INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,

MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)

ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XII, December 2025

CASP-CUSUM Schemes Based on Truncated Gompertz Family of

Distribution

Dr. G. Venkatesulu¹, Dr. P. Mohammed Akhtar², B. Sainath ³, Dr. B.R. Narayana Murthy⁴

¹Lecturer in Statistics Govt. (U.G&P. G) College, Ananthapuramu-515001

²Professor Dept. of Statistics, Sri Krishanadevaraya University, Ananthapuramu-515003

³Assistant Professor(C), Dept. of OR&SQC, Rayalaseema University, Kurnool-518007

⁴Lecturer in Statistics Govt. (U.G&P. G) College, Ananthapuramu-515001

DOI : https://doi.org/10.51583/IJLTEMAS.2025.1412000113

Received: 29 December 2025; Accepted: 03 January 2026; Published: 12 January 2026

ABSTRACT:

acceptance sampling plan was adopted to study mainly for valid conclusions with regard to consideration accept

or reject of the finished products. In this way numbers of optimal techniques were developed to increase and

control the quality of the products. Basing on the assumption the variable with regard to quality characteristic

is distributed accordingly to certain probability law. In our study we optimized CASP-CUSUM Schemes based

on the assumption that the continuous variable which is under the consideration follows a Truncated

Expoentiated Gompertz distribution utilized in Statistical Quality Control and Reliability analysis. In particular

the distribution is meant for estimating the optimal truncated point and probability of acceptance of lot. The

operating characteristic and Average run length values are presented. The results are illustrated by figures.

Keywords: CASP-CUSUM Schemes, Optimal Truncated point, Truncated Expoentiated Gompertz

Distribution.

INTRODUCTION

Acceptance sampling is an inspecting procedure applied in statistical quality control. Acceptance sampling is a

part of operations management and services quality maintenance. It is important for industrial, but also for

business purposes helping the decision making a process for the purpose of quality management. Producers are

very careful about the quality of their products so that they do not face any difficulty in the acceptance when the

consumer comes to buy them.

Acceptance sampling is most likely to be useful in the situations when testing is destructive, or when the cost of

100% inspection is extremely high, or when 100% inspection is not technologically feasible or would require so

much calendar time that the production schedule would be seriously impacted.

It is well known that the exponential distribution having the constant failure rate (hazard function is constant)

whereas the Gompertz, and Generalized exponential distributions having either, increasing or decreasing failure

rate (hazard rate). Further, in these distributions failure rate depends upon the shape parameter of the respective

distributions. Thus, these distributions are very flexible for modeling lifetime components by selecting the

appropriate value of the shape parameter. Thus, these distributions have been used to make optimal decisions

with regard to quality, reliability and quality management.

The Gompertz distribution is one of the classical mathematical models that represent survival functions based

on laws of mortality. This distribution plays an important role in modeling human mortality and fitting actuarial

tables. The Gompertz distribution was first introduced by Gompertz [02]. It has been as a growth mode and

also used to fit the tumor growth.

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Gupta and Kundu proposed a generalized exponential distribution. Gupta and Kundu provided a gentle

introduction of the generalized exponential distribution and discussed some of its recent developments in the

acceptance sampling plans to determine Operating Characteristics such as producer’s risk, consumer risk and

sample size required to ensure mean lifetime. Mudholkar et al.Introduced the exponentiated Weibull family is

an extension of the Weibull family obtained by adding an additional shape parameter. Its properties studied in

detail by Gera , Mudholkar and Hutson and Nassar and Eissa. Nadarajah and Gupta [90] introduced the different

closed form for the moments with no restrictions imposed on the parameters of the Exponentiated Weibull

distribution.

Shanmugapriya and Lakshmi applied the Exponentiated Weibull model for analyzing bathtub failure rate data.

Nadarajah et al.reviewed the Exponentiated Weibull distribution and included some of its properties. Gupta et

al. Studied the exponentiated gamma (EG) distributions and the Exponentiated Pareto (EP) distribution.

Nadarajah considered five kinds of EP distributions and some of their properties. El-Gohary et al. introduced

the three-parameter generalized Gompertz distribution by exponentiation the Gompertz distribution to the model

life of the components.

S. Poetrodjojo et.al Proposed Optimal CUSUM schemes for monitoring variability in the mean level of the

process rather than process variability. They studied the use of Markov chain approach in calculating the average

run length (ARL) of CUSUM schemes when controlling variability. They considered ‘S’ and ‘S²’, where ‘S’ is

the standard deviation of a normal process to determine CUSUM schemes. The control statistic ‘S^2’is used to

prove that the CUSUM scheme is superior to Exponentially Weighted Moving Average (EWMA) by means of

any large or small increase in the variability of the normal process. Finally, they proved that the control statistic

‘S^2’and ‘S’ are uniformly better than the control statistic log S².

Shangli Zhang and Wenhao Gui proposed acceptance sampling plan based on the truncated life test for the

Gompertz distribution at different acceptance numbers, consumer's risk, confidence levels and values of the ratio

of the experimental time to be specified mean lifetime, the minimum sample size required to ensure the specified

mean lifetime are obtained. The operating characteristic function values and associated producer's risk are also

determined. Finally, real examples are provided to illustrate the acceptance sampling plans.

All these research work related to evaluating reliability indices, Operating Characteristics under certain

acceptance sampling plans. Based on this understanding, in this study, a new three-parameter distribution called

as an Exponentiated Gompertz distribution and study some of the statically properties.

Exponentiated Gompertz Distribution

The non-negative random variable X is said to have an Exponentiated Gompertz distribution if its P.D.F is

given by

θ−1

αx−1)

αx −λ(e

αx−1

−λ(e

)

(

)

[

]

f x; λ, α, θ = θλαe e

1 − e

Where λ, α, θ, x > 0

…... (3.2.1)

Properties of Exponentiated Gompertz Distribution

The probability density function of the Exponentiated Gompertz distribution is

θ−1

αx−1)

αx −λ(e

αx−1

−λ(e

)

(

)

[

]

f x; λ, α, θ = θλαe e

1 − e

.….. (3.2.2)

The Median of the Exponentiated Gompertz distribution is

1

휃

( )

푀푒푑_퐸퐺퐷푋 = _훼푙푛 1 − _휆ln(1 − 푞 ) , 0 <q<1

[

]

…… (3.2.3)

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The Mode of the Exponentiated Gompertz distribution is

푀표푑_Gom(x) ₌_훼¹푙푛 (¹_휆) 푎푡 ꢀ = 1

...... (3.2.4)

The survival function of the Exponentiated Gompertz distribution is

ꢄ

ꢂꢃ−1)

−휆(ꢁ

( )

푆 푥 = 1 − (1 − 푒

)

…… (3.2.5)

The cumulative density function of the Exponentiated Gompertz distribution is

ꢄ

ꢂꢃ−1)

−휆(ꢁ

( )

퐹 푥 = (1 − 푒

)

…… (3.2.6)

The hazard function of the Exponentiated Gompertz distribution is

θ−1

αx−1)

αx −λ(e

θλαe

αx−1

−λ(e

)

e

[1−e

ꢂꢃ−1)

]

( )

ℎ 푥 =

…… (3.2.7)

휃

)

−ꢅ(ꢆ

1−(1−ꢁ

Truncated Exponentiated Gompertz Distribution

It is the ratio of probability density function of the Exponentiated Gompertz distribution to their corresponding

cumulative distribution function at the point B.

The random variable X is said to follow a truncated Exponentiated Gompertz Distribution as

θ−1

αx−1)

αx −λ(e

θλαe

αx−1

−λ(e

)

e

[1−e

ꢂꢃ−1)

]

f_B(x) 

λ >0, α and θ > 0

휃

)

…... (3.2.3)

−ꢅ(ꢆ

1−(1−ꢁ

Where’ B’ is the upper truncated point of the Exponentiated Gompertz Distribution.

Description of The Plan and Type- C OC Curve

Battie [3] has suggested the method for constructing the continuous acceptance sampling plans. The procedure,

suggested by him consists of a chosen decision interval namely, “Return interval” with the length h’, above the

decision line is taken. We plot on the chart the sum S 

(X k )X 's(i 1,2,3........) are distributed



m

i

1

i

independently and k₁is the reference value. If the sum lies in the area of normal chart, the product is accepted

and if it lies of the return chart, then the product is rejected, subject to the following assumptions.

When the recently plotted point on the chart touches the decision line, then the next point to be plotted at the

maximum, i.e., h+h’

When the decision line is reached or crossed from above, the next point on the chart is to be plotted from the

baseline.

When the CUSUM falls in the return chart, network or a change of specification may be employed rather than

outright rejection.

The procedure in brief is given below.

1. Start plotting the CUSUM at 0.

2. The product is accepted when S 

(X k)  h; when S_m< 0, return cumulative to 0.



m

i

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3. When h <S_m< h+h’ the product is rejected: when S_mcrossed h, i.e., when S_m>h+h’ and continue rejecting

product until S_m>h+h’ return cumulative to h+h’

The type-C, OC function, which is defined as the probability of acceptance of an item as function of incoming

quality, when sampling rate is same in acceptance and rejection regions. Then the probability of acceptance P

(A) is given by

L(0)

P(A) 

…… (2.1)

L(0)  L'(0)

Where L (0) = Average Run Length in acceptance zone and

L’ (0) = Average Run Length in rejection zone.

Page E.S. [8] has introduced the formulae for L (0) and L’ (0) as

N(0)

L(0) 

…… (2.2)

1 P(0)

N'(0)

L'(0) 

…… (2.3)

1 P'(0)

Where P (0) =Probability for the test starting from zero on the normal chart,

N (0) = ASN for the test starting from zero on the normal chart,

P’ (0) = Probability for the test on the return chart and

N’ (0) = ASN for the test on the return chart

He further obtained integral equations for the quantities

P (0), N (0), P’ (0), N’ (0) as follows:

h

.….. (2.4)

…… (2.5)

,

^{P(z)  F(k  z) }^{P(y) f (y  k  z)dy}

1

0

h

,

^{N(z) 1}^{N(y) f (y  k  z)dy}

1

0

B

h

P'(z) 

.….. (2.6)

…… (2.7)

1



^{f (y)dy }^{P'(y) f (y  k  z)dy}

k₁z

0

h

^{N'(z) 1}^{N'(y)f (y  k  z)dy,}

1

0

h

^{F(x) 1} ^{f (x)dx :}

A

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k₁z

F(k₁ z) 1

f (y)dy



A

and z is the distance of the starting of the test in the normal chart from zero.

METHOD OF SOLUTION

We first express the integral equation (2.4) in the form

d

…… (3.1)

^{F(X )  Q(X ) }^R(x,t)F(t)dt

c

where

F(X )  P(z),

Q(X )  F(k  z),

R(X ,t)  f (y  k  z)

d

Let the integral

be transformed to

^{I } ^{f (x)dx}

c

d c ^d

d c

..…. (3.2)

I 

f (y)dy 

a f (t )



i



2

c

2x  (c  d)

y 

Where

where a_i’s and t_i’s respectively the weight factor and abscissa for the Gass-Chibyshev

d  c

polynomial, given in Jain M.K. and et al [4] using (3.1) and (3.2),(2.4) can be written as

d  c

F(X )  Q(X )

a R(x,t )F(t )



i

2

..…. (3.3)

Since equation (3.3) should be valid for all values of x in the interval (c, d), it must be true for x=t_i, i = 0 (1) n

then obtain.

.….. (3.4)

d  c

F(t_i)  Q(t_i) 

a R(t ,t )F(t )



i

j

i

2

j  0(1)n

Substituting

F(t_i)  F ,Q(t_i)  Q_i,i  0(1)n,in (3.4), we get

i

d  c

F₀ Q₀

[a₀R(t₀,t₀)F₀ a₁R(t₀,t₁)F  ...........a_nR(t₀,t_n)F_n)]

1

2

d c

F  Q₁

[a₀R(t₁,t₀)F₀ a₁R(t₁,t₁)F ...........a_nR(t₁,t_n)F_n)]

1

2

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…………

………..

………

………….

………..

……….

d c

F_n Q_n

[a₀R(t_n,t₀)F₀ a₁R(t_n,t₁)F ...........a_nR(t_n,t_n)F_n)]

…… (3.5)

1

2

In the system of equations except F_i, i= 0,1,2……………n are known and hence can be solved for F_i, we

solved the solved the system of equations by the method of Iteration. For this we write the system (3.5) as

[1Ta₀R(t₀,t₀)]F₀ Q₀T[a₀R(t₀,t₀)F₀ a₁R(t₀,t₁)F .........a_nR(t₀,t_n)F_n)]

1

[1Ta₁R(t₁,t₁)]F  Q T[a₀R(t₁,t₀)F₀ a₁R(t₁,t₁)F .........a_nR(t₁,t_n)F_n)]

1

…………..

..…………

…………….

……………..

…………

[1Ta_nR(t_n,t_n)]F_n Q_nT[a₀R(t_n,t₀)F₀ a₁R(t_n,t₁)F .........a_nR(t_n,t_n)F_n)]

….. (3.6)

1

d  c

T 

Where

2

To start the Iteration process, let us put F  F₂ ....  F_n 0 in the first equation of (3.6), we then

1

obtain a rough value of

F₀. Putting this value of F₀and F  F₂ ....  F_n 0 on the second equation, we get

1

the rough value and so on. This gives the first set of values

F_ii= 0,1,2,...,n which are just the refined values

F

1

of

F

i

i= 0,1,2,…,n. The process is continued until two consecutive until two consecutive sets of values are

obtained up to a certain degree of accuracy. In the similar way solutions P’ (0), N (0), N’ (0) can be obtained.

COMPUTATION OF ARL AND P (A)

We developed computer programs to solve these equations and we get the the following results given in the

tables (4.1) to (4.18).

TABLE-4.1

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1, θ=2, k=2, h=0.06, h’=0.06

B

L(0)

L’(0)

P(A)

2.5

2.4

2.3

2.2

2.1

1447.3428

1569.5541

1795.8468

2287.8896

3870.2703

1.7892110

1.7893583

1.7895780

1.7899058

1.7903954

0.9987653494

0.9988612533

0.9990044832

0.9992182851

0.9995375872

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TABLE-4.2

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1, θ=2, k=2, h=0.08, h’=0.08

B

L(0)

L’(0)

P(A)

2.5

2.4

2.3

2.2

2.1

1621.8329

1759.4933

2014.6801

2571.1130

4372.3262

2.4279270

2.4282882

2.4288278

2.4296329

2.4308357

0.9985052347

0.9986218214

0.9987958670

0.9990559220

0.9994443655

TABLE-4.3

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1, θ=2, k=2, h=0.10, h’=0.10

B

L(0)

L’(0)

P(A)

2.5

2.4

2.3

2.2

2.1

1842.8021

2000.3245

2292.4983

2931.0557

5015.4434

3.7758284

3.7769217

3.7785528

3.7809896

3.7846324

0.9979552031

0.9981154203

0.9983544946

0.9987117052

0.9992460012

TABLE-4.4

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.2, θ=2, k=3, h=0.06, h’=0.06

B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

106973.5391

122075.7188

153725.0469

253084.1250

1.5458745

1.5458764

1.5458788

1.5458828

0.9999855757

0.9999873638

0.9999899268

0.9999939203

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3.1

5188230.0000

1.5458885

0.9999997020

TABLE-4.5

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.2, θ=2, k=3, h=0.08, h’=0.08

B

L(0)

L’(0)

P(A)

3.6

3.5

3.4

3.3

3.2

134861.6719

148170.4375

175952.4844

242171.3281

563048.9375

1.8897232

1.8897254

1.8897289

1.8897340

1.8897420

0.9999859929

0.9999872446

0.9999892712

0.9999921918

0.9999966621

TABLE-4.6

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.2, θ=2, k=3, h=0.10, h’=0.10

B

L(0)

L’(0)

P(A)

3.7

3.6

3.5

3.4

3.3

186483.0625

203436.1250

232224.3750

300192.6875

523740.9375

2.4302924

2.4302957

2.4303002

2.4303079

2.4303186

0.9999869466

0.9999880791

0.9999895096

0.9999918938

0.9999953508

TABLE-4.7

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.4, θ=2, k=3, h=0.06, h’=0.06

B

L(0)

L’(0)

P(A)

3.5

3.4

76613.9688

85126.6563

1.4064300

1.4064312

0.9999816418

0.9999834895

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3.3

3.2

3.1

101630.8438

143306.9063

362175.9688

1.4064331

1.4064358

1.4064401

0.9999861717

0.9999901652

0.9999961257

TABLE-4.8

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.4, θ=2, k=3, h=0.08, h’=0.08

B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

3.1

90764.4609

102110.0547

124934.7266

186306.3438

732377.6250

1.6268270

1.6268291

1.6268327

1.6268378

1.6268451

0.9999820590

0.9999840856

0.9999870062

0.9999912977

0.9999977946

TABLE-4.9

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.4, θ=2, k=3, h=0.10, h’=0.10

B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

3.1

109864.8047

125646.5547

159035.2656

264442.6875

11371056.0000

1.9291357

1.9291395

1.9291455

1.9291544

1.9291679

0.9999824166

0.9999846220

0.9999878407

0.9999927282

0.9999998212

TABLE-4.10

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.6, θ=2, k=3, h=0.06, h’=0.06

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B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

3.1

62140.7734

68048.5391

79530.0078

106772.3984

214731.3281

1.3156629

1.3156638

1.3156652

1.3156675

1.3156708

0.9999788404

0.9999806881

0.9999834299

0.9999876618

0.9999938607

TABLE-4.11

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.6, θ=2, k=3, h=0.08, h’=0.08

B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

3.1

68987.7500

75938.0313

90050.4609

123341.9297

275019.5000

1.4703773

1.4703790

1.4703815

1.4703853

1.4703906

0.9999786615

0.9999806285

0.9999836683

0.9999880791

0.9999946356

TABLE-4.12

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.6, θ=2, k=3, h=0.10, h’=0.10

B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

3.1

77300.7266

85615.9922

102331.5625

145200.3438

377011.9688

1.6663281

1.6663307

1.6663349

1.6663407

1.6663494

0.9999784231

0.9999805093

0.9999837279

0.9999884963

0.9999955893

TABLE-4.13

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.8, θ=2, k=3, h=0.06, h’=0.06

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B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

3.1

53335.7734

58094.9766

67191.7188

87818.0547

161375.1719

1.2517955

1.2517964

1.2517977

1.2517995

1.2518021

0.9999765158

0.9999784827

0.9999813437

0.9999857545

0.9999922514

TABLE-4.14

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.8, θ=2, k=3, h=0.08, h’=0.08

B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

3.1

57269.0625

62541.4727

72965.0859

96100.4219

185854.7813

1.3664875

1.3664888

1.3664907

1.3664936

1.3664979

0.9999761581

0.9999781251

0.9999812841

0.9999857545

0.9999926686

TABLE-4.15

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =1.8, θ=2, k=3, h=0.10, h’=0.10

B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

3.1

61840.4883

67739.7734

79203.4688

106149.0859

214509.8750

1.5043156

1.5043176

1.5043206

1.5043250

1.5043316

0.9999756813

0.9999777675

0.9999809861

0.9999858141

0.9999929667

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TABLE-4.16

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =2, θ=2, k=3, h=0.06, h’=0.06

B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

3.1

47165.3398

51346.2109

59031.8008

76279.8203

134318.9063

1.2044592

1.2044599

1.2044609

1.2044624

1.2044647

0.9999744892

0.9999765158

0.9999796152

0.9999842048

0.9999910593

TABLE-4.17

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =2, θ=2, k=3, h=0.08, h’=0.08

B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

3.1

49751.5820

54248.9102

62554.1445

81373.4453

146596.5781

1.2925504

1.2925514

1.2925529

1.2925553

1.2925588

0.9999740124

0.9999761581

0.9999793172

0.9999841452

0.9999911785

TABLE-4.18

Values of ARL’s AND TYPE-C OC CURVES when

b=4, η =2, θ=2, k=3, h=0.10, h’=0.10

B

L(0)

L’(0)

P(A)

3.5

3.4

3.3

3.2

52561.4922

57243.7109

66402.7500

86612.3359

1.3945440

1.3945454

1.3945478

1.3945513

0.9999734759

0.9999756217

0.9999790192

0.9999839067

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3.1

160652.1094

1.3945563

0.9999912977

NUMERICAL RESULTS AND CONCLUSIONS

At the hypothetical values of the parameters b, η ,θ , k, h and h’ given at the top of each table, we determined

optimum truncated point B at which P(A) is the probability of accepting an item is maximum and also obtained

ARL’s values which represent the acceptance zone L(0) and rejection zone L’(0) values. The values of truncated

point ‘B’ of random variable X, L(0), L’(0) and the values for Type-C OC Curve, i.e. P(A) are given in columns

I, II, III, and IV respectively.

From the above tables (4.1) to (4.18) we made the following conclusions:

1. From the tables (4.1) to (4.3), it is observed that the value of L (0) and P (A) is increased as the value

of truncated point decreases. Thus, the truncated point of the random variable and the various

parameters for CASP-CUSUM are related.

2. And also we observe that it could minimize the truncated point B by decreasing the value of k.

3. From tables (4.1) to (4.3), it is observed that the truncated point B of the random variable X decreases

from 2.5 to 2.1 as h→0.10, while the value of L (0) increases from 3870.2703 to 5015.4434 and the

probability of acceptance P (A) Changes from 0.9995375872to 0.9992460012. Thus at constant

hypothetical value h and truncated point B are positively related, while the values of L (0) and P (A) are

inversely related.

4. From tables (4.4) to (4.6), it is observed that truncated point B of the random variable X decreases from

3.5 to 3.1 as h→ 0.10, while the value of L (0) increases from 5188230.0000 to 523740.9375 and whereas

the probability of acceptance P (A) changes from 0.9999997020 to 0.9999953508. Thus hypothetical

value h and truncated point B are positively related, while the values L (0) and P (A) are positively

related.

5. From tables (4.7) to (4.9), it is observed that truncated point B of the random variable X decreases from

3.5 to 3.1 as h→ 0.10, while the value of L (0) increases from 362175 to 11371056.00 whereas the

probability of acceptance P (A) changes from 0.9999961257to 0.9999998212. Thus hypothetical value

h and truncated point B are positively related, while the values L (0) and P (A) are positively related.

6. From tables (4.10) to (4.12), it is observed that the truncated point B of the random variable X decreases

from 3.5 to 3.1 as h→0.10, while the value of L (0) increases from 214731.3281 to 377011.9688 and the

probability of acceptance in P (A) Changes from 0.9999938607 to 0.9999955893 at different truncated

points of B. Thus the values of L (0) and P (A) are positively related.

7. From tables (4.13) to (4.15), it is observed that the truncated point B of the random variable X decreases

from 3.5 to 3.1 as h→0.10, while the value of L (0) increases from 161375.1719 to 214509.8750 and the

probability of acceptance in P (A) Changes from 0.9999922514 to 0.9999929667. Thus the values of L

(0) and P (A) are positively related.

8. From tables (4.16) to (4.18), it is observed that the truncated point B of the random variable X decreases

from 3.9 to 3.1 as h→0.10, while the value of L (0) increases from 134318.9063 to 160652.1094 and the

probability of acceptance in P (A) Changes from 0.9999910593 to 0.9999912977. Thus the values of L

(0) and P (A) are positively related.

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TABLE – 5.1

Consolidated Table

B

b

η

θ

k

h

0.06

0.08

0.10

0.06

0.08

0.10

0.06

0.08

0.10

0.06

0.08

0.10

0.06

0.08

0.10

0.06

0.08

0.10

h’

0.06

0.08

0.10

0.06

0.08

0.10

0.06

0.08

0.10

0.06

0.08

0.10

0.06

0.08

0.10

0.06

0.08

0.10

L(0)

L’(0)

P(A)

2.1

3.1

3.2

3.3

3.1

3.2

3.1

4

1

2

3870.2703

1.7903954

2.4308357

3.7846324

1.5458885

1.8897420

2.4303186

1.4064401

1.6268451

1.9291679

1.3156708

1.4703906

1.6663494

1.2518021

1.3664936

1.5043316

1.2044647

1.2925588

1.3945563

0.9995375872

0.9994443655

0.9992460012

0.9999997020

0.9999966621

0.9999953508

0.9999961257

0.9999977946

0.9999998212

0.9999938607

0.9999946356

0.9999955893

0.9999922514

0.9999857545

0.9999929667

0.9999910593

0.9999911785

0.9999912977

4

1

2

3

4372.3262

1

5015.4434

1.2

1.4

1.6

1.8

2.0

5188230.000

563048.9375

523740.9375

362175.9688

732377.6250

11371056.00

214731.3281

275019.5000

377011.9688

161375.1719

96100.4219

214509.8750

134318.9063

146596.5781

160652.1094

By observing the Table-5.1, we can conclude that the optimum CASP-CUSUM Schemes which have the values

of ARL and P (A) reach their maximum i.e. 11371056.00, 0.9999998212 respectively, is

B  3.1









b  4.0



  1.4



  2.0



k  3.0

h  0.10

h' 0.10









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REFERENCES

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