NEW APPROXIMATE INFERENTIAL METHODS FOR THE RELIABILITY PARAMETER IN A STRESS-STRENGTH MODEL: THE NORMAL CASE

Transcription

1 Communications in Statistics-Theory and Methods 33 (4) NEW APPROXIMATE INFERENTIAL METODS FOR TE RELIABILITY PARAMETER IN A STRESS-STRENGT MODEL: TE NORMAL CASE uizhen Guo and K. Krishnamoorthy Department of Mathematics University of Louisiana at Lafayette Lafayette, LA krishna@louisiana.edu Key Words: explanatory variables; generalized p-value; lognormal distribution; noncentral t distribution; parametric bootstrap; quantile; Satterthwaite approximation; size; tolerance limits. ABSTRACT This article considers the problem of testing and interval estimation of the reliability parameter P (Y 1 > Y ), where Y 1 and Y are independent normal random variables with unknown means and variances. A new approximate method is proposed, and is compared with two existing approximate methods and a generalized variable method. Simulation studies indicate that the sizes of the existing approximate tests exceed the nominal level considerably in some situations while the new method controls the sizes satisfactorily in all situations considered. The studies also indicated that the generalized variable test is too conservative for small samples. A confidence limit for the reliability parameter based on the new approximate test is also given. The results are extended to the case where Y 1 and Y depend on some explanatory variables. The methods are illustrated using three examples, and some recommendations are made regarding what method should be used for practical applications. 1

2 1. INTRODUCTION The classical stress-strength reliability problem concerns the proportion of the times the strength Y 1 of a component exceeds the stress Y to which it is subjected. If Y 1 Y, then either the component fails or the system that uses the component may malfunction. If both Y 1 and Y are random, then the reliability R of the unit can be expressed as R = P (Y 1 > Y ), where P denotes the probability. To assess the reliability of the component, inference on R is desired. This reliability problem arises in the areas of aeronautical, civil, mechanical and nuclear engineering. Some specific examples are as follows: all (1) provided an example of a system application where the breakdown voltage Y 1 of a capacitor must exceed the voltage output Y of a transverter (power supply) in order for a component to work properly. Guttman et. al. () presented a rocket-motor experiment data where Y 1 represents the chamber burst strength and Y represents the operating pressure. These two examples along with another example are illustrated in Section 3 of this article. In this article, we consider some approximate inferential procedures for R when Y 1 and Y are independent normal random variables with unknown means and variances. Relevant references for the present problem can be found in the papers by Reiser and Guttman (3) and Weerahandi and Johnson (4). To formulate the present problem, let Y 1 N(µ 1, σ 1) independently of Y N(µ, σ ). Then the reliability parameter R can be expressed as R = P (Y 1 > Y ) = Φ µ 1 µ σ1 + σ, (1.1) where Φ is the standard normal distribution function. Let us consider the problem of testing : R R vs a : R > R, (1.) where R is a specified probability which is usually close to 1. Let z R quantile of the standard normal distribution. Then denote the R th η = µ 1 µ z R σ 1 + σ. (1.3) is the (1 R )th quantile of the distribution of Y 1 Y. Note that R > R if and only if η >, and hence the hypotheses in (1.) are equivalent to : η vs a : η >. (1.4) The null hypothesis in (1.4) is rejected at the nominal level α whenever a lower 1 α confidence limit η L for η is greater than. The lower limit η L is referred to as the R content - (1 α) coverage one-sided lower tolerance limit for the distribution of Y 1 Y. That is, the interval (η L, ) would contain at least R proportion of data from the distribution of Y 1 Y with confidence level 1 α. Thus, a test for the reliability parameter R can be obtained using a one-sided tolerance limit for the distribution of Y 1 Y. The problem of constructing one-sided tolerance limit for the distribution of Y 1 Y has been addressed in

3 all (1). Specifically, all developed an exact one-sided tolerance limit when the variance ratio q 1 = σ1/σ is known, and an approximate limit when q 1 is unknown. Although, all has argued that the two problems are related, the above description is simple and clearly points out the relation. The theoretical developments in all (1) can be used to construct lower confidence limits for R in (1.1). In particular, the results given in Reiser and Guttman (3) and Guttman et. al. () can be easily deduced from the results in all (1). The approximate methods suggested in the above articles are essentially based on the well-known Aspin-Welch approximate test (see Casella and Berger (5), p. 49) for the Behrens-Fisher problem. Let x i and s i denote respectively the mean and variance of a sample of n i observations from N(µ i, σi ) population, i = 1,. all s approximate tolerance limit is obtained by first developing a limit which depends on the variance ratio q 1, and then replacing q 1 by a suitable estimate. all recommended using the unbiased estimator ˆq 1 = (n 3)s 1/((n 1)s ) whereas, in the problem of reliability estimation, Reiser and Guttman (3) suggested the usual estimator s 1/s. Although, the procedures based on these two estimates may not differ significantly for large n, they do differ for small n. See the sizes of the tests based on these two estimates in Figures 1-3 ( and tests). Guttman et. al. () extended the results of Reiser and Guttman (3) to the case where Y 1 and Y depend on some covariates. Weerahandi and Johnson (4) proposed inferential procedures for the reliability parameter R based on the generalized p-value approach. As claimed by Weerahandi and Johnson (4), the sizes of the generalized p-value test are very close to the nominal level, and it performs better than the approximate test due to Church and arris (6) when Y 1 and Y are normal and the distribution of Y is known. owever, the generalized p-value test is approximate, often conservative, when the distribution of Y is unknown (see Test in Figures 1 and 3). It is clear from the above discussion that the all s (1984) approach depends on the labeling of the populations. Specifically, if the variance ratio is defined as σ/σ 1, and is estimated by the unbiased estimator (n 1 3)s /((n 1 1)s 1), then the resulting test will be different from the one due to all. The two tests may be combined to overcome the label dependency. In this article, we propose a new approximate test that is obtained by combining the two tests. This article is organized as follows. For the sake of completeness and easy reference, we present in Section the exact test for the reliability parameter based on the all s approach when the variance ratio is known. In Section 3, we describe the existing approximate tests, the new test mentioned in the preceding paragraph, and the generalized variable test due to Weerahandi and Johnson (4). The tests are compared with respect to the size using the Monte Carlo simulation. The simulation studies indicate that the new test performs better than the existing tests, and behaves like an exact test when the sample sizes are equal and the common sample size is 15 or more. Furthermore, we observed from our simulation studies that the generalized variable test is often very conservative when sample sizes are moderate and close to each other. Interval estimates for R based on the approximate tests are given. Construction of the tests and interval estimates are illustrated using three examples. In Section 4, we extend the results of Section 3 to the reliability problem when Y 1 and Y have 3

4 covariates. Some concluding remarks are given in Section 5.. AN EXACT TEST FOR R WEN q 1 = σ 1/σ IS KNOWN We shall now develop an exact test for the reliability parameter R using the all s approach for constructing tolerance limits for Y 1 Y. Letting s d = (1 + q 1 1 )((n 1 1)s 1 + (n 1)q 1 s )/(n 1 + n ), (.1) it can be shown that s d/(σ 1 + σ ) χ n 1 +n /(n 1 + n ), where χ b denotes the chi-square random variable with df = b. Using this result and standardizing ȳ d = ȳ 1 ȳ, we see that ȳ d η s d = Z + z R m1 m1 χ n1 +n /(n 1 + n ) 1 m1 t n1 +n (z R m1 ), (.) where Z is the standard normal random variable, η is given in (1.3), m 1 = n 1(1+q 1 ) q 1 +n 1 /n and t n (ξ) denotes the noncentral t variable with df = n and noncentrality parameter ξ. This result in (.) leads to an exact 1 α lower limit for η (or R content - (1 α) coverage lower tolerance limit for the distribution of Y 1 Y ), and is given by ȳ d t n1 +n,1 α(z R m1 ) s d m1, (.3) where t n,p (c) denotes the 1pth percentile of a noncentral t distribution with df = n and nocentrality parameter c. The test based on (.3) rejects the null hypothesis in (1.) when the lower tolerance limit in (.3) is greater than zero or equivalently, when the p-value ( ) m1 ȳ d P t n1 +n (z R m1 ) > < α. (.4) 3. TESTS FOR R WEN q 1 = σ 1/σ IS UNKNOWN If q 1 is unknown, an approximate test can be obtained by substituting an estimate for q 1 in (.4). This approximate test, as pointed out by all (1) in the problem of constructing tolerance limits for Y 1 Y, is too liberal in some situations. all also suggested an alternative approach that can be used to develop an approximate test for R. Using the usual moment approximation (popularly known as the Satterthwaite s (7) approximation), it can be shown that (s 1 + s )/(σ1 + σ) is approximately distributed as χ f 1 /f 1, where f 1 is given in (3.1). Using this result and standardizing ȳ d, it can be readily verified that ȳ d η s 1 + s 1 m1 Z + z R m1 χ f1 /f 1 s d approximately, where m 1 = n 1(1 + q 1 ) q 1 + n 1 /n and f 1 = (n 1 1)(q 1 + 1) q 1 + (n 1 1)/(n 1). (3.1) 4

5 Noticing that Z+z R m1 χ f 1 /f 1 t f1 (z R m1 ), we get a 1 α lower confidence limit for η as s ȳ d t f1,1 α(z R m1 ) 1 + s. (3.) m 1 Thus, if q 1 is known, then (3.) is an approximate (R, 1 α) lower tolerance limit for the distribution of Y 1 Y. At the level of significance α, the null hypothesis in (1.) will be rejected whenever the limit in (3.) is greater than zero or equivalently, whenever the p-value P t f1 (z R m1 ) > m1 ȳ d s 1 + s < α. (3.3) It can be verified that the test based on (3.3) is exact only when n 1 = n and σ1 = σ. Nevertheless, when q 1 is unknown, the following approximate tests based on (3.3) perform better than the corresponding ones based on (.4). We shall now present three approximate tests, and the generalized p-value test due to Weerahandi and Johnson (4) for the case of unknown variance ratio. The Test: If q 1 is unknown, replacing q 1 in (3.1) by its unbiased estimator ˆq 1 = s 1(n 3)/(s (n 1)), we get an approximate test that rejects the in (1.) when the p-value P 1 = P (t ˆf1 (z R ˆm 1 ) > ˆδ ) ˆm 1 < α, (3.4) where ˆδ = ȳ d / s 1 + s, ˆm 1 and ˆf 1 can be obtained from (3.1) by replacing q 1 with ˆq 1. As we already mentioned, this unbiased estimate of q 1 is used in all (1) for constructing tolerance limit for the distribution of Y 1 Y. For this reason, we refer to this test as the test. The Test: The estimate ˆq = s 1/s is used to construct a lower bound for R in the paper by Reiser and Guttman (3). This choice of estimate yields a test that rejects the in (1.) when the p-value P ( ) t ˆf(z R ˆm) > ˆδ ˆm < α, (3.5) where ˆm and ˆf can be obtained from (3.1) by replacing q 1 with ˆq = s 1/s. The Test: As we already mentioned, the p-value in (3.4) depends on the definition of the variance ratio. If the variance ratio is defined as q = σ/σ 1, and it is estimated by ˆq = s (n 1 3)/(s 1(n 1 1)), then the test based on ˆq rejects the in (1.) when P = P (t ˆf (z R ˆm ) > ˆδ ) ˆm < α, (3.6) where ˆm = n (1 + ˆq ) ˆq + n /n 1, and ˆf = (n 1)(ˆq + 1) ˆq + (n 1)/(n 1 1). (3.7) 5

6 It is clear that the p-values P 1 in (3.4) and P in (3.6) are different. Furthermore, our preliminary numerical studies indicated that the sizes of the test based on (3.4) are very close to the nominal level when q 1 = σ1/σ is small and n 1 n, and the sizes of the test based on P in (3.6) are close to the nominal level when q = σ/σ 1 is small and n n 1. In particular, we found if one of the tests is too liberal for some sample size and parameter configurations, then the other test performs satisfactorily at the same configurations. These findings suggest that the test that rejects in (1.) whenever P a = max{p 1, P } < α (3.8) should be less liberal than the all s test and the one based on P in (3.6). Notice that the value of P a does not depend on labels of the samples. In other words, P a is a symmetric function of (n 1, s 1) and (n, s ). The test based on (3.8) is referred to as the test. The Test: The generalized variable statistic due to Weerahandi and Johnson (4) is given by T = (n ȳ d Z 1 1)s 1 n 1 U 1 + (n 1)s n U, (3.9) (n 1 1)s 1 U 1 + (n 1)s U where Z, U 1 and U are independent random variables with Z N(, 1) and U i χ n i 1, i = 1,. For given sample sizes, ȳ d, s 1 and s, the test based on T rejects the null hypothesis in (1.) whenever the generalized p-value P Z,U1,U ( T < Φ 1 (R ) ) < α. (3.1) 3.1 Simulation Studies of the Sizes The sizes of the approximate tests given in the preceding section are estimated using Monte Carlo simulation consisting of 1, runs. From the development of the tests, we see that the p-values depend only on the variance ratio q 1. The sizes are computed when η =, or equivalently when µ 1 µ = z R σ 1 + σ, and R =.95. To estimate the size( of the test, for example, the test based on (3.4), we computed the probability P 1 = P t ˆf1 (z R ˆm1 ) > ) ˆm 1 ȳ d for each simulated statistic (ȳ s 1 +s d, s 1, s ), and then found the proportion of the values of P 1 which are less than the nominal level α. For a good test, this proportion should be close to α. To compute the noncentral t probabilities, we used the algorithm due to Benton and Krishnamoorthy (8), which is an enhanced version of the algorithm due to Lenth (9). To estimate the sizes of the generalized p-value test in (3.1), we used 5 simulated values of ( x 1, s 1, x, s ), and for each such simulated value, we used 5 simulated values of (Z, U 1, U ) to estimate the probability in (3.1). The percentage of the 5 probabilities which are less than α is an estimate of the size of the generalized p-value test. IMSL subroutines RNCI and RNNOA are used to generate respectively the chi-square random numbers and the normal random numbers. The following tests are considered in our simulation studies. 6

7 1. The test based on P 1 in (3.4); ˆq 1 = (n 3)s 1 (n 1)s. The test based on (3.5); ˆq = s 1 s 3. The test based on (3.8); the new test 4. The test based on the generalized p-value in (3.1) The estimated sizes of the tests are plotted in Figure 1 when the sample sizes are equal,.5 σ1/σ, and the nominal level α =.5. When n 1 = n = 5, we see that the tests and are liberal when the ratio q 1 = σ1/σ 3 and the new test is either slightly conservative or very close to the nominal level.5. When the common sample size is 1 there is only a slight difference between the sizes of tests and ; the new test is almost exact except in the neighborhood of q 1 = 1. We observe from Figures 1(a-d) that the test is almost exact when the common sample size greater than or equal to 15. The generalized p-value test is in general conservative; it is too conservative when σ1/σ is small. In Figures (a-e), we plotted the sizes of the tests when the sample sizes are unequal. It is clear that the sizes of the tests and could be as high as 1 (see Figure (a)) when n 1 is much smaller than n and σ1/σ is small. The differences among the sizes are not appreciable when n 1 n is not too large; still the test controls the sizes very satisfactorily. The sizes of the test range from.3 to.7 while the sizes of the tests and range from to 1 for all the cases considered. The generalized p-value test is slightly liberal in some cases (see Figure (b)), and in other situations its sizes are close to the nominal level.5. Since we noticed that the difference between the sample sizes has impact on the sizes, we plotted the sizes in Figures 3(a-h) as a function of n (ranging from 5 to 14) while n 1 = 7 and 15, and q 1 =.,.8, and 8. We again observe from these graphs that the test performs better than the other two approximate tests. The sizes of the new test ranges from to.7 where as the sizes of the other two approximate tests could go as high as. Thus, we see that the test is clearly preferable to other two approximate tests for practical applications. The generalized p-value test is again either conservative or almost exact for all the cases considered in Figure 3. Our overall conclusions based on Figures 1-3 are as follows. The new test is certainly preferable to the other approximate tests when the sample sizes are close to each other. Our numerical studies (not reported here) showed that if both sample sizes are at least 5, and their ratio is between.7 and 1.3, then the sizes of the new test are very close to the nominal level. The generalized p-value test can be recommended for practical use if the ratio of the sample sizes is less than.7 or greater than

8 (a) n 1 = n = 5 (b) n 1 = n = σ1/σ (c) n 1 = n = 15 σ1/σ (d) n 1 = n = σ1/σ σ1/σ Figure 1. Sizes of the Tests When Sample Sizes are Equal and α =.5; Test is based on (3.4), Test is based on (3.5), Test is based on (3.8) and Test in (3.1) 8

9 (a) n 1 = 5, n = 5 (b) n 1 = 1, n = σ1/σ (c) n 1 = 15, n = 5 σ1/σ (d) n 1 =, n = σ1/σ (e) n 1 = 5, n = 5 σ1/σ (f) n 1 = 3, n = Figure. Sizes of the Tests σ1/σ When Sample Sizes are Unequal and α =.5; σ Test 1/σ is based on (3.4), Test is based on (3.5), Test is based on (3.8) and Test Test in (3.1) 9

10 .... (a) n 1 = 7, σ 1/σ = n (c) n 1 = 7, σ1/σ = n (e) n 1 = 7, σ1/σ = n (g) n 1 = 7, σ1/σ = n.... (b) n 1 = 15, σ 1/σ = n (d) n 1 = 15, σ1/σ = n (f) n 1 = 15, σ1/σ = n (h) n 1 = 15, σ1/σ = n Figure 3. Sizes of the Tests as a Function of n ; Test is based on (3.4), Test is based on (3.5), Test is based on (3.8) and Test Test in (3.1); α =.5. 1

11 3. Lower Confidence Bounds for R We shall now consider interval estimation of R on the basis of the tests in the previous section. Let δ = (µ 1 µ )/ σ1 + σ. Since R = Φ(δ) is a one-one function of δ, it suffices to find a confidence bound for δ. To get confidence bound for δ, we note that ˆδ = ȳ d s 1 + s = ȳ d µ d + δ m σ 1 /n 1 +σ 1 /n s m1 1 +s σ1 +σ Z + m 1 δ m1 χ f1 /f 1 approximately, (3.11) where m 1 and f 1 are given in (3.1). Note that ˆδ t f1 ( m 1 δ)/ m 1 approximately. Let F (x; n, ξ) denote the noncentral t cdf with df = n, and noncentrality parameter ξ. For any given x and n, the noncentral t cdf is strictly decreasing with respect to its noncentrality parameter, and hence a 1-α lower bound δ L for δ can be obtained as the solution of P (t f1 ( m 1 δ L ) < m 1ˆδ) = F ( m1ˆδ; f1, m 1 δ L ) = 1 α. (3.1) If the variance ratio q 1 is unknown, then replacing m 1 and f 1 by their estimates, we get the lower bound δ 1L as the solution of F ( ˆm 1ˆδ; ˆf1, ˆm 1 δ 1L ) = 1 α, (3.13) where ˆm 1 and ˆf 1 are defined as in (3.4). The lower bound corresponding to ˆm and ˆf in (3.7) is the δ L that satisfies Thus, the combined lower bound is given by F ( ˆm ˆδ; ˆf, ˆm δ L ) = 1 α. (3.14) δ L = min{δ 1L, δ L }. (3.15) If q 1 is known, then the exact lower bound for δ based on the test (.4) is given by δ EL which satisfies F ( m 1 ˆδ ; n 1 + n, m 1 δ EL ) = 1 α, (3.16) where ˆδ = ȳ d /s d, s d is defined in (.1) and m 1 is given in (3.1). The generalized 1 α lower limit for δ on the basis of T in (3.9) is given by the αth quantile T α of T, which leads to the lower limit Φ(T α ) for R. 11

12 3.3 Examples We shall now illustrate the methods using the examples given in all (1), Reiser and Guttman (3) and Weerahandi and Johnson (4). The noncentral t probabilities and critical values are calculated using the software StatCalc, and it can be downloaded for free from or using an online calculator at The generalized p-values and limits are computed using 1,, simulation runs. Example 1. (all 1984) A sample of n 1 = 5 capacitors yielded mean breakdown voltage ȳ 1 = 6.75 kv and s 1 = 3. The voltage output from n = transverters (power supplies) produced ȳ = 4. kv and s =.53. We have ȳ d = ȳ 1 ȳ =.75. Using the methods in the preceding sections, we like to compute 95% lower limits for R. To compute δ cl in Section., we note that ˆq 1 =.77, ˆm 1 =.37, ˆf 1 = 7.543, ˆq = , ˆm =.6471, ˆf = and ˆδ = The equation (3.13) is F ( ˆm 1ˆδ; f1, ˆm 1 δ 1L ) = F (16.77; 7.543, 4.74δ 1L ) =.95. Solving this for δ 1L, we found δ 1L =.556. Similarly, solving the equation (3.14), we found δ L = Thus, δ L = min{δ 1L, δ L } =.556. The 95% lower limit for R is Φ(δ L ) = Φ(.556) = The 95% lower limit using ˆq = s 1/s =.31 is Φ(.5693) = This limit, due to Reiser and Guttman (3), can be obtained from (3.1) with (m 1, f 1 ) replaced by ( ˆm, ˆf) defined in (3.5). The generalized limit given in Section. is Φ(T α ) = Φ(.5118) =.994. All these estimates are very close to each others. Example. The summary statistics of this example are taken from Reiser and Guttman (3). The data are pertaining to mechanical component that yielded ȳ 1 = 17, psi, s 1 = 5, psi, ȳ = 144, 5 psi, s = 8, 9 psi and n 1 = n = 3. Using the formulas of the preceding sections, we computed ˆq 1 =.955, ˆf 1 = , ˆq =.964, ˆf = 49.78, ˆm = ˆm 1 = 3 and ˆδ =.498. We are interested in obtaining a 9% lower bound for R. Using ˆm 1, ˆf1, ˆδ and 1 α =.9 in (3.13), we obtained δ 1L = Similarly, using (3.14) we got δ L =.951. The 9% lower bound based on (3.15) for R is Φ(.8946) = The 9% lower bound using the approach of Reiser and Guttman (3) is The 9% generalized lower limit for this example is.984. Suppose we want to test : R.95 vs a : R >.95. Then using (3.4), we have P 1 = P ( t ˆf1 (z R ˆm1 ) > ( ˆm 1ˆδ) = P t ( ) > ) =.7. Using (3.6), we have P =.4. Thus, the p-value of the test is.7. The p-value of the test in (3.5) is.4. The generalized p-value in (3.1) is.4. The p-values of the approximate tests are very close to each other while the generalized p-value is slightly higher than the others. Example 3. In this example, we consider the rocket-motor experiment data set given in Guttman et. al. () which was later used by Weerahandi and Johnson (4) for illustration purpose. We are interested in making inferences on the reliability of the rocket motor at the 1

13 highest operating temperature of 59 degrees centigrade. At this temperature, the operating pressure Y distribution tends to be closest to the chamber burst strength Y 1 distribution. Assumption of normality has been verified in the paper just cited. The hypotheses considered by Weerahandi and Johnson are : R and a : R > This is equivalent to testing : δ Φ 1 ( ) = vs a : δ > A sample of n 1 = 17 and a sample of n = 4 observations yielded the statistics ȳ 1 = , s 1 =.349, ȳ = 7.789, s =.5414, ˆq 1 = , ˆm 1 = , ˆf 1 = ˆq = 3896, ˆm = 17.67, ˆf =.486 and ˆδ = Using ˆm 1, ˆf1, ˆδ and Z R = in (3.4), we computed P 1 =.4. Similarly, using (3.6), we computed P =.5. Thus, the p-value of the test is.5. The generalized p-value based on our own simulation is.4 (the reported value in Weerahandi and Johnson (4) is.4). Even though both tests provided strong evidence in favor of the alternative hypothesis a, we see that the p-value of the new test is much smaller than the generalized p-value. 4. INFERENCE ON TE RELIABILITY PARAMETER WEN Y 1 AND Y DEPEND ON COVARIATES We shall now extend the results of the preceding sections to the case where Y 1 and Y depend on some covariates. Specifically, we consider the models and Y 1i = β 1X 1i + ɛ 1i, i = 1,..., n 1, Y i = β X i + ɛ i, i = 1,..., n, where ɛ 1i s are independent N(, σ 1) random variables, ɛ i s are independent N(, σ ) random variables, X 1i s are p 1 1 vectors and X i s are p 1 vectors. Let X i = (X i1,..., X ini ), and Y i = (Y i1,..., Y ini ), i = 1,. The least squares estimator of β i is ˆβ i = (X ix i ) 1 X iy i, i = 1,, and the residual sums of squares based on the models are (n 1 p 1 )s 1 and (n p )s. All these variables are independent with The reliability parameter is defined by ˆβ i N pi (β i, σ 1(X ix i ) 1 ), i = 1,, (n i p i )s i /σ i χ n i p i, i = 1,. R(x, x ) = P (Y 1 > Y ) = Φ β 1x 1 β x σ 1 + σ where x 1 and x are given values for the explanatory variables. Writing c i = x i(x ix i ) 1 x i, i = 1,, we see that ˆβ ix i N(β ix i, c i σ i ), i = 1,. Because we deal with the same distributions as those in Section 3, the results of Section 3 can be extended to the present problem in 13,

14 a straightforward manner. For the sake of completeness, we shall present the test equivalent to the new approximate test in (3.8). Let ˆq 1 = (n p )s 1 (n p )s, ˆm 1 = c 1 1 (1 + ˆq 1 ) ˆq 1 + c /c 1 and ˆf 1 = (n 1 p 1 )(1 + ˆq 1 ) ˆq 1 + (n 1 p 1 )/(n p ). (4.1) The terms ˆq, ˆm and ˆf can be defined by interchanging the subscripts 1 and in (4.1). Define ˆδ = ( ˆβ 1x 1 ˆβ x )/ s 1 + s. The p-value P 1 similar to the one in (3.4) is given by P (t ˆf1 (z R ˆm1 ) > ˆm 1ˆδ), and the one similar to P in (3.6) is given by P (t ˆf (z R ˆm ) > ˆmˆδ). The null hypothesis : R(x, x ) R will be rejected whenever the maximum of these two p-values is less than α. The lower bounds δ 1L and δ L can be computed using the above ( ˆm 1, ˆf1 ) and ( ˆm ˆf ) respectively in (3.13) and (3.14). The combined lower bound for R(x, x ) is then given by δ L = min{δ 1L, δ L }. 5. CONCLUDING REMARKS In this article, we explored the all s (1) approximate method of constructing tolerance limits for the distribution of Y 1 Y, and proposed an approximate inferential procedures for the reliability parameter R. The proposed methods require computations of the noncentral t cumulative probabilities, critical points and the noncentrality parameter (given other parameters of the noncentral t) with non-integer degrees of freedom. All these quantities can be computed using freely available PC calculators (see example section for a URL). If the data are from lognormal distributions, then the proposed methods can be applied to the logged data. Even though the proposed methods are accurate enough for practical purpose still they are not exact. Some readers may want to explore the applicability of the parametric bootstrap (PB) method. Nowadays, such numerically intensive methods are commonly used to tackle non-standard problems such as the present one. We indeed numerically investigated the properties of the test based on the PB method. Our extensive simulation studies showed the test based on the PB method is liberal (in some situations, the size of the test exceeds. when the nominal level is.5). For these reasons, we have not discussed the PB test here. Our overall conclusion is that, among the methods considered in this article, the test and the confidence limit based on it (the δ L in (3.15)) are preferable for practical applications when the sample sizes are close to each other, and the generalized variable method is preferable to others when one of the sample sizes is drastically different from the other. 14

15 BIBLIOGRAPY (1) all, I. J. Approximate One-Sided Tolerance Limits for the Difference or Sum of Two Independent Normal Variates. Journal of Quality Technology, 1984, 16, () Guttman, I.; Johnson, R. A.; Bhattacharya, G. K.; Reiser, B. J. Confidence Limits for Stress-Strength Models With Explanatory Variables. Technometrics, 1988, 3, (3) Reiser, B. J.; Guttman, I. Statistical inference for Pr(Y < X): The normal case. Technometrics, 1986, 8, (4) Weerahandi, S.; Johnson, R. A. Testing Reliability in a Stress-Strength Model When X and Y are Normally Distributed. Technometrics, 199, 34, (5) Casella, G.; Berger, R. L. Statistical Inference, Duxbury, CA: Pacific Grove,. (6) Church, J. D.; arris, B. The Estimation of Reliability From Stress-Strength Relationships. Technometrics, 197, 1, (7) Satterthwaite, F. E. An Approximate Distribution of Estimates of Variance Components. Biometric Bulletin, 1946,, (8) Benton, D.; Krishnamoorthy, K. Computing Discrete Mixtures of Continuous Distributions: Noncentral Chisquare. Noncentral t and the Distribution of the Square of the Sample Multiple Correlation Coefficient. Computational Statistics and Data Analysis, 3, 43, (9) Lenth, R. V. Cumulative Distribution Function of the Noncentral t Distribution. Applied Statistics, 1989, 38,