Soft Comput (2015) 19:113–119 DOI 10.1007/s00500-014-1235-7
METHODOLOGIES AND APPLICATION
Hypothesis testing for the mean of inverse Gaussian distribution using α-cuts ˙ Duygu Içen · Sevil Bacanlı
Published online: 22 February 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract In this study, we modify the method proposed by Buckley to testing statistical hypothesis for the mean of an inverse Gaussian distribution. In order to obtain fuzzy test statistic, we use confidence intervals by the help of α-cuts. Then the method is applied to test the hypothesis for the mean of inverse Gaussian distribution when the scale parameter is known. Also a comparison is made between the fuzzy and non-fuzzy test procedure for the inverse Gaussian distribution. Keywords Inverse gaussian distribution · Fuzzy test statistic · α-cuts
1 Introduction Testing of statistical hypothesis has been developed in different approaches since fuzzy sets were introduced by Zadeh (1965). Buckley (2005, 2006) introduced and developed an approach to the estimation of unknown parameters in statistical models. Buckley’s method uses a set of confidence intervals producing a triangular shaped fuzzy number for the estimation of the parameter of interest. This method has been applied to test the parameter of a normal distribution (mean and variance), the parameter of a Binomial distribution, the mean of a Poisson distribution and the mean of an exponential distribution. Falsafin et al. (2008) developed Buckley’s approach. They introduced a method to find the explicit forCommunicated by G. Acampora. D. ˙Içen (B) · S. Bacanlı Department of Statistics, Hacettepe University, 06800 Ankara, Turkey e-mail:
[email protected] S. Bacanlı e-mail:
[email protected]
mula for membership functions of the fuzzy estimations in statistical models not known before. Taheri and Arefi (2009) introduced a credit level to decide to accept or reject fuzzy hypothesis based on fuzzy test statistic. Hryniewicz (2006) proposed another way of looking at the results of statistical tests which are similar in the spirit to that of Buckley. It is well known that the inverse Gaussian (IG) distribution is a very useful alternative to the popular life time distributions such as Weibull, gamma and log-normal. The distribution has a wide application area in clinical trials, quality and reliability theory, industrial engineering applications and life tests. The probability density function (PDF) of the two parameter IG distribution ( IG(μ,λ)) random variable X is −λ(x − μ)2 λ 1/2 (1) exp f (x; μ, λ) = 2π x 3 2μ2 x where μ > 0 and λ > 0. The parameter μ is the mean of the distribution and λ is a scale parameter. The shape parameter for the IG(μ,λ) is λ/μ. The variance, coefficients of skew√ ness and kurtosis are μ3 /λ, 3 μ/λ and 15μ/λ respectively (Chhikara and Folks 1989). For a given random sample of size n, X 1 , X 2 , . . . , X n from IG(μ,λ). Maximum likelihood estimators of μ and λ are μˆ = X¯
n 1 1 1 1 − = n Xi X¯ λˆ i=1
where X¯ =
n 1 (X i ). n i=1
Here μˆ is distributed as IG(μ, nλ), λˆ −1 is distributed as (nλ)/λˆ ∼ χ 2 n−1 and χ 2 n−1 is Chi-square distribution with (n − 1) degrees of freedom. The two statistics μˆ and λˆ −1 are independent.
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IG is related to the normal distribution. This relation was given in the following Theorem 1 in Chhikara and Folks (1989) which establishes a basic relationship between IG and the normal. √ √ Theorem 1 Let Y = λ(x − μ)/μ X . Then the PDF of Y is given by 2 √1 e−y /2 , −∞ < y < ∞. f (y) = 1− √ y 2 2π
4λ/μ+y
(2) √ The transformation Y = λ(x − μ)/μ x is one-to-one, x varies from 0 to ∞ and y varies −∞ to ∞. Then the cumulative distribution function of F(y) of Y is √
F(y) = (y)+e2λ/μ (− 4λ/μ + y 2 ),
−∞ < y < ∞ (3)
where is the standard normal distribution function. In this case F(y) → (y) as φ = μ/λ → ∞. Because of this and because of the one-to-one relationship between x and y, one finds that the distribution of X is asymptotically normal with mean μ and variance μ3 /λ. In this study we show that the α- cuts approach proposed by Buckley can be used within the hypothesis testing of the mean of inverse Gaussian distribution. In this way we used more information than just a point estimate or a single confidence interval. Finally, we illustrated our work by a numerical example. This paper is organized as follows: in Sect. 2 testing statistical hypothesis using α-cuts for the mean of a normal distribution with known variance is briefly reviewed. In Sect. 3 and Sect. 4 fuzzy test statistic for the mean of an inverse Gaussian distribution is described. Also a comparison is made between the applied method and crisp hypothesis testing procedure for the inverse Gaussian distribution.
Buckley’s approach uses a set of confidence intervals producing a triangular shaped fuzzy number for the estimator. Then this approach produces a fuzzy test statistic and fuzzy critical values in hypothesis testing. In classical hypothesis testing, based on a random sample size of n from N (μ, σ 2 ), variance (σ 2 ) is known. The following hypothesis H0 : μ = μ0 against H1 : μ = μ0 is going to be tested at significance level (0 < γ < 1). Under H0 , the crisp test statistic X¯ − μ0 √ σ/ n
is distributed by the standard normal distribution.
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X¯ − μ P −z γ /2 ≤ √ ≤ z γ /2 = 1 − γ σ/ n
(5)
Equation (5) can be written for μ, √ √ P( X¯ − z γ /2 σ/ n ≤ μ ≤ X¯ + z γ /2 σ/ n) = 1 − γ .
(6)
This interval can be arranged according to the method proposed by Buckley (2005) that is to find the (1 − γ )100 % confidence interval for all 0.01 ≤ γ ≤ 1. In this method, starting at 0.01 is arbitrary and it is possible to begin at 0 or 0.05 or 0.005, etc. So, Eq. (6) is defined by the following equation, √ √ [μ1 (γ ), μ2 (γ )] = [ X¯ − z γ /2 σ/ n, X¯ + z γ /2 σ/ n].
(7)
Then (1 − γ )100 % confidence intervals for μ is obtained for 0.01 ≤ γ ≤ 1. By placing these confidence intervals, one on top of the other, a triangular shaped fuzzy number X whose α-cuts are the confidence intervals as X [α] = [μ1 (α), μ2 (α)] is given
2 Testing statistical hypothesis using α-cuts
Z0 =
Let 0.01 < γ < 1 which denotes the significance level of the test. Under the null hypothesis H0 , Z 0 is N (0, 1) and decision rule is (a) reject H0 if Z 0 ≥ z (γ /2) and Z 0 ≤ −z (γ /2) ; and (b) do not reject H0 when −z (γ /2) < Z 0 < z (γ /2) . In the above decision rule ±z (γ /2) are called critical values (C V ) for the test. In the above decision rule z (γ /2) is the z value so that probability of a random variable having the N (0, 1) probability density, exceeding z is γ /2 . √ It is known that X¯ is N (μ, σ 2 /n). Thus ( X¯ − μ)/(σ/ n) is N (0, 1). Therefore
(4)
√ √ X˜ [α] = [ X¯ − z α/2 σ/ n, X¯ + z α/2 σ/ n]
(8)
for 0, 01 ≤ α ≤ 1. Hence we obtain the fuzzy parameter estimation of μ. When σ 2 is known, the fuzzy statistic for the mean of normal distribution is defined as X˜ − μ0 Z= √ σ/ n
(9)
Substituting the bounds of X˜ , Eq. (8), into Eq. (9) and using the interval arithmetic, (Moore 1979) α-cuts of the fuzzy test statistic are obtained as
Hypothesis testing for the mean of inverse Gaussian distribution
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Fig. 1 Decision criteria for testing H0 : μ = μ0 versus H1 : μ = μ0 using α-cuts
X˜ [α] − μ0 Z [α] = √ σ/ n
√ √ X¯ − z α/2 σ/ n − μ0 X¯ + z α/2 σ/ n − μ0 = , √ √ σ/ n σ/ n ¯ X − μ0 X¯ − μ0 (10) = , − z + z √ √ α/2 α/2 σ/ n σ/ n = [Z 0 − z α/2 , Z 0 + z α/2 ] Since test statistic is fuzzy the critical values will also be fuzzy. There will be two fuzzy critical value sets: (a) let V 2 go with z γ /2 . C V 1 correspond to −z γ /2 ; and (b) let C Set C V i [α] = [cvi1 (α), cvi2 (α)], i = 1, 2. The end points of an α-cut of C V 2 are computed from the end points of the corresponding α-cut of Z˜ with the equations given below Pr ob(Z 0 + z > cv22 ) = γ /2
(11)
Hence under H0 , Z 0 is N (0, 1) so cv22 (α) = z γ /2 + z α/2 . By using the left end point of Z˜ [α] in Eq. (11) ,we have cv21 (α) = z γ /2 − z α/2 . Hence the α-cuts of C V 2 and C V 1 are given by C V 2 [α] = [z γ /2 − z α/2 , z γ /2 + z α/2 ] C V 1 [α] = −C V 2 [α] C V 1 [α] = [−z γ /2 − z α/2 , −z γ /2 + z α/2 ]
(12) (13)
respectively (Buckley 2005; Taheri and Arefi 2009). Both C V 1 and C V 2 are triangular shaped fuzzy numbers. Since the V2 crisp test statistic has a normal distribution, C V 1 = −C because this density is symmetric with respect to zero (Buckley 2006). The final decision depends on the positions between tri V 2 . All posangular shaped fuzzy numbers Z˜ , C V 1 and C ˜ sible positions between Z , C V 1 and C V 2 are illustrated
with Fig. 1. Hence Fig. 1 represents test results with α-cuts approach proposed by Buckley. Test procedure is as follows: V1 (a) if Z >C V 2 then reject H0 (Fig. 1a); (b) if Z˜ < C then reject H0 (Fig. 1b); (c) if C V1 < Z < C V 2 then V1 ≈ Z < C V 2 or do not reject H0 (Fig. 1c); (d) if C ˜ C V 1 < Z ≈ C V 2 then no decision about H0 (Fig. 1d). These situations are explained in detail as follows. For exam V 2 then find the ple, if Z˜ > C V 2 , draw Z˜ to the right of C height of the intersection as, Z (x), C V 2 (y))|x ≤ y} v( Z ≤C V 2 ) = max{min( which measures how much C V 2 (y) is less than or equal to V 2 ) < η, where η is Z (x). Thus, C V 2 < Z if v( Z ≤ C some fixed fraction in (0, 1) (Fig. 1a). Other situations are summarized in Fig. 1b–d. In this figure, the height of the intersection is y0 and η = 0.8 as Buckley (2006) . Now the Z > C V2 results can be given as: (a) if y0 < 0.8, then (Fig. 1a), and (b) if y0 ≥ 0.8 then Z ≈ C V i , i = 1, 2 (Fig. 1di, dii). Similar results hold for Z versus C V 1 (Fig. 1b–d). V 2 , if It is interesting that after evaluating Z and C V 1 ,C Z ≈ C V i , i = 1, 2 (Fig. 1di, dii) then the final decision is “no decision” on H0 . This is because of the fuzzy numbers that incorporate all uncertainty in confidence intervals (Buckley 2006). In addition, Buckley’s approach for testing P for a Binomial population is described as follows. Fuzzy estimator for P is triangular shaped fuzzy number given as ⎡ = ⎣ Pˆ − z α/2 P[α]
Pˆ Qˆ ˆ ; P + z α/2 n
⎤ Pˆ Qˆ ⎦ n
(14)
ˆ The where Pˆ is the point estimate of P and Qˆ = 1 − P. classical hypothesis test for P is H0 : P = P0 , Hs : P = P0 ,
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⎡ ⎛ ⎛ ⎞−1 ⎞−1 ⎤ X¯ X¯ ⎥ ⎢ P ⎣ X¯ ⎝1 + z 1−γ /2 ⎠ ≤ μ ≤ X¯ ⎝1 − z 1−γ /2 ⎠ ⎦ nλ nλ
then determine the statistic Pˆ − P0 Z0 = . P0 Q 0 n
=1−γ
Here α-cuts of the fuzzy statistic for P can be defined as ⎡ ⎤ ˆ ˆ ˆ ˆ Pˆ − z α/2 PnQ − P0 Pˆ + z α/2 PnQ − P0 ⎦ Z [α] = ⎣ ; P0 Q 0 n
⎡
z α/2 Pˆ − P0 = ⎣ − P0 Q 0 n
⎡ = ⎣ Z 0 − z α/2
P0 Q 0 n
Pˆ Qˆ n
P0 Q 0 n
z α/2 Pˆ − P0 ; + P0 Q 0 n
Pˆ Qˆ ; Z 0 + z α/2 P0 Q 0
Pˆ Qˆ n
⎤ ⎦
P0 Q 0 n
⎤
(15)
3 Testing for the mean of an inverse Gaussian distribution using α-cuts We will assume that a random sample X 1 , X 2 , . . . , X n has been taken from the inverse Gaussian distribution. Consider the testing of the following hypothesis H0 : μ = μ0 against H1 : μ = μ0 . When λ is known, the crisp test statistic for the mean is defined as Y =
nλ(x¯ − μ0 ) √ μ0 x¯
X˜ [α] ⎡ ⎛ ⎛ ⎞−1 ⎞−1 ⎤ X¯ X¯ ⎥ ⎢ z 1−α/2 ⎠ , X¯ ⎝1+ z 1−α/2 ⎠ ⎦ = ⎣ X¯ ⎝1+ nλ nλ
for 0, 01 ≤ α ≤ 1. The fuzzy statistic for the mean of inverse Gaussian distribution is given in Eq. (20) = Y
√ nλ( X − μ0 ) √ μ0 X
(17)
Under H0 , the crisp test statistic Y is distributed as N (0, 1) (Chhikara and Folks 1989). When λ is known, its (1 − γ )100% confidence interval for μ is
(20)
Substituting the α-cuts of X , Eq. (19), into Eq. (20) and using the interval arithmetic (Moore 1979), α-cuts of the fuzzy test statistic are obtained as follows, by which we can test the hypothesis of interest √ nλ( X [α]) − μ0 Y [α] = μ0 X [α] ⎡ ⎛ ⎞ ⎢√ ⎢ nλ 1 X ⎝ − μ0 ⎠ , =⎢ ⎢ μ X ⎣ 0 X 1 + nλ z α/2 X nλ z α/2
1−
⎛
√
nλ μ0
1 X 1+
since X˜ ∼ I G(μ0 , nλ), it follows from the Theorem 1 given in Sect. 1 that the critical region of uniformly most powerful unbiased (UMPU) level γ test of the hypothesis given above, λ known, corresponds to γ . |Y | = Z 1−γ /2 = −1 1 − 2
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In the sense of the information given in Sect. 2, we modify hypothesis testing using α-cuts for the inverse Gaussian distribution. Therefore, we obtain the fuzzy parameter estimation of inverse Gaussian mean
(19)
Pˆ Qˆ ⎦ . P0 Q 0
By using the right hand side of this equation, α-cuts of critical values can be obtained. Pˆ Qˆ P Z 0 + z α/2 > C V22 = γ /2 P0 Q 0 Pˆ Qˆ = γ /2 (16) P Z 0 > C V22 − z α/2 P0 Q 0 Pˆ Qˆ C V22 = z γ /2 + z α/2 P0 Q 0
√
(18)
X nλ z α/2
⎝ 1−
X
X nλ z α/2
⎤ ⎞ ⎥ ⎥ − μ0 ⎠⎥ ⎥ ⎦
(21)
where z (α/2) is the z value so that probability of random variable having N (0, 1) probability density, exceeding z is α/2 . There will be two fuzzy critical value sets for V I )2 ) which inverse Gaussian distribution ((C V I )1 and (C are defined as k1 (α), cvi k2 (α)], C V I k [α] = [cvi
k = 1, 2 .
(22)
In the sense of the information given in Sect. 2 with Eq. (11) we defined (C V I )2 [α] and (C V I )1 [α] for inverse Gaussian distribution
Hypothesis testing for the mean of inverse Gaussian distribution ⎡
⎢ √nλ P⎢ ⎣ μ0 ⎡ =P
=P
1
X
X X nλ z α/2
X nλ z α/2
1−
− μ0
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⎤ ⎥ ≥ cvi 22 (α)⎥ ⎦=
γ 2
1+
⎤ X X (X −(1− nλ z α/2 )μ0 ) 1+ nλ z α/2 nλ(X −μ ) 0 ⎣ ≥ cvi 22 (α)⎦ √ (X −μ0 ) X μ0 X z α/2 1− nλ ⎡ ⎤ X X X − 1− nλ z α/2 μ0 1+ nλ z α/2 ⎣ ≥ cvi 22 (α)⎦ = γ Y 2 (X −μ0 ) X 1− nλ z α/2 √
=
γ 2
(23) Therefore cvi 22 (α) can be given from Eq. (23) cvi 22 (α)
= z γ /2
4 Application
X X X − 1 + nλ z α/2 μ0 1 + nλ z α/2 . (X − μ0 ) X 1 − nλ z α/2 (24)
[α] given with Eq. (21) and By using the left end point of Y cvi 21 (α) instead of cvi 22 (α) into Eqs. (23) and (25) can be derived as cvi 21 (α) = z γ /2
X X X − 1 + nλ z α/2 μ0 1 − nλ z α/2 (X − μ0 ) X 1 + nλ z α/2 (25)
Equation (26) is defined by Eqs. (24) and (25) ⎡
⎢ C V I 2 [α] = ⎢ ⎣z γ /2
z γ /2
X X − 1+ nλ z α/2 μ0
(X −μ0 ) X X − 1− nλ z α/2 μ0 (X −μ0 )
X z α/2 1− nλ , X z α/2 1+ nλ
⎤
respectively. Then the final decision for the mean of inverse Gaussian distribution depends on the relationship between > (C < (C , (C V I )2 . If Y V I )2 or Y V I )1 Y V I )1 and (C then reject H0 . If (C V I )1 < Y < (C V I )2 then accept H0 . Graphical demonstration for the situations above is similar instead of V )1 to Fig. 1. If we take Y Z ,(C V I )1 instead of (C and (C V I )2 instead of (C V )2 in Fig. 1, decision rules for testing the mean of inverse Gaussian distribution using α-cuts can be shown graphically.
X z α/2 ⎥ 1+ nλ ⎥ ⎦ X z α/2 1− nλ
In this section, hypothesis testing using α-cuts for the mean (μ) of an inverse Gaussian distribution have been examined on an example. We consider the data set that fits the inverse Gaussian distribution in Chhikara and Folks (1989) on page 73. They reported certain test data on the endurance of deep groove ball bearings. The data consists of the number of million revolutions before failure for each of 23 ball bearings used in the life test. In this example, suppose γ is the significance level of the test, λ = 231.63, α represents the alphacuts of triangular shaped fuzzy number for 0.01 ≤ α ≤ 1 and z (γ /2) is the z value so that the probability of exceeding it equals to γ /2 (γ = 0.05). In order to do the following hypothesis test, calculations are performed by α-cuts with the help of package program Maple 10 (2005). We want to test H0 : μ = 70 against H1 : μ = 70. From the sample we compute its mean x¯ = 72.22, with this in mind we proceed to the fuzzy situation by using Eq. (18). According to Fig. 2, we estimate the parameter μ to be almost 72.22 with considering the scale parameter is known. Almost 72.22 is represented by the membership function in is given Fig. 2 (Maple 10 2005). Then fuzzy test statistic Y with Fig. 3. We calculate all α-cuts (0.01 ≤ α ≤ 1.00) to draw the membership values of fuzzy parameter estimation of inverse
(26) C V I 1 [α] = −C V I 2 [α] ⎡
⎢ C V I 1 [α] = ⎢ ⎣−z γ /2
−z γ /2
(27)
X X − 1− nλ z α/2 μ0 (X −μ0 )
X X − 1+ nλ z α/2 μ0 (X −μ0 )
X z α/2 1+ nλ , X 1− nλ z α/2
⎤
X z α/2 ⎥ 1− nλ ⎥ ⎦ X z α/2 1+ nλ
(28)
Fig. 2 Fuzzy parameter estimation of inverse Gaussian mean, X
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and critical values (C Fig. 4 Fuzzy test statistic Y V I )1 , (C V I )2 Table 2 Fuzzy critical values of C V I 1 [α] and C V I 2 [α]
Fig. 3 Fuzzy test statistic Y Table 1 Fuzzy parameter estimation of inverse Gaussian mean X¯ [α] and fuzzy test statistic Y¯ [α] α
X˜ [α]
Y˜ [α]
0.01
[55.6 ; 99.9]
[−1.97 ; 3.32]
0.20
[63.0 ; 84.9]
[−0.95 ; 1.68]
0.40
[65.8 ; 80.1]
[−0.55 ; 1.16]
0.60
[68.1 ; 77.0]
[−0.25 ; 0.83]
0.80
[70.2 ; 74.4]
[0.07 ; 0.54]
1.00
[72.2 ; 72.2]
[0.28 ; 0.28]
(Fig. 3). Gaussian mean X (Fig. 2) and fuzzy test statistic Y Moreover, Table 1 gives more detailed information about [α] for the minimum and maximum values of X [α] and Y α = 0.01, 0.2, 0.4, 0.6, 0.8, 1.00 than Fig. 3. For instance, minimum value is 70.2 and maximum value is 74.4 for X [α = 0.80]. Similarly, minimum value is 0.07 and maximum value [α = 0.80]. It is also important that we obtain is 0.54 for Y crisp parameter estimation of μ and crisp test statistic for α = 1.00. We calculate fuzzy critical values (C V I )k , k = 1, 2 after obtaining fuzzy test statistic Y . Now we may compare Y to (C V I )1 and then (C V I )2 to determine final conclusion on H0 . Because our decision depends on the relationship and (C V I )2 . The results, using Maple between Y V I )1 , (C 10 (2005), are represented in Fig. 4 with α-cuts (0.01 ≤ α ≤ and (C V I )2 . 1.00) of Y V I )1 , (C just to and (C In Fig. 4, Y V I )2 are very close, with Y the left of (C V I )2 , so (C V I )2 ≈ Y . The final result is V I )2 . Hence our final conclusion in (C V I )1 < Y ≈ (C hypothesis testing for the mean of inverse Gaussian distribution using α-cuts is “no decision on H0 : μ = 70”. On the other hand the crisp test would decide to accept H0 .
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α
C V I 1 [α]
C V I 2 [α]
0.01
[−29.99 ; 11.99]
[−11.99 ; 29.99]
0.20
[−13.22 ; 5.835]
[−5.835 ; 13.22]
0.40
[−9.439 ; 3.402]
[−3.402 ; 9.439]
0.60
[−6.398 ; 1.848]
[−1.848 ; 6.398]
0.80
[−4.104 ; −0.170]
[0.170 ; 4.104]
1.00
[−1.937 ; −1.937]
[1.937 ; 1.937]
Table 2 gives more detailed information about the mini V I )2 [α] for mum and maximum values of (C V I )1 [α] and (C α = 0.01, 0.20, 0.40, 0.60, 0.80, 1.00 than Fig. 4. In addition, it is seen in Table 2 that crisp critical values are obtained for α = 1.00. Consequently, we test the mean of inverse Gaussian distribution using α-cuts when scale parameter is known. Fuzzy test statistic and fuzzy critical values are calculated based on Buckley’s approach. Hence not only one value but also all confidence intervals are taken into account to test the mean of inverse Gaussian distribution.
5 Conclusion This article presents the modification of hypothesis testing using α-cuts for inverse Gaussian distribution. We use crisp data, through placing confidence intervals on top of each other to construct fuzzy location parameter estimator for the inverse Gaussian distribution, hence we use much more information from classical approach, like all the confidence intervals between 0 and 99 %. As a result of this study, hypothesis test approach for some other statistical distributions: Poisson, Normal, Binomial, Uniform that was proposed by Buckley can also be used for the inverse Gaussian distribution. Thus, it is shown that the advantages of Buckley’s approach are also valid for inverse Gaussian distribution by this study.
Hypothesis testing for the mean of inverse Gaussian distribution
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