A method for constructing fuzzy test statistics with application

Mathematical Theory and Modeling
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.13, 2013

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A Method for Constructing Fuzzy Test Statistics with Application
Assel S. Mohammad
Al – Kindy Medical Collage, University of Baghdad,Iraq
nbrhms@yahoo.com
Abstract:
The scale parameter ( q ) play an important role in statistical inference and in statistical testing of hypotheses
,but sometimes when the available observation about this parameter ,especially when ( q ) represent the
percentage of defective in production ,or mean time to failure of certain product ,imprecise information about
parameter of the distribution under hypotheses testing lead to test the hypothesis with fuzzy concepts .this paper
deals with problem of testing hypothesis when the hypotheses are fuzzy and data are crisp. We first introduce the
approach to testing fuzzy hypothesis ,when the parameter q represent percentage of defective in production and
it is considered random variable having prior distribution g (q ) ,and each inspected item is considered either
good or defective so the distribution of ( n ) inspected units is binomial with parameters (n,q ) .The data used
are taken from certain industrial company in Baghdad ,Iraq. The fuzzy hypothesis test here were done by two
methods, the first is the fuzzy Bayesian sampling test and the second depend on the procedure of using the

~
[a - cut ] of confidence interval of the parameter q , X (a ) .first the introduction is given and second some

preliminaries about fuzzy ,then we explain fuzzy hypothesis and finally the application.
Keywords: Testing fuzzy hypothesis, Membership function, Prior distribution,Posterior distribution.
1. Introduction
The statistical analysis in its normal form depend on crispness of data, random variables, hypothesis, decision
rules and parameters estimations and testing.
The Bayesian approach is one of the important methods which is used in statistical analysis in estimation and
testing hypothesis. Thomas Bayes in 1967 discussed the estimation of the parameter q which represent the
percentage of defective in products and considered to be random variable varied from lot to lot. Casals &Gil in
1986 studied the problem of testing statistical hypothesis with fuzzy sets and vague data. Also Casals and Gill in
1994 introduce Bayesian sequential test for fuzzy parametric problem.
Arnold in 1996 gives an approach of fuzzy hypothesis testing .In 2001 Holenam gives a fuzzy logic
generalization of a data mining.Viertl in 2006 introduce some methods to construct confidence intervals and
statistical tests for fuzzy data. Many papers deals with problems of testing fuzzy hypothesis using fuzzy data and
work on constructing a test statistic like Arefi&Taheri 2011and Bukley 2004 and Denoeus& Masson 2005.Our
research deals with introducing a method for testing fuzzy hypothesis about the percentage of defective in
production process and how to take a decision for acceptance or rejection of H 0 (q ) and H1 (q ) depending on
a statistical test for fuzzy Bayesian statistical hypothesis which to be solved using numerical integral method.
2. Fuzzy hypothesis
Any hypothesis about the parameter

~

q ,written in the form ( H : q ) ,H is called a fuzzy hypothesis ,where
parameter space Q ,with membership function H .The ordinary

H : q ® [0,1] is a fuzzy subset of
hypothesis is ( H 0 : q Îq0 ) is called a fuzzyhypothesis with the membership function ( H 0 : Iq 0 ) .here are
some required definitions.
Definition1:

~

The fuzzy hypothesis ( H : q is H ) be such that
1- H is a monotone function of q
2- There exists q1 Î Q such that

H (q ) = 1 for q ³ q1 or ( q £ q1 )
~
3The range of H contains the interval (0,1] , then H is called a one sided fuzzy hypothesis. If fuzzy
~
hypothesis ( H : q is H ) be such that there exist an interval [q1 ,q 2 ] Ì Q such that H (q ) = 1 for
q Î [q1,q2 ] and inf[q : q Î Q] < q1 £ q2 < sup[q : q Î Q] . And H is an increasing function for
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Vol.3, No.13, 2013

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~
q £ q1 and is decreeing for q ³ q1 and the range of H contains the interval (0,1] ,then H is called two –
sided fuzzy hypothesis.
Definition2:
The ordinary hypothesis ( H 0 : q Îq0 ) is a fuzzy hypothesis with membershipfunction
.
In testing simple or composed hypothesis about parameter and there is indicator define the range parameter
space we can use NeymanPearson or maximum likelihood ratio test to obtain the test statistics based on sample
information to take a decision for accept or reject. But in case of fuzzy hypothesis the hypothesis about q is
H : q H (q ) a membership function on q ,i.e. it is a function from q to [0,1] .
Let X = ( X1 , X 2 ,..., X n ) be a random sample that observed values

x = ( x1 , x2 ,..., xn ) and X i have
probability function
is unknown parameter and have prior distribution p (q ) .
Let H 0 (q ) , H1 (q ) be two membership functions, and the range of H contains the interval (0,1] .It is requires
to test;

H 0 : q is H 0 (q )
H1 : q is H1 (q )

L(q , a), a Î H is the set of all possible decision and
L(q , a) : q ´ A ® R is a loss function from taking decision a according to state of nature (q ) .
n
Let D be the set of all decision functions which define ( R on A ), then the risk function due to the wrong

Using Bayesian test we need the Loss function

decision is defined to be the expected Loss:
R(q , d ) = E(L(q , d ( X ))) (1)
Is the risk function from taken d about q .
When q is random variable, have p (q ) , then the posterior distribution of

q

given the sample

x = ( x1 , x2 ,..., xn ) is;
And
And Bayes Risk due to

(d )

R(p , d ) = E(R(q , d )) (2)
Here we need to define the Bayes test according to;
1- Bayes test without Loss function
Here we want to test the fuzzy hypothesis H 0 (q ) against H1 (q ) ,according to sample information

p (q ) ,the rule here depend on using membership function instead of indicator function.
Let q = q0 È q1 be the space of parameter,and
ì1 if q Î q 0
(3)
H 0 (q ) = í
î0 if q Î q1
ì0 if q Î q0
(4)
H1 (q ) = í
î1 if q Î q1
And

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Vol.3, No.13, 2013

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H 0 is accepted when
Some researcher introduce a factor called degree of certainty about our decision defined as

D=

a0
a 0 + a1

Where;

ii- Bayes test of fuzzy hypothesis using loss function
The test depend on loss function which defined on membership function H 0 (q ) , H1 (q ) ,i.e:

L(q , a0 ) = a(q )[1 - H 0 (q )] (5)
L(q , a1 ) = b(q )[1 - H1 (q )] (6)
a(q ) , b(q ) ordinary positive function defined on q and choosing of it depend on the sensitivity of wrong
decision for rejection or acceptance.
For this test H 0 is accepted if
Let a(q ) = c11 , b(q ) = c1 be constants.
Then the fuzzy Bayes test

H 0 is accepted if
(7)

c1 may represent the cost or the loss when type I error is happen (rejecting true hypothesis)
Table (1): percentage of defectives
and c11 is the loss or cost due to type II error (which is the probability of accepting false hypothesis).
The distribution of percentage of defective of 120 Lots ,each of size production (2000) ,from certain product of
Iraqi –industrial are;
0.1
6

0.2
20

0.3
34

0.4
29

0.5
18

0.6
9

0.7
4

a = 3, b = 4 with estimated p = 0.4285
which is represented by g (q ) or p (q )

Which found to be Beta with
of

The

ì60q 2 (1 - q )3
0
î

p (q ) = í

0 £q £1
o.w

And we find the estimated average of percentage of defective in this industrial state is ( p = 0.0425) ,this
percentage does not consist with (AOQL:Average outgoing quantity level,equal 2%)and the produced units are
important,so the company work on applying the testing of fuzzy hypothesis were percentage of defective is about
(0.2 £ q £ 0.43)% .
Therefore the fuzzy hypothesis is;

H 0 : (0.2 £ q £ 0.4)%
H1 : q is away from 0.4

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Vol.3, No.13, 2013

Now we shall find the posterior distribution of
f

www.iiste.org

,since each produced unit inspected and it is either good or
n

defective so the distribution of number of defective is Bernoulli and the

( y = å xi ) (sum of defectives in the
i =1

sample n ) is random variable follow binomial distributing (n,q ) .
Therefore;

=

n
C yq y (1 - q ) n - y 60q 2 (1 - q )3
1

ò 60C q
n
y

y+2

(1 - q ) n + 3- y dq

0

=

q y + 2 (1 - q ) n + 3- y
Beta ( y + 3, n + 4 - y )

This indicates that;
Now the membership function under H 0 and under H1 is;

ì5q - 1 0.2 £ q £ 0.4
H 0 (q ) = í
î3 - 5q 0.4 £ q < 0.6

According to fuzzy hypothesis and membership function,

H 0 is rejected if;

After solving I1 under H 0 (q ) and I 2 under H1 (q )
If I1 £ I 2 then reject H 0 (q ) otherwise accept H 0 (q )

(5q - 1)q y + 2 (1 - q ) n + 3- y
(3 - 5q )q y + 2 (1 - q ) n + 3- y
=ò
dq + ò
dq
Beta ( y + 3, n + 4 - y)
Beta ( y + 3, n + 4 - y)
0.4
0.2
0.4

0.6

And

(1 - 2.5q )q y + 2 (1 - q )n + 3- y
1 / 3(5q - 2)q y + 2 (1 - q )n + 3- y
=ò
dq + ò
dq
Beta ( y + 3, n + 4 - y )
Beta ( y + 3, n + 4 - y )
0.4
0
0.4

1

Now for I1

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Vol.3, No.13, 2013

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for n = 5,10 , I1 can be solved numerically
Also for I 2
0.4

0.4

1

1

1
I 2 = [ ò q y+2 (1 - q ) n+3- y dq - 2.5 ò q y+3 (1 - q ) n+3- y dq + ò 5 / 3q y+3 (1 - q ) n+3- y dq - ò 2 / 3q y+2 (1 - q ) n+3- y dq ]
k 0
0
0.4
0.4

k = Beta ( y + 3, n + 4 - y)
Also I 2 can be solved numerically using trapezoidal method or Simpsons.
Then the results for I1 , I 2 are computed ,when I1

£ I 2 ,then the Bayesdecision is work on rejecting H 0 and

accepting H1 otherwise is true.
Table (2):results of integral
5

I1
0
1
2
3
4
5

0.2894
0.4391
0.4843
0.4044
0.2575
0.1224

I2
0.4115
0.2814
0.2253
0.2479
0.3318
0.4508
10

I1
0
1
2
3
4
5
6
7
8
9
10

I2

0.1239
0.25816
0.40779
0.51868
0.54853
0.48969
0.37035
0.23587
0.12461
0.053215
0.017639

0.5624
0.42672
0.31054
0.22817
0.19087
0.2008
0.25008
0.3256
0.41511
0.51055
0.60797

decision
Reject
Accept
Accept
Accept
Reject
Reject
decision
Reject
Reject
Accept
Accept
Accept
Accept
Accept
Reject
Reject H0
Reject H0
Reject H0

Conclusion
In practical problems, we may face fuzzy information about observation and about parameter, so we may face
fuzzy hypothesis rather than crisp hypothesis, for example, when we are interested in evaluating average of
percentage of defectives for testing product produced by a factory, and the information are uncertain, this lead to
perform the fuzzy test rather than crisp. The present work introduce a fuzzy test for the hypothesis about the
percentage of defective in product of (120) lot of some product produced by some Iraqi industries. The
,
, where the underling distribution is Beta –
hypothesis tested depend on membership function
Binomial. After formulating the test statistics, which depend on the fuzzy Bayes test
against
. The values
of integral
and
are computed numerically using Trapezoidal method, the results are explained for
, when
, the Bayes decision work on rejecting
and accepting
, the results are
explained in table (2). Also it is necessary to remark that the membership function, here under
, are
constructed depend on the mean value of estimated average of defective (
), which is indicated by
for
and
References
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Journal of Uncertain Systems, 5, 45-61.
28

Vol.3, No.13, 2013

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[2] Arnold B.F (1996),"An approach to fuzzy hypothesis testing",metrika,4,4:119-126.
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29

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