Institut f. Statistik u. Wahrscheinlichkeitstheorie 1040 Wien, Wiedner Hauptstr. 8-10/107 AUSTRIA http://www.statistik.tuwien.ac.at
Testing statistical hypotheses based on fuzzy confidence intervals J. Chachi, S. Taheri, and R. Viertl
Forschungsbericht SM-2012-2 Februar 2012
Kontakt:
[email protected]
Testing Statistical Hypotheses Based on Fuzzy Confidence Intervals Jalal Chachi , Seyed Mahmoud Taheri and Reinhard Viertl Isfahan University of Technology, Iran Technische Universit¨at Wien, Austria
Abstract: A fuzzy test for testing statistical hypotheses about an imprecise parameter is proposed for the case when the available data are also imprecise. The proposed method is based on the relationship between the acceptance region of the statistical tests at level and confidence intervals for the parameter of interest at confidence level . First, a fuzzy confidence interval is constructed for the fuzzy parameter of interest. Then, using such a fuzzy confidence interval, a fuzzy test function is constructed. The obtained fuzzy test, contrary to the classical approach, leads not to the binary decision (i.e. to reject or to accept the given null hypothesis) but to a fuzzy decision showing the degrees of acceptability of the null and alternative hypotheses. Numerical examples in lifetime testing are given to clarify the theoretical results, and show the possible applications in testing hypotheses for fuzzy observations. Zusammenfassung: Aufbauend auf der Beziehung zwischen Konfidenzintervallen f¨ur Parameter von stochastischen Modellen und statistischen Tests f¨ur Parameterhypothesen, wird eine Verallgemeinerung f¨ur den Fall unscharfer Daten und zugeh¨origen unscharfen Konfidenzintervallen vorgeschlagen. Die zugeh¨origen verallgemeinerten Tests liefern unscharfe Entscheidungen mit Graden von Annahmen bzw. Ablehungen von Hypothesen. Numerische Beispiele aus der Lebensdaueranalyse zeigen die Anwendbarkeit von solchen statistischen Tests f¨ur unscharfe Beobachtungen. Keywords: Confidence interval, Fuzzy parameter, Fuzzy test, Fuzzy random variable, Lifetime testing, Testing statistical hypotheses.
1
Introduction
In the classical theory of parametric statistical inference there is a relationship between the totality of parameter values for which the null hypothesis is accepted and the structure of the confidence intervals. Namely, a family of acceptance regions for a statistical test about a parameter , at level , is equivalent to a certain family of confidence intervals for the parameter, at confidence level . If the value of the parameter specified by the null hypothesis is contained in the confidence interval then the null hypothesis cannot be rejected at level , and if it is not contained in the confidence interval then the null hypothesis can be rejected at level . In this paper, we wish to apply this point of view to fuzzy environment to propose a fuzzy test for testing hypotheses about a fuzzy parameter of a statistical model, based on a fuzzy confidence interval for the fuzzy parameter. The problems of testing fuzzy hypotheses and fuzzy confidence intervals have been devised independently by many authors. But, up till now, these two problems, i.e. testing
statistical hypotheses and confidence intervals in fuzzy environment, have been separately considered in the literature. Some of the main approaches to such problems are briefly reviewed below. Grzegorzewski and Hryniewicz (1997) reviewed some methods in testing statistical hypotheses in fuzzy environment, point out their advantages or disadvantages and practical problems. Grzegorzewski (2000, 2009) suggested some fuzzy tests for testing statistical hypotheses based on vague data in parametric and non-parametric populations. Arnold and Gerke (2003) studied testing fuzzy linear hypothesis in linear regression models. Filzmoser and Viertl (2004) presented an approach for testing hypotheses at the basis of fuzzy values by introducing the fuzzy p-value, (see also Parchami et al. (2010) for a value based approach to the problem of testing fuzzy hypotheses). Montenegro et al. (2004), using a generalized metric for fuzzy numbers, proposed a method to test the fuzzy mean of a fuzzy random variable (here after FRV). Parchami et al. (2005) studied a fuzzy version of some process capability indices when specification limits are fuzzy rather than precise, and obtained fuzzy confidence intervals for such indices. Wu (2005) proposed decision rules based on FRVs that are used to accept or reject the null and alternative hypotheses about a fuzzy parameter using the concepts of degrees of optimism and pessimism. Gonz´alez-Rodr´ıguez et al. (2006) introduced a bootstrap approach to the one-sample and multi-sample tests of means for imprecisely valued sample data. Hryniewicz (2006) investigated the problem of the interpretation of the results of statistical tests in terms of the theory of possibility. In this work, the concept of -value was given a new possibilistic interpretation and was generalized for the case of imprecisely defined statistical hypotheses and vague statistical data. Viertl (2006, 2011) investigated some methods to construct confidence intervals and statistical tests for fuzzy data. Akbari and Rezaei (2009) investigated a bootstrap method for inference about the variance based on fuzzy data. Wu (2009) and Chachi and Taheri (2011) proposed some approaches to construct fuzzy confidence intervals for the unknown fuzzy parameter. Arefi and Taheri (2011) developed an approach to test fuzzy hypotheses upon fuzzy test statistic for vague data. The reader is referred to the work by Taheri (2003) for a general review on statistical methods in fuzzy environment. The aim of this work is to introduce a new approach to the problem of testing statistical hypotheses for fuzzy data using the relationship between confidence intervals and testing hypotheses. To do this we employ the method of constructing fuzzy confidence intervals for fuzzy parameters investigated by Chachi and Taheri (2011). The rest of this paper is organized as follows. In the next section, some basic concepts that will be used in the sequel are recalled. Sections 3 and 4 provide statement and formalization of the problem of constructing fuzzy tests for vague data and imprecise parameters, respectively. Numerical examples are given in Section 5 to clarify the theoretical results, and to show possible applications of testing hypotheses about a fuzzy parameter for fuzzy observations. In the final section, we make some concluding remarks.
2
Preliminaries
2.1 Fuzzy arithmetic In this paper let , the set of all real numbers, be the universal set which is endowed with a topological structure. by its A fuzzy subset (briefly, a fuzzy set) of is defined is defined
membership function . For each , the -level set of " !$#%'& ( &*),+ *- , and " . is the closure of the set %'& ( &*),/ 0- . The by " ! is a nonempty closed bounded interval for fuzzy set is called a fuzzy number if each " !2# 35!4 6387! , all 1 . The -level set of each fuzzy number is usually denoted by ! #:9%'& ? @ &*)A+ *- and 387! #:BDCFEG%'& ? @ &*)A+ *- . is called where 354 &*)#JI'K LNM &*) , a pointI(crisp number) with the value H if its membership function is O stands for the indicator function of a set . where ) A wide class of fuzzy sets in PQ R (the set of all fuzzy numbers of ), which is rich and flexible to cover most of the applications, is the class of so-called SUT -fuzzy enough # V numbers XWY6Z[]\ )_^8` with central value Wa b , left and right spreads )Q# Z[]TA\c :)Q # , decreasing left and right shape functions , S 6 T ,
e d , with f S . Typically, the SUTg fuzzy number V could be shown by the following membership function (Zimmermann , 2001)
V Rh U) #Ji fS 6jlknm ) ) if hpoq WY / 4 T 6mrkns j if h WYt A V We can easily obtain the -level sets of as follows V !$# W !4 ]W 7! # Wu S kwv x ) Z[]Wzy{T kwv x ) \|x} 1 xt V # XWY6Z[]\ )_^8` with S # T and Z # \ #c~ is called symmetric An SUT -fuzzy number V # XWY ~)_^ . and abbreviated by of SUT -fuzzy numbers are the triangular fuzzy numbers denoted by # A special s kind ) 3w 6 3 4 63 . The membership function and the -level sets of a triangular fuzzy number are as follows &*) # & q 33 4 ) Il
µ j is a fuzzy random sample of# Definition 1 (Wu (2009)) We say that µ size W from ¹wº Ó , if µp× ’s are independent and identically distributed FRVs from ¹*º Ó for Ø ×µ j is a normal fuzzy random sample of size W and fuzzy parameters and ´ Ô , if the µp× ’s are independent identically dis# 8ttt]and tributed normal FRVs with fuzzy parameters and ´ Ô for Ø W . In this case, we ×µ × . $ ) . % , which for which the hypothesis is accepted pothesis á .>) .)fvalues . exhibits )f ÿz the / 0- and with degree ÿz , i.e. ÿz xµ those for it is rejected with .)â/ which . ) % . â ) ) degree ÿz , i.e. ÿz xµ ÿz 0- , where ÿz xµ is a fuzzy confidence interval for the fuzzy parameter of interest, and different types of statistical hypotheses about an imprecise parameter are defined in the following definition.
#
)
Definition 2 . Let ¼ PQ x¼ be the class of all fuzzy numbers on the parameter space ¿ ¼ and be a known fuzzy number in ¼ . Then
# . ” is called a simple hypothesis. ® # © . ” is called a two-sided hypothesis. Any hypothesis of the form “ á ° . ” is called a right one-sided hypothesis. Any hypothesis of the form “ á ¬ . ” is called a left one-sided hypothesis. Any hypothesis of the form “ á
1. Any hypothesis of the form “ á 2. 3. 4.
Note that, in the above framework, the possible values of the parameter of interest are expressed as linguistic variables (Zimmermann , 2001). The unknown fuzzy parameter may be treated as a fuzzy perception of the usual unknown parameter (Kruse and Meyer , 1987; Grzegorzewski , 2009). Example 2 Let µ . Then
¼ # Q P R )
1. The hypothesis “ á
be the parameter space for the fuzzy mean of a normal FRV
# _88 ò8)
” is a simple hypothesis.
#
ò8)
2. The hypothesis “ á Â © _88 hypothesis. This hypothesis is ” is ! a # two-sided Ò # % # ò GÇGÅ 1 - , where, ! 4 7 equivalent to Dá ¼ ¼ © G © , ÷ v of the triangular fuzzy number _88 ò8)_ . G6÷$ ò is the -level set
A ° _88 ò8) ” is a right one-sided hypothesis. This hypothe Dá ¼ v #Ò% ¼ l!4 / GN|!7 / ÷$ ò GÇÅ 1 - . ¬ _88 ò8) ” is a left one-sided hypothesis. This hypothesis The hypothesis “ á ¼ #Ò% ¼ l!4 q GN|!7 ÷$ ò GÇÅ 1 - . is equivalent to Dá v
3. The hypothesis “ á sis is equivalent to 4.
4
The proposed procedure
In this section, we investigate a procedure to provide a fuzzy test for testing hypotheses of a fuzzy parameter for fuzzy data, based on fuzzy confidence intervals. In order to derive the degrees of acceptability of the null and the alternative hypotheses, we introduce the following procedure. Note that, for simplicity, we explain the procedure for the case that the data available are observations of a normal fuzzy random sample with the unknown ×)â # . á v .)â u# © . (3) into a set of crisp testing problems concerning -levels of the fuzzy parameter. For) each # ) # ! ! ! ! -level, based on the samples X4 R¸Qv 4 tttD¸Qj 4 and X7 R¸(v7 ! tttD¸(j 7 ! , the following classical testing problems are solved at level á .f.f !4 ## .[4 ! á v !4 ## © .[4 ! (4) ! [ . ! ! [ . ! á 7 7 á v 7 © 7 (5) !$# ! ! where 4 6 7 and .[!# .[4 ! 6 .[7 ! . ! ! Step 2. We obtain the "ì confidence intervals for the crisp parameters 54 and |7 ) ) ) ) ! ! ! ! for each 1 , denoted by S X 4 6S X 4 and X7 X7 , respectively. Ô v andÔ (5) by investigating v Step 3. We test the hypotheses (4) if ) the observed two-sided ) ) ) ¦ confidence intervals S v x!4 6S Ô x!4 and v x7! Ô x7! contain l.[4 ! and |.[7 ! , respectively. In fact, the test functions can be shown to be the following:
ê X!4 )U#
ê X7! )N#
I
^ è Í r ^ Þ è Í ö l.[4 ! ) X X . .[! ) Accept á 4 I
è ÎÍ r Þ è ÎÍ ö |.[7 ! ) X X . .[! ) Accept á 7
I
^ è Í r ^ Þ è Í ö l.[4 ! ) X X . .[! ) Reject á 4 I
è ÎÍ r Þ è ÎÍ ö |.[7 ! ) X X . t .[! ) Reject á 7
h ¯− X
1
¯+ X
√σ z1− α n 2
¯u − X h ¯l − X h
¯l − X 0
√σ z1− α n 2
√σ z1− α n 2
¯ hl + X
¯l + X 0
X¯ Conf. Bounds
√σ z1− α n 2
√σ z1− α n 2
¯ hu + X
√σ z1− α n 2
√σ z1− α n 2
√σ z1− α n 2
¯u − X 0
√σ z1− α n 2
¯u + X 0
√σ z1− α n 2
Figure 1: Graphical representation of the confidence bound constructed from the class of ) two-sided _ confidence intervals in Step 4
×
#ED . Ù. ü ÿ 4 Ó % B º C A % # t ø 6 t ô = ø x ÿ v7 . . # ò 5õ0töõ08 v % ºBÓ AC% ã Ù 7 ü # "7
# x . Ù. ü ÿ 4 Ó % B º & A % # t ô = ø xÿ Ô 7 . . # 8 =t @8 Ô % ºBÓ A&% Ù ã ü # ÷ ø t ÷5õ0 "
#ED # 6t ø x # '÷ ò úFt =8 # ÷ ø t ÷5õ0t
Table 1: Triangular fuzzy numbers for tire lifetime in Example 4 ÷8÷@5õ øø >õ0 òò >@8 )) ÷ òò =ô8|ôõ0 ô8ò úFø >=8ô ÷ ø5)) ÷8÷8 ô8ò ]ú3=5øõ0>õé÷ ø5ô8)) ÷ òò =88 øô8@ò 8 ø8ø = )) ÷8÷éúFò >ø8=ø ]ú3ø @8 ) ÷ ú ]únõ ø õ8@ø5) ÷8÷éú3=8ò ÷6÷= ò >== ô8) ÷ ú3ô == ÷>õlú3=ò8) ÷=ò8ò úF ô 8 ÷ø = ) ÷8÷85ø8øõé>@8ô8ô 8 ô @ ) ÷8÷ò8ô õ0>õ0 ò >@éôéúø ) ÷8÷ ô ú5÷ô >=éú5÷ø >õ8@ ò8) ÷8÷ úF ÷5õ0>= úô8) ÷8 ô 8 ø úFô ø5=éò ú ) ÷ ò8ôé@5ø õ0]úF ô ôé@ø5) ÷ ø = ø 6÷5ø õ >=5õ ) ÷éú5ò88ò ÷ => ='÷>õé÷ ) ÷éú5òéø ÷ ø ú ÷ô8) ÷ ø úF>@éúø >@ ô8) ÷ ÷ô @8÷>@8ø ) ÷ @8>õ8õ8@>õ8õlú ÷ 8 ==>=éú ÷8÷ ú8úF>õ úF>=8 ÷éúF õ0>=@8÷ |õ
h 1
X¯ θ˜ Conf. Bounds
32541.21
33232.79
×103
30
32
31
33
34
Figure 5: The membership functions and confidence bound in Example 4 part (i)
#
ò
ò
)
The degree of membership of 88 88 in the two-sided fuzzy confidence )Q# ÷88t ò88ò 8÷ . With interval is obtained as ô ÿ other words, based on the fuzzy observations and at level t @ , the amount of possibility that the true mean of population .f #is # ò ÷888ò 8 ò 8ò 8 ò )8 8 ) , is t ò8ò ÷ . Therefore, at level # t ô , # the hypothesis á ò 88 ò 88 ò 88 ) ÷ 88 88 88 is accepted against the hypothesis á Ë © ÷ .)U# t ò8ò ÷ . Briefly, the fuzzy v test function is with degree of acceptability ÿ
t ò ÷8÷ töõ8=5õ ë t ê xµ N) #bi . .) Reject á . .) Accept á .f # . versus á g ° . ii) Testing á ô v ! ! The right one-sided t @ confidence intervals for parameters 54 and |7 are obtained as ò ø ) ø ô ) ÷@ @töõ F y ==5õ8 GH G and ÷8÷8÷éúF8töõ ¨õlú GH G , respectively. In this case we have (see also Chachi and Taheri (2011))
# D # ú5 ô tù|õ0 . Ù. ü E . Ù . ü # x 4 ÿ 4 ã Ó Ó º B & A % B º C A % # ô8ò # 8ô ò # |õ ò t =8t $ ºB7Ó A&% . Ù . ü 6t úéx(ÿ$ºB7Ó AC% . Ù . ü t Fú x4" # ô So, based on the fuzzy observations and at level t @ , the degree of membership of ` )"# tù' @ . ÷8888 ò 88 ò 88 ) in# the right one-sided fuzzy confidence interval is ÿ .c # ÷ ò 88 ò 88 ò 88 ) is acTherefore, at level t ô the hypothesis á ò 88 ò 88 ò 88 ) with degree of acceptability cepted against the hypothesis á g ° ÷ v test function is ÿ ` .)N# tù' @ . Briefly, the fuzzy tù'@ t ø @ ë t ê xµ )N# i . .) Reject á . .) Accept á . The confidence bounds and the membership functions of and µ í are shown in Fig. 6. .f # . versus á ¬ . iii) Testing á ô ò8ô 't ò8ò y v ! ! The left one-sided t @ ò8ò confidence intervals for parameters 4 and 7 are _IG6÷ 8 ò ò ô ==5õ8 and _I G6÷8÷ @ t õlú , respectively, and we have # D # ò t úF8 $ . Ù . ü # x . Ù. ü E 4 ÿ 4 ã Ó Ó º B C A % º B C A % $ . . # t = ò xÿ$Bº 7Ó AC% . Ù . ü # 6t = ò8) 4 " # |õ ò t =8t Bº 7 Ó AC% Ù ü $
h 1
X¯ θ˜0 Conf. Bounds
32.59678
∞
0.524
h ×103
30
32
31
31.92978 + 0.667h
33
34
33.34178 − 0.745h
Figure 6: Confidence bound and the membership functions of (ii)
.
and µ
í
in Example 4 part
ô
So, based on the fuzzy observations and at level t @ , the degree of membership ^ )¨of# fuzzy confidence interval is ÿ # ÷8888 ò 88 ò 88 ) .?in the# left ò one-sided t @ò @@ . The hypothesis á ÷ 88 ò 88 ò 88 ) versus ^ hypothesis á v ¬ .)z# ò ò ) ÷ # 88 ô 88 88 is accepted with degree of acceptability ÿ t @@@ at level t . The fuzzy test function, therefore, is obtained as
ê xµ )N# i
t @@@ t 8 ë t . .) Reject á . .) Accept á
Example 5 (See also Arefi and Taheri (2011)) Assume ) that we have taken a random . sample ¸ tttD¸ from an exponential distribution Õ with the following density,
v
þ
* & U ) # . ¹|º| KJ k
&(/ ¼ # HG ) t & The recorded data (the centers of fuzzy numbers in Table 2, i.e. × ) show the lifetimes (in '88ML0H ) of front disk brake pads on a randomly selected set of ú5 cars (same model)
that were monitored by a dealer network (Lawless , 2003, pp. 337). But, in practice measuring the lifetime of a disk may not yield an exact result. A disk may work perfectly over a certain period but be braking for some time, and finally be unusable at a certain time. So, such data may be reported as imprecise quantities. Assume that the lifetimes of front disk brake pads are reported numbers in Table 2. In fact, imprecision # as & fuzzy is formulated by fuzzy numbers µâ× ×[ × )` , × # t ô|& × , Ø # 8 ò ttt]ú5 , with the following membership function
HN
& × o?hâo & F× y × P'hD \RQ"Ø t J J . are observations of In this example, we could assume that the fuzzy numbers µ ttt> µ ) v þ an exponential fuzzy random variable with unknown fuzzy mean PQ x¼ . µp×_ Rh )U# i mrk O
N
for
Now suppose that the company claims that the average lifetime of the new disks is longer than the well-known average lifetime of the previous brand, which is known to be
Table 2: The lifetimes fuzzy data (in S,TU8V W8XY3V Z[ S\Y ^`V dXCW8V_^ [ S\YRW8V W8XCW8V_^ [ S,ZT8V T8XH^`V c[ S,]`dV bRXCW8V ][ Sfb`Z8V_^`XCZ8V b[ S,Z^`V bRXH^`V U[
`
`
`
`
`
`
`
S\YR]8V_^`XCW8V Z[ S,]U8V Y3XCW8V T[ S,]^`V U8XCW8V U[ S,]`dV W8XCW8V ][ S\YRW8V T8XCW8V_^ [ S,WT8V Y3XH^`V Y8[ S,ZZ8V c8XH^`V b[
`
'88IL0H ) of front disk brake pads in Example 5 ` S,]aY3V W8XCW8V b[ ` S,]c8V dXCZ8V dR[ ` S,ZT8V Y3XH^`V c[ ` ` S,UW8V Y3XCZ8V_^ [ ` S\YR]8V ]8XCW8V Z[ ` S,ZU8V bRXH^`V T[ ` ` S,WW8V bRXH^`V_^ [ ` S,WW8V U8XH^`V_^ [ ` S\YdV dXCW8V dR[ ` ` S,T^`V bRXY3V_^ [ ` S,U^`V ]8XCZ8V_^ [ ` S,]Z8V U8XCW8V b[ ` ` S\YRW8V bRXCW8V_^ [ ` S,T`dV U8XY3V dR[ ` S,UaY3V ]8XCZ8V W[ ` ` S\YR]8V c8XCW8V Z[ ` S,ZZ8V T8XH^`V b[ ` S,]c8V T8XCZ8V dR[ ` ` S,]U8V bRXCW8V T[ `
S,]W8V_^`XCW8V U[ S,T^`V Z8XY3V_^ [ S,ZaY3V Y3XH^`V b[ S\YRT8V T8XCW8V Y8[ Se^>dW8V ]8XC]8V_^ [ S\YRU8V c8XCW8V Z[ S,]`dV U8XCW8V ][
` ` ` ` ` ` .,#
ô ô
ô )
the triangular fuzzy number should .>)â consider . .)â ÷ # t ÷ 8ô t t ô 8t ô .Therefore, ) versus we ô t ô the ô prob) , lem of testing hypothesis á á ° ÷ 8 t # tù' to test the claim. Note that, using fuzzy arithmetic, v at level we have
µ !## µ .[!A#
& ô 8t ò & õ ô # ò tô ô8ô õò ô8)ô ` ô ò ô )Dò t ô8ô õ ô x _
!4 7! # 8ôt õô 8t õ ) ,y .[4 ! 6 .[7 ! ÷ t q _ { 8t ô 6÷ ô t ô y _,{ ) x t
! # R ¸Q4 ! tttD¸Q4 ! ) v j _ )) 8t ô (8) y _ { (9)
According to the procedure introduced in Section 4, based on the samples X 4 # ) ! ! ! and X7 R¸(7 tttD¸(7 , we test the following hypotheses
v j á f. f. !4 # ÷ # ô t ô ô ô _ ) 8t ô ) á v !4 // ÷ ôô t ôô á !7 ÷ t y _,{ á v !7 ÷ t # tù' , by using the corresponding confidence intervals. We see that for at level ! ! exponentially distributed fuzzy random sample µ ttt>µ , the samples ¸ 4 tttD¸ 4 v j ! ! v j and ¸(7 tttD¸(7 are independent and identically distributed random from N i A j variables N Í v j Í !) !) exponential distributions Õ 4 and Õ 7 , respectively. Since 8Ô gIh º ½ k@Ôý . and l N i A j N Í Ô gIh º ÍÎ Î m 8 ½ k@Ôý . the usual right one-sided confidence intervals at confidence level # t @8 for l!4 and |!7 are, respectively, . & ! ò þ ã ` x!4 ) # !4 !4 k Ô ×.D . v . ×4 H G ) # ÷ ô t ò =éú ø H G ) ý . Ùn ò `ã x7! ) # !4 !4 þ× v & 7× ! H G ) # ø w ô 8t ò õ ô ,y _ ò { )Dò t ô8ô õ ô8) H G ) .D . . k Ôý Ù n 8 =t ÷ where k Ô ð is the È quantail of the Chi-square distribution with W degrees of freedom. j
Therefore, the following test functions are obtained for testing the hypotheses (8) and (9) at level 1
I
) ô8) I
p ô ô p ô ê x!4 ) # i [û üDÙ Ô o þ ý . ÷ ô t ô _ { ) 8t ô8) û[üDÙ Ôo þ ý . ô ÷ ô Accept á ÷ . t q _ { 8t Reject á ÷ t è è ! Ù ü Rx è è ! Ù ü # s tv r q u Accept w A û[üDÙ ü A û[. v üDÙ ü Reject w èA û[üDÙ ü k è v[k ! v Ù ü Rx èA û[v üDÙ ü k è v[k ! v Ù ü u Accept w k v[k v k v[k v Reject w
t ô q _ {) ) ô88) t ô8) ë q _ { 8t 1 6t ø ú5÷ ò8) 1 t ø ú5÷ ò x
I çay0è ÎÍ ÷ ô t ô y _,{ )D) I çayè ÎÍ ÷ ô t ô y _ )D)
) # i x ê x 7! . ô ô )D) Reject á x . ÷ ô t ô y _,{ )D) ë Accept á ÷ . t y _ ) #rs qtvu Accept w A è û[üDÙ ü d è v[k !ù Reject w A è .v û[üDÙ ü d è v[k !ùRx * ® { 6t =@ è è !ùRx p 1 t =@xt è è !ù u Reject w A û[üDÙ ü d v[k Accept w A v û[üDÙ ü d v[k We can easily obtain
. Ù . # t ø ú5÷ ò xÿºB4Ó AC% . Ù . # 6t ø ú5÷ ò8) ã # töõ0'÷@ 4 Ó º B C A % ) # t ô ú ø t $ v # v # ºB7Ó AC% . Ù v . t =@x ÿ$ºB7Ó AC% . Ù v . 6t =@ " .1# ô ô ô ) So, the degree of membership of .)® ÷ # t ! 8t ç # in the right one-sided fuzzy con` fidence interval is calculated as ÿ we obtain the ! .>)â t # 5õ0'÷ ô . ô Therefore, d . ô ) . .)â following fuzzy test for testing the hypothesis á ÷ t 8 t versus á ° ÷ ô t ô 8t ô ) at level # tù' by . t 5õ0ô '÷ô ô )) . t @ òéô ø ô õ ô )) ë t ê xµ )N#bi Accept á D ÷ t 8t Reject á D ÷ t 8t $
6
Conclusion
In the present work, based on the concept of fuzzy confidence interval, we introduced the so called fuzzy test for testing statistical hypotheses about an imprecise parameter when the data are reported as fuzzy numbers. In the proposed approach, the available data are assumed to be the observations of FRVs. We investigated a procedure to construct the fuzzy test for testing a simple hypothesis versus one-sided and two-sided hypotheses. A well known method of constructing fuzzy confidence intervals is used to determine the degree of membership of each fuzzy parameter in the fuzzy confidence interval and then to make inference in testing a hypothesis about the fuzzy parameter of interest. The proposed fuzzy test, contrary to the classical crisp test, does not lead to a binary decision, i.e. to accept or to reject the null hypothesis, but to a fuzzy decision. Such a fuzzy test is a natural generalization of the traditional significance tests, i.e. if the data and the parameter(s) of interest are precise, we get classical statistical tests with the binary decision as a special case. Based on the proposed approach, the decision makers actually accept or reject the given hypothesis about a fuzzy parameter, however an index, called the degree of acceptability, would support the decision.
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Authors’ addresses: Jalal Chachi and Seyed Mahmoud Taheri Department of Mathematical Sciences Isfahan University of Technology Isfahan 8415683111, Iran E-Mail: z|{}8~ ~RR
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R* |{ `{|{ Tel.: +98 311 3913615 Fax: +98 311 3912602 Reinhard Viertl (Corresponding author) Technische Universit¨at Wien Institut f¨ur Statistik und Wahrscheinlichkeitstheorie Wiedner Hauptstrasse 8-10/107 A-1040 Wien, Austria E-Mail: {B83 38R{|{8
