Methods based on Chhikara and Folks

3.2.1 Inferences on μ

Suppose X=

(

X X1, 2,...,X_n

)

is from ^IG

(

^{μ λ,}

)

, the joint density function of when λ is unknown can equivalently be stated as follow:

'

and

Moreover, this critical region is n-1 degrees of freedom. (Chhikara and Folks, 1976)

It is interesting to note that the critical region in (3.18) is equivalent to

2

which is the same as our result in (3.6). Thus we can conclude that our procedure is easily applicable.

On the other hand, according to Chhikara and Folks (1989), the confidence intervals for the parameter μ can be obtained by inverting the acceptance regions.

Therefore, when λ is unknown, it follows from (3.18) that the ^{100 1}

(

^−α

)

^percent

3.2.2 Inferences on λ

Roy and Wasan (1968) derived the UMP-unbiased test for ₀

0

− denotes the chi-square distribution function with n-1 degrees of freedom, and then C₁ and C₂ are uniquely determined from using tables of the chi-square distribution. Thus, for the equal tail test, C₁ and C₂ can be obtained by solving is the rth quantile of chi-square distribution with n-1 degrees of freedom. Therefore the p-value is ^{p=2* min}

{

^P⎡⎣^{χn2−1>λ0v⎤}⎦^,P⎡⎣^{χn2−1<λ0v⎤}⎦ and the

}

^{100 1}

(

^−α

)

^%

We note that these results are equivalent to our results in (3.11) and (3.12).

Chapter 4 Inferences on two populations of Inverse Gaussian

Although there has been a rapid growth in IG, the problem about making inference to the ratio of two IG means still need to be investigated. As the scale parameters λ and ₁ λ of two independent populations are the same, i.e. ₂ λ λ₁= ₂, the two-sided exact confidence interval of θ μ μ= _{1 2} has been discussed by Chhikara and Folks (1989). However, it is not practical to expect two IG populations to have the identical scale parameter all the time. Recently, Tian and Wilding (2005) presented an approximate approach to construct the confidence interval of θ μ μ= _{1 2} of two independent IG populations based on the modified

directed likelihood ratio method (Barndorff-Nielsen, 1986). Nevertheless, the exact property of θ μ μ= _{1 2} deserves further study. Therefore, in this chapter we will provide an exact and convenient method based on generalized p-value and generalized confidence interval to perform the hypothesis testing and then construct confidence intervals for θ μ μ= _{1 2} and the ratio of two scale parameters δ λ λ= _{1 2}. In this chapter, we will also briefly introduce some methods in the literature which will be utilized to compare with our procedure in numerical examples and simulation studies.

4.1 Methods based on the generalized test variable and generalized

X X X be independent random samples from IG

(

μ λ and 1, 1

)

IG

(

μ λ , respectively, where 2, 2

)

μ_i and λ_i are unknown and possible unequal with i=1, 2. The independent sufficient statistics are given by

( )

Suppose we are interested in making inference in the parameter θ μ μ= _{1 2}, consider the following hypothesis testing:

1 that the generalized test variable for two populations of inverse Gaussian is parallel that it for one population of IG in (3.5). In fact, the generalized test variable (3.5)

and its observed value

2 Therefore we find another more flexible generalized test variable

(

^{1, 2, ,1 2; ,1 2, ,1 2, 1, 2, 1, 2}

)

G X X V V x x v v μ μ λ λ ≡G which is constructed by two independent statistics ^{G X V x v1}

(

^{1, ; , ,1 1 1 μ λ1, 1}

)

^{≡G1 and G2}

(

^{X V x v2, 2; 2, 2,μ λ2, 2}

)

^≡G2.

Above all we deliberate the statistic G_i based on the random independent which has been mentioned in (2.5) and (2.6), respectively. Since ~ ² ₁

i ni

B χ ₋ and

2

~ 1

Ui χ , then one part of the generalized test variable for testing (4.2) can be deduced as following equation: used as a generalized test variable in one population case and the result is equivalent to what we got in Chapter 3.

Eventually, since G₁ and G₂ are independent generalized test quantities, the generalized test variable G can be defined as follows:

1 2 noted that the distribution of G is independent of the nuisance parameters λ or ₁

λ , and the observed value 2

(

1 2 1 2 1 2 1 2 1 2 1 2

)

¹

(2.8), G is a generalized test variable which can be applied for testing the hypothesis ₀ ¹ ₀

1 1

We next consider the problem of interval estimation for μ μ based on _{1 2} generalized pivotal quantity. Since the observed value of G is μ μ , the _{1 2} parameter of interest, and the properties of G fulfill the requirements in (2.10), thus

G in (4.5) is indeed a generalized pivotal quantity which can be used to construct a generalized confidence interval. Therefore the ^{100 1}

(

^−α

)

^% equal tail confidence interval for μ μ can be computed by _{1 2} necessary. Since the property of IG does not hold for the location change, it is hard to make inferences for the mean difference without any restriction. On the contrary, our procedure is readily applicable and easy to use to deal with mean difference problem without any restriction.

4.1.2 Inferences on λ λ 1 2

It is also an interesting problem concerning the parameter δ λ λ= _{1 2}. Consider the hypothesis employed in one population case can be applied to the two populations’ case as well.

Similarly, since distribution of random variable T is free of nuisance parameters, the observed value ^*

( )

variable which satisfies the three conditions in (2.8). Therefore T is indeed a ^* generalized test variable and can be used to test the hypothesis in (4.8). The generalized p-value for testing (4.8) can be computed by

{

^{* * 0 * * 0}

}

Furthermore, in order to construct confidence interval of λ λ , _{1 2}

( )

where T^*

(

v v1, 2;γ stands for the th

)

γ quantile of T^*

(

V V v v1, 2; ,1 2,ψ which is

)

defined in (4.10).

4.2 Methods based on Chhikara and Folks (1989)

4.2.1 Inferences on μ μ_{1 2}

Under the restriction of λ λ₁= ₂ =λ and ¹₂ ²₂

1 2

λ λ ξ

μ ⁼ μ ^{= ,} ξ is a constant, Chhikara and Folks (1989) derived a UMP-unbiased tests by constructing critical points of their rejection regions using percentage points of Student’s t distribution.

For the significance size α test of H₀:μ₁=μ₂ versus H₁:μ₁≠μ₂ ,

The test can be extended to compare the two inverse Gaussian means in terms of their ratio. This follows because of the property that density function

(

^{; ,}

)

¹

(

^;1,

)

It is straightforward to express the UMP-unbiased test procedures in terms of θ₀ obtained by inverting the acceptance regions of these tests at level α . When λ is unknown, the confidence interval for θ₀ is given by

and ^{1 1}

(

^{11 11}

)

For more details, we refer to the paper and the book by Chhikara and Folks (1975, 1989), respectively.

( )

It is straightforward to construct a confidence interval for ¹

2 degrees of freedom. Let

1 2

confidence interval for ¹

2

It is interesting to note that the result in (4.18) is the same as our result in (4.14). In our procedure, the pivotal quantity (4.10) of ¹

2

− , the quantile points which satisfys

*

for ¹

The signed log likelihood ratio has been discussed by many authors, McCullagh (1982), Petersen (1981), Pierce and Schafer (1986), and Barndorff-Nielsen (1986) etc., to obtain a statistic which is asymptotically standard normally distributed with error of order O n( ^{−3/ 2}) by repeated sampling. Tian and Wilding (2005) provided an estimating approach for constructing a confidence interval of μ μ based on the _{1 2} directed likelihood ratio method. The procedure is as follows.

Suppose the ratio of the two means is the parameter of interest, that is

1 2

θ μ μ= and the vector of nuisance parameters is η=

(

μ λ λ2, 1, 2

)

and ^ζ=

(

^θ,η

)

^.

Let Y_ij =1 X_ij, 1,..., ; 1, 2j= n i_i = , then Y and _{1 j} Y are two independent samples _{2 j} from RRIG

(

μ λ and 1, 1

)

RRIG

(

μ λ , respectively, where RRIG means the 2, 2

)

reciprocal root IG distribution. The log-likelihood function is

( ) (

1 2

)

¹ 1 ² 2 ^{1 1}

where

The maximum likelihood estimates of the parameters of (4.19) are

(

S n1 1

) (

S n2 2

)

θ)=

, μ)₂ =S₂ n₂ , λ)1=1 T n

(

1 1−n S1 1

)

and λ)2 =1 T n

(

2 2−n S2 2

)

. For a given value of θ , the constrained maximum likelihood estimates of the

nuisance parameters η=

(

μ λ λ2, 1, 2

)

can be obtained by solving

Chapter 5 Numerical Examples and Simulation Studies

Some IG data are given to compare our procedure with other methods with respect to their confidence intervals and confidence lengths. Several simulation studies are also presented to compare the performances of three methods, (1) Chhikara and Folks (2) Tian and Wilding (3) the generalized approaches, in terms of their coverage probabilities, expected lengths and the Type I error.

5.1 Numerical examples

Example 1.

Gacula and Kubala (1975) reported certain sensory failure data for two refrigerated food products, M and K as these were called, and studied their shelf life which fit the IG distribution well. The summary data are given in Table 1 and the 95%

confidence intervals for three methods are presented in Table 2.

Table 1. Summary data

Product size μˆ _λ ^ˆ

M 26 42.885 18.622

K 17 56.941 14.881

Table 2. 95% confidence intervals and lengths for ¹

2

θ μ

=μ

Method

θ ˆ

95% confidence interval length

Chhikara 0.771 ( 0.635 , 0.905 ) 0.270

Directed 0.753 ( 0.562 , 0.962 ) 0.400

Generalized 0.755 ( 0.554 , 0.977 ) 0.422

Example 2.

Four sets of IG data presented in Folks and Chhikara (1978) who judged that the data are very well described by the Inverse Gaussian distribution. The first set, data (1), gives fracture toughnesses of MIG welds. The second set, data (2), gives data of precipitation (inches) from Jug Bridge, Maryland. The third set, data (3), gives runoff amounts at Jug Bridge, Maryland. Additionally, Gacula and Kubala (1975) gave data (4) on shelf-life of a food product. The summary data for four sets of IG data are shown in Table 3. For investigating the ratio of means of two independent populations when the scale parameters are more different than those in Example 1, we will compare the means of these four data sets mutually and show the results in Table 4.

Table 3. The summary data for four data sets

data size μˆ λ ˆ

( 1 ) 19 74.300 4924.070

( 2 ) 25 2.160 8.080

( 3 ) 25 0.800 1.440

( 4 ) 26 42.885 484.253

Table 4. 95% confidence intervals and lengths for ¹

2

θ μ

=μ

(2)/(1)

θ ˆ

95% confidence interval length

Chhikara 0.0303 ( 0.020 , 0.040 ) 0.020

Directed 0.0290 ( 0.024 , 0.036 ) 0.012 Generalized 0.0294 ( 0.024 , 0.037 ) 0.014

(3)/(1)

θ ˆ

95% confidence interval length

Chhikara 0.0118 ( 0.006 , 0.017 ) 0.011

Directed 0.0108 ( 0.008 , 0.016 ) 0.008 Generalized 0.0111 ( 0.008 , 0.016 ) 0.008

(4)/(1)

θ ˆ

95% confidence interval length

Chhikara 0.5857 ( 0.469 , 0.702 ) 0.233

Directed 0.5771 ( 0.509 , 0.661 ) 0.152 Generalized 0.5796 ( 0.505 , 0.667 ) 0.162

(3)/(2)

θ ˆ

95% confidence interval length

Chhikara 0.4046 ( 0.154 , 0.655 ) 0.501

Directed 0.3726 ( 0.262 , 0.558 ) 0.296 Generalized 0.3812 ( 0.258 , 0.564 ) 0.306

(2)/(4)

θ ˆ

95% confidence interval length

Chhikara 0.0521 ( 0.034 , 0.070 ) 0.036

Directed 0.0503 ( 0.040 , 0.066 ) 0.026 Generalized 0.0509 ( 0.040 , 0.065 ) 0.025

(3)/(4)

θ ˆ

95% confidence interval length

Chhikara 0.0201 ( 0.011 , 0.029 ) 0.018

Directed 0.0187 ( 0.014 , 0.028 ) 0.014 Generalized 0.0193 ( 0.014 , 0.028 ) 0.014

From Example 1 and Example 2, the results show that the confidence lengths obtained by the generalized methods are the smallest or close to the smallest confidence lengths no matter what the scale parameters perform when two IG populations are non-homogeneous. Some simulation studies are also worth to be inspected, and we will make discussion in next subsection.

5.2 Simulation studies

Some simulation studies are performed to compare the 95% coverage probabilities, expected lengths and type I errors of three procedures for the ratio of two means, θ μ μ= _{1 2} . We will choose different combinations of sample sizes

(

n n1, 2

) (

= 5,10 , 10,5 and 10,10

) ( ) ( )

, respectively, and various values of the ratio of

scale parameters, λ λ , with 1,000 replicates for each combination. The results _{1 2} appear in Tables 5-9. In addition, we will present powers of the tests obtained by the generalized method in Table 10.

Table 5. Coverage probabilities (CP) and expected lengths (length) of 95% confidence intervals of ¹

○1 Generalized methods ○2 Chhikara and Folks (1989) ○3 Directed likelihood ratio statistic

Table 6. Coverage probabilities (CP) and expected lengths (length) of 95% confidence

○1 Generalized methods ○2 Chhikara and Folks (1989) ○3 Directed likelihood ratio statistic

Table 7. Coverage probabilities (CP) and expected lengths (length) of 95% confidence

○1 Generalized methods ○2 Chhikara and Folks (1989) ○3 Directed likelihood ratio statistic

Table 8. Type I error for testing H₀:θ θ= ₀ versus H₁:θ θ≠ , ₀ ¹

λ Generalized○¹ Chhikara○² Directed○³

( 5 , 10 ) 0.5 0.04 0.05 0.08

○1 Generalized methods ○2 Chhikara and Folks (1989) ○3 Directed likelihood ratio statistic

Table 9. Type I error for testing H₀:θ θ= ₀ versus H₁:θ θ≠ , ₀ ¹

λ Generalized○¹ Chhikara○² Directed○³

( 5 , 10 ) 0.5 0.05 0.17 0.09

○1 Generalized methods ○2 Chhikara and Folks (1989) ○3 Directed likelihood ratio statistic

Table 10. Simulated powers for testing H₀:θ =1 versus H₁:θ≠ , 1 ¹

From Table 5 to Table 10, we can conclude that the coverage probabilities obtained by generalized methods are very close to the nominal level 95% and the Type I error are exact or close to the nominal level 0.05. On the other hand, the coverage probabilities obtained by directed likelihood ratio are too small and the type I errors exceed the 5% in all cases. Besides, Chhikara and Folks (1989)’s procedure performs well under λ λ₁ = ₂ and ⁱ₂ , a constant, 1, 2

i

λ ξ i

μ ^{= =} , but its performance

becomes worse when the heteroscedasticity is increasing. In fact, the procedure based on generalized methods is readily applicable and easy to perform even under the sample sizes are quite small. The simulation results show that its results are better than the other two methods with respect to having almost exact coverage probabilities and the type I errors.

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