Large Sample and Bootstrap Intervals for the Gamma Scale Parameter ...

Journal of Modern Applied Statistical Methods Volume 5 | Issue 2

Article 10

11-1-2005

Large Sample and Bootstrap Intervals for the Gamma Scale Parameter Based on Grouped Data Ayman Baklizi Yarmouk University, Irbid, Jordan, [email protected]

Amjad Al-Nasser Yarmouk University, Irbid, Jordan, [email protected]

Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Baklizi, Ayman and Al-Nasser, Amjad (2005) "Large Sample and Bootstrap Intervals for the Gamma Scale Parameter Based on Grouped Data," Journal of Modern Applied Statistical Methods: Vol. 5 : Iss. 2 , Article 10. DOI: 10.22237/jmasm/1162354140 Available at: http://digitalcommons.wayne.edu/jmasm/vol5/iss2/10

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Journal of Modern Applied Statistical Methods November, 2006, Vol. 5, No. 2, 356-366

Large Sample and Bootstrap Intervals for the Gamma Scale Parameter Based on Grouped Data Ayman Baklizi

Amjad AL-Nasser

Department of Statistics Yarmouk University

Interval estimation of the scale parameter of the gamma distribution using grouped data is considered in this article. Exact intervals do not exist and approximate intervals are needed Recently, Chen and Mi (2001) proposed alternative approximate intervals. In this article, some bootstrap and jackknife type intervals are proposed. The performance of these intervals is investigated and compared. The results show that some of the suggested intervals have a satisfactory statistical performance in situations where the sample size is small with heavy proportion of censoring. Key words: bootstrap, gamma distribution, grouped data, interval estimation.

In a recent article, Chen and Mi (2001) provided a general method for constructing intervals for the unknown parameters in the distribution using grouped data. Assume that the data are grouped in the classes [0, t1 ), [t1 , t 2 ),..., [t k −1 , t k ), [t k , ∞) . The i-th

Introduction In many practical studies, the collected data may not be complete observations, they may be in a form of counts of observations in certain intervals; such data is often called grouped data. Grouped data arise frequently in life testing experiments when inspecting the test units intermittently for failure, this procedure is frequently used because it requires less testing effort than continuous inspection. The data obtained from intermittent inspection consists only of the number of failures in each inspection interval. Other examples of natural occurrences of grouped data are given in Pettitt and Stephens (1977).

interval is [ti −1 , ti ) . Assume that t 0 = 0 and

t k +1 = ∞ . Let ri be the number of failures in the i-th interval. Define the random variable k

ζ n = ∑ ri t i + rk +1t k .

It

i =1

ζ n − ng (λ ) nsn where

follows

that

→ N (0,1) in law as n → ∞ , k

g (λ ) = ∑ t i pi + t k p k +1 i =1

and 2

Ayman Baklizi is an Associate Professor of statistics, and currently is at the department of mathematics and physics at Qatar University. He is an elected-member at the ISI. Amjad AlNasser is an Assistant Professor of statistics at Yarmouk University, Jordan.

k k s n = ⎛⎜ ∑ t i2 pˆ i + t k2 pˆ k +1 ⎞⎟ − ⎛⎜ ∑ t i pˆ i + t k pˆ k +1 ⎞⎟ . ⎝ i =1 ⎠ ⎝ i =1 ⎠

It

follows

that,

asymptotically,

ζ s s ⎞ ⎛ζ p⎜⎜ n − zα 2 n < g (λ ) < n + zα 2 n ⎟⎟ = 1 − α n n n⎠ ⎝ n When the function g (λ ) is monotone, an approximate (1 − α )% confidence interval for λ , call it the CM interval, can be obtained as

356

BAKLIZI & AL-NASSER ⎡ −1 ⎛ ζ n s ⎞⎤ s ⎞ ⎛ζ − zα 2 n ⎟⎟, g −1 ⎜⎜ n + zα 2 n ⎟⎟⎥ . ⎢ g ⎜⎜ n ⎠⎦ n⎠ ⎝ n ⎣ ⎝ n However, the above interval possesses exact coverage probabilities and symmetry probabilities only for sufficiently large sample sizes. In this article, the properties of these intervals are investigated and some bootstrap based intervals that use the result of Chen and Mi (2001) are proposed. A similar problem has been investigated for the Burr type X distribution by Al-Nasser and Baklizi (2004). Bootstrap Intervals Let x1 , … , x n be a random sample from the gamma distribution whose probability density function is given by

f ( x, λ , c ) =

λc

Γ(c )

[ (

(t i −1 , t i ) , i = 1,…, k + 1 .

The

r1 ,… , rk +1 is multinomial with parameters n and p1 ,..., p k +1 . The following confidence intervals are based on the Bootstrap approach (Efron & Tibshirani, 1993). There are several Bootstrap based intervals discussed in the literature, the most common ones are the bootstrap –t interval, the percentile interval and the bias corrected and accelerated ( BC a ) interval. The Bootstrap – t Interval (BTS Intervals) Let ζ n be the random variable defined k

ζ n = ∑ ri t i + rk +1t k calculated from the i =1

original data and let ζ n* be calculated from the bootstrap sample. Let zα* be the α quantile of the bootstrap distribution of Z * =

(

ζ n*

−ζ n s n*

)

)]

(

zα* is determined by simulation The Percentile Interval (PRC Interval) Here, the bootstrap distribution of ζ n* are simulated by resampling repeatedly from the parametric model of the original data and calculating ζ n*,i , i = 1, … , B where B is the number of bootstrap samples. Let Gˆ be the cumulative distribution function of ζ n* , then the

1 − α interval is given by

⎡ ˆ −1 ⎛ α ⎞ ˆ −1 ⎛ α ⎞⎤ ⎢G ⎜ 2 ⎟ , G ⎜1 − 2 ⎟⎥ . ⎝ ⎠⎦ ⎝ ⎠ ⎣

joint probability function of

as

where s n* is estimated variance of ζ n calculated from the bootstrap sample. The bootstrap-t λ is given by interval for −1 * * −1 * * g ζ n − zα 2 s n , g ζ n + zα 2 s n where

x c −1e −λx , x > 0.

Let ri be the number of observations falling in the i-th interval

357

),

The Bias Corrected Interval (BC Interval) The bias corrected interval (Efron, 1982) is calculated using the percentiles of the bootstrap distribution of ζ n* . The determination of the appropriate percentiles depends on a number ( zˆ 0 ) called the bias correction. The

1 − α interval is given by

[Gˆ

−1

(α 1 ) ,

]

Gˆ −1 (α 2 )

where

α 1 = Φ (2 zˆ 0 + zα 2 ) ,

α 2 = Φ (2 zˆ 0 + z1−α 2 ) , Φ (.)

is the standard normal cumulative distribution function, zα is the α quantile of the standard normal distribution. The value of

{

⎛ # ζ n* < ζ n ⎜ B ⎝

calculated as zˆ 0 = Φ −1 ⎜

}⎞⎟ . ⎟ ⎠

zˆ 0 are

358 LARGE SAMPLE AND BOOTSTRAP INTERVALS FOR GAMMA PARAMETER The Bias Corrected and Accelerated Interval (BCa Interval) The bias corrected and accelerated interval is calculated also using the percentiles of the bootstrap distribution of. The percentiles depend on two numbers aˆ and zˆ 0 called the acceleration and the bias correction. The 1 − α interval is given by

(Gˆ

−1

(α1 ) ,

where

)

seˆ 2 =

⎛ zˆ 0 + zα 2 ⎞ ⎟, ⎜ ⎟ ˆ ˆ a z z − + 1 0 α 2 ⎠ ⎝ ⎛ zˆ 0 + z1−α 2 ⎞ ⎟, α 2 = Φ⎜ zˆ 0 + ⎜ ⎟ ˆ ˆ a z z − + 1 0 1 − α 2 ⎝ ⎠ Φ (.)

α 1 = Φ⎜ zˆ 0 +

(

)

(

)

is the standard normal cummulative distribution function, zα is the α quantile of the standard normal distribution. The values of aˆ and zˆ 0 are calculated as follows; n

∑ (ζ n (.) − ζ n (i ))

3

i =1

2 6⎧⎨ ∑ (ζ n (.) − ζ n (i )) ⎫⎬ ⎩i =1 ⎭ n

32

where ζ n (i ) is calculated using the original data excluding the i-th observation and n

ζ n (.) =

ζ n (.) ± zα 2 seˆ .,

Gˆ −1 (α 2 )

where

aˆ =

Jacknife Intervals (JAC Intervals) An interval based on the jacknife (Efron & Tibshirani, 1993) can be constructed as follows;

∑ ζ n (i )

i =1

n

is the jackknife estimate of the variance of ζ n . Intervals Based on the Bootstrap Standard Deviation (BSD Intervals) An interval similar in form to the based on the jacknife can be constructed as follows;

ζ n ± zα 2 se~ ., where

se~ 2 =

.

⎛ # ζ n* < ζ n zˆ 0 = Φ −1 ⎜⎜ B ⎝

}⎞⎟ . ⎟ ⎠

(

1 B * ζ i , n − ζ n* ∑ B − 1 i =1 1 B ζ n* = ∑ ζ i*, n B i =1

)

2

,

is the bootstrap estimate of the variance of ζ n . Small Sample Performance of the Intervals For the confidence intervals with nominal confidence coefficient (1 − α) , the criterion of attainment of lower and upper error probabilities (Jennings, 1987) is used, which are both taken equal to

The value of zˆ 0 is given, as before, by

{

n −1 n 2 ∑ (ζ n (.) − ζ n (i )) n i =1

α . A simulation study is 2

conducted to investigate the performance of the intervals. The sample sizes chosen are n = 20,30,50, 100 . The number of groups k + 1 is taken as 3, 5 and 9. The censoring proportion (cp) is taken as 0.6, 0.4, and 0.2. The confidence coefficient is taken as 95%, and the shape parameter r is taken as 0.4, 0.8, 1.2, 1.6, and 2. For each combination of the simulation indices 2000 samples were generated from the

BAKLIZI & AL-NASSER gamma distribution with λ = 1 The intervals are calculated, B = 2000 was used for bootstrap calculations. The following quantities are simulated for each interval using the results of the 2000 samples; 1. Lower error rates (L): The fraction of intervals that fall entirely above the true parameter. 2. Upper error rates (U): The fraction of intervals that fall entirely below the true parameter. 3. Total error rates (T): The fraction of intervals that did not contain the true parameter value. The results are given in Tables 1-3.

359

the nominal sizes. As the censoring proportion is light to moderate, the PRC and the BTS intervals tend to be highly conservative while the BC and BCa intervals tend to be grossly anticonservative. For larger sample sizes (n = 50, 100) all intervals attain their nominal sizes except for the BC and BCa intervals where they remain anticonservative. In situations where k = 2 and small sample size, all intervals are asymmetric. As k increases, the intervals tend generally to be more symmetric. The performance of the BC and BCa intervals improves considerably for larger values of k. Also their performance improves for higher values of r, that is, the more symmetric the parent gamma distribution, the more symmetric the BC and BCa intervals tend to be.

Results

Conclusion

From the simulation results, it appears that for k = 2, small sample size (n = 20, 30) and heavy censoring (cp = 0.4,0.6), the CM intervals tend to be anti-conservative. This is also true for PRC, BC and BCa intervals. On the other hand, the BTS, BSD, and JAC intervals tend to attain

It appears that the intervals proposed by Chen and Mi (2001) have a good performance except for situations of small sample size and heavy censoring. In this case the BTS, JAC, and especially BSD intervals provide better alternatives.

360 LARGE SAMPLE AND BOOTSTRAP INTERVALS FOR GAMMA PARAMETER

Table 1. Results for K=2 R 0.4

CP 0.6

0.4

0.2

0.8

0.6

0.4

0.2

1.2

0.6

0.4

n M CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD

L1 0.045 0.043 0.013 0.027 0.015 0.045 0.013 0.019 0.019 0.018 0.116 0.042 0.019 0.019 0.007 0.007 0.037 0.006 0.005 0.007 0.007 0.033 0.033 0.000 0.005 0.001 0.033 0.005 0.029 0.025 0.005 0.010 0.005 0.029 0.029 0.013 0.013 0.013 0.002 0.002 0.013 0.013 0.010 0.010 0.000 0.010 0.000 0.010 0.010 0.020 0.020 0.004 0.004 0.003 0.020 0.020

20 U1 0.038 0.012 0.012 0.104 0.104 0.038 0.038 0.050 0.017 0.004 0.173 0.160 0.050 0.050 0.067 0.020 0.000 0.224 0.184 0.067 0.064 0.037 0.013 0.042 0.178 0.178 0.037 0.037 0.035 0.024 0.024 0.168 0.168 0.035 0.035 0.056 0.019 0.007 0.214 0.214 0.056 0.056 0.038 0.014 0.014 0.089 0.089 0.038 0.038 0.033 0.032 0.032 0.163 0.163 0.033 0.033

T1 0.083 0.055 0.025 0.130 0.119 0.083 0.051 0.069 0.036 0.022 0.289 0.202 0.069 0.069 0.074 0.027 0.037 0.230 0.188 0.074 0.070 0.070 0.045 0.042 0.182 0.178 0.070 0.042 0.064 0.049 0.029 0.178 0.173 0.064 0.064 0.069 0.032 0.020 0.216 0.216 0.069 0.069 0.048 0.024 0.014 0.099 0.089 0.048 0.048 0.053 0.052 0.036 0.166 0.166 0.053 0.053

L2 0.033 0.032 0.010 0.022 0.021 0.033 0.017 0.027 0.026 0.026 0.011 0.009 0.027 0.027 0.023 0.032 0.032 0.005 0.005 0.023 0.023 0.013 0.013 0.003 0.013 0.008 0.013 0.013 0.026 0.024 0.011 0.011 0.005 0.026 0.026 0.025 0.025 0.025 0.010 0.008 0.025 0.025 0.015 0.015 0.002 0.017 0.015 0.051 0.015 0.013 0.032 0.012 0.013 0.011 0.033 0.013

30 U2 0.039 0.016 0.016 0.079 0.082 0.039 0.039 0.028 0.009 0.002 0.101 0.100 0.028 0.028 0.085 0.017 0.005 0.238 0.234 0.085 0.038 0.037 0.003 0.020 0.080 0.087 0.037 0.037 0.032 0.022 0.022 0.137 0.137 0.032 0.032 0.046 0.014 0.006 0.183 0.173 0.046 0.046 0.031 0.024 0.027 0.063 0.063 0.031 0.031 0.050 0.017 0.017 0.099 0.100 0.050 0.050

T2 0.071 0.047 0.026 0.100 0.103 0.071 0.055 0.055 0.035 0.028 0.112 0.109 0.055 0.055 0.108 0.049 0.037 0.242 0.239 0.108 0.060 0.050 0.016 0.023 0.093 0.095 0.050 0.050 0.057 0.045 0.032 0.147 0.141 0.057 0.057 0.071 0.039 0.031 0.193 0.180 0.071 0.071 0.046 0.039 0.029 0.080 0.078 0.081 0.046 0.062 0.049 0.029 0.111 0.111 0.083 0.062

L3 0.017 0.018 0.007 0.017 0.012 0.017 0.017 0.014 0.018 0.018 0.007 0.007 0.030 0.017 0.011 0.026 0.026 0.005 0.005 0.011 0.012 0.030 0.013 0.006 0.028 0.012 0.030 0.030 0.019 0.021 0.011 0.014 0.013 0.019 0.019 0.021 0.034 0.034 0.008 0.008 0.021 0.021 0.017 0.017 0.006 0.018 0.017 0.017 0.017 0.024 0.024 0.011 0.016 0.012 0.024 0.024

50 U3 0.025 0.018 0.018 0.079 0.083 0.025 0.027 0.034 0.014 0.014 0.132 0.124 0.034 0.034 0.055 0.012 0.004 0.234 0.230 0.055 0.041 0.022 0.011 0.024 0.047 0.050 0.022 0.022 0.030 0.015 0.015 0.078 0.078 0.030 0.030 0.041 0.010 0.003 0.105 0.100 0.041 0.041 0.024 0.004 0.018 0.050 0.056 0.024 0.021 0.028 0.014 0.014 0.094 0.094 0.028 0.034

T3 0.042 0.036 0.025 0.096 0.095 0.042 0.044 0.048 0.032 0.032 0.139 0.131 0.064 0.051 0.066 0.038 0.030 0.239 0.235 0.066 0.053 0.051 0.023 0.030 0.075 0.061 0.051 0.052 0.049 0.036 0.026 0.092 0.091 0.049 0.049 0.062 0.044 0.036 0.113 0.108 0.062 0.062 0.041 0.021 0.024 0.068 0.073 0.041 0.038 0.051 0.037 0.025 0.109 0.105 0.051 0.058

L4 0.023 0.018 0.009 0.017 0.013 0.023 0.023 0.020 0.030 0.030 0.015 0.014 0.020 0.020 0.015 0.049 0.049 0.005 0.005 0.015 0.017 0.035 0.029 0.009 0.027 0.020 0.035 0.031 0.022 0.030 0.017 0.014 0.011 0.022 0.022 0.022 0.047 0.047 0.009 0.008 0.022 0.022 0.017 0.016 0.003 0.023 0.018 0.017 0.017 0.019 0.022 0.014 0.019 0.016 0.019 0.019

100 U4 0.027 0.010 0.026 0.059 0.060 0.027 0.027 0.035 0.014 0.009 0.079 0.079 0.035 0.035 0.034 0.003 0.002 0.173 0.172 0.034 0.034 0.016 0.016 0.029 0.035 0.040 0.029 0.017 0.025 0.014 0.014 0.079 0.079 0.025 0.026 0.030 0.009 0.004 0.098 0.097 0.030 0.030 0.024 0.020 0.039 0.024 0.027 0.024 0.024 0.026 0.021 0.021 0.065 0.066 0.041 0.034

0.049 0.028 0.035 0.076 0.073 0.049 0.050 0.055 0.043 0.038 0.093 0.092 0.055 0.055 0.048 0.052 0.051 0.178 0.177 0.048 0.051 0.051 0.045 0.038 0.061 0.059 0.063 0.048 0.047 0.044 0.031 0.092 0.089 0.047 0.048 0.051 0.056 0.051 0.106 0.105 0.051 0.051 0.041 0.036 0.041 0.047 0.044 0.041 0.041 0.045 0.042 0.035 0.084 0.082 0.059 0.053

BAKLIZI & AL-NASSER

361

Table 1. Continued R

1.6

CP 0.2

0.6

0.4

0.2

2.0

0.6

0.4

0.2

n M CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD

L1 0.016 0.016 0.016 0.003 0.002 0.016 0.016 0.034 0.034 0.000 0.016 0.009 0.034 0.000 0.012 0.048 0.012 0.013 0.010 0.012 0.012 0.015 0.015 0.014 0.005 0.005 0.015 0.015 0.043 0.043 0.000 0.000 0.000 0.043 0.000 0.025 0.025 0.005 0.008 0.005 0.025 0.025 0.033 0.031 0.009 0.005 0.003 0.033 0.016

20 U1 0.050 0.017 0.006 0.167 0.165 0.050 0.050 0.046 0.008 0.044 0.066 0.068 0.046 0.046 0.034 0.010 0.010 0.073 0.073 0.034 0.034 0.063 0.021 0.008 0.225 0.211 0.063 0.063 0.019 0.014 0.018 0.038 0.043 0.019 0.019 0.049 0.013 0.013 0.095 0.097 0.049 0.049 0.035 0.015 0.015 0.197 0.188 0.081 0.036

T1 0.066 0.033 0.022 0.169 0.167 0.066 0.066 0.080 0.041 0.044 0.081 0.076 0.080 0.046 0.045 0.058 0.022 0.086 0.083 0.045 0.045 0.078 0.036 0.022 0.230 0.215 0.078 0.078 0.062 0.057 0.018 0.038 0.043 0.062 0.019 0.074 0.038 0.017 0.103 0.102 0.074 0.074 0.068 0.046 0.023 0.201 0.190 0.114 0.052

L2 0.018 0.018 0.017 0.008 0.004 0.018 0.018 0.023 0.023 0.000 0.019 0.005 0.023 0.006 0.037 0.036 0.015 0.015 0.015 0.037 0.015 0.018 0.018 0.018 0.010 0.007 0.018 0.018 0.007 0.007 0.000 0.007 0.005 0.007 0.007 0.029 0.029 0.009 0.024 0.014 0.029 0.027 0.024 0.021 0.021 0.009 0.008 0.024 0.024

30 U2 0.059 0.024 0.011 0.188 0.188 0.059 0.047 0.021 0.004 0.021 0.052 0.061 0.021 0.021 0.024 0.009 0.009 0.051 0.051 0.024 0.024 0.043 0.018 0.007 0.130 0.124 0.043 0.043 0.021 0.010 0.036 0.055 0.091 0.021 0.021 0.023 0.012 0.012 0.052 0.052 0.023 0.023 0.047 0.020 0.008 0.157 0.153 0.047 0.047

T2 0.077 0.041 0.028 0.195 0.191 0.077 0.065 0.044 0.026 0.021 0.071 0.066 0.044 0.027 0.060 0.045 0.024 0.066 0.066 0.060 0.038 0.061 0.035 0.024 0.140 0.130 0.061 0.061 0.028 0.017 0.036 0.061 0.096 0.028 0.028 0.052 0.041 0.020 0.076 0.066 0.052 0.050 0.070 0.040 0.028 0.165 0.161 0.070 0.070

L3 0.024 0.035 0.035 0.012 0.011 0.024 0.024 0.017 0.017 0.001 0.029 0.016 0.017 0.017 0.019 0.019 0.010 0.018 0.016 0.019 0.019 0.023 0.031 0.031 0.009 0.008 0.023 0.023 0.015 0.015 0.000 0.015 0.015 0.015 0.015 0.036 0.018 0.009 0.019 0.015 0.036 0.032 0.014 0.016 0.016 0.011 0.010 0.014 0.014

50 U3 0.051 0.013 0.008 0.145 0.143 0.051 0.036 0.018 0.017 0.042 0.054 0.057 0.018 0.019 0.038 0.017 0.017 0.051 0.052 0.038 0.031 0.048 0.016 0.008 0.108 0.103 0.048 0.046 0.023 0.016 0.042 0.045 0.045 0.023 0.022 0.021 0.009 0.021 0.079 0.083 0.021 0.022 0.048 0.022 0.007 0.099 0.098 0.048 0.041

T3 0.075 0.048 0.043 0.157 0.154 0.075 0.060 0.035 0.034 0.043 0.083 0.073 0.035 0.036 0.056 0.036 0.027 0.069 0.068 0.056 0.049 0.071 0.047 0.039 0.117 0.111 0.071 0.069 0.038 0.030 0.042 0.059 0.060 0.038 0.037 0.057 0.027 0.030 0.098 0.097 0.057 0.053 0.062 0.038 0.023 0.109 0.107 0.062 0.055

L4 0.020 0.028 0.028 0.015 0.014 0.020 0.023 0.025 0.012 0.005 0.030 0.026 0.025 0.022 0.030 0.025 0.014 0.024 0.019 0.030 0.018 0.017 0.033 0.033 0.015 0.014 0.017 0.017 0.023 0.010 0.001 0.035 0.025 0.023 0.023 0.022 0.013 0.006 0.013 0.012 0.022 0.015 0.018 0.031 0.031 0.014 0.011 0.018 0.019

100 U4 0.026 0.009 0.009 0.113 0.111 0.026 0.027 0.024 0.028 0.050 0.038 0.043 0.024 0.026 0.036 0.020 0.020 0.049 0.050 0.036 0.034 0.038 0.013 0.007 0.068 0.067 0.038 0.037 0.021 0.022 0.041 0.023 0.025 0.021 0.021 0.027 0.019 0.036 0.062 0.067 0.027 0.027 0.039 0.009 0.009 0.071 0.071 0.039 0.033

0.045 0.037 0.037 0.127 0.125 0.045 0.049 0.049 0.039 0.054 0.067 0.068 0.049 0.047 0.066 0.045 0.034 0.073 0.068 0.066 0.052 0.055 0.045 0.040 0.083 0.081 0.055 0.054 0.044 0.032 0.042 0.057 0.050 0.044 0.044 0.048 0.031 0.042 0.075 0.079 0.048 0.041 0.057 0.040 0.040 0.084 0.082 0.057 0.051

362 LARGE SAMPLE AND BOOTSTRAP INTERVALS FOR GAMMA PARAMETER

Table 2: Results for K=4 R 0.4

CP 0.6

0.4

0.2

0.8

0.6

CM BTS PRC BC BCA JAC BSD

0.4

CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD

0.2

1.2

n M CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD

0.6

0.4

CM BTS PRC BC BCA JAC BSD

L1 0.017 0.013 0.002 0.027 0.020 0.018 0.014 0.015 0.011 0.007 0.027 0.013 0.015 0.016 0.015 0.014 0.023 0.012 0.010 0.018 0.017 0.018 0.008 0.000 0.024 0.015 0.018 0.008 0.018 0.009 0.003 0.022 0.016 0.021 0.013 0.014 0.008 0.009 0.009 0.006 0.014 0.013 0.014 0.007 0.000 0.023 0.011 0.014 0.004 0.010 0.008 0.004 0.013 0.011 0.011 0.009

20 U1 0.035 0.012 0.030 0.077 0.084 0.040 0.039 0.048 0.022 0.015 0.113 0.111 0.055 0.049 0.068 0.034 0.007 0.182 0.170 0.072 0.053 0.045 0.018 0.047 0.070 0.076 0.048 0.047 0.050 0.019 0.018 0.087 0.085 0.055 0.051 0.056 0.019 0.004 0.142 0.135 0.063 0.044 0.031 0.014 0.040 0.052 0.058 0.035 0.034 0.047 0.016 0.022 0.074 0.074 0.054 0.048

T1 0.052 0.025 0.031 0.104 0.104 0.058 0.053 0.063 0.032 0.022 0.140 0.124 0.070 0.065 0.083 0.048 0.030 0.194 0.180 0.090 0.069 0.063 0.025 0.047 0.094 0.091 0.066 0.055 0.067 0.028 0.021 0.109 0.101 0.075 0.064 0.070 0.027 0.013 0.151 0.141 0.076 0.056 0.045 0.021 0.040 0.075 0.069 0.049 0.038 0.057 0.024 0.025 0.087 0.085 0.065 0.056

L2 0.021 0.014 0.003 0.028 0.022 0.023 0.018 0.022 0.018 0.015 0.022 0.019 0.022 0.021 0.017 0.020 0.030 0.008 0.008 0.017 0.017 0.020 0.013 0.001 0.032 0.025 0.021 0.015 0.017 0.010 0.005 0.026 0.020 0.023 0.018 0.019 0.017 0.020 0.014 0.013 0.020 0.019 0.018 0.006 0.000 0.034 0.028 0.021 0.007 0.015 0.010 0.003 0.024 0.018 0.015 0.013

30 U2 0.032 0.015 0.033 0.062 0.063 0.033 0.035 0.046 0.022 0.019 0.097 0.090 0.049 0.047 0.054 0.019 0.006 0.152 0.142 0.058 0.041 0.032 0.016 0.046 0.054 0.061 0.033 0.035 0.043 0.020 0.024 0.090 0.089 0.045 0.045 0.053 0.020 0.007 0.133 0.128 0.057 0.045 0.028 0.016 0.049 0.030 0.039 0.029 0.028 0.034 0.020 0.024 0.054 0.055 0.035 0.035

T2 0.053 0.028 0.036 0.090 0.085 0.056 0.052 0.068 0.040 0.034 0.119 0.108 0.071 0.068 0.071 0.039 0.035 0.160 0.149 0.075 0.058 0.052 0.029 0.047 0.086 0.086 0.054 0.050 0.060 0.029 0.029 0.115 0.109 0.068 0.062 0.072 0.036 0.027 0.147 0.140 0.076 0.063 0.046 0.022 0.049 0.064 0.067 0.050 0.035 0.049 0.030 0.026 0.078 0.072 0.050 0.048

L3 0.016 0.009 0.002 0.026 0.021 0.016 0.014 0.019 0.017 0.013 0.018 0.017 0.020 0.021 0.016 0.024 0.026 0.009 0.009 0.017 0.017 0.019 0.009 0.002 0.034 0.030 0.019 0.014 0.015 0.012 0.006 0.022 0.018 0.016 0.015 0.013 0.015 0.015 0.006 0.005 0.013 0.012 0.013 0.005 0.001 0.035 0.030 0.014 0.009 0.019 0.011 0.006 0.029 0.025 0.021 0.018

50 U3 0.030 0.021 0.034 0.048 0.052 0.030 0.030 0.035 0.017 0.017 0.081 0.079 0.036 0.037 0.045 0.011 0.006 0.141 0.134 0.046 0.037 0.025 0.019 0.037 0.029 0.033 0.026 0.026 0.038 0.021 0.028 0.064 0.064 0.038 0.039 0.044 0.012 0.009 0.105 0.101 0.044 0.037 0.028 0.027 0.065 0.031 0.039 0.029 0.027 0.039 0.020 0.029 0.055 0.057 0.040 0.039

T3 0.046 0.029 0.036 0.074 0.073 0.046 0.044 0.054 0.034 0.030 0.099 0.096 0.055 0.057 0.061 0.034 0.032 0.150 0.142 0.062 0.054 0.044 0.027 0.038 0.062 0.062 0.045 0.040 0.053 0.033 0.034 0.086 0.082 0.054 0.054 0.056 0.027 0.023 0.111 0.106 0.056 0.049 0.041 0.032 0.065 0.066 0.069 0.043 0.036 0.058 0.031 0.035 0.084 0.082 0.061 0.056

L4 0.028 0.011 0.007 0.040 0.037 0.028 0.026 0.019 0.019 0.017 0.016 0.015 0.019 0.018 0.018 0.036 0.039 0.006 0.006 0.018 0.018 0.020 0.012 0.004 0.050 0.044 0.020 0.018 0.024 0.020 0.015 0.029 0.027 0.024 0.024 0.014 0.023 0.026 0.009 0.009 0.015 0.015 0.018 0.008 0.002 0.050 0.049 0.018 0.014 0.020 0.012 0.006 0.035 0.031 0.020 0.019

100 U4 0.028 0.025 0.036 0.037 0.038 0.029 0.029 0.040 0.019 0.021 0.073 0.073 0.040 0.040 0.038 0.010 0.006 0.130 0.126 0.038 0.035 0.030 0.037 0.055 0.029 0.033 0.031 0.030 0.028 0.015 0.018 0.043 0.044 0.030 0.028 0.037 0.010 0.007 0.103 0.100 0.038 0.035 0.029 0.044 0.065 0.021 0.026 0.029 0.029 0.030 0.025 0.030 0.038 0.040 0.030 0.031

0.056 0.036 0.043 0.077 0.074 0.057 0.055 0.059 0.037 0.037 0.088 0.088 0.059 0.058 0.055 0.046 0.045 0.136 0.132 0.055 0.052 0.049 0.048 0.058 0.078 0.077 0.051 0.048 0.052 0.035 0.033 0.072 0.071 0.054 0.052 0.051 0.033 0.032 0.111 0.108 0.052 0.050 0.047 0.052 0.067 0.071 0.074 0.047 0.042 0.050 0.036 0.036 0.073 0.070 0.050 0.050

BAKLIZI & AL-NASSER

363

Table 2. Continued R

1.6

CP 0.2

0.6

0.4

0.2

2

0.6

0.4

0.2

n M CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD

CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD

L1 0.020 0.015 0.014 0.018 0.014 0.022 0.019 0.005 0.002 0.000 0.015 0.009 0.005 0.002 0.013 0.007 0.000 0.013 0.008 0.013 0.009 0.013 0.011 0.007 0.012 0.011 0.014 0.013 0.002 0.000 0.000 0.013 0.002 0.002 0.000 0.014 0.005 0.000 0.017 0.013 0.014 0.006 0.008 0.008 0.003 0.008 0.006 0.009 0.008

20 U1 0.058 0.019 0.006 0.128 0.122 0.070 0.052 0.030 0.012 0.063 0.047 0.057 0.034 0.030 0.040 0.016 0.035 0.071 0.078 0.043 0.040 0.052 0.017 0.013 0.125 0.121 0.058 0.043 0.041 0.019 0.081 0.055 0.070 0.043 0.038 0.040 0.013 0.026 0.057 0.061 0.042 0.040 0.068 0.025 0.018 0.137 0.134 0.075 0.059

T1 0.078 0.033 0.020 0.146 0.136 0.092 0.071 0.034 0.014 0.063 0.062 0.065 0.039 0.032 0.052 0.023 0.035 0.083 0.086 0.056 0.049 0.064 0.028 0.020 0.137 0.132 0.072 0.055 0.043 0.019 0.081 0.068 0.072 0.045 0.038 0.054 0.018 0.026 0.074 0.074 0.056 0.046 0.076 0.033 0.020 0.144 0.140 0.083 0.066

L2 0.017 0.016 0.015 0.014 0.013 0.018 0.016 0.014 0.009 0.000 0.032 0.025 0.014 0.007 0.019 0.012 0.001 0.021 0.018 0.020 0.015 0.017 0.011 0.010 0.016 0.013 0.017 0.015 0.011 0.004 0.000 0.025 0.020 0.011 0.002 0.011 0.005 0.001 0.020 0.013 0.011 0.007 0.012 0.008 0.006 0.013 0.011 0.015 0.012

30 U2 0.053 0.021 0.009 0.113 0.110 0.056 0.049 0.024 0.014 0.043 0.026 0.029 0.027 0.025 0.042 0.023 0.048 0.073 0.076 0.045 0.043 0.049 0.015 0.011 0.106 0.103 0.050 0.043 0.036 0.031 0.079 0.035 0.042 0.038 0.035 0.039 0.021 0.037 0.054 0.058 0.044 0.039 0.048 0.017 0.013 0.094 0.092 0.049 0.043

T2 0.070 0.036 0.024 0.127 0.123 0.074 0.065 0.037 0.023 0.043 0.057 0.053 0.040 0.031 0.060 0.034 0.049 0.094 0.094 0.064 0.058 0.066 0.026 0.021 0.121 0.116 0.067 0.058 0.047 0.035 0.079 0.060 0.062 0.049 0.036 0.049 0.026 0.038 0.074 0.071 0.055 0.046 0.060 0.024 0.019 0.106 0.103 0.064 0.055

L3 0.019 0.019 0.018 0.019 0.019 0.019 0.019 0.018 0.006 0.000 0.048 0.041 0.018 0.009 0.019 0.011 0.002 0.030 0.027 0.020 0.014 0.022 0.018 0.013 0.018 0.018 0.022 0.022 0.020 0.008 0.000 0.057 0.049 0.020 0.009 0.011 0.006 0.003 0.024 0.021 0.013 0.009 0.015 0.012 0.007 0.014 0.012 0.016 0.015

50 U3 0.044 0.020 0.017 0.096 0.092 0.046 0.042 0.029 0.035 0.075 0.029 0.034 0.030 0.027 0.026 0.020 0.028 0.040 0.043 0.028 0.027 0.037 0.015 0.015 0.097 0.096 0.040 0.035 0.027 0.046 0.108 0.025 0.033 0.031 0.025 0.042 0.034 0.051 0.049 0.052 0.044 0.042 0.046 0.021 0.021 0.090 0.088 0.047 0.041

T3 0.063 0.038 0.035 0.115 0.111 0.065 0.061 0.046 0.040 0.075 0.076 0.074 0.047 0.036 0.045 0.031 0.030 0.069 0.070 0.048 0.041 0.059 0.033 0.028 0.115 0.114 0.062 0.057 0.047 0.054 0.108 0.082 0.082 0.051 0.034 0.053 0.039 0.053 0.073 0.073 0.056 0.051 0.061 0.033 0.027 0.104 0.100 0.063 0.056

L4 0.020 0.025 0.024 0.014 0.014 0.022 0.022 0.024 0.007 0.002 0.066 0.061 0.024 0.018 0.020 0.011 0.007 0.032 0.030 0.020 0.018 0.015 0.014 0.011 0.013 0.012 0.015 0.015 0.022 0.007 0.001 0.074 0.069 0.022 0.013 0.020 0.006 0.003 0.039 0.033 0.020 0.016 0.017 0.015 0.014 0.018 0.017 0.017 0.016

100 U4 0.038 0.018 0.017 0.081 0.079 0.038 0.037 0.029 0.053 0.078 0.021 0.024 0.029 0.026 0.031 0.030 0.039 0.036 0.037 0.031 0.032 0.035 0.015 0.015 0.065 0.064 0.036 0.032 0.027 0.070 0.101 0.015 0.019 0.027 0.024 0.034 0.038 0.063 0.034 0.037 0.036 0.032 0.042 0.021 0.017 0.062 0.062 0.042 0.039

0.058 0.043 0.040 0.095 0.092 0.060 0.059 0.053 0.060 0.080 0.086 0.085 0.053 0.044 0.050 0.041 0.046 0.067 0.067 0.051 0.050 0.049 0.029 0.026 0.078 0.076 0.050 0.047 0.049 0.076 0.101 0.089 0.087 0.049 0.036 0.054 0.044 0.066 0.073 0.070 0.056 0.048 0.059 0.036 0.031 0.080 0.079 0.059 0.055

364 LARGE SAMPLE AND BOOTSTRAP INTERVALS FOR GAMMA PARAMETER

Table 3: Results for K=8 R 0.4

CP 0.6

0.4

0.2

0.8

0.6

0.4

0.2

1.2

0.6

N M

CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD

L1 0.022 0.010 0.001 0.042 0.034 0.022 0.014 0.012 0.010 0.007 0.027 0.016 0.013 0.012 0.011 0.007 0.012 0.011 0.009 0.013 0.013 0.016 0.010 0.000 0.038 0.030 0.017 0.011 0.015 0.005 0.001 0.022 0.019 0.015 0.011 0.012 0.007 0.007 0.014 0.014 0.015 0.012 0.009 0.003 0.000 0.023 0.018 0.009 0.003

20 U1 0.048 0.024 0.046 0.073 0.074 0.051 0.053 0.046 0.024 0.022 0.087 0.080 0.049 0.047 0.075 0.031 0.005 0.156 0.144 0.082 0.059 0.040 0.024 0.063 0.053 0.057 0.043 0.044 0.050 0.027 0.032 0.082 0.080 0.058 0.055 0.062 0.021 0.007 0.121 0.107 0.065 0.053 0.038 0.024 0.072 0.042 0.049 0.042 0.040

T1 0.069 0.033 0.047 0.114 0.108 0.073 0.066 0.058 0.033 0.028 0.114 0.096 0.062 0.059 0.086 0.038 0.016 0.166 0.153 0.095 0.071 0.056 0.034 0.063 0.090 0.087 0.060 0.055 0.064 0.032 0.033 0.104 0.099 0.072 0.066 0.074 0.027 0.014 0.135 0.121 0.080 0.065 0.047 0.027 0.072 0.065 0.067 0.051 0.043

L2 0.019 0.009 0.001 0.045 0.036 0.022 0.014 0.019 0.011 0.008 0.027 0.023 0.021 0.020 0.014 0.013 0.018 0.011 0.011 0.015 0.015 0.021 0.008 0.001 0.042 0.037 0.023 0.015 0.017 0.005 0.002 0.032 0.027 0.018 0.016 0.013 0.010 0.011 0.014 0.014 0.013 0.014 0.013 0.006 0.000 0.035 0.030 0.014 0.008

30 U2 0.039 0.023 0.047 0.054 0.056 0.040 0.040 0.041 0.022 0.020 0.081 0.077 0.044 0.044 0.059 0.025 0.007 0.149 0.139 0.062 0.050 0.033 0.025 0.053 0.034 0.037 0.038 0.038 0.037 0.022 0.028 0.055 0.054 0.039 0.040 0.052 0.016 0.007 0.112 0.104 0.055 0.047 0.034 0.032 0.091 0.031 0.035 0.037 0.037

T2 0.057 0.032 0.048 0.099 0.092 0.062 0.054 0.060 0.032 0.028 0.108 0.099 0.064 0.064 0.072 0.038 0.025 0.159 0.149 0.077 0.065 0.054 0.033 0.054 0.076 0.074 0.060 0.053 0.054 0.027 0.030 0.087 0.081 0.057 0.055 0.065 0.025 0.018 0.126 0.117 0.068 0.060 0.047 0.037 0.091 0.066 0.065 0.051 0.045

L3 0.021 0.008 0.003 0.049 0.044 0.021 0.020 0.021 0.012 0.007 0.026 0.024 0.023 0.022 0.011 0.015 0.024 0.009 0.009 0.012 0.014 0.023 0.007 0.001 0.059 0.055 0.024 0.017 0.021 0.012 0.006 0.033 0.031 0.021 0.021 0.013 0.011 0.012 0.012 0.011 0.013 0.013 0.013 0.006 0.000 0.059 0.053 0.013 0.011

50 U3 0.033 0.028 0.042 0.036 0.038 0.033 0.037 0.036 0.020 0.023 0.065 0.065 0.038 0.039 0.045 0.018 0.005 0.124 0.117 0.046 0.043 0.032 0.041 0.073 0.032 0.034 0.034 0.034 0.038 0.028 0.036 0.054 0.054 0.039 0.041 0.038 0.018 0.015 0.103 0.099 0.038 0.036 0.031 0.052 0.097 0.020 0.025 0.034 0.033

T3 0.053 0.036 0.044 0.085 0.081 0.054 0.056 0.057 0.032 0.029 0.091 0.088 0.060 0.061 0.056 0.033 0.028 0.132 0.125 0.058 0.057 0.055 0.047 0.074 0.090 0.089 0.058 0.051 0.058 0.040 0.042 0.086 0.085 0.060 0.062 0.050 0.029 0.026 0.115 0.110 0.051 0.049 0.044 0.057 0.097 0.079 0.077 0.047 0.044

L4 0.021 0.010 0.005 0.052 0.048 0.022 0.020 0.020 0.012 0.010 0.025 0.024 0.021 0.021 0.016 0.028 0.036 0.010 0.010 0.016 0.017 0.020 0.007 0.001 0.067 0.063 0.020 0.016 0.018 0.009 0.006 0.030 0.028 0.018 0.018 0.017 0.019 0.022 0.014 0.014 0.017 0.018 0.024 0.007 0.002 0.082 0.077 0.024 0.021

100 U4 0.021 0.025 0.040 0.022 0.024 0.022 0.023 0.024 0.015 0.015 0.044 0.044 0.024 0.027 0.040 0.013 0.008 0.111 0.105 0.040 0.038 0.039 0.060 0.076 0.031 0.032 0.041 0.040 0.036 0.032 0.038 0.043 0.044 0.037 0.038 0.034 0.011 0.011 0.082 0.079 0.034 0.033 0.028 0.062 0.094 0.012 0.016 0.028 0.026

0.042 0.035 0.045 0.074 0.072 0.043 0.043 0.044 0.027 0.025 0.068 0.068 0.045 0.048 0.055 0.041 0.043 0.121 0.115 0.055 0.055 0.059 0.067 0.077 0.097 0.095 0.061 0.056 0.054 0.041 0.044 0.073 0.071 0.055 0.056 0.050 0.030 0.033 0.096 0.092 0.050 0.051 0.052 0.069 0.096 0.094 0.092 0.052 0.047

BAKLIZI & AL-NASSER

365

Table 3. Continued R

CP 0.4

0.2

1.6

0.6

0.4

0.2

2

0.6

0.4

n M CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD CM BTS PRC BC BCA JAC BSD

L1 0.012 0.005 0.001 0.024 0.018 0.012 0.009 0.011 0.006 0.004 0.016 0.011 0.014 0.012 0.005 0.001 0.000 0.029 0.023 0.007 0.000 0.011 0.006 0.000 0.025 0.018 0.011 0.009 0.013 0.007 0.005 0.020 0.015 0.014 0.012 0.002 0.000 0.000 0.019 0.013 0.002 0.000 0.006 0.001 0.000 0.018 0.015 0.007 0.003

20 U1 0.056 0.029 0.045 0.075 0.073 0.063 0.059 0.059 0.024 0.014 0.115 0.107 0.062 0.052 0.035 0.026 0.080 0.033 0.038 0.039 0.036 0.046 0.029 0.048 0.056 0.056 0.050 0.049 0.055 0.024 0.018 0.091 0.083 0.062 0.050 0.037 0.030 0.112 0.036 0.049 0.046 0.036 0.040 0.027 0.059 0.053 0.056 0.045 0.040

T1 0.068 0.033 0.046 0.099 0.090 0.075 0.067 0.070 0.030 0.018 0.131 0.118 0.076 0.064 0.040 0.027 0.080 0.062 0.060 0.046 0.036 0.056 0.035 0.048 0.081 0.074 0.060 0.058 0.068 0.031 0.022 0.111 0.098 0.076 0.061 0.039 0.030 0.112 0.054 0.062 0.048 0.036 0.046 0.028 0.059 0.071 0.070 0.052 0.043

L2 0.011 0.005 0.001 0.029 0.026 0.012 0.009 0.017 0.011 0.011 0.021 0.019 0.017 0.016 0.011 0.002 0.000 0.036 0.032 0.011 0.002 0.014 0.004 0.001 0.032 0.028 0.015 0.009 0.010 0.006 0.004 0.014 0.012 0.011 0.010 0.010 0.004 0.000 0.046 0.040 0.011 0.003 0.006 0.002 0.000 0.021 0.017 0.006 0.004

30 U2 0.039 0.027 0.038 0.052 0.052 0.042 0.042 0.043 0.021 0.015 0.075 0.072 0.044 0.039 0.031 0.038 0.112 0.026 0.032 0.033 0.032 0.041 0.029 0.053 0.045 0.046 0.045 0.041 0.047 0.021 0.016 0.081 0.076 0.049 0.045 0.032 0.053 0.108 0.017 0.025 0.037 0.027 0.052 0.045 0.078 0.052 0.056 0.056 0.051

T2 0.049 0.032 0.039 0.081 0.078 0.054 0.050 0.059 0.031 0.026 0.095 0.090 0.061 0.055 0.042 0.040 0.112 0.062 0.064 0.044 0.034 0.055 0.033 0.054 0.076 0.074 0.059 0.050 0.057 0.027 0.020 0.095 0.088 0.060 0.055 0.041 0.057 0.108 0.063 0.064 0.048 0.030 0.057 0.047 0.078 0.072 0.073 0.062 0.055

L3 0.020 0.008 0.002 0.050 0.045 0.021 0.018 0.016 0.010 0.011 0.017 0.017 0.016 0.015 0.011 0.002 0.000 0.051 0.046 0.011 0.004 0.012 0.005 0.002 0.032 0.028 0.012 0.009 0.017 0.006 0.005 0.021 0.020 0.017 0.015 0.013 0.004 0.000 0.066 0.065 0.014 0.006 0.018 0.008 0.001 0.047 0.040 0.019 0.013

50 U3 0.034 0.029 0.043 0.041 0.042 0.036 0.036 0.046 0.022 0.019 0.082 0.079 0.048 0.044 0.033 0.070 0.136 0.019 0.023 0.034 0.031 0.036 0.038 0.055 0.033 0.035 0.037 0.035 0.044 0.026 0.022 0.068 0.063 0.045 0.040 0.029 0.071 0.135 0.016 0.019 0.030 0.024 0.034 0.045 0.067 0.030 0.034 0.035 0.034

T3 0.054 0.037 0.045 0.090 0.086 0.057 0.054 0.062 0.032 0.030 0.099 0.096 0.064 0.059 0.044 0.072 0.136 0.070 0.069 0.045 0.034 0.047 0.043 0.056 0.064 0.063 0.048 0.044 0.061 0.032 0.027 0.089 0.083 0.062 0.055 0.042 0.075 0.135 0.082 0.084 0.044 0.030 0.052 0.053 0.067 0.076 0.074 0.053 0.047

L4 0.020 0.010 0.005 0.045 0.042 0.021 0.019 0.015 0.013 0.015 0.017 0.017 0.016 0.016 0.020 0.004 0.000 0.095 0.089 0.021 0.011 0.017 0.006 0.002 0.042 0.040 0.018 0.014 0.014 0.008 0.008 0.019 0.018 0.014 0.013 0.019 0.004 0.000 0.102 0.098 0.019 0.011 0.018 0.007 0.002 0.049 0.044 0.018 0.015

100 U4 0.030 0.035 0.043 0.033 0.033 0.030 0.030 0.030 0.019 0.018 0.059 0.058 0.031 0.030 0.031 0.088 0.121 0.009 0.014 0.031 0.029 0.033 0.047 0.057 0.027 0.028 0.033 0.033 0.035 0.022 0.021 0.055 0.052 0.035 0.034 0.029 0.118 0.156 0.013 0.015 0.030 0.024 0.030 0.049 0.073 0.020 0.022 0.031 0.028

0.050 0.044 0.048 0.078 0.075 0.051 0.049 0.045 0.032 0.033 0.076 0.074 0.046 0.045 0.051 0.091 0.121 0.104 0.103 0.052 0.040 0.050 0.053 0.059 0.069 0.068 0.051 0.047 0.048 0.030 0.029 0.074 0.070 0.049 0.047 0.048 0.122 0.156 0.115 0.113 0.048 0.035 0.047 0.056 0.075 0.069 0.066 0.049 0.043

366 LARGE SAMPLE AND BOOTSTRAP INTERVALS FOR GAMMA PARAMETER

Table 3. Continued R

CP 0.2

n M CM BTS PRC BC BCA JAC BSD

L1 0.014 0.006 0.001 0.018 0.015 0.015 0.011

20 U1 0.065 0.032 0.029 0.100 0.091 0.069 0.061

T1 0.078 0.038 0.030 0.117 0.105 0.084 0.072

L2 0.010 0.003 0.001 0.015 0.011 0.010 0.008

References Al-Nasser, A., & Baklizi, A. (2004). Large sample and bootstrap intervals for the scale of burr type x distribution based on grouped data. Journal of Modern Applied Statistical Methods, 3(2), 386 – 392. Chen, Z., & Mi, J. (2001). An approximate confidence interval for the scale parameter of the gamma distribution based on grouped data. Statistical Paper, 4, 285-299. Cohen, A., & Whitten, B. (1988). Parameter estimation in reliability and life span models. New York, N.Y: Marcel Dekker. Efron, B. (1982). The jackknife, the bootstrap and other resampling plans. Philadelphia, PA: SIAM.

30 U2 0.048 0.028 0.029 0.086 0.083 0.051 0.048

T2 0.057 0.031 0.030 0.101 0.094 0.061 0.056

L3 0.014 0.007 0.004 0.022 0.021 0.014 0.012

50 U3 0.048 0.028 0.027 0.071 0.067 0.050 0.048

T3 0.062 0.034 0.031 0.093 0.088 0.064 0.059

L4 0.018 0.009 0.007 0.023 0.022 0.018 0.017

100 U4 0.037 0.027 0.031 0.049 0.048 0.038 0.038

0.055 0.036 0.038 0.071 0.070 0.056 0.055

Efron, B. & Tibshirani, R. (1993). An introduction to the bootstrap. New York, N.Y.: Chapman and Hall. Jennings, D. (1987). How do we judge confidence intervals adequacy? The American Statistician, 41(4), 335-337. Kulldorff, G. (1961). Estimation from grouped and partially grouped samples. New York, N.Y.: Wiley. Meeker, Jr, W. (1986). Planning Life tests in which units are inspected for failure. IEEE Trans. on Reliability R-35, 571-578. Pettitt, A. N., & Stephens, M. A. (1977). The Kolmogrov-Smirnov goodness-of-fit statistic with discrete and grouped data. Technometrics, 19, 205 – 210.