
estimatorswithlargemhavegoodperformancewhenγ andl arefixed.
6 ARealExample
The following data reported by Dyer (1981) are the annual wage data (in multiplies of
100 US dollars) of a random sample of 30 production-line workers in a large industrial

56 L.BarmoodehandM.NaghizadehQomi
0.0 0.2 0.4 0.6 0.8 1.0
2 4 6 8 10
m=2
l
γ=0.01
γ=0.05
γ=0.1
0.0 0.2 0.4 0.6 0.8 1.0
2 4 6 8 10
m=3
l
γ=0.01
γ=0.05
γ=0.1
0.0 0.2 0.4 0.6 0.8 1.0
2 4 6 8 10
m=4
l
γ=0.01
γ=0.05
γ=0.1
0.0 0.2 0.4 0.6 0.8 1.0
2 4 6 8 10
m=5
l
γ=0.01
γ=0.05
γ=0.1
Figure3: TheREbetweenthePTSestimatorandtherisk-unbiasedestimatorforselected
valuesof α
?
= 1, m = 2(1)5, γ = 0.01,0.05,0.1withrespectto l
firm:
112 154 119 108 112 156 123 103 115 107
125 119 128 132 107 151 103 104 116 140
108 105 158 104 119 111 101 157 112 115
He determined that Pareto distribution provided an adequate fit for these data. If we
considerm = 3,thentheobservedrecorddataareobtainedinTable1.
We get T
3
= 0.441 and then the MLE of α is ˆ α =
3
T
3
= 6.804. We consider the
estimate of α when the guess value is α
0
= 6. Also, d
1
= e
Ψ(2)
/3 = 0.509 and ˆ α
u
=
3.4632 with risk R(α, ˆ α
u
) = 0.64493. The estimate of α
?
is ˆ α
?
=
6
6.804
= 0.89. We
considerfourvaluesofshrinkagefactorasfollows:
1. The value of l
1
= 0.013, which is obtained from minimizing the risk of shrinkage
estimator ˆ α
s
givenin(4.2).

PretestShrinkageEstimators... 57
0.0 0.2 0.4 0.6 0.8 1.0
2 4 6 8 10
γ=0.01
l
m=2
m=3
m=4
0.0 0.2 0.4 0.6 0.8 1.0
1.0 2.0 3.0
γ=0.05
l
m=2
m=3
m=4
0.0 0.2 0.4 0.6 0.8 1.0
1.0 1.5 2.0 2.5
γ=0.1
l
m=2
m=3
m=4
0.0 0.2 0.4 0.6 0.8 1.0
1.0 1.2 1.4
γ=0.2
l
m=2
m=3
m=4
Figure4: TheREbetweenthePTSestimatorandtherisk-unbiasedestimatorforselected
valuesof α
?
= 1, γ = 0.01,0.05,0.1,0.2and m = 2(1)4withrespectto l
2. Thevalueofl
2
= d
1
= 0.509.
3. TheteststatisticfortestingH
0
: α = 6isχ
2
= 2α
0
T
3
= 5.292andthecorrespond-
ingpvalueisP(χ
2
> 5.292) = 0.259. Alargepvalueindicatesthatαisclosetothe
guessα
0
= 6(TseandTso,1996). Thenwecanconsiderl
3
= 1−pvalue = 0.741.
4. The root of p-value support α
0
more strongly. Thus, the final shrinkage factor can
bel
4
= 1−
√
pvalue = 0.491.
The risks and RE’s of the risk-unbiased estimator and the PTS estimators ˆ α
(i)
st
corre-
spondingtotheshrinkagefactorsl
i
, i = 1,2,3,4aresummarizedinTable2.
From Table 2, we observe that all of the PTS estimators are better than the estimator
ˆ α
u
. Also, the estimator ˆ α
(1)
st
corresponding to the shrinkage factor l
1
= 0.013 is more
efficientthanotherestimators.

58 L.BarmoodehandM.NaghizadehQomi
Table1: Recorddataarisingfromannualwagedata
i 1 2 3
R
i
112 108 103
K
i
3 4 1
Table2: RisksandREsoftherisk-unbiasedestimatorandthePTSestimators
Estimator ˆ α
u
ˆ α
(1)
st
ˆ α
(2)
st
ˆ α
(3)
st
ˆ α
(4)
st
Risk 0.64493 0.21233 0.33857 0.45729 0.33080
R.E. — 3.03738 1.90482 1.41033 1.94961
7 SummaryandSomeRemarks
ThepresentpaperdealingwiththeconstructionofPTSestimatorsfortheshapeparameter
ofaParetomodelbasedonlowerrecordvaluesundertheSLEL.Arisk-unbiasedestimator
oftheshapeparameterisderivedundertheSLEL.WeproposedPTSestimatorsbasedon
Thompson method and compute their risks numerically. The RE of these estimators and
therisk-unbiasedestimatoriscalculatedandplottedforvarioussettings. Theseplotsshow
thattheproposedPTSaremoreefficientwhentheexperimenterhasapointguesscloseto
thetrue. Inthiscase, theREisdecreasinginshrinkagefactorforfixedotherparameters.
Also, a PTS estimator constructed by a smaller level of significance is preferable. In a
real example, we used four shrinkage factors for constructing the PTS estimators. We
observed that the PTS estimator with the shrinkage factor obtained by minimizing the
shrinkageestimator ˆ α
s
hasasmallerriskthanotherPTSestimators.
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