Control Charts For The Log-logistic Distribution

c Heldermann Verlag
ISSN 0940-5151

Economic Quality Control Vol 21 (2006), No. 1, 77 – 86

Control Charts for the Log-Logistic Distribution R.R.L. Kantam, A.Vasudeva Rao and G.Srinivasa Rao

Abstract: This paper deals with the log-logistic distribution as a life time model. Control charts and the corresponding control limits are developed analogous to Shewhart-charts for the process mean and process range. The proposed control limits are compared with those of Shewhart in detecting out of control signals for various sample sizes.

1

Introduction

In classical Statistics confidence intervals for unknown parameters of a statistical population play an important role. An application of confidence interval, when the random variable follows approximately a normal distribution, is the origin of the well-known Shewhart control charts. Construction of control charts using the theory of confidence intervals, when the random variable follows Inverse Gaussian distribution is considered by Edgemen [3]. Edgemen justifies his assumption by noticing that relying on the central limit theorem for non-normal processes is hazardous as the sample size in control charting is usually less than 10. Moreover quality characteristics such as product life are always better modeled by a probability distribution with non-negative support rather than a normal distribution. Therefore, Kantam and Sriram [6] developed control charts to be used when the process characteristic follows a gamma distribution. Log - logistic distribution is another distribution of a positive valued random variable often applied in survival analysis. A number of problems with special reference to log -logistic distribution have been studied in recent times. Some of these are - Ragab and Green [7, 8], Balakrishnan and Malik [1], Balakrishnan et al [2], Guptha et al [4], and Kantam et al [6]. In this paper control charts are developed for averages and ranges when the quality characteristic follows a log-logistic distribution. The details of control charts for averages are presented in Section 2. The construction of range charts follows in Section 3. Comparisons with Shewhart’s control limits are performed in Section 4.

2

Control Chart for Averages

¯ We know that the Shewhart X-chart is based on the 3σ limits of the normal
distribution.  σ ¯ √ If (X1 , X2 , . . . , Xn ) is a random sample of size n from N (µ.σ) then X ∼ N µ, n , where

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R.R.L. Kantam, A.Vasudeva Rao and G.Srinivasa Rao

µ is the process average, σ is the process standard deviation and the probability of the event   σ σ (1) x | µ − 3√ ≤ x ≤ µ + 3√ n n is 99.73%. For fixing the limits µ is sometimes estimated by the mean of the sample and σ is estimated in different ways either by the sample standard deviation or the sample range applying various estimation methods. The estimated limits are used for deciding, whether the variability of a process characteristic is still within the tolerable region or not. In fact, the 99.73% confidence interval for the process mean of a normal distribution has become the most often used tolerance region for a sample process variable. When the process characteristic cannot be approximated by a normal random variable, the above outline proceedings may nevertheless be useful for deriving estimators and control limits for nonnormal cases. In this paper the log-logistic distribution is a regarded as an appropriate distribution for a random lifetime. Let (X1 , X2 , . . . , Xn ) be a random sample of size from the log-logistic distribution with shape parameter β and scale parameter σ. Then the density function and the distribution function are given by:   β x β−1 f (x) =
σ σ  2 β 1 + σx  x β

F (x) =

Setting Z =

σ

1+ X σ

 x β

for x ≥ 0, β > 1, σ > 0

for x ≥ 0, β > 1, σ > 0

(2)

(3)

σ

gives a standard log-logistic distribution:

βz β−1 for z ≥ 0, β > 1 (1 + z β )2 zβ for z ≥ 0, β > 1 F (z) = 1 + zβ f (z) =

(4) (5)

The mean and variance of standard log-logistic distribution for β = 3 can be obtained by the following relation:   k k (6) E Z k = Γ(1 + ) Γ(1 − ) for k = 1, 2 β β yielding E[Z] = 1.2167 V [Z] = 0.9529 Thus, the bias-corrected moment estimator of σ is immediately obtained:

(7)

Control Charts for the Log-Logistic Distribution

¯ X 1.2167

σ ˆmom =

79

(8)

¯ is not analytically available. However, by the definition of scale The distribution of X ¯ is independent of σ. Hence, we take 1 X ¯ as pivotal parameter the distribution of σ1 X σ ¯ quantity to simulate the sampling distribution of X, thereby finding its percentiles. Let L, U be two limits such that   ¯ < L = 0.00135 P  σ1 X  (9) ¯ > U = 0.00135 P σ1 X yielding

1 ¯ P L ≤ X ≤ U = 0.9973 σ

(10)

The following procedure is used for determining L and U . We have generated 10,000 random samples of size n = 3, 4, . . . , 10 from standard log-logistic distribution with shape parameter β = 3. For each sample we have calculated ¯ • The mean of the sample X. • The BLUE of σ (coefficients taken from Srinivasa Rao [9]). • The MLE of σ by solving the following equation iteratively n  i=1

ziβ 1+

ziβ

−

n =0 2

where zi =

xi σ

(11)

¯ BLUE, MLE over 10,000 samples are given in Table 5. The calculated percentiles of X, ¯ The percentiles of X corresponding to 0.00135 and to 0.99865 are made use of to get the ¯ control limits of an X-chart. Let these be respectively denoted by L and U . From (9) using L and U we get the following probability statement: ¯ ≤ U σ) = 0.9973 P (Lσ ≤ X (12) Lσ and U σ are taken as the 99.73% control limits (analogous to the 3σ limits of the Shewhart charts) provided the unknown parameter σ is known by specification or by estimation. In this paper, three estimators are applied namely moment estimator, best linear unbiased estimator (BLUE), and maximum likelihood estimator (MLE). The control limits to be used are obtained by repeated sampling, computing the corresponding estimates and subsequently calculating some constants by the mean of the estimates for specified method of estimation. For example, if σ is estimated by the bias corrected moment estimator for a sample of size n the control limits in (12) may be obtained for β = 3 and β = 4 by the constants A∗2 and A∗∗ 2 , respectively, given in Table 1.

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R.R.L. Kantam, A.Vasudeva Rao and G.Srinivasa Rao

For example for β = 3 the constants are defined as follows: L A∗2 = 1.2167 U = A∗∗ 2 1.2167

(13) (14)

¯ Table 1: Constants for X-charts based on the moment estimator. β=3 β=4 ∗ ∗∗ ∗ n A2 A2 A2 A∗∗ 2 3 0.3118 3.8611 0.4341 2.6525 4 0.3794 3.8290 0.4950 2.4718 5 0.4208 3.2426 0.5390 2.2335 6 0.4472 2.9127 0.5627 2.0500 7 0.4645 2.7607 0.5781 1.9590 8 0.4774 2.6919 0.5895 1.8848 9 0.5037 2.7426 0.6100 1.9145 10 0.5263 2.5692 0.6287 1.8671 ¯ of estimates of the Given a number of subgroups of a fixed size, say n, the overall mean x ¯ sub groups is calculated. The control limits for a X-chart for the log-logistic distribution ¯ and A∗∗ ¯ are obtained by means of the constants in Table 1 as A∗2 x 2 x. ¯ ¯ ¯ is If σ is estimated by the MLE the control limits are L Mc and U Mc , respectively, where M the mean of the sub groups MLEs and c is the simulated sampling mean of MLE given in L U last column of Table 5. Let A∗2M and A∗∗ 2M represent the quantities c and c , respectively, ∗ ∗∗ ∗ ¯ and A M ¯ . The constants A and A∗∗ are then the control limits are given by A2M M 2M 2M 2M given below in Table 2.

¯ Table 2: Constants for X-charts based on MLE. β=3 β=4 ∗ ∗∗ ∗ n A2M A2M A2M A∗∗ 2M 3 0.3496 4.3294 0.4609 2.8166 4 0.4360 4.3787 0.5323 2.6586 5 0.4907 3.7361 0.5841 2.4206 6 0.5325 3.5904 0.6142 2.2378 7 0.5504 3.4387 0.6320 2.1419 8 0.5953 3.3339 0.6456 2.0644 9 0.6123 3.1249 0.6690 2.0996 10 0.6206 2.9595 0.6904 2.0505 BLUE is an unbiased estimator of σ and, hence, a bias-correction in (8) is not necessary. ¯ and U B. ¯ The constants L and U in case of BLUE are The control limits are given by LB given below in Table 3.

Control Charts for the Log-Logistic Distribution

81

¯ Table 3: Constants for X-charts based on BLUE. β=3 β=4 n L U L U 3 0.3794 4.6979 0.4822 2.9464 4 0.4616 4.6358 0.5498 2.7457 5 0.5120 3.8981 0.5987 2.4810 6 0.5494 3.7046 0.6250 2.2772 7 0.5659 3.5357 0.6421 2.1761 8 0.6101 3.4167 0.6548 2.0937 9 0.6258 3.1937 0.6776 2.1266 10 0.6329 3.0181 0.6983 2.0740 ¯ Tables 1,2,3 can be used for obtaining the control limits of X-charts for three estimators in case the log-logistic distribution is an appropriate one. Moreover, motivated by ¯ ± 3SE[X] ¯ as another possibility of defining control the Shewhart chart, we suggest E[X] ¯ limits for X-charts, although these limits do not guarantee that with 99.73% probability there will be no false alarms. The limits are straightforward calculated from the simu¯ They are compared with those obtained in the earlier lated sampling distribution of X. methods as well as are compared with the earlier obtained ones and also with the normal distribution situation. The results of the comparison are displayed in Table 7.

3

Range Chart

The control limits for a range chart assuming the standard density are determined by the equation P (L ≤ z(n) − z(1) ≤ U ) = 0.9973

(15)

P (σL ≤ Ri σ ≤ U ) = 0.9973

(16)

or

where Ri stands for the range of the ith sub group of size n. Provided σ is known, L and U are given by the percentiles of the simulated distribution of range, and we get the control limits from equation (16). As described in Section 2, we have simulated the sampling distribution of the range of samples from the log-logistic distribution with shape parameter β and sample sizes n = 3, 4, . . . 10. The percentiles of the range are given in Table 6. The scale parameter σ in Equation (16) was estimated by the mean of all ranges divided by (αn − α1 ), where αi stands for the value of the first moment of the ith order statistic of the corresponding sample of size n from the log-logistic distribution with ¯ ¯ and αnRU , respectively. shape parameter β. Thus, the control limits in (16) become αnRL −α1 −α1 Taking the αi s from Srinivasa Rao [9], and L and U from Table 6, the constants D3∗ and D4∗ with

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R.R.L. Kantam, A.Vasudeva Rao and G.Srinivasa Rao

D3∗ = D4∗ =

L αn −α1 U αn −α1

(17)

¯ have to be calculated from the are displayed in Table 4 below. The values of Ri and R ¯ and D4∗ R ¯ for the range given sample data allowing to compute the control limits D3∗ R chart. Table 4: Constants for the Range Chart n 2 3 4 5 6 7 8 9 10

β=3 D3∗ D4∗ 0.0012 12.0824 0.0270 9.6728 0.0663 8.7207 0.1060 8.2868 0.1551 8.0536 0.1848 7.7324 0.1985 7.2560 0.2091 7.0406 0.2229 6.8429

β=4 D3∗ D4∗ 0.0013 8.9319 0.0308 6.9528 0.0733 6.1802 0.1280 5.9604 0.1691 5.3344 0.2080 5.0877 0.2176 4.9862 0.2344 4.9846 0.2606 4.8490

Note that the control limits given by Shewhart for range chart in the case of a normal distribution are based on E[R] ± 3SE[R], although this interval has not a probability of 99.73%, when operating the in-control state. Following Shewhart’s suggestion, we have studied the same control limits for the range chart of a log-logistic process distribution. The limits are calculated taking the necessary moments of the order statistics from Srinivas Rao [9]. The performance of the resulting range chart is compared with the Shewhart, where the results are listed in Table 8.

4

Comparison with Shewhart-Charts

If the normal approximation is used as process distribution, the constants for computing ¯ chart and and an R - chart are available in all standard textthe control limits of an X books on Statistical Quality Control with the notations A2 ,D3 and D4 . The performance of the control limits presented in Sections 2 and 3 depends on the power, with which the charts detect out of control state. This question is investigated here by a Monte - Carlo simulation study. 10,000 samples of size n = 3, 4, . . . , 10 are generated from a log-logistic distribution with shape parameters β = 3 and β = 4. For each sample the ¯ for the X-chart, ¯ traditional control limits of the Shewhart control charts namely x¯ ± A2 R ¯ ¯ and D3 R, D4 R for the R-chart are computed. The control limits under the assumption of log-logistic distribution are obtained using the constants in Tables 1,2,3,4. Out of 10,000 samples the number of samples, for which the mean and the range fall outside the control

83

Control Charts for the Log-Logistic Distribution

limits of Shewhart and outside the control limits of our charts are counted. These counts are given in Tables 7 and 8. ¯ From Table 7, we see that the out of control signals for X-chart are less in the case of loglogistic distribution with limits (9), when compared with the chart based on the normal distribution, but largest for the charts with 3SE limits assuming log-logistic distribution, which is due to the skewness of the log-logistic distribution. The control limits for the range chart assuming the log-logistic distribution with the 99.73% probability limits as well as the 3SE limits lead both to less out of control signals, when compared with the charts based on the normal approximation. This may be explained by the exactness of the moments of the range in log-logistic case.

5

Tables of Percentiles and Chart Performance Table 5a: Percentile of σ-estimators for β = 3. n 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10

Estimator Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE

0.00135 0.3794 0.4723 0.2916 0.4616 0.4937 0.3573 0.5120 0.4859 0.4108 0.5441 0.5074 0.4597 0.5652 0.5190 0.4789 0.5808 0.5361 0.5012 0.6128 0.5499 0.5322 0.6403 0.5798 0.5557

0.025 0.4977 0.5366 0.3992 0.5529 0.5495 0.4692 0.6025 0.5597 0.5077 0.6453 0.5813 0.5401 0.6714 0.6005 0.5729 0.6917 0.6161 0.5996 0.7196 0.6306 0.6149 0.7357 0.6481 0.6304

0.01 0.5694 0.5783 0.461 0.6324 0.5996 0.5252 0.6750 0.6116 0.5668 0.7024 0.6295 0.5981 0.7280 0.6510 0.6281 0.7487 0.6666 0.6512 0.7730 0.6851 0.6677 0.7898 0.6964 0.6794

0.05 0.6327 0.6188 0.5269 0.6925 0.6437 0.5873 0.7359 0.6648 0.6229 0.7532 0.6762 0.6481 0.7840 0.6993 0.6754 0.8032 0.7116 0.6931 0.8207 0.7280 0.7116 0.8367 0.7410 0.7242

0.1 0.7194 0.6879 0.6005 0.7727 0.7084 0.6590 0.8049 0.7297 0.6939 0.8278 0.7392 0.7130 0.8521 0.7553 0.7337 0.8702 0.7727 0.7534 0.8832 0.7812 0.7660 0.8950 0.7891 0.7754

0.5 1.1081 1.0157 0.9508 1.1234 1.0074 0.9617 1.1386 1.0061 0.9721 1.1410 1.0015 0.9742 1.1482 1.0003 0.9773 1.1557 1.0013 0.9817 1.1604 1.0034 0.9856 1.1661 1.0032 0.9856

0.95 2.1590 1.7774 1.6717 2.0228 1.6485 1.5661 1.9443 1.5435 1.4958 1.8543 1.4768 1.4396 1.8202 1.4418 1.4054 1.7865 1.4104 1.3819 1.7596 1.3746 1.3508 1.7324 1.3576 1.3340

0.975 2.5789 2.0205 1.8866 2.3520 1.8252 1.7339 2.2119 1.6931 1.6298 2.0980 1.6120 1.5689 2.0577 1.5528 1.5170 2.0080 1.5157 1.4794 1.9441 1.4773 1.4459 1.9060 1.4442 1.4196

0.99865 4.6979 3,0883 2.7765 4.6587 2.6305 2.5056 3.9453 2.3009 2.2148 3.5439 2.1183 2.0437 3.3589 2.0982 2.0351 3.2753 1.9615 1.9130 3.3369 1.8491 1.8325 3.1260 1.8105 1.7890

mean 1.2173 1.0851 1.0055 1.2192 1.0587 1.0054 1.2170 1.0434 1.0044 1.2085 1.0318 1.0012 1.2108 1.0282 1.0025 1.2104 1.0249 1.0029 1.2118 1.0220 1.0028 1.2115 1.0198 1.0025

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R.R.L. Kantam, A.Vasudeva Rao and G.Srinivasa Rao

Table 5b: Percentile of σ-estimators for β = 4. n 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10

Estimator Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE Moment MLE BLUE

0.00135 0.4822 0.5082 0.4070 0.5498 0.5603 0.4701 0.5987 0.5666 0.5217 0.6250 0.5936 0.5623 0.6421 0.5965 0.5793 0.6548 0.6220 0.5999 0.6776 0.6385 0.6263 0.6983 0.6559 0.6458

0.025 0.5832 0.6025 0.5126 0.6326 0.6175 0.5743 0.6723 0.6379 0.6075 0.7046 0.6564 0.6315 0.7267 0.6786 0.6618 0.7435 0.6946 0.6848 0.7633 0.7077 0.6971 0.7735 0.7200 0.7102

0.01 06438 0.6440 05753 0.6944 0.6640 0.6230 0.7307 0.6862 0.6591 0.7520 0.7035 0.6866 0.7717 0.7229 0.7101 0.7882 0.7367 0.7272 0.8067 0.7530 0.7422 0.8211 0.7615 0.7510

0.05 0.6987 0.6813 0.6304 0.7465 0.7122 0.6792 0.7800 0.7336 0.7081 0.7941 0.7425 0.7272 0.8163 0.7637 0.7498 0.8309 0.7745 0.7634 0.8449 0.7881 0.7783 0.8557 0.7983 0.7881

0.1 0.7688 0.7446 0.6947 0.8076 0.7673 0.7393 0.8352 0.7881 0.7672 0.8511 0.7958 0.7812 0.8692 0.8095 0.7966 0.8816 0.8238 0.8119 0.8914 0.8309 0.8218 0.8996 0.8371 0.8289

0.5 1.0627 1.0089 0.9739 1.0698 1.0034 0.9791 1.0788 1.0042 0.9857 1.0784 1.0008 0.9856 1.0838 0.9999 0.9873 1.0863 1.0009 0.9902 1.0888 1.0026 0.9925 1.0919 1.0024 0.9929

0.95 1.7229 1.5337 1.4844 1.6280 1.4545 1.4138 1.5700 1.3848 1.3617 1.5183 1.3396 1.3218 1.4920 1.3157 1.2976 1.4738 1.2942 1.2800 1.4529 1.2695 1.2561 1.4367 1.2577 1.2450

0.975 1.9411 1.6907 1.6282 1.8029 1.5700 1.5254 1.7184 1.4843 1.4540 1.6554 1.4306 1.4073 1.6259 1.3910 1.3733 1.5908 1.3649 1.3460 1.5528 1.3400 1.3217 1.5270 1.3174 1.3054

0.99865 2.9464 2.2325 21771 2.7457 2.0655 2.0240 2.4810 1.8682 1.8258 2.2772 1.7558 1.7071 2.1761 1.7433 1.7186 2.0937 1.6575 1.6342 2.1266 1.5857 1.5775 2.0740 1.5608 1.5518

mean 1.1167 1.0461 1.0041 1.1168 1.0328 1.0040 1.1156 1.0250 1.0035 1.1110 1.0176 1.0009 1.1123 1.0160 1.0019 1.1121 1.0142 1.0020 1.1125 1.0128 1.0020 1.1123 1.0115 1.0018

Table 6: Percentile estimates of the sample range. β 3

n 0.00135 0.025 0.01 0.05 0.1 0.5 0.95 0.975 0.99865 mean 2 0.0010 0.0087 0.0240 0.0446 0.0912 0.5290 2.4666 3.3275 9.7408 0.8156 3 0.0327 0.0985 0.1556 0.2184 0.3244 0.9165 3.1706 4.1719 11.6964 1.2201 4 0.0990 0.2148 0.3055 0.3877 0.5120 1.1692 3.6111 4.8259 13.2436 1.5154 5 0.1820 0.3477 0.4406 0.5352 0.6631 1.3602 4.0380 5.1771 14.5616 1.7361 6 0.2828 0.4257 0.5374 0.6445 0.7818 1.5215 4.2920 5.5557 14.0863 1.8946 7 0.3687 0.5249 0.6428 0.7483 0.8883 1.6555 4.7052 6.0334 14.7530 2.0620 8 0.4179 0.6087 0.7273 0.8395 0.9796 1.7790 4.8882 6.2280 15.7062 2.1988 9 0.4823 0.6852 0.8459 0.9200 1.0700 1.9077 5.0890 6.5468 16.7458 2.3410 10 0.5545 0.7528 0.8855 0.9888 1.1533 2.0032 5.3066 6.8314 17.0936 2.4612 4 2 0.0007 0.0067 0.0180 0.0353 0.0704 0.4033 1.6168 2.0535 4.9608 0.5611 3 0.0257 0.0752 0.1177 0.1696 0.2496 0.6857 2.0004 2.5341 5.7924 0.8409 4 0.0751 0.1662 0.2362 0.3056 0.3977 0.8663 2.2703 2.8589 6.3279 1.0352 5 0.1500 0.2667 0.3443 0.4159 0.5117 1.0015 2.5291 3.1085 6.9826 1.1815 6 0.2186 0.3306 0.4183 0.5001 0.6058 1.1125 2.6685 3.2944 6.8958 1.2906 7 0.2904 0.4069 0.4999 0.5801 0.6835 1.2016 2.8655 3.4961 7.1029 1.3968 8 0.3235 0.4697 0.5603 0.6510 0.7543 1.2854 2.9833 3.6397 7.4135 1.4831 9 0.3674 0.5375 0.6291 0.7132 0.8159 1.3660 3.0852 3.7563 7.8138 1.5678 10 0.4275 0.5875 0.6861 0.7637 0.8812 1.4301 3.1899 3.9237 7.9558 1.6419

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Control Charts for the Log-Logistic Distribution

¯ Table 7: Out of control signals of X-charts.. ¯ x β n 3 3 1.2173 4 1.2192 5 1.2170 6 1.2085 7 1.2108 8 1.2104 9 1.2118 10 1.2115 4 3 1.1167 4 1.1168 5 1.1156 6 1.1110 7 1.1123 8 1.1121 9 1.1125 10 1.1125

Log-logistic distribution LCL UCL out Prop 0.3795 4.7003 27 0.0027 0.4625 4.6685 27 0.0027 0.5121 3.9461 27 0.0027 0.5404 3.5200 23 0.0023 0.5624 3.3427 26 0.0026 0.5778 3.2583 26 0.0026 0.6103 3.3235 24 0.0024 0.6376 3.1126 27 0.0027 0.4848 2.9621 28 0 0028 0.5528 2.7604 27 0.0027 0.6013 2.4918 32 0.0032 0.6251 2.2775 28 0.0028 0.6430 2.1791 27 0.0027 0.6555 2.0960 26 0.0026 0.6787 2.1299 29 0.0029 0.6993 2.0769 27 0.0027

Normal distribution LCL UCL out Prop 0.0000 2.4656 297 0.0297 0.1190 2.3144 273 0.0273 0.2189 2.2111 251 0.0251 0.2932 2.1232 229 0.0229 0.3440 2.0772 233 0.0233 0.3861 2.0371 231 0.0231 0.4233 1.9985 208 0.0208 0.4553 1.9657 195 0.0195 0.2565 1.9769 222 0.0222 0.3621 1.8715 193 0.0193 0.4339 1.7973 169 0.0169 0.4876 1.7344 155 0.0155 0.5270 1.6976 167 0.0167 0.5589 1.6653 162 0.0162 0.5842 1.6408 134 0.0134 0.6066 1.6180 142 0.0142

LCL 0.2406 0.4855 0.6311 0.7237 0.7945 0.8462 0.8877 0.9199 0.5333 0.6792 0.7659 0.8208 0.8634 0.8942 0.9188 0.9380

3 S.E. Limits UCL out Prop 2.1940 479 0.0479 1.9529 616 0.0616 1.8029 868 0.0868 1.6933 1192 0.1192 1.6271 1524 0.1524 1.5746 1878 0.1878 1.5359 2200 0.2200 1.5031 2511 0.2511 1.7001 578 0.0578 1.5544 890 0.0890 1.4653 1256 0.1256 1. 4012 1689 0.1689 1.3613 2048 0.2048 1.3300 2408 0.2408 1.3062 2708 0.2708 1.2866 3013 0.3013

Table 8: Out of control signals of range charts.. ¯ β n x 3 2 0.8156 3 1.2201 4 1.5154 5 1.7361 6 1.8946 7 2.0620 8 2.1988 9 2.3410 10 2.4612 4 2 0.5611 3 0.8409 4 1.0352 5 1.1815 6 1.2906 7 1 3968 8 1.4831 9 1.5678 10 1.6419

Log-logistic distribution LCL UCL out Prop 0.0010 9.8544 27 0.0027 0.0329 11.8028 26 0.0026 0.0998 13.1308 29 0.0029 0.1830 14.3055 29 0.0029 0.2938 15.2567 26 0.0026 0.3822 15.9921 25 0.0025 0.4393 16.0583 28 0.0028 0.4887 16.4539 30 0.0030 0.5465 16.7774 26 0.0026 0.0007 5.0117 26 0.0026 0.0259 5.8466 26 0.0026 0.0759 6.3977 28 0.0028 0.1512 7.0422 26 0.0026 0.2182 6.8846 27 0.0027 0.2905 7.1065 27 0.0027 0.3227 7.3950 27 0.0027 0.3675 7.8149 27 0.0027 0.4279 7.9616 27 0.0027

Normal distribution 3 S.E. Limits LCL UCL out Prop LCL UCL out Prop 0.0000 2.6646 427 0.0427 0 4.2259 149 0.0149 0.0000 3.1420 508 0.0508 0 5.2385 144 0.0144 0.0000 3.4360 571 0.0571 0 5.9893 143 0.0143 0.0000 3.6511 637 0.0637 0 6.5425 146 0:0146 0.0000 3.7964 683 0.0683 0 6.9193 138 0.0138 0.1572 3.9792 725 0.0725 0 7.3638 142 0.0142 0.3076 4.1252 748 0.0748 0 7.7213 135 0.0135 0.4300 4.2440 812 0.0812 0 8.1130 136 0.0136 0.5468 4.3568 855 0.0855 0 8.4390 137 0.0137 0.0000 1.8331 350 0.0350 0 2.3935 168 0.0168 0.0000 2.1653 396 0.0396 0 2.9479 165 0.0165 0.0000 2.3623 444 0.0444 0 3.3157 151 0.0151 0.0000 2.4989 513 0.0513 0 3.5869 159 0.0159 0.0000 2.5864 539 0.0539 0 3.7799 161 0.0161 0.1062 2.6874 607 0.0607 0 3.9853 164 0.0164 0.2062 2.7645 645 0.0645 0 4.1479 162 0.0162 0.2885 2.8471 669 0.0669 0 4.3162 156 0.0156 0.3661 2.9177 697 0.0697 0 4.4624 154 0.0154

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R.R.L. Kantam, A.Vasudeva Rao and G.Srinivasa Rao

References [1] Balakrishnan, N. and Malik, H.J. (1987): Moments of order statistics from truncated log-logistic distribution. J.Statist. Plann. & Inf. 17, 251- 267. [2] Balakrisnan, N., Malik, H.J. and Puthenpura, S. (1987): Best linear unbiased estimation of location and scale parameters of the log-logistic distribution. Commun. Statist. Theor. Meth. 16, 3477-3495. [3] Edgeman, R.L. (1989): Inverse Gaussian control chart. Australian Journal of Statistics 31, 435-446. [4] Gupta, R.C., Akman, O. and Lvin, S. (1999): A study of log-logistic model in survival analysis. Biometrical Journal 41, 431-443. [5] Kantam, R.R.L., Rosaiah, and Srinivasa Rao, G. (2001): Acceptance sampling based on life tests: log - logistic model. Journal of Applied Statistics 28, 121-128. [6] Kantam, R.R.L. and Sriram, B. (2001): Variable control charts based on gamma distribution. IAPQR Transaction 26, 63-78. [7] Ragab, A. and Green, J. (1984): On order statistics from the log-logistic distribution and their properties. Common. Statist. Theor.Meth. 13, 2713-2724. [8] Ragab, A. and Green, J. (1987): Estimation of the parameters of the log-logistic distribution based on order statistics. Amer. J. Math. Mangag. Sci. 7, 307-324. [9] Srinivasa Rao, G. (2001): Some problems of statistical Inference with applications in log-logistic distribution. Unpublished Ph.D. Thesis awarded by Nagarjuna University.

R.R.L. Kantam and A.Vasudeva Rao G.Srinivasa Rao Department of Statistics Department of Science and Humanities Nagarjuna University DVR & Dr. HS MIC College of Technology Guntur-522510 Kancikacherla-522006 India India