Generalization of usual capability indices for unilateral tolerances

HAL Id: hal-00133777

https://hal.archives-ouvertes.fr/hal-00133777

Preprint submitted on 27 Feb 2007

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Generalization of usual capability indices for unilateral tolerances

Daniel Grau

To cite this version:

Daniel Grau. Generalization of usual capability indices for unilateral tolerances. 2007. �hal-00133777�

GENERALIZATION OF USUAL CAPABILITY INDICES FOR UNILATERAL TOLERANCES

DANIEL GRAU

Laboratory of Applied Mathematics, CNRS UMR 5142 IUT de Bayonne, Université de Pau et des Pays de l’Adour,

3 Av. Jean Darrigrand, 64100 Bayonne, France Email: [email protected] Abstract :

Capability indices are dimensionless quantities measuring the aptitude of a process to manufacture items whose characteristics must be within a specified tolerance ranges.

The usual indices C

_p

, C

_pk

, C

_pm

, and C

_pmk

are used for a process of normal distribution and a target located at the center of the tolerance interval. Various indices derived from the

previous family allow to consider more complex situations when asymmetrical tolerances and non-normal distributions are taken into account. In this paper we study the case where a single tolerance is imposed because the shifts in the direction of this tolerance appear much more serious than in the opposite direction. We propose a family of four indices having

interpretations and properties similar to those of the usual family.

Key words: Process Capability Indices, Unilateral Tolerances, Non-normal Processes.

1. Introduction

The quality of an industrial process is assured by the monitoring of one or more variables of interest. The process will be considered of good quality if it is able to produce items whose variable of interest is within the tolerances L and U (Lower and Upper tolerance limits) and close to a specified target value T. Accordingly, the capability indices used to measure the quality of the process are linked to the location and the dispersion of the supervised variable.

For a target centered in its tolerance interval and a variable of normal distribution, the first work of Kane [9] was completed to lead to a family of four indices generally used. X being the supervised characteristic of normal distribution N( μ , σ ), one defines

-

6 3

p

U L d

C ⁼ σ ⁼ σ ^{, where d =} ( ^{U L -} ) ² is the half-length of the tolerance interval,

( )

min - , -

= 3

pk

U L

C μ μ

σ ^,

( )

²

2

-

6 -

pm

C U L

σ μ T

=

+

,

( )

( )

²

2

min - , -

3 -

pmk

U L

C

T μ μ

σ μ

=

+ .

When asymmetrical tolerances and normal data are concerned, many proposals have been

made, the most coherent of which as the generalization of the usual family, is given by the

family C

^''_p

, C

^"_pk

, C

^''_pm

, and C

^"_pmk

introduced by Pearn, Chen, and Lin in various articles [3, 4,

12, 16, 17].

In the case of non-normal distributions, Clements [5] proposes to generalize the indices C

_p

and C

_pk

replacing the mean μ by the median M and the dispersion 6 σ by Up – Lp where Up and Lp are the 99.865 and 0.135 percentiles. In the same way, Pearn and Kotz [15] provide a extension of the indices C

_pm

and C

_pmk

. However let us note that considering the difficulty to obtain reliable estimations of Up and Lp, many other proposals have been provided the

bibliography of which can be found in the articles of Tang and Than [21], Mac Cormack, Harris, Hurwitz, and Spagon [11], or Ding [6].

The literature related to the processes limited by two tolerance limits is important, but it is not the same in the less frequent situation where only one limit is imposed. In this case two situations can be considered. The first one occurs when the variable of interest, because of its nature, cannot exceed a certain level which represents the target of the process. It is the case for example for concentricity or circularity where the observed measure is obviously positive or null, the target being equal to 0. The second one occurs when a drift of the mean in a direction appears much less serious to the user, so that he is induced to define only one single tolerance. It is this second situation which is the subject of this paper. We recall the rare indices which one finds in the literature, then propose a family of four indices having interpretations and properties, similar to the ones of the usual family C

_p

, C

_pk

, C

_pm

, and

C

pmk

. For normal data, the densities and the moments of the natural estimators are given. The expressions used in the case of a normal distribution can be easily generalized to the case of non-normal distributions by replacing the mean μ by the median M, and the natural variation 3 σ by Up – M or M – Lp. The estimation of the extreme percentiles Up and Lp is however far from reliable on samples of reasonable size when one uses the traditional moment estimation, as Clements does [5]. The work of Shore [19,20] enables us to obtain estimations much more reliable which will be developed in an example.

2. Existing indices

Starting from the definition of ^min ( ^{- , -} )

= 3

pk

U L

C μ μ

σ , Kane [8] defines ( ) / 3

CPU = U − μ σ and CPL = ( μ − L ) / 3 σ in order to measure the capability of a process in the unilateral tolerance situation. Let us note that the CPU and CPL indices do not take into account the existence of a target which, for the index C

_pk

, is supposedly, implicitedly located at the center of the tolerance interval. When the target is not centered, Kane [8] suggests an index referred to as C

^*_pk

from which he gets the indices ^CPU

^*

⁼ ( ^{U − − − T T μ} ) ^{3 σ and}

( )

*

3

CPL = T − − − L T μ σ , in the case of a unilateral tolerance. In the same way as with the usual indices, the value of these indices is equal to 0 if the previous calculus leads to a negative value. Chan, Cheng and Spiring [2], in the case where one-sided tolerance is required, have suggested generalizing C

_pm

to

( )

*

2 2

3

pmu

U T C

σ μ T

= −

+ − and

( )

*

2 2

3

pml

T L C

σ μ T

= −

+ − .

On the same principle, Vänmann [23] suggests generalizing C

_pmk

to

( )

²

3

2 pmku

C U

T μ

σ μ

= −

+ − and

( )

²

3

2 pmkl

C L

T μ

σ μ

= −

+ − . Moreover in order to generalize all the previous suggestions, Vänmann [23] puts forward two families of indices,

2 2

( , )

3 ( )

pau

U u T

C u v

v T

μ μ

σ μ

− − −

= + − and

2 2

( , )

3 ( )

pal

L u T

C u v

v T

μ μ

σ μ

− − −

= + − on the one hand,

2 2

( , )

3 ( )

pvu

U T u T

C u v

v T

μ

σ μ

− − −

= + − and

2 2

( , )

3 ( )

pvl

T L u T

C u v

v T

μ

σ μ

− − −

= + − on the other hand. We have C

_pau

(0, 0) = CPU , (0, 0) C

_pal

= CPL , (0,1) C

_pau

= C

_pmku

, (0,1) C

_pal

= C

_pmkl

,

(1, 0)

*

C

pvu

= CPU , C

_pvl

(1, 0) = CPL

^*

, C

_pvu

(0,1) = C

^*_pmu

, and C

_pvl

(0,1) = C

^*_pml

. As Vänmann points out, the fact that there is an unilateral tolerance can be interpreted in such a way that a shift of μ away from T towards that tolerance is more serious than a shift of μ towards the opposite side. The indices derived from C

_pvu

( , ) u v and C

_pvl

( , ) u v , which are symmetrical around the target are thus of little interest. From the properties of the estimators of the indices

( , )

C

pau

u v and C

_pal

( , ) u v , Vänmann [23] suggests using C

_pau

(0, 4) and C

_pal

(0, 4) , although these indices are not maximum when µ is on the target T (fig 1). This drawback is not deemed too serious by the author, since a shift of μ away from T to the left (for C

_pau

( , ) u v ) is less important considering the expected percentage of nonconforming than a shift of μ away from T towards U.

3. Suggestions of indices for a normal distribution

Our objective is to build four indices generalizing the usual properties and interpretations of the indices C

_p

, C

_pk

, C

_pm

, and C

_pmk

. Let us mention that for a normal distribution and a target centered in the tolerance interval, C

_p

is linked to the proportion of non conforming

0 0,2 0,4 0,6 0,8 1

-18 -15 -12 -9 -6 -3 0T 3 U6

(0,1) Cpau

(0, 4) Cpau

Figure 1. C_pau(0, 4)

and

C_pau(0,1)

as a function of μ for U = 6, T = 0, and σ = 2.

items when the mean is on the target, and is equal to 1 for a proportion of 0,27% of non conforming. Moreover it is meant as the potential capability of the process, that is to say the maximum capability which one can obtain for a given dispersion, when the mean is on the target. The modern standard of quality deems that a process should not be considered capable if μ is far away from T, even if σ is small. C

_p

which is unrelated to T does not satisfy this requirement. For this reason the indices C

_pk

, C

_pm

, and C

_pmk

, which take into account the location of the process mean as well as the process variability, have been introduced. The 3 of them are maximum and equal to C

_p

when the mean is on the target. The deviation of the mean is taken linearly into account by C

_pk

, so that it is null when the mean reaches the tolerance limits. Thus, the ratio C

_pk

C

_p

allows to determine the position of the mean between the target and the tolerance towards which it deviates. C

_pm

takes the deviation into account in a quadratic way. Thus, it is not null at the tolerance limits, but takes the same value. C

_pmk

being the combination of C

_pk

and C

_pm

allows to take the deviation into account in a "quadratic" way and to obtain a null index when the mean reaches one of the tolerances.

Thus, in the case of unilateral tolerance, in order to keep interpretations similar to the bilateral case, we will require

a) that the potential capability indices C

_pu

and C

_pl

take value 1 for a 0,135% proportion of non conforming, when the mean is on the target

b) that the indices of capability taking the position of the mean related to the target into account, are maximum and equal to potential capability when the mean is on the target.

c) that the indices C

_pku

and C

_pkl

decrease linearly and take value 0 when the mean reaches the tolerance limit

d) that the indices C

_pmu

and C

_pml

decrease in a quadratic way

e) that the indices C

_pmku

and C

_pmkl

decrease in a "quadratic" way and take value 0 when the mean reaches the tolerance limit.

The main difficulty in the building of indices lies in the fact that we have no knowledge of the risk taken when the mean moves away from the target in the opposite direction to the tolerance limit. However, even if there is no tolerance limit, obviously, a production manager cannot accept a too large deviation in a direction even if, a priori, it does not seem too serious to him. Thus, it appears fundamental to us that he should quantify the "not too serious". Is this twice, five times, ten times less serious? We consider thereafter that it has been decided that the risk is k time less serious. The choice of the constant k (>1) being rather approximate we require a last condition:

f) when the means deviates towards the tolerance limit, the capability indices must be independent of the choice of k.

Let us consider for the moment the case of an upper tolerance U, and note

( ) ( ) ( ) ( ( ) )

( )

max / , /

u

T U T T k U T

α = μ − − − μ − , and δ

u

= ( U − T ) α σ

u

. The new indices suggested are defined by :

pu

3

U T

C σ

= − ,

( ¹ )

pku u pu

C = − α C ^max ( ^, ( ) ^/ )

3

U T μ T T μ k

σ

− − − −

= ,

( ¹

²

)

¹²

pmu u pu

C = + δ

⁻

C

( )

( )

²

3

2

max , /

U T

T T k

σ μ μ

= −

⎡ ⎤

+ ⎣ − − ⎦

,

( ¹ ) ( ¹

²

)

¹²

^{( 1 )} ( ¹

²

)

¹²

pmku u pmu u pku u u pu

C = − α C = + δ

⁻

C = − α + δ

⁻

C

( )

( )

( )

( )

²

2

max , /

3 max , /

U T T T k

T T k

μ μ

σ μ μ

− − − −

=

⎡ ⎤

+ ⎣ − − ⎦

.

By using notations similar to Vänmann’s [23], we can write the general formula

( )

( )

( )

( )

²

2

max , /

( , )

3 max , /

pu

U T u T T k

C u v

v T T k

μ μ

σ μ μ

− − − −

=

⎡ ⎤

+ ⎣ − − ⎦

*

2 *2

3

u u

U T uA σ vA

= − − +

, where

( )

( )

*

max , /

A

u

= μ − T T − μ k . This expression gives the four indices for the couples ( , ) u v = (0,0), (1,0), (0,1) and (1,1) again. Note that the letter u used in subscript is an abbreviation of the word upper and is independent of the first parameter of the indices family. The indices

( , )

C

pu

u v are identical Vännman’s C

_pvu

( , ) u v indices [23] when μ − > T 0 , but are different when 0 μ − < T .

It is obvious that when μ = T and C

_pu

= 1, U − = T 3 σ , and a proportion of 0.135% of items is thus beyond U, that satisfies a). If μ = T , then α

_u

= δ

_u

= 0 , thus C

_pku

= C

_pmu

= C

_pmku

=

C

pu

, which satisfies b). If μ = U , C

_pku

= C

_pmku

= 0. In addition, from the previous algebraic expressions, when μ moves away from the target, it is obvious that C

_pku

decreases linearly and C

_pmu

in a quadratic way, hence the conditions c), d), and e). For μ − > T 0 , μ moves away towards U, and C

_pu

( , ) u v is thus independent of k, which satisfies the condition f).

Finally, let us notice some additional properties identical to those of the usual family. From the previous algebraic relations, we have obviously C

_pu

≥ C

_pku

≥ C

_pmku

,

pu pmu pmku

C ≥ C ≥ C , and

_pmku ^{pku pmu}

pu

C C

C = C . The C

_pku

suggested, linked to C

_pu

, gives a precise idea of the position of the mean in the [ ^{T U ;} ] interval. Indeed, if C

_pku

/ C

_pu

= h and μ > T , then ^{U − = μ h U} ( ^{− T} ) . Thus for h = ½ by example, the mean is halfway between the target and the tolerance limit. For μ > T , C

_pmu

< ( U − T ) 3( μ − T ) . In particular for C

_pmu

= 1, ( μ − T ) < ( U − T ) / 3 . Thus a C

_pmu

value of 1 and μ > T , implies that the process mean μ is in the middle third of the interval [T;U]. For μ > T , C

pmku

< [ ⁽ U − T ^{) 3(} μ − T ⁾ ] − ^{1 3} . Thus a C

_pmku

value of 1 and μ > T , implies that the process mean μ is in the middle fourth of the interval [T;U].

To visualize the properties of the four indices, figure 2 represents the evolution of C

_pu

, C

_pku

, C

pmu

, and C

_pmku

according to the variations of the mean μ , in the case where the deviation to the left is considered to be three times less important than to the right.

In a similar way, if the single tolerance is a lower limit L, we obtain the general formulation

( )

( )

( )

( )

²

2

max / ,

( , )

3 max / ,

pl

T L u T k T

C u v

v T k T

μ μ

σ μ μ

− − − −

=

⎡ ⎤

+ ⎣ − − ⎦

*

2 *2

3

l l

T L uA σ vA

= − − +

. Assume that

( ) ( ( ) ) ( ) ( )

( )

max / , /

l

T k T L T T L

α = μ − − − μ − , and δ

l

= ( T − L ) α σ

l

, we find similar algebraic expressions, C

pkl

= − ( ¹ α

l

) C

pl

, ^C

^pml

^{= +} ( ^{1 δ}

^l2

)

⁻¹²

^C

^pl

^{, and C}

^pmkl

^{= − ( 1 α}

^l

^{) C}

^pml

( ^{1 δ}

^l2

)

⁻¹²

^C

^pkl

^{( 1 α}

^l

⁾ ( ^{1 δ}

^l2

)

⁻¹²

^C

^pl

= + = − + , as well as the same properties stated in the case of an upper tolerance.

4. Suggestions of indices for a non-normal distribution

As in the bilateral case, we replace the mean μ by the median M and the natural variation 3 σ by U

_p

− M or M − L

_p

according to each case. Hence the formulas

( )

( )

( )

( )

2 2

max , /

( , )

3 max , /

3

pu

p

U T u M T T M k

C u v

U M

v M T T M k

− − − −

=

⎛ − ⎞ + ⎡ − − ⎤

⎜ ⎟ ⎣ ⎦

⎝ ⎠

,

0 0,2 0,4 0,6 0,8 1

-18 -15 -12 -9 -6 -3 T 0 3 U 6

Figure 2.

C_pu

,

C_pku

,

C_pmu

, and

C_pmku

as a function of μ for U = 6, T = 0, σ = 2, and k = 3.

C

pmku

C

pmu

C

pku

C

pu

and ( ( ) )

( )

( )

2 2

max / ;

( , )

3 max / ;

3

pl

p

T L u M T k T M

C u v

M L

v M T k T M

− − − −

=

⎛ − ⎞ + ⎡ − − ⎤

⎜ ⎟ ⎣ ⎦

⎝ ⎠

.

When ( , ) u v = (0,0), (1,0), (0,1) and (1,1), the previous formulas give generalizations of the indices C

_p

, C

_pk

, C

_pm

, and C

_pmk

, in the case of non-normal distributions. Assuming that

( ) ( ) ( ) ( )

( )

max / , /

u

M T U T T M k U T

α = − − − − , δ

u

= ³ ( U − T ) α

u

⁽ U

p

− M ⁾ ,

( ) ( ) ( ) ( )

( )

max / , /

l

M T k T L T M T L

α = − − − − , and δ

l

= ³ ( T − L ) α

l

⁽ M − L

p

⁾ , we find the same algebraical expressions and thus the same properties as those stated in the case of a normal distribution. Practically, to be able to use these results we need to estimate the percentiles M, U

_p

, and L

_p

. The determination of a percentile is easy when the distribution from which the observations ensue is known. To identify this distribution, the most usual method consists in fitting the distribution to a member of a family covering a great number of usual distributions. To achieve this adjustment the method of the moments requires the identification of 3 or 4 parameters according to the family being used. If the estimate of the mean and of the standard deviation is rather reliable, it is no longer the same for skewness and kurtosis, which are manifestly known for their great dispersion. Thence, the resulting

estimates are not at all reliable for the extreme percentiles U

_p

and L

_p

. Thus for a non negative variable, Shore [19] suggests approaching the p percentile Q

_p

by the relation

1

2 2

1

when 1 2 1

ln when 1 2

1

B

p

A p p

Q p

A p B p

p

⎧ ⎛ ⎞

⎪ ⎜ − ⎟ <

⎪ ⎝ ⎠

= ⎨ ⎪ ⎪ ⎩ ⎛ ⎜ ⎝ − ⎞ ⎟ ⎠ + ≥ .

By identification of the complete moments

¹

0 ⁱ

( )

ⁱ

( ) ( )

i

x dF x x f x d x

μ = ∫ = ∫

and partial

1

1 2 ⁱ

( )

ⁱ

( ) ( )

i x M

M x dF x x f x d x

= ∫ = ∫

≥

of order i smaller or equal to 2, he obtains the coefficients A

1

= exp 2 { [ μ

1

( ) 0.6931 Z + B

1

] } , ^B

¹

^{= 1.7099 0.5} { ^μ

²

^{( ) Z −} [ ^μ

¹

^{( ) Z} ]

²

}

^0.5

^,

[ ]

²

2 1

2 2

( ) 2 ( )

0.6840

M Y M Y

A −

= , and B

2

= 2 [ M Y

1

( ) 0.6931 − A

2

] where ln( ) pour

0 pour

Y Y M

Z Y M

⎧ <

= ⎨ ⎩ ≥ ,

Y the subjacent distribution, μ

_i

( ) Z the ith moment of Z, and M Y

_i

( ) the ith partial moment of Y. Shore’s method leads to better estimates of the percentiles than those obtained by

Clements‘s method [4], insofar as the expected values being similar, the mean squared error is much lower by Shore‘s method [19,20,21].

5. Distribution and moments of the estimators of the indices suggested for a normal distribution.

The studied characteristic of the process is supposed normally distributed with mean μ and

variance σ ² . Two natural estimators of C

_pu

( , ) u v can be considered, differing in the way the

variance σ ² is estimated. We define the estimators C ˆ

_{pu n,}

( , ) u v and C ˆ

_{pu n, −1}

( , ) u v as

*

, 2 *2

ˆ ( , ) ˆ 3 ˆ

u pu n

n u

U T uA C u v

S vA

= − −

+ , and

*

, 1 2 *2

1

ˆ ( , ) ˆ 3 ˆ

u pu n

n u

U T uA

C u v

S vA

−

−

= − −

+ ,

where A ^ˆ

u^*

= ^max { X − T T ^{, (} − X ^{) /} k } , ^{X =} ( ^∑

ⁿⁱ⁼¹

^X

ⁱ

) ^{n , S}

ⁿ²

^{= ∑}

ⁿⁱ⁼¹

^{( X}

ⁱ

^{− X )}

²

^{n , and}

( )

²

( )

2

1 ⁿ1

1

n i i

S

₋

= ∑

₌

X − X n − ^.

The two estimators are related by ^{C ˆ}

^{pu n,} −¹

^{( , ) u v} = ( ( ⁿ − ¹ ) ⁿ )

^{1 2}

^{C ˆ}

^{pu n,}

( ^{u v n , (} − ^{1 ) n} ) . The study of the statistical properties of these estimators is facilitated insofar as C

_pu

( , ) u v can be expressed according to Chen’s and Pearn’s C u v

^''_p

( , ) index [4]. Let us recall that for an interval [L;U] and a target T not centered, they introduce the family

* *

''

2 2

( , )

p

3

d uA C u v

σ vA

= −

+ , u ≥ 0 , v ≥ 0 ,

where ^A

^*

^{= max} { ^d

^*

^{( μ − T ) D d T}

^u

^,

^*

^{( − μ ) D}

^l

} ^,

{ }

max ( )

_u

, ( )

_l

A = d μ − T D d T − μ D = dA d

^{* *}

, D

u

= − U T , D

_l

= − T L , and d

^*

= ^min { D D

u

^,

l

} .

In the case of a single upper tolerance U, considering that the risk of a deviation towards the left of the target is k time less serious than towards the upper limit U can be interpreted as the positioning of a lower limit L so that T − = L D

_l

= kD

_u

. In these conditions d = + ( ¹ k D )

u

² ,

*

d = D

u

, A

u^*

= A

^*

= ^max { μ − T T ^, ( − μ ) k } , ^{A = dA d}

^{* *}

^{= +} ( ^{1 k A} )

^*

^{2 , and}

( ) ( )

* * * 2

" *

2 2

2 * 2 * 2

( , ) ,

3 3

u u

pu p

u

D uA d uA

C u v C u v d d

vA v d d A

σ σ

− − ⎛ ⎞

= + = + = ⎜ ⎝ ⎟ ⎠

( )

(

²

)

"

, 4 1 C

p

u v k

= + . (1) Assuming that ^{δ = n} ( ^{μ − T} ) ^{σ , λ δ =}

²

^{, and D}

^*

^{= nd}

^*

^σ , Grau [8] gives the r-th

moment of C ˆ

^''_{p n,}

( , ) u v in the form

(

^'',

)

^{* 2 2}

⁽ _{( )} ^{) ( )}

0 0

1 2

ˆ ( , ) ( ) ( , )

3 2 2 !

r i j j

r r

i

p n r n

i j

r D e c a b

E C u v u i j

i j c

λ

δ γ

π

− − ∞

= =

Γ − Γ

⎛ ⎞

= ∑ − ⎛ ⎞ ⎜ ⎟ ⎝ ⎠⎝ ⎜ ⎟ ⎠ × ∑ Γ , (2) with ( , )

^* 2 1

( , ; ; ) ( ) 1

^* 2 1

( , ; ; )

i i

j

n u l

u l

d d

i j F a b c z F a b c z

D D

γ = ^⎡ ⎢ ^⎛ ⎜ ^⎞ ⎟ + − ^⎛ ⎜ ^⎞ ⎟ ^⎤ ⎥

⎢ ⎝ ⎠ ⎝ ⎠ ⎥

⎣ ⎦

,

where ( ) ^{− u}

ⁱ

should be interpreted as 1 when i = 0, also for the case u = 0, and

2

F a b c z

1

( , ; ; ) is the Gaussian hypergeometric function (Abramowitz and Stegun [1]) with parameters

2

a = r , ^{b = + +} ( ^{1 i j} ) ^{2 , c =} ( ^{n i + + j} ) ^{2 ,} z

u

= − ¹ ( d D

u

)

²

v , and z

l

= − ¹ ( d D

l

)

²

v . In order to distinguish the properties of the estimators of the indices C

_pu

( , ) u v and ( , ) C

_pl

u v subsequently, we assume B

u

= n U ( − T ) ^/ σ , which in that case is equal to D

^*

. From relations (1) and (2) we deduce the r-th moment of C ˆ

_{pu n,}

( , ) u v ,

(

^,

)

^{2 2}

⁽ _{( )} ^{) ( )}

^,

0 0

1 2

ˆ ( , ) ( ) ( , )

3 2 2 !

r i j j

r r

i u

pu n r u n

i j

r B e c a b

E C u v u i j

i j c

λ

δ γ

π

− − ∞

= =

Γ − Γ

⎛ ⎞⎛ ⎞

= ∑ − ⎜ ⎟⎜ ⎝ ⎠ ⎝ ⎟ ⎠ × ∑ Γ , (3)

with ^γ

^{u n,}

^{( , ) i j = ⎡} _⎣

²

^{F a b c}

¹

( ^{, ; ;1 − + − v} ) ( ) ¹

^j

^k

⁻ⁱ²

^{F a b c}

¹

( ^{, ; ;1 − v k /}

²

) ^⎤ _⎦ ^.

In the same way we obtain ^{E C} ( ^ˆ

^{pl n,}

^{( , ) u v} )

^r

by replacing B

_u

by B

l

= n T ( − L ) σ and

,

( , )

u n

i j

γ by ^γ

^{l n,}

^{( , ) i j = ⎡} _⎣ ^k

⁻ⁱ²

^{F a b c}

¹

( ^{, ; ;1 − v k /}

²

) ^{+ − ( ) 1}

^{j 2}

^{F a b c}

¹

^{( , ; ;1 − v ) ⎤} _⎦ ^.

5.1 Estimation and distribution of C

_pu

and C

_pl

As previously, if we consider that the choice of the constant k can be interpreted as the positioning of a lower limit L so that T − = L D

_l

= kD

_u

, then C

_pu

can be expressed according to the usual index C

_p

by the relation C

pu

= ( ^{2 1} ( + k ) ) C

p

. From the density of probability and moments of ˆ

_{, 1}

C

p n₋

given by Kotz and Lovelace [10], we obtain

( )

^{( )}

^{( )}

2

, 1

( 1) / 2

( 1) / 2

ˆ ( 3) / 2

( 1)

( ) , 0

( 1) 2 2

pu pu n

n n

n C x

pu n

C

pu

n C

f x e x

C n x

−

− − −

−

⎛ ⎞

= Γ − − ⎜ ⎝ ⎟ ⎠ > ,

and ^{E C} ( ^ˆ

^{pu n, −1}

)

^r

^{= ( ( n − 1) 2 )}

^r2

^Γ _Γ ^{( (} ₍ ⁿ _{( n} ^{− − 1} _{− 1 2 )} ^{r )} ₎ ^{2 ) C}

^pur

^.

In particular ^{E C} ( ^ˆ

^{pu n, −1}

) ^{= b C}

^{−f1 pu}

^{and V C} ( ^ˆ

^{pu n, −1}

) ^{= ⎛ ⎜} _{⎝ n} ⁿ ₋ ^{− 1} ₃ ^{− b}

^−f2

^{⎞ ⎟} _⎠ ^C

^2pu

^where

( )

( )

( )

( ^{1 2} )

2

1 2 2

f

b n

n n

Γ −

= − Γ − .

Moreover, since ^{C ˆ}

^{pu n,}

= ( ^{n n} ( − ¹ ) )

^{1 2}

^{C ˆ}

^{pu n,} −¹

,

( )

( )

^{( )}

^{( )}

2

,

1 2

/ 2

ˆ

( )

( 3) / 2

, 0

( 1) 2 2

pu pu n

n n

n C x

pu C n

pu

n C

f x e x

C n x

− −

−

⎛ ⎞

= Γ − ⎜ ⎝ ⎟ ⎠ > ,

and ( ) ^ˆ

^,

^{( ) 2}

²

^{( (} _{( (} ¹ ₎ ⁾ ₎ ^{2 )}

1 2

r r r

pu n pu

n r

E C n C

n

Γ − −

= Γ − .

In particular ^{E C} ( ) ^ˆ

^{pu n,}

^{= c C}

^{−f1 pu}

^{and V C} ( ) ^ˆ

^{pu n,}

^{= ⎛ ⎜} _{⎝ n} ⁿ _{− 3} ^{− c}

^−f2

^{⎞ ⎟} _⎠ ^C

^2pu

^where

( )

( )

( )

( ^{1 2} )

2

f

2 2 c n

n n

Γ −

= Γ − .

In a similar way we obtain the density and the moments of ˆ

_,

C

pl n

and ˆ

_{, 1}

C

pl n₋

replacing C

_pu

by C

_pl

.

5.2 Estimation and distribution of C

_pku

and C

_pkl

If we consider that the choice of the constant k can be interpreted as the positioning of a lower limit L so that T − = L D

_l

= kD

_u

, then C

_pku

= C

_pu

(1, 0) = C

^''_p

(1, 0) = C

^"_pk

according to (1).

Since D

^*

= B

_u

, assuming that D = n

^{1 2}

d σ , according to the appendix, we obtain

,

1 1

ˆ 1

0 1

( , ) , 0

( )

( , ) , 0

pku n

C

J x t dt x

f x

J x t dt x

⎧−

∞

<

= ⎨ ⎪

⎪ >

⎩

∫

∫ ^,

where ( )

²

_{( )}

2 ²

( )

²

1

1 2 1

( , )

3 3

u u

u

K Yu u

B t B B t

J x t f f B t

x x x

⎛ ⎛ − ⎞ ⎞ ⎛ − ⎞

⎜ ⎜ ⎟ ⎟ ⎜ ⎟

= ⎜ ⎜ ⎝ ⎜ ⎝ ⎟ ⎠ ⎟ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠ , and

( ) ( )

( )

( ) 1

Yu

2

f y y k k y

y φ δ φ δ

= − + + when y > 0, with ( ) φ x the probability density of a N(0,1) distribution. By substitution, the density can still be written

2 , 2

' 1 ˆ

' 0 1

( , ) , 0

( )

( , ) , 0

u

pku n u

B

C B

J x t dt x

f x

J x t dt x

⎧−

∞

<

= ⎨ ⎪

⎪ >

⎩

∫

∫ ^,

where ( )

2 2

' 1

( , ) 2

3 3

u u

K Yu

B t B t

J x t f f t

x x x

⎛ ⎛ − ⎞ ⎞ ⎛ − ⎞

⎜ ⎟

= ⎜ ⎝ ⎜ ⎜ ⎝ ⎟ ⎟ ⎠ ⎟ ⎠ ⎜ ⎜ ⎝ ⎟ ⎟ ⎠ . Since C ˆ

_{pku n}, ₋1

= ( ( n − 1 ) n )

^{1 2}

C ˆ

_{pku n},

, we deduce

, 1

1 1

ˆ 1

0 1

( , ) , 0

( )

( , ) , 0

pku n

C

L x t dt x

f x

L x t dt x

−

⎧−

∞

<

= ⎨ ⎪

⎪ >

⎩

∫

∫ ^,

where ( )

²

_{( )}

2 ²

( )

²

1

1 2 1

1 1

( , )

3 3

u u u

K Yu u

B t B B t

n n

L x t f f B t

n x x n x

⎛ − ⎛ − ⎞ ⎞ − ⎛ − ⎞

⎜ ⎜ ⎟ ⎟ ⎜ ⎟

= ⎜ ⎜ ⎝ ⎜ ⎝ ⎟ ⎠ ⎟ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠

. Since D

^*

= B

_u

, we find the density of ˆ

^'' _{, 1}

ˆ

_{, 1}

pk n pku n

C

₋

= C

₋

given by Pearn, Lin and Chen [18].

Only the moments of order 1 and 2 of ˆ

^'' _{, 1}

C

pk n₋

are explicitly given by Pearn and Chen [13].

Thus, we use the equation (3) for the r-th moment which, in addition, leads to expressions of the first two moments simpler than those given by Pearn and Chen [13].

(

^,

)

²

^{( (} _{( ( )} ⁾ ₎ ⁾

0

1 1 2

ˆ ( 1)

3 2 2 1 2

r r i

r i u

pku n r

i

r B e n r

E C i n

λ

π

− −

=

Γ − −

⎛ ⎞⎛ ⎞

= ∑ − ⎜ ⎟⎜ ⎝ ⎠ ⎝ ⎟ ⎠ × Γ −

( )

( )

2

0

2 1 2 ( , )

!

j j

u j

i j i j

j

δ γ

∞

=

× ∑ Γ + + ^,

where γ

u

^{( , ) 1} i j = + − ( ) ¹

^j

k

⁻ⁱ

. In particular, Grau [8],

( ^ˆ

^,

)

¹

¹

¹

^{( )}

²²

3 2

pku n f pku

k e

E C c C

n

δ δ

δ

π

− −

−

⎡ + ⎛ ⎞ ⎤

= ⎢ ⎢ ⎣ + ⎜ ⎜ ⎝ Φ − − ⎟ ⎟ ⎠ ⎥ ⎥ ⎦ ,

( ^ˆ

^,

)

²

_{( 3)}

²

^{2 ( ) ( 1}

¹

⁾

²²

^{( )}

9 2

pku n pku

U T k

n e

E C C

n n

δ

δ δ

σ π

− −

⎡ − + ⎛ ⎞

⎢

= − ⎢ ⎣ − ⎜ ⎜ ⎝ − Φ − ⎟ ⎟ ⎠

(

²

)

²

_{( ) { ( ) }}

2

1

2

1 1

18 9 2 2 1 2

k k e

n n

δ

δ

δ δ δ

π δ

−

−

−

⎛

−

⎞ ⎤

+ ⎥

+ + ⎜ ⎜ ⎝ − Φ − + − Φ − ⎟ ⎟⎥ ⎠⎦ .

( ^ˆ

^{pku n, 1}

)

^r

E C

₋

, ^{E C} ( ^ˆ

^{pku n, −1}

) ^{and E C} ( ^ˆ

^{pku n, −1}

)

²

are obtained without difficulty since

( ^{C ˆ}

^{pku n,−1}

)

^r

⁼ ( ^{( n − 1 ) n} )

^r2

( ^{C ˆ}

^{pku n,}

)

^r

^.

For ˆ

_,

pkl n

C and ˆ

_{, 1}

pkl n

C

₋

we obtain similar results replacing f

_Yu

( ) y by

( ) ( )

( )

( ) 1

Yl

2

f y k k y y

y φ δ φ δ

= − + + , B

_u

by B

_l

, and γ

_u

( , ) i j by γ

l

^{( , )} i j = k

⁻ⁱ

+ − ( ) ¹

^j.

5.3 Estimation and distribution of C

_pmu

and C

_pml

If we consider that the choice of the constant k can be interpreted as the positioning of a lower limit L so that T − = L D

_l

= kD

_u

, then ^C

^pmu

^{= C}

^pu

^{(0,1) = C}

^''p

^{⎛ ⎜} _⎝ ^0, ( ) ^d

^*

^d

²

^{⎞ ⎟} _⎠ according to (1).

From the appendix,

,

1

ˆ 2

( )

0

( , ) , 0

pmu n

f

C

x = ∫ J x t dt x > ^,

where J x t

2

( , ) = f

_K

( G x ( ) 1 ( − t ) ) f

_Yu

( G x t ( ) 2 ) x

⁻¹

( G x ( ) )

²

, and G x ^{( )} = ( B

u

( ) ³ x )

²

, or

,

( ) '

ˆ 2

( )

0

( , ) , 0

pmu n

G x

f

C

x = ∫ J x t dt x > ^,

where J x t

2^'

( , ) = f

_K

( G x ( ) − t f ) ( )

_Yu

t 2 x G x

⁻¹

( ) .

Since ^{C ˆ}

^{pmu n,} −¹

= ( ( ⁿ − ¹ ) ⁿ )

^{1 2}

^{C ˆ}

^{pu n,}

( ^{0, ( n} − ^{1 ) n} ) , we deduce

, 1

1

ˆ

( )

0 2

( , ) , 0

pmu n

f

C

x L x t dt x

−

= ∫ >

where L x t

2

( , ) = f

_K

( ( ( n − 1 n G x ) ( ) 1 ( − t ) ) f

_Yu

( G x t ( ) 2 ( ) ( n − 1 n x ) (

⁻¹

G x ( ) )

²

. The moments are obtained from the equation (3):

(

^,

)

^{2 2}

⁽ _{( )} ^{) ( )}

^,

0

1 2

ˆ ( )

3 2 2 !

r j j

r

u

pmu n r u n

j

c a b

B e

E C j

j c

λ

δ γ

π

− ∞

=

Γ − Γ

⎛ ⎞

= ⎜ ⎝ ⎟ ⎠ × ∑ Γ ^,

(

^{, 1}

)

^{2 2 2}

⁽ _{( )} ^{) ( )}

^{, 1}

0

1 1 2

ˆ ( )

3 2 2 !

r r j j

r u

pmu n r u n

j

c a b

B

n e

E C j

n j c

λ

δ γ

π

− ∞

− −

=

Γ − Γ

− ⎛ ⎞

⎛ ⎞

= ⎜ ⎝ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠ × ∑ Γ ^,

where a = r 2 , b = + (1 j ) 2 , c = ( n + j ) 2 , ^γ

^{u n,}

^{( ) j = + − 1} ( ) ¹

^{j 2}

^{F a b c}

¹

( ^{, ; ;1 − k}

⁻²

) ^{, and}

(

¹

) ^{( )} (

^{1 2}

)

, 1

( )

2 1

, ; ; 1

^j 2 1

, ; ;1 ( 1)

u n

j F a b c n F a b c n n k

γ

₋

=

⁻

+ − − −

^{− −}

.

For ˆ

_,

pml n

C and ˆ

_{, 1}

pml n

C

₋

, we obtain similar results replacing f

_Yu

( ) y by f

_Yl

( ) y , B

_u

by B

_l

,

,

( )

u n

j

γ by ^γ

^{l n,}

^{( ) j =}

²

^{F a b c}

¹

( ^{, ; ;1 − k}

⁻²

) ^{+ − ( ) 1}

^j

^{, and γ}

^{u n, −1}

^{( ) j by}

(

^{1 2}

) ^{( )} (

¹

)

, 1

( )

2 1

, ; ;1 ( 1) 1

^j 2 1

, ; ;

l n

j F a b c n n k F a b c n

γ

₋

= − −

^{− −}

+ −

⁻

.

5.4 Estimation and distribution of C

_pmku

and C

_pmkl

If we consider that the choice of the constant k can be interpreted as the positioning of a lower limit L so that T − = L D

_l

= kD

_u

, then ^C

^pmku

^{= C}

^pu

^{(1,1) = C}

^''p

^{⎛ ⎜} _⎝ ^1, ( ) ^d

^*

^d

²

^{⎞ ⎟} _⎠ . From the

appendix, we obtain

,

1

ˆ 1

0

( , ) , 1 0

( ) 3

( , ) , 0

pmku n

C

J x t dt x

f x

J x t dt x

⎧−

∞

− < <

= ⎨ ⎪

⎪ >

⎩

∫

∫