Paper

Statistical analysis related to impulse tests for self-restoring insulation
Yoshio Komori∗MemberFor dielectric tests with impulse voltage, some statistical methods with the step-up procedure have been
recently proposed. In the present paper, the methods are reconsidered from the viewpoint of statisticalindependence concerning disruptive discharges. As a result, modiﬁed methods with the step-up procedureor with the up-and-down procedure are proposed for self-restoring insulation, and it is shown that they havegood properties in the maximum likelihood estimation for the mean and the standard deviation of a normal
distribution.
Keywords: Maximum likelihood estimation; Normal distribution; Dielectric breakdown; Impulse testing; Up-anddown test; Step-up test

is to disclose merits and demerits of the step-up method
as a method for self-restoring insulation.
In the papers (5) (9) , Hirose has pointed out that withthe recent improvement of voltage measuring instruments, we can observe a voltage at the moment whena disruptive discharge occurs. For this, he has proposed the new step-up method and the new up-anddown method and has shown the superiority of themto the step-up method and the up-and-down method,respectively. The new methods, however, have a technical defect. Although a disruptive discharge can typicallyoccur after the peak of an impulse voltage, it is not dealt
with appropriately in the methods. (We will see details
later.) Thus, our third aim is to propose a modiﬁed stepup method and a modiﬁed up-and-down method, which
inherit merits of the new step-up method and the new
up-and-down method and avoid the technical defect inthem.
Throughout the present paper we use the word
“method” to mean a combination of a test procedureand the way of analysis of test results, such as the upand-down method.
In Section 2 we will introduce the step-up, the upand-down, the new step-up and the new up-and-down
methods and give some remarks about them. In Section
3 the step-up method and the up-and-down method will
be compared in the asymptotic or empirical errors. In
Section 4 a modiﬁed step-up method and a modiﬁed upand-down method will be proposed and investigated. In
Section 5, conclusions will be given. An iterative formulato get estimates and a formula to calculate an observed
information matrix will be given in Appendix.

1. Introduction
We are concerned with statistical methods dealing
with dielectric tests with impulse voltage. The conventional methods are given by International Electrotechnical Commission (IEC) (1) or IEEE (2) . On the other
hand, two diﬀerent types of methods with the step-up
procedure have been recently proposed in a series of thepapers by Hirose (3)∼(8) .
The two types of methods are the step-up method,which was originally proposed in the papers (3) (4), andthe new step-up method (5)∼(8) . In the methods, eachresult in voltage stress applications is supposed to bestatistically independent under an underlying probability distribution, regardless of whether a disruptive discharge occurs or not. This assumption is the same asthat in the up-and-down method (1) (2) , which is for selfrestoring insulation. In spite of that, these methods havebeen proposed for non-self-restoring insulation.
Self-restoring insulation like liquid or gaseous electrical insulation is the insulation that completely recovers its insulating properties after a disruptive discharge
caused by a voltage stress application (1) (2) . In addition, after it recovers, another disruptive discharge canbe caused by the application of another voltage stressindependent of the voltage stress when the previous disruptive discharge occurred. On the other hand, nonself-restoring insulation like solid electrical insulation is
the insulation that loses its insulating properties, or does
not recover them completely, after a disruptive discharge
caused by a voltage stress application (1) (2) . Thus, we
can see that there is a big diﬀerence in inﬂuence by a
voltage application between self-restoring insulation andnon-restoring insulation.
Now, a question arises: ”is the step-up method (notthe step-up procedure itself) really appropriate for nonself-restoring insulation?” Our ﬁrst aim is to consider
this question. After the consideration, our second aim∗

2. Preliminaries
We introduce the test procedures and the likelihood
functions of the step-up, the up-and-down, the new stepup and the new up-and-down methods, respectively, andstate some remarks.
2.1 Step-up method
The test procedure proceeds as follows (3)∼(8) .
i) Decide the ﬁrst voltage level U1 , a small amount

Department of Systems Design and Informatics, Kyushu
Institute of Technology, 680-4, Kawazu, Iizuka 820-8502,
Japan

IEEJ Trans. XX, Vol.xxx, No.xx, xxxx

1

4U and a maximum number m of voltage stressapplications at a voltage level. Here, U1 should
be suﬃciently low such that almost no disruptive
discharge occurs at the level.
ii) If no disruptive discharge occurs when substantially equal voltage stresses are applied m times
at the voltage level U1 , set the next voltage levelU2 at U1 + 4U . Similarly increase the succeeding voltage levels U3 , U4 , ... until a disruptive
discharge occurs.
iii) Perform ii) n times.
The likelihood function has been given as follows(3)∼(8). For the jth disruptive discharge (1 ≤ j ≤ n),denote by δj and mj an integer for which a disruptive
discharge occurs at Uδj and the number of voltage stressapplications at the level. If F (U ; θ) is the discharge
probability distribution function (θ denotes a vector ofparameters), the likelihood function becomes:

n 
∏
L=
F (Uδj ; θ)[1 − F (Uδj ; θ)]mj −1

j=1

δj −1

∏
×
[1 − F (Uk ; θ)]m . · · · · · · · (1)in (2), the maximum likelihood (ML) estimates of theparameters µ and σ are the sample mean and the sample
standard deviation. Hence, the diﬀerence between the
step-up method and the engineers’ estimation method
is the assumption of statistical independence. In a nonself-restoring specimen like solid insulation, it is hard toaccept that the voltage at which a disruptive discharge
occurs can vary entirely independently of the specimen
itself for every voltage stress application. The voltage
could rather depend on the specimen itself. For non-selfrestoring specimens, thus, the engineers’ assumption ismore natural than that in the step-up method.
In addition, it should be noted that, in the papers(3) (5), the superiority of the step-up method is indicatedby using data simulated under the assumption of statistical independence in the step-up method. Under theassumption, the probability that a disruptive discharge
does not occur before the k0 th voltage level, is
k∏
0 −1
k=1

which goes to 0 as k0 → ∞. Thus, by noting that k0
such that Uk0 > u for a u > U1 becomes larger as 4U
becomes smaller, we can see that the statement “thesmaller the 4U1 , the smaller the distribution of U4j ” (5)comes from the assumption.
2.2 Up-and-down method
The up-and-down
50% disruptive discharge voltage test is deﬁned as follows (1, p. 91) (2, p. 118) .
i) Decide the ﬁrst voltage level U1 , a relatively
small value 4U and the total number N of voltage stress applications.
ii) When a voltage stress is applied at the voltage
level U1 , and no disruptive discharge occurs, set
the next voltage level U2 at U1 + 4U . If it occurs,
set that at U1 − 4U .
iii) Perform similar tests at the succeeding voltage
levels U2 , U3 , ..., UN .
The likelihood function for the test is given as follows (1, p. 99) (2, p. 122) . Denote by di the number of discharges found in a voltage application at a voltage level
Ui . Since di = 1 or 0, the number of withstands is given
by 1 − di at Ui . Hence, the likelihood function L becomes:

k=1

All the test results are expressed by U1 , 4U and{(δj , mj )}nj=1 As seen in (1), each result (discharge orwithstand) in voltage stress applications is dealt withstatistically independently. This method has been proposed for non-self-restoring insulation (3)∼(8) .
Electrical engineers sometimes employ the sample
mean and the sample standard deviation as estimatesof the mean parameter and the standard deviation parameter for a normal distribution. This approach has
been criticized in the papers (3) (5) , in which the authors
claim that the approach is incorrect and insist that thelikelihood analysis via (1) is necessary (3) (5) . However,in this paper, we argue that the method commonly employed by electrical engineers is in fact correct, for thefollowing reasons.
In order to see the reasons, let us consider the following likelihood function:L=n∏[1 − F (Uk ; θ)]m ,f (Ũj ; θ), · · · · · · · · · · · · · · · · · · · · · · · · · · · · (2)

j=1

L=

where Ũj denotes the voltage level at which a disruptive discharge occurs and f (Ũ ; θ) denotes the discharge
probability density function. In (2), note that only voltage levels Ũj ’s at which disruptive discharges occur aredealt with statistically independently and the other voltage levels Ui ’s before disruptive discharges occur do notappear in (2). In other words, disruptive discharge voltages on test objects are expressed by a independent random variable Ũ , and Ũj corresponds to the disruptive
discharge voltage of the jth test object.
When a normal distribution is assumed, that is,√
f (Ũ ; θ) = exp[−(Ũ − µ)2 /2σ 2 ]/ 2πσN∏[F (Ui ; θ)]di [1 − F (Ui ; θ)]1−di . · · · · · · · · (3)i=1All the test results are expressed by U1 , 4U and {di }Ni=1 .
As seen in (3), each result di in voltage stress applications is dealt with statistically independently. Now, we
can see that the assumption of statistical independence
is the same as that in the step-up method and the diﬀerence between the up-and-down method and the step-up
method is only the way of obtaining data. Hence, byremembering that the ML methods permit estimationof any statistical quantities once the value of θ has been
determined, we can say that the up-and-down method
and the step-up method are competitors each other in
2

IEEJ Trans. XX, Vol.xxx, No.xx, xxxx

obtaining good estimate of θ. Eventually, since the upand-down method is well-known as a method for selfrestoring insulation, we can see that the step-up method
is a method for self-restoring specimens rather than for
non-self-restoring specimens.
Up to this subsection, we have introduced the methods that deal with censored data only. That is, all theinformation they use is whether a disruptive discharge
occurs at a voltage level or not. Censored data are akind of incomplete data (10, p. 24) . In the next two subsections, let us introduce methods dealing with observation values, which are called complete data. As said in
Section 1, the recent improvement of voltage measuring
instruments has made these methods available.
2.3 New step-up method
The new step-up
method is a counterpart of the step-up method that
deals with complete data. In the test procedures, thediﬀerence between them is only whether a voltage, say
uj , at the moment when a disruptive discharge occurs is
supposed to be recorded or not.
The likelihood function is given as follows (5)∼(8) :

n 
∏
L=
f (uj ; θ)[1 − F (Uδj ; θ)]mj −1

j=1

δj −1

∏
×
[1 − F (Uk ; θ)]m . · · · · · · · · (4)All the test results are expressed by 4U and{(δj , mj , uj )}nj=1 .
2.4 New up-and-down method
The new upand-down method is a counterpart of the up-and-down
method that deals with complete data. In the test procedures, the diﬀerence between them is the same as thediﬀerence between the step-up method and the new stepup method.
The likelihood function is given as follows (9) :N∏

Fig. 2.

Disruptive discharge on the tail.

3. Comparison of methods
We ﬁrst show the asymptotic errors of parameter estimators in the step-up method and the up-and-down
method, and second investigate how many times an experimenter needs to test to obtain good estimates whose
errors are close to the asymptotic errors. In the sequelthe discharge probability distribution is supposed to bea normal distribution with mean µ and standard deviation σ.
3.1 Asymptotic unit errors
We deﬁne theasymptotic unit errors of the ML estimators of µ andσ by [n(I −1 )11 ]1/2 and [n(I −1 )22 ]1/2 , where I stands
for the Fisher information matrix. In order to seek thematrix in the step-up method, let us rewrite (1). Denote by λi and νi the total numbers of discharges andwithstands in the voltage applications at a voltage level
Ui , respectively. These are expressed by[f (ui ; θ)]di [1 − F (Ui ; θ)]1−di . · · · · · · · · (5)i=1

2.5 Remarks on methods
In the papers (5)∼(9) ,it is reported that the new step-up method and the
new up-and-down method give more precise estimation
than that by the step-up method and the up-and-down
method, respectively. This is one of the virtues of including complete data. These methods, however, havea technical problem that is not resolved. A disruptive
discharge causes a rapid collapse on an impulse voltage
to zero or nearly to zero. The collapse can occur before
or after the peak of the impulse voltage. Let us call theformer or latter case a disruptive discharge on the front
or tail of the impulse, respectively, (1, pp. 119-120) . For example, Figure 1 or 2 shows a disruptive discharge on thefront or tail of an impulse. These collapses are diﬀerent phenomenon. Although it is not clearly understood
which voltage should be used as a voltage datum for a
disruptive discharge when it occurs on the tail of an impulse, some voltage is used as a complete datum in themethods (5) (9) . (For example, the peak voltage is used
IEEJ Trans. XX, Vol.xxx, No.xx, xxxx

Disruptive discharge on the front.

in the paper (5).) This defect will be removed by somemodiﬁed methods in Section 4.
The up-and-down method has been proposed by
Dixon and Mood (11) and they have given the following
explanation concerning U1 : if U1 is poorly chosen, theearly observations from U1 to some Ui will be spent ingetting from U1 to the region of the mean; they will obviously contribute little to the more precise location of
the mean. According to this, the step-up method seems
to be inferior to the up-and-down method. In the next
section we will see that it is not always true.

k=1

L=

Fig. 1.

λi =
νi =

n
∑

Ii (δj ),

j=1
n (
∑

)
mI˜i (δj ) + (mj − 1)Ii (δj ) ,

j=1
def
def
where Ii (k) = 1 if i = k or 0 otherwise, and I˜i (k) = 1
if i < k or 0 otherwise. By utilizing these expressions,
we can obtain

3

4

2

es HΣL, eu HΣL

es HΜL, eu HΜL3
eu (µ)

10

es (µ)

0

1

2

3

3

1
0

4

eu (σ)2es (σ)

0

1

2

DUΣ

3

4

DUΣ

Fig. 3. Asymptotic unit errors of the step-up method or the up-and-down method.
(Solid lines: Case A; dotted lines: Case B.)2ēs (µ); n = 20

XXX
H
Hz
H
H
j

1

es HΣL, es HΣL

es HΜL, es HΜL

1.5

es (µ)

0.5
0

0

0.5

1

1.5

ēs (σ); n = 20ēs (σ); n = 80

1.5

@
@
R
XXX
z
X

1
es (σ)

0.5
0

2

0

0.5

1

1.5

2

DUΣ

DUΣ

Fig. 4. Empirical or asymptotic unit errors of the step-up method.
(Solid lines: Case A; dotted lines: Case B.)
def

where I¯i (k) = 1 if Ui = U1 + k4U or 0 otherwise (12) .
In the step-up method, let es (µ) be the asymptotic
unit error of the ML estimator of µ and es (σ) that of σ.
Similarly, let us denote by eu (µ) and eu (σ) the asymptotic unit errors of the ML estimators in the up-anddown method.
Suppose that µ = 0 and σ = 1. We investigate thetwo cases, Case A: the mean falls on a voltage level andCase B: the mean is midway between two voltage levels.
In the step-up method, U1 needs to be suﬃciently low.
When U1 ≤ µ − 3.5σ and m = 1, each graph of es (µ)
and es (σ) in the interval of 4U/σ from 0.2 to 4 keeps
the same shape in each case. For this, we set m = 1 andU1 = maxi (µ − i4U ) or maxi (µ − (i + 0.5)4U ) under
the condition that U1 ≤ µ − 3.5σ.
On the other hand, in the up-and-down method, U1
does not need to be low, but it rather needs to be close
to µ. For this, let us suppose U1 = maxi (µ − i4U )or maxi (µ − (i + 0.5)4U ) under the condition thatU1 ≤ µ − σ. Then, each graph of eu (µ) and eu (σ) also
keeps almost the same shape if N ≥ 40. This means thatthe sample size 40 is large enough for us to know how
the errors asymptotically behave in the up-and-down
method. Thus, we set N = 40.
We show es (µ), eu (µ), es (σ) and eu (σ) in Figure
3. The thick or normal lines correspond to the stepup method or the up-and-down method, respectively.
Throughout the present paper, the solid or dotted lines
correspond to Case A or B, respectively. The ﬁguretells us that the step-up method is quite superior to the
up-and-down method in the asymptotic unit errors. Especially, it is remarkable that es (µ) and es (σ) are small,and in the interval of 4U/σ from 0.2 to 1.8, each offrom (1). ∏Now, let us seek
the Fisher information
matrixλi∏
i νi · ·m· · · · (6)
[F
(U
;
µ,
σ)]
[1
−
F
(U
;
µ,
σ)]
L
=
i
i
for (6). Denote F (Ui ; µ, σ), 1 − pi and k=1 qk by pi , qii≥1
and ri , respectively.
In addition, introduce the following
symbols:
( 2)
1
x
def
def
xi = (Ui − µ)/σ,zi = √ exp − i ,
2
2π
]
[
1 xi
def
Ai =
.
xi x2i
From (6) the Fisher information matrix in the step-up
method becomes:(m−1 )
∑
n ∑
zi2
I= 2
Ai · · · · · · · · · · · · (7)
ri−1
qik
σ
pi qi
i≥1

k=0

since the expectations E[λi ] and E[νi ] are expressed by
(
)
n
m−1
∑
∑
k
ri−1
qi pi = nri−1 (1 − qim )· · · · · · · · (8)
j=1

and
n
∑

k=0

(
mri + ri−1

j=1

m−1
∑
k=0

)
kqik pi= nri−1m∑qik , · (9)k=1

respectively.
From (3) the Fisher information matrix in the up-anddown method becomes:z2
1 ∑ ∑P [I¯i (k) = 1] k Ak , · · · · · · · (10)
2
σ i=1
pk qk
N

I=

i

k=−i

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IEEJ Trans. XX, Vol.xxx, No.xx, xxxx

Table 1. Average number of voltage stressapplications.
4U/σCase A
Case B

0.4
9.0
7.5

0.8
5.1
4.6

1.2
4.4
3.9

1.6
3.4
2.9

charges on the tails in the same way as the up-and-down
method does. That is, these methods use a censored datum if a disruptive discharge occurs on the tail. We callthem the modiﬁed step-up method and the modiﬁed upand-down method.
4.1 Asymptotic unit errors
First, we seek thelikelihood function and the Fisher information matrix
in the modiﬁed step-up method. For the jth disruptive
discharge (1 ≤ j ≤ n), deﬁne{
1 (it occurs on the tail),defτj =0 (it occurs on the front).

2.0
2.5
3.0

them is almost the same in Cases A and B.
3.2 Evaluation in small samples
We haveseen that the step-up method can provide asymptotically good estimations in the interval [0.2, 1.8]. This is
not, however, suﬃcient to approve the performance ofthe method. In this subsection, let us evaluate it insmall samples by means of Monte Carlo simulation.
We deﬁne the empirical unit error of an ML estimatorby n1/2 times its root mean square error, and denote theempirical unit errors of µ and σ in the step-up methodby ēs (µ) and ēs (σ). When n = 20 or 80, 10000 setsof independent pseudo-random samples were considered
for each value of 4U/σ(= 0.2, 0.3, . . . , 2.0).
Figure 4 gives ēs (µ) and ēs (σ) for n = 20, 80 as wellas es (µ) and es (σ) for comparison. The thick lines correspond to the asymptotical unit errors, and normal orgray lines correspond to the empirical unit errors forn = 20 or 80, respectively. From the ﬁgure we can see
the following: when n = 20, the empirical unit errors goaway from the asymptotic unit errors as 4U/σ becomesclose to 2; when n = 80, the empirical unit errors arealmost the same as the asymptotic unit errors exceptēs (σ) in Case A.
Next, let us seek the average of the number of voltage stress applications necessary to obtain one disruptive discharge. By adding (8) and (9), substituting n = 1
into it and taking a summation
over ∑
possible values of
∑
m−1 k
r
(
i, we obtain the average:i−1
k=0 qi ). Tablei≥1
1 shows its values when m = 1. From the table, forexample, we can see that it may be necessary to apply
voltage stresses 272 times when n = 80 and 4U/σ = 1.6
in Case A. (Note that the number n = 80 comes fromthe last sentence in the previous paragraph.)
Summarizing what we have seen in this section, we
can say the following:• The step-up method is superior to the up-and-down
method in the asymptotic unit errors. This is a newﬁnding, which is diﬀerent from the explanation concerning U1 in the paper (11). (For details, remember
the last paragraph in Subsection 2.4.)• In order to attain the empirical unit errors as small
as the asymptotic unit errors, however, the step-up
method requires a much larger number of voltage
stress applications than the up-and-down method
does. This is also a new ﬁnding, which has not beenshown in the papers (3) (5).

The likelihood function becomes:

n 
∏
L=
[F (Uδj ; θ)]τj [f (uj ; θ)]1−τj

j=1

δj −1

×[1 − F (Uδj ; θ)]mj −1[1 − F (Uk ; θ)]m

k=1





.

As in Subsection 3.1, this can be rewritten into]Ii (δj )
n [
∏ ∏
τj
1−τj
L=
[F (Uδj ; µ, σ)] [f (uj ; µ, σ)]

j=1
i≥1


× [1 − F (Ui ; µ, σ)]νi . · · · · · · · · · · · ·(11)For ease of analysis, let us suppose that disruptive discharges occur on the tails independently of voltage stresslevels. Then, we can set
P [τj = 1|Ii (δj ) = 1] = γ,
where γ is a constant. By similar calculations to those in
Subsection 3.1, from (11) the Fisher information matrix
in the modiﬁed step-up method is given as follows:(m−1 )
∑
n ∑
I= 2
ri−1
qik Ci , · · · · · · · · · · · · · · (12)
σ
i≥1

where

[

k=0

−zi − x2i zi
Bi =−xi zi − x3i zi + 2pi()(pi + γqi )zi2
def
Ci =Ai + (1 − γ)Bi .
pi qi
def

−xi zi + pi
−zi − x2i zi],

On the other hand, by similar calculations to thosein the up-and-down method and the modiﬁed step-up
method, the Fisher information matrix in the modiﬁed
up-and-down method becomes:
1 ∑ ∑P [I¯i (k) = 1]Ck . · · · · · · · · · · · · (13)σ 2 i=1
NI=

4. Modiﬁed methods

i

k=−i

In Section 2, we have seen that the new step-up
method and the new up-and-down method contain thetechnical problem. To avoid it, let us consider new othermethods that use a disruptive discharge voltage as acomplete datum only if a disruptive discharge occurs on
the front of an impulse and that deal with disruptive disIEEJ Trans. XX, Vol.xxx, No.xx, xxxx∏

The following are remarkable:• The expressions in the right-hand side of (7) and(12) or (10) and (13) are the same except the difference between zi2 Ai /pi qi and Ci or zk2 Ak /pk qk andCk .
5

ems HΣL, emu HΣL

ems HΜL, emu HΜL2
emu (µ)1.5
1ems (µ)

0.5
0

0

1

2

3

4

2
1.5

emu (σ)1ems (σ)

0.5
0

0

1

2

DUΣ

3

4

DUΣ

1.5

ēms (µ); n = 20

HH
j
H

1

ems HΣL, ems HΣL

ems HΜL, ems HΜL

Fig. 5. Asymptotic unit errors of the modiﬁed step-up method or the modiﬁed up-and-down method
when γ = 0.7.
(Solid lines: Case A; dotted lines: Case B.)

ems (µ)

0.5
0

0

0.5

1

1.5

2

1.5

ēms (σ); n = 20

@
@
R

1
ems (σ)

0.5
0

0

0.5

1

1.5

2

DUΣ

DUΣ

Fig. 6. Empirical or asymptotic unit errors of the modiﬁed step-up method when γ = 0.7.
(Solid lines: Case A; dotted lines: Case B.)• When γ = 1, (12) is equivalent to (7), whereas (13)is equivalent to (10).
We denote the asymptotic unit errors of the ML estimators in the modiﬁed step-up method or the modiﬁed
up-and-down method by ems (µ) and ems (σ) or emu (µ)
and emu (σ).
In addition to the setting of µ, σ, m, U1 and N in
Subsection 3.1, we set γ = 0.7. Then, Figure 5 givesems (µ), emu (µ), ems (σ) and emu (σ). The thick or normal lines correspond to the modiﬁed step-up method or
the modiﬁed up-and-down method, respectively. The
ﬁgure shows that the modiﬁed step-up method is superior to the modiﬁed up-and-down method in the asymptotic unit errors, and in the interval of 4U/σ from 0.2
to 2.0, each of ems (µ) and ems (σ) is almost the same inCases A and B.
By comparing Figures 3 and 5, we can see that the errors in Figure 5 are smaller as a whole. This is because
more precise estimation can be given by using complete
data even only for disruptive discharges on the fronts ofimpulses.
4.2 Evaluation in small samples
We evaluate
the performance of the modiﬁed step-up method in small
samples by means of Monte Carlo simulation. We denote by ēms (µ) and ēms (σ) the empirical unit errors of
the ML estimators of µ and σ in the modiﬁed step-up
method. Under the setting that n = 20 and γ = 0.7,
10000 sets of independent pseudo-random samples wereconsidered for each simulation. Figure 6 shows ēms (µ)
and ēms (σ) by normal lines as well as ems (µ) and ems (σ)
by thick lines for comparison. From this we can see thatthe empirical unit errors are very close to the asymp-

totic unit errors throughout the interval [0.2, 1.4]. Sincen = 20, for example, Table 1 tells us that it is necessary to apply voltage stresses about 80 or 90 times
when 4U/σ = 1.2.
Compared with the step-up method, the modiﬁed
step-up method can reduce the number of voltage stressapplications necessary to attain such small empiricalunit errors. This is one of the virtues of including complete data.
4.3 Concrete example
We give an example toshow the diﬀerence between the up-and-down method
and the modiﬁed up-and-down method. The modiﬁed
up-and-down method is the same as the up-and-down
method in the test procedure except the following: whena disruptive discharge occurs on the front of an impulse,the voltage at the moment when it occurs needs to berecorded in the modiﬁed up-and-down method. Let us
give such data in Table 2, which have been simulated
under the setting that µ = 40, σ = 1.6 and γ = 0.7.
The table shows that disruptive discharges occurred on
the front of the impulse when i = 4, 10, 13, 14, 31, 33
and 40. Each voltage ui at the moment when a disruptive discharge occurred on the front is given in theparenthesis.
Noting that ui ’s are utilized to seek the estimates of µ
and σ in the modiﬁed up-and-down method, we obtainµ̂ = 39.7 and σ̂ = 2.09 as the ML estimates by means of
the formula in Appendix, and then obtain 0.44 and 0.48
as the estimates of the standard errors of them, respec−1−1tively, by [(Iob)11 ]1/2 and [(Iob)22 ]1/2 in Appendix. On
the other hand, noting that ui ’s are not utilized in the
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IEEJ Trans. XX, Vol.xxx, No.xx, xxxx

up-and-down method, we obtain µ̂ = 39.5, σ̂ = 2.57 andthe estimates of the standard errors 0.56 and 0.87 in asimilar way. Here, it should be remembered that theerrors of parameter estimates in the modiﬁed up-anddown method are less inﬂuenced by an unknown value
4U/σ than the up-and-down method. (See Figures 3
and 5.)

curs on the tail of an impulse, some voltage is used
as a complete datum in the methods. The modiﬁed step-up method or the modiﬁed up-and-down
method removes this defect.
• Whereas the step-up procedure requires choosing a
suﬃciently low voltage level for every object as U1 ,the up-and-down procedure allows us to choose it
roughly. In addition, the modiﬁed up-and-down
method makes it possible to choose 4U roughly
since the errors of the ML estimators are less sensitive to its value than the up-and-down method.
Consequently, we can roughly choose U1 and 4U in
the modiﬁed up-and-down method.

5. Conclusions
First, we have stated that, from the viewpoint of statistical independence, the step-up method is a method
for self-restoring insulation. Second, we have investigated the performance of the step-up, the up-and-down,
the modiﬁed step-up and the modiﬁed up-and-down
methods as a method for self-restoring insulation. Our
conclusions are as follows.
• The asymptotical unit errors of the ML estimators in the step-up method or the modiﬁed step-up
method are smaller than those in the up-and-down
method or the modiﬁed up-and-down method, respectively. This is a merit which is obtained byusing the step-up procedure under the statistical independence of all voltage stress applications.
• In order to attain the empirical unit errors as small
as them, however, the step-up method demands amuch larger number of voltage stress applications
than 40, which is suﬃcient in the up-and-down
method. This disadvantage is reduced in the modiﬁed step-up method. This is one of the virtues ofincluding complete data. Even the method, however, still demands a larger number of voltage stressapplications than 40. That is, only in the case thatthe number is allowed to be about 100, it is eﬀective.
• The asymptotical unit errors of the ML estimatorsin the modiﬁed step-up method or the modiﬁed upand-down method are less inﬂuenced by an unknown
value 4U/σ than those in the step-up method or the
up-and-down method. This is also one of the virtues
of including complete data.
• As shown in the papers (5) (9) (12), the asymptotical unit errors in the new step-up method or the
new up-and-down method are also not inﬂuenced
so much by 4U/σ. The methods, however, havethe technical problem. That is, although it is notclearly understood which voltage should be used as a
voltage datum for a disruptive discharge when it ocTable 2.
i

di

1
2
3
4

1
0
0
1

5
6
7
8
9
10

0
1
1
0
0
1

Ui
(ui )
38.0
36.0
38.0
40.0
(37.1)
38.0
40.0
38.0
36.0
38.0
40.0
(39.6)

i

di

11
12
13

0
0
1

14

1

15
16
17
18
19
20

0
1
0
1
0
1

Appendix
By utilizing the expectation-maximization algorithm
as in the paper (12), we give an iterative formulato get the ML estimates and a formula to calculate anobserved information matrix. Let nt , n and w denotethe number of disruptive discharges on the tails of impulses, the number of disruptive discharges and the number of withstands, respectively. Of all the voltage levelsw{Ui }w+ni=1 , pick up the voltage levels {U(i) }i=1 for withw+nt
stands and {U(i) }i=w+1 for discharges on the tails. On
the other hand, of all the discharge voltages {ui }ni=1 ,tpick up the discharge voltages on the fronts {u(i) }n−ni=1 .
Before we see formulas, we should note that each of themhas the same in both of the modiﬁed step-up method andthe modiﬁed up-and-down method. That is, whereas theformulas give results in the modiﬁed step-up method if
data in the modiﬁed step-up method are given, they give
results in the modiﬁed up-and-down method if data inthe modiﬁed up-and-down method are given.
The iterative formula for the ML estimation is
(10)

n−n
∑t
w + nt (k)
1
µ +
u(i)
w+n
w + n i=1
( w
)
w+n
∑ (k)
∑t (k)
1+
D +Ei,
w + n i=1 i
i=w+1
 w+n (
)2 ( (k) )2t[ σ (k) + 4µ1
σ (k+1) =] w+nµ(k+1) =

n−n
∑t (
)2
1
u(i) − µ(k+1)
w + n i=1
[ w∑(
1(k) ) (k)+U(i) + 4µ2 Di
w + n i=1
] 1/2
w+n

∑t (
)
(k)
(k)
+
U(i) + 4µ2 Ei,
+Data example.

Ui
(ui )
38.0
40.0
42.0
(41.7)
40.0(39.0)
38.0
40.0
38.0
40.0
38.0
40.0

i

di

21
22
23
24
25
26
27
28
29
30
31

1
0
0
1
1
0
0
0
0
1
1

IEEJ Trans. XX, Vol.xxx, No.xx, xxxx

Ui
(ui )
38.0
36.0
38.0
40.0
38.0
36.0
38.0
40.0
42.0
44.0
42.0
(39.8)

i

di

32
33

0
1

34
35
36
37
38
39
40

1
0
1
0
0
0
1

Ui
(ui )
40.0
42.0
(40.8)
40.0
38.0
40.0
38.0
40.0
42.0
44.0
(38.5)

i=w+1

where
(

)2

f (U(i) ; µ(k) , σ (k) ),1 − F (U(i) ; µ(k) , σ (k) ))2 f (U ; µ(k) , σ (k) )((i)(k) def,
Ei = − σ (k)F (U(i) ; µ(k) , σ (k) )(k) def

Di

7

=

σ (k)

(k) def

4µ1= µ(k) − µ(k+1) ,(k) def

4µ2= µ(k) − 2µ(k+1) .

(8)When µ(0) and σ (0) are properly given, the formulaprovides two series of approximates {µ(k) }k≥1 and{σ (k) }k≥1 to the ML estimates of µ and σ. One of
the good selections for µ(0) and σ (0) is a pair of thesample mean and the sample standard deviation of
w+ntt{u(i) }n−ni=1 ∪ {U(i) }i=w+1 .
Each element of the observed information matrix Iob
is given as follows:
[n−n
]
w+n
w
∑
∑t
1 ∑t 2
2
2(Iob )11 = 4
ū(i) +
Di +
Ei ,
σ̂
i=1
i=1
i=w+1
(Iob )12 = −

(10)
(11)

(12)

w
]
1 ∑ [
Di Ū(i) (Ū(i) − Di ) + σ̂ 25σ̂ i=1−(Iob )22 =(9)(2004).
H. Hirose, “More accurate breakdown voltage estimation for
the new step-up test method in the gumbel distribution
model”, European J. Oper. Res. 177 (1), pp. 406-419 (2007).
H. Hirose and K. Kato, “A new analysis of the up-anddown method”, Trans. IEE Japan 118-A (10), pp. 1087–1093(1998) (in Japanese).
G.J. McLachlan and T. Krishnan, The EM Algorithm andExtensions, John Wiley & Sons, New York (1997).
W.J. Dixon and A.M. Mood, “A method for obtaining andanalyzing sensitivity data”, J. Amer. Statist. Assoc. 43, pp.
109–126 (1948).
Y. Komori and H. Hirose, “An easy parameter estimation by
the EM algorithm in the new up-and-down method”, IEEE
Trans. Dielect. Elect. Insul. 7 (6), pp. 838–842 (2000).

w+n
[
]
1 ∑t
Ei Ū(i) (Ū(i) − Ei ) + σ̂ 2 ,5σ̂ i=w+1

2(n − nt )
σ̂ 2
w
[
]
1 ∑
2
− 6
Ū(i) Di Ū(i)− Ū(i) Di + σ̂ 2
σ̂ i=1
−

w+n
[
]
1 ∑t
2
2
Ū
E
Ū
−
Ū
E
+
σ̂
,
i
i
(i)
(i)
(i)
σ̂ 6 i=w+1

where (Iob )21 = (Iob )12 ,defū(i) = u(i) − µ̂,
defEi = −σ̂ 2

defDi = σ̂ 2f (U(i) ; µ̂, σ̂),F (U(i) ; µ̂, σ̂)f (U(i) ; µ̂, σ̂),
1 − F (U(i) ; µ̂, σ̂)
defŪ(i) = U(i) − µ̂.

Lastly, it is also remarkable that these formulas give
results in the step-up method or the up-and-down
method if n = nt , whereas they give results in the
new step-up method or the new up-and-down method
if nt = 0.

References
(1)

(2)

(3)

(4)

(5)

(6)

(7)

IEC Std 60-1, High-voltage Test Techniques Part 1: General Deﬁnitions and Test Requirements, IEC Central Oﬃce,
Geneva (1989).
IEEE Std 4-1995, IEEE Standard Techniques for High-voltage
Testing, The Institute of Electrical and Electronics Engineers,
Inc., New York (1995).
H. Hirose, “Estimation of impulse dielectric breakdown voltage by step-up method”, Trans. IEE Japan 104-A (8), pp.
464–470 (1984) (in Japanese).
H. Hirose, “Estimation of impulse dielectric breakdown voltage by step-up method when underlying breakdown voltage
follows the Weibull distribution function”, Trans. IEE Japan
106-A (11), pp. 511–518 (1986) (in Japanese).
H. Hirose, “More accurate breakdown voltage estimation for
the new step-up test method”, IEEE Trans. Dielect. Elect.
Insul. 10 (3), pp. 475–482 (2003).
H. Hirose, “More accurate breakdown voltage estimation for
the new step-up test method in the Weibull model”, IEEE
Trans. Dielect. Elect. Insul. 11 (3), pp. 418–423 (2004).
H. Hirose, “More accurate breakdown voltage estimation for
the new step-up test method for various probability distribution models”, Trans. IEE Japan 124-A (7), pp. 587–592

8

IEEJ Trans. XX, Vol.xxx, No.xx, xxxx