# Estimating the reliability function for a family of exponentiated distributions.

1. Introduction

The reliability function R(t) is defined as the probability of failure-free operation until time t. Thus, if the random variable (rv) X denotes the lifetime of an item, then R(t) = P(X > t). Another measure of reliability under stress strength setup is the probability P = P(X > Y),which represents the reliability of an item of random strength X subject to random stress Y.

A lot of work has been done in the literature to deal with inferential problems related to various exponentiated distributions. In particular, Mudholkar and Srivastava [1] considered exponentiated Weibull family for analyzing bathtub failure-real data. The exponentiated Weibull family was also applied to the bus-motor-failure data in Davis [2], to head-and-neck cancer clinical trial data in Efron [3] [see[4]], and in analyzing flood data [see [5]]. Jiang and Murthy [6] considered exponentiated Weibull family and illustrated some of its properties by using graphical approach. Nassar and Eissa [7, 8] studied a two-parameter exponentiated Weibull distribution and they gave some of its properties and estimated the parameters by using the maximum likelihood and Bayes methods based on type II censored data. They used the squared-error and linear in exponential (LINEX) loss functions and an informative prior to obtain the Bayes estimates. Pal et al. [9] showed that the failure rate of exponentiated Weibull behaves more like the failure rate of the Weibull distribution than that of the

Gamma distribution. They also obtained maximum likelihood estimators, Fisher's information matrix, and confidence intervals for related parameters. Gupta et al. [10] introduced exponentiated exponential distribution in a series of papers. Gupta and Kundu [11-17], Gupta et al. [18], and Kundu et al. [19] concentrated on the study of the exponentiated exponential distribution. Kundu and Gupta [20] stated that exponentiated exponential distribution is an alternative to the well-known two-parameter Gamma, two-parameter Weibull or two-parameter lognormal distributions. Kundu and Gupta [20] estimated the reliability of the stress-strength model P(X<Y),when X and Y are independent exponentiated exponential random variables. Raqab and Ahsanullah [21] estimated the parameters of the exponentiated exponential based on ordered statistics. Inference for exponentiated Weibull distribution based on record values was made by Raqab [22]. Abdel-Hamid and AL-Hussaini [23] obtained the maximum likelihood estimates of the parameters when step-stress accelerated life testing is applied for exponentiated exponential distribution. Al-Hussaini and Hussein [24, 25] fitted the exponentiated Burr model to the data of breaking strength of single carbon fibers of particular length (data in [26]) in the complete sample case and when type II censoring is imposed on data. SELF and LINEX loss functions are used in the Bayes estimation. Gupta et al. [10] introduced a new distribution, called the exponentiated Pareto distribution. Shawky and Abu-Zinadah [27] obtained maximum likelihood estimators of the different parameters of the exponentiated Pareto distribution. They also considered five other estimation procedures and compared their performances through numerical simulations.

The purpose of the present paper is many fold. We propose a family of exponentiated distributions, which covers as many as ten exponentiated distributions as specific cases. We consider the problems of estimating R(t) and "P." Uniformly minimum variance unbiased estimators (UMVUES) and maximum likelihood estimators (MLES) are derived. In order to obtain these estimators, the major role is played by the estimator of the probability density function (pdf) at a specified point, and the functional forms of the parametric functions to be estimated are not needed. It is worth mentioning here that, in order to estimate "P," in the literature, the authors have considered the case when X and Y follow the same distributions, may be with different parameters. We have generalized the result to the case when X and Y may follow any distribution from the proposed exponentiated family of distributions. Simulation study is carried out to investigate the performance of the estimators.

In Section 2, we introduce the family of exponentiated distributions. In Section 3, we derive the UMVUES of R(t) and "P." In Section 4, we obtain the MLES of R(t) and "P." In Section 5, analysis of a simulated data has been presented for illustrative purposes. Finally, in Section 6, conclusions have been provided.

2. The Family of Exponentiated Distributions

Let the rv X follow the distribution having the pdf

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Here, g(x; a, [[theta].bar]) is a function of X and may also depend on (may be vector-valued) parameter [[theta].bar].Moreover, g(x; a, [[theta].bar]) is a real-valued, monotonically increasing function of X with g(a; a, [[theta].bar]) = 0, g([infinity]; a, [[theta].bar]) = [infinity],and g'(x; a, [[theta].bar]) denotes the derivative of g(x; a, [[theta].bar]) with respect to x. Our case is a special model of the beta extended Weibull family proposed by Cordeiro et al. [28] and can be obtained from equation (7) when b = 0. We note that (1) represents a family of exponentiated distributions, as it covers the following exponentiated distributions as special cases.

(i) For g(x; a, [[theta].bar]) = x and a = 0, we get the exponentiated exponential distribution [see [10]].

(ii) For g(x; a, [[theta].bar])[x.sup.2] and a = 0, it gives exponentiated Rayleigh distribution [see [29]].

(iii) For g(x; a, [[theta].bar]) = [x.sup.[delta]] [delta] > 0 and a = 0, we get the exponentiated Weibull distribution [see [1]].

(iv) For g(x; a, [[theta].bar]) = log(1+[x.sup.[delta]]), [delta] > 0 and a = 0, it leads us to pdf of exponentiated Burr distribution [see [24]].

(v) For g(x; a, [[theta].bar] = log(x/a), it turns out to be exponentiated Pareto distribution [see [10]].

(vi) For g(x; a, [[theta].bar] = (1 + (x/[xi])), [xi] > 0 and a = 0, it gives us exponentiated Lomax distribution [see [30]].

(vii) For g(x; a, [[theta].bar] = (1 + ([x.sup.[xi]]/[xi])), [xi] > 0, v > 0 and a = 0, it turns out to be exponentiated Burr distribution with scale parameter v [see [31]].

(viii) For g(x; a, [[theta].bar] = [x.sup.[gamma]] exp(vx), [gamma] > 0, v > 0 and a = 0, it gives the exponentiated form of modified Weibull distribution of Lai et al. [32].

(ix) For g(x; a, [[theta].bar] exp[(x/[gamma]).sup.v]-1], [gamma] > 0, v > 0 and a = 0, we get the exponentiated form of a modification of Weibull distribution [see [33]].

(x) For g(x; a, [[theta].bar] = (x - a) + (v/[lambda]) log((x + v)/(a + v)), v > 0, it gives exponentiated form of the generalized Pareto distribution [see [34]].

3. UMVUES of R(t) and "P"

Throughout this section, we assume that a is unknown, but "a", [lambda],and [[theta].bar] are known. Let [X.sub.1],[X.sub.2], ..., [X.sub.n] be a random sample of size n from (1).

Lemma 1. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, T is complete and sufficient for the family of distributions given in (1). Moreover, the pdf of T is

g(t;[alpha],[lambda], [[theta].bar])= [[alpha].sup.n]/[GAMMA](n) [(-t).sup.n-1] [e.sup.[alpha]t];-[infinity] < t < 0. (2)

Proof. Denoting by h([x.sub.1],[x.sub.2], ..., [x.sub.2];a,[alpha],[lambda], [[theta].bar]) the joint pdf of [X.sub.1],[X.sub.2], ..., [X.sub.n],we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

It follows from (3) and Fishers-Neyman factorization theorem [see [35, p. 341]] that T is sufficient for the family of distributions f(x;a,[alpha],[lambda][theta].bar], (9).It is easy to see that the rv V = -2[alpha] log(1 - [e.sup.-[lambda]g(x;a[[theta].bar])] follows [[chi square].sub.(2)] distribution. Thus, from the well-known additive property of Chi-square distribution [see Johnson and Kotz, [36,p. 170]], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an [chi square](2n) rv and the result follows. Since the distribution of T belongs to exponential family, it is also complete [see [35, p. 341]].

The following lemma provides the UMVUES of the powers of a.

Lemma 2. For q [member of] (-[[infinity], [infinity]), the UMVUE of [[alpha].sup.q] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Proof. The result follows from Lemma 1, Lehmann-Scheffe theorem [see [35, p. 357]], and the fact that

E[(-2[alpha]T).sup.-q] = [GAMMA](n - q)/[2.sup.q][GAMMA](n); q < n. (5)

In the following lemma, we provide the UMVUE of the sampled pdf (1) at a specified point "x"

Lemma 3. The UMVUE of f(x; a, [alpha], [lambda], [[theta].bar]) at a specified point "x" is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Proof. Since T is complete and sufficient for the family of distributions f(x; a, [alpha], [lambda], [[theta].bar]),any function H(T) of T satisfying E[H(T)] = f(x; a, [alpha], [lambda], [[theta].bar]) will be the UMVUE of f(x; a, [alpha], [lambda], [theta].bar]). To this end, from (1) and Lemma 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Equation (8) is satisfied if we choose

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

and the lemma holds.

Remark 4. We can write (1) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Using Lemma 1 of Chaturvedi and Tomer [37] and Lemma 2, the UMVUE of f(x; a, [alpha], [lambda], [[theta].bar]) at a specified point 'Vis

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

which coincides with Lemma 3. Thus, the UMVUE of the power of a can be used to derive the UMVUE of [??](x; a, [alpha], [lambda], [[theta].bar]) at a specified point.

In the following theorem, we obtain the UMVUE of R(t).

Theorem 5. The UMVUE of R(t) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Proof. Let us consider the expected value of the integral [[integral].sup.[infinity].sub.t] [??](x; a, [alpha], [lambda], [[theta].bar])dx with respect to T; that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

We conclude from (14) that the UMVUE of R(t) can be obtained simply integrating f(x; a, [alpha], [lambda], [[theta].bar]) from t to [infinity]. Thus, denoting by [g.sup.-1](*), the inverse function of g(x;a, [[theta].bar]), from Lemma 3, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

and the theorem follows.

Let X and Y be two independent rvs following the families of distributions [f.sub.1](x;[a.sub.b1],[[alpha].sub.1],[lambda].sub.1], [[[theta].bar].sub.1]) and [f.sub.2](y;[a.sub.2],[a.sub.2],[X.sub.2], [[[theta].bar].sub.2]), respectively. We assume that a1 and [alpha]2 are unknown but [a.sub.1],[a.sub.2],[[lambda].sub.1],[[lambda].sub.2], [[[theta].bar].sub.1],and [[theta].bar].sub.2] are known. Let [X.sub.1],[X.sub.2], ..., [X.sub.n] be a random sample of size n from [f.sub.1](x; [a.sub.1], [[alpha].sub.1], [[lambda].sub.1], [[[theta].bar].sub.1]) and let [Y.sub.1],[Y.sub.2], ..., [Y.sub.m] be a random sample of size m from [f.sub.2](y;[a.sub.2],[[alpha].sub.2],[[lambda].sub.2], [[[theta].bar].sub.2]). Let us denote by[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In what follows, we obtain the UMVUE of "P".

Theorem 6. The UMVUE of "P" is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Proof. From the arguments similar to those adopted in proving Theorem 5, it can be shown that the UMVUE of "P" is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII (17)

Thus, using Lemma 3,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Let us first consider the case when [h.sup.-1]{-l[[lambda].sup.-1.sub.2] log(1 - [e.sup.s])} > [g.sup.-1] {-[[lambda].sup.-1.sub.1] log(1 - [e.sup.T])}. In this case, from (18),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Now we consider the case when [h.sup.-1]{-[[lambda].sup.-1.sub.2] log(1 - [e.sup.s])} < [g.sup.-1] {-[lambda]-1 log(1 - [e.sup.T])}. In this case, from (18),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

The theorem now follows on combining (19) and (20).

Corollary 7. For the case when [a.sub.1] = [a.sub.2] = a, say, [[lambda].sub.1] =[[lambda].sub.2] = [lambda], say, [[[theta].bar].sub.1] = [[[theta].bar].sub.2] = [[theta].bar],say, and g(x; a, [[theta].bar]) = h(y; a, [[theta].bar]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Remarks 1.

(i) In the literature, the researchers have dealt with the estimation of R(t) and "P" separately. If we look at the proofs of Theorem 5 and Theorem 6, we observe that the UMVUE of the sampled pdf is used to obtain the UMVUES of R(t) and "P" Thus, we have established interrelationship between the two estimation problems.

(ii) In the literature, the researchers have derived the UMVUES of "P" for the case when X and Y follow the same distribution (maybe with different parameters). We have obtained UMVUES of "P" for all the three situations, when X and Y follow the same distribution having all the parameters same other than a's, when X and Y have the same distribution with different parameters and when X and Y have different distributions.

(iii) It follows from Lemma 2 that Var ([??]) = [[alpha].sup.2]/(n - 1) [right arrow] 0 as n [right arrow] [infinity]. Thus, [??] is a consistent estimator of a. Since [??](x; a, [alpha], [lambda], [[theta].bar]), [??](t), and [??] are continuous functions of consistent estimators, they are also consistent estimators of f(x; a, [alpha], [lambda], [[theta].bar]), R(t),and "P," respectively.

4. MLES of R(t) and "P" When All the Parameters Are Unknown

Looking at the distributions belonging to the family (1), we notice that, except for exponentiated Pareto and generalized exponentiated Pareto distributions [(V) and (X)], a = 0; that is, "a" is known. For these two distributions [[theta].bar] contains "a."

Irrespective of the distribution, the MLE of a is [??] [X.sub.(1)=] [min.sub.1[less than or equal to]i[less than or equal to]n] [X.sub.i].From (3), the log-likelihood function is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

First of all, were place "a" by [X.sub.(1)] in (22); then we differentiate (22) with respect to all the unknown parameters and equate these differential equations to zero. The MLES of unknown parameters are obtained on solving these equations simultaneously. Let [??], [??], and [[??].bar] be the maximum likelihood estimators of [alpha], [lambda],and [[theta].bar] respectively.

The following lemma provides The MLE of f (x; a, [alpha], [lambda], [[theta].bar]) at a specified point "x".

Lemma 8. The MLE of f(x; a, [alpha], [lambda], [[theta].bar]) at a specified point "x" is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

Proof. The proof follows from (1) and the one-to-one property of the MLE.

In the following theorem, we derive the MLE of R(t).

Theorem 9. The MLE of R(t) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Proof. Using Lemma 8 and invariance property of the MLES,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

and the theorem follows.

In the following theorem, we obtain the maximum likelihood estimator of "P."

Theorem 10. The MLE of "P "is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

Proof. Using Lemma 8 and the one-to-one property of the MLES

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

and the result follows.

Corollary 11. For the case when [a.sub.1] =[a.sub.2] = a, say,[[lambda].sub.1] = [[lambda].sub.2] = [lambda], say, [[[theta].bar].sub.1] = [[[theta].bar].sub.2] = [[theta].bar],say, and g(x; a, [[theta].bar]) = h(y; a, [[theta].bar]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

Remarks 2.

(i) All the comments made under Remarks 1 for UMVUES are tenable for MLES.

(ii) In the present approaches of obtaining UMVUES and MLES, one does not need the expressions of R(t) and "P."

(iii) Since the UMVUES and MLES of powers of a are obtained under the same conditions, we compare their performances. For q = -1 the UMVUE and MLE of a are, respectively, [??] =(n - 1)[(-T).sup.-1] and [??] = (n)[(-T).sup.-1]. For these estimators

V([??]) = [[alpha].sup.2]/(n - 2), V([??]) = [n.sup.2]/(n-1)[sup.2] (n - 2). (29)

Hence,

V([??]) - V([??]) = (2n - 1)/(n - 1) (n - 2) [[alpha].sup.2]. (30)

Thus, the UMVUE of [alpha] is more efficient than its MLE. Similarly, we can compare the performances of these estimators for other powers of [alpha].

5. Simulation Studies

In order to verify the consistency of estimators, we have drawn samples of sizes n = 5, 20 and 30 from (1) with g(x; a, [[theta].bar]) = x, a = 0, [alpha] = 3, and [lambda] = 1. In Figure 1, we have plotted f(x; [alpha], [lambda] corresponding to these samples. We conclude that as the sample size increases, the curves of [??] (x; [alpha], [lambda]) come closer to the curve of f(x; a, A),which is plotted in the same figure only for n = 30. For n = 30, the curves overlap. A similar pattern follows for the curves of [??](x; [alpha],[lambda]).

For the case when a is unknown but other parameters are known, we have conducted simulation experiments using bootstrap resampling technique for sample sizes n = 5, 10, 20, and 50. The samples are generated from (1), with, g(x;a,[theta].bar]) = x, a = 0, [alpha] = 3, and [lambda] = 1. For different values of t, we have computed [??](t), [bar.R](t),their corresponding bias, variance, 95% confidence length, and corresponding coverage percentage. All the computations are based on 500 bootstrap replications and results are reported in Table 1.

In order to estimate "P," for the case when [[alpha].sub.1] and [[alpha].sub.2] are unknown but other parameters are known, we have conducted simulation experiments using bootstrap resampling technique for sample sizes (n,m) = (5, 5),(10,10),(15,15), (25, 25), and (30, 30). The samples are generated from (1), with g(x;[a.sub.1], [[[theta].sub.2].bar]) = x,h(y;[a.sub.2], [[[theta].sub.2].bar]) = y, [a.sub.1] =[a.sub.2] = 0, [X.sub.1] = [[lambda].sub.2] =1, [[alpha].sub.1] = 2, and [[alpha].sub.2] = 0.5(0.25)1.25. The computations are based on 500 bootstrap replications. We have computed -[??], [??], bias, variance, 95% confidence length, and corresponding coverage percentage. The results are presented in Table 2.

In order to demonstrate the application of the theory developed in Section 4, we consider the following example.

Here, the first row indicates the estimate, the second row indicates the bias, the third row indicates variance, the fourth row indicates 95% bootstrap confidence length, and the fifth row indicates the coverage percentage. Let the rv follow exponentiated exponential distribution with pdf

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

where [alpha],[lambda],and [delta] are unknown. Denoting, [L.sub.1] ([alpha],[lambda],[delta]|x),the likelihood, the log-likelihood is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Considering negative log-likelihood, then differentiating it with respect to all unknown parameters, and equating these differential coefficients to zero, from (32),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

Here, the first row indicates the estimate, the second row indicates the bias, the third row indicates variance, the fourth row indicates 95% bootstrap confidence length, and the fifth row indicates the coverage percentage.

The following sample of size 50 is generated from (31), for [alpha] = 3.5, [lambda] = 1, and [delta] = 1 as follows.

Sample 1. 1.4904, 0.5808, 2.2326, 0.9756, 2.3062, 1.4692,

1.0433, 2.9527, 0.9241, 2.5901, 1.2021, 1.3022, 3.6532, 0.8146, 1.7083, 0.3126, 2.4925, 0.9462, 0.4557, 2.0120, 1.6554, 1.1651, 0.9703, 1.0305, 0.4618, 0.7465, 1.9249, 1.5763, 1.9127, 0.9742, 2.7401, 1.3686, 1.8546, 2.3296, 1.4526, 1.1719, 4.4671, 1.1218, 1.1901, 0.8248, 3.0091, 0.9511, 1.3308, 2.2145, 2.0216, 1.6740, 1.5620, 2.0339, 0.6882, 3.1335.

Assuming that the data represents life spans of items in hours, R(0.35) = 0.986, and solving (33) simultaneously, we get [??] = 3.711821, [??] = 1.170381, [??] = 1.098677, and [??](0.35) = 0.9872.

In order to obtain the maximum likelihood estimator of "P," we have generated one more sample of size 50 from (31), for [alpha] = 3, [lambda] = 1, and [delta]= 1 as follows.

Sample 2. 1.2271, 0.9467, 1.9800, 0.6613, 0.4526, 1.5701, 1.0414, 1.9008, 1.2378, 3.7137, 1.4012, 0.5640, 1.6704, 2.0615, 1.4360, 1.6696, 0.7443, 0.9568, 1.6306, 0.2327, 1.0024, 0.8178, 0.7704, 2.1587, 0.8460, 2.3833, 1.0702, 1.1584, 3.1261, 3.7417, 3.0401, 2.4465, 0.6874, 3.2289, 2.4881, 1.6636, 0.4313, 2.4856, 1.2428, 1.5750, 1.1564, 1.4370, 3.1483, 1.1472, 1.4889, 0.8209, 0.6693, 1.2684, 3.2650, 3.2285.

For this sample, we have a = 2.932394, [lambda] = 1.044041, and [??] = 1.106363. Using these results (obtained from Samples 1 and 2) and Theorem 10, we get [??] = 0.5384615 and [??] = 0.5586546.

6. Conclusions

We propose a class of distributions, which covers as many as ten exponentiated distributions as special cases. The problems of estimating R(t) and "P" are considered. UMVUES and MLES are derived. A comparative study of the two methods of estimation is done. The estimators of "P" are derived, which cover the cases when X and Y may follow the same, as well as, different distributions. By estimating the sampled pdf to obtain the estimators of R(t) and "P," an interrelationship between the two estimation problems is established. Simulation study is performed, and a real-data example is presented.

http://dx.doi.org/10.1155/2014/563093

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are thankful to the referee for his valuable comments.

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Ajit Chaturvedi and Anupam Pathak

Department of Statistics, University of Delhi, Delhi 110007, India

Correspondence should be addressed to Anupam Pathak; pathakanupam24@gmail.com

Received 4 June 2013; Revised 17 December 2013; Accepted 22 March 2014; Published 29 April 2014

```
TABLE 1: Simulation results for R(t).

n = 5

t         R(t)        [??](t)         [??](t)

0.9628          0.9429
-0.0014         -0.0213
0.40     0.9642     0.001071326     0.001390097
0.1211          0.1182
86.6741         92.6193

0.8576          0.8546
-0.0815         -0.0845
0.50     0.9391     0.009229477     0.008767263
0.1912          0.1524
93.9509         94.0809

0.8314          0.8338
-0.0016          7e-04
0.80     0.8330     0.004528085     0.002894221
0.2552          0.2057
94.1964         94.3942

0.729          0.7499
-0.0184         0.0025
1.00     0.7474     0.008217478     0.005415503
0.3395          0.2812
94.0502         94.2394

0.4442          0.4933
-0.087          -0.0378
1.50     0.5311     0.01260328      0.00618332
0.2803          0.2719
94.2755         94.5729

0.4971          0.5442
0.1435          0.1906
2.00     0.3535     0.02667172       0.0416806
0.3125          0.2928
94.3591         94.7634

0.2396          0.2813
0.013          0.0547
2.50     0.2266     0.001733384     0.004863281
0.1576          0.1725
94.0777         94.3206

n = 10

t         R(t)        [??](t)         [??](t)

0.9726          0.9638
0.0084         -4e - 04
0.40     0.9642    0.0003646232    0.0003069262
0.0647          0.0668
91.428          93.112

0.8789          0.8769
-0.0601         -0.0622
0.50     0.9391     0.007167494     0.006830551
0.2262          0.2086
94.1651         94.4641

0.8355          0.8374
0.0025          0.0044
0.80     0.8330     0.002485577     0.002035995
0.191          0.1725
94.4812         94.5182

0.7936          0.7994
0.0461           0.052
1.00     0.7474     0.005849085     0.005767851
0.2359          0.2142
94.5787         94.6276

0.5094           0.532
-0.0218          9e-04
1.50     0.5311     0.00520108      0.004460047
8 0.2721         0.2648
94.464          94.653

0.3349          0.3573
-0.0187         0.0037
2.00     0.3535     0.004913515     0.004757415
0.2712          0.2765
94.5513         94.7147

0.2699          0.2906
0.0433           0.064
2.50     0.2266     0.002956535     0.005265369
0.1312          0.1364
94.2939         94.3724

n = 20

t         R(t)        [??](t)         [??](t)

0.9639          0.9598
-2e - 04         -0.0044
0.40     0.9642    0.0002067265    0.0002223408
0.0555          0.0552
93.5019         93.8349

0.9282          0.9252
-0.0109         -0.0139
0.50     0.9391     0.001421963     0.001408017
0.1412          0.1371
94.2804         94.6018

0.8408          0.8417
0.0077          0.0087
0.80     0.8330     0.002223323     0.002035827
0.1796          0.1711
94.6424         94.6596

0.7279          0.7338
-0.0195         -0.0137
1.00     0.7474     0.004407818     0.003903546
0.2505          0.2407
94.9352         94.9729

0.5283          0.5392
-0.0028         0.0081
1.50     0.5311     0.004379108     0.004317541
0.26           0.2564
94.7342         94.7791

0.3429           0.354
-0.0106          5e-04
2.00     0.3535     0.00239676      0.002330965
0.1923          0.1943
94.7592         94.8151

0.2047          0.2131
-0.0219         -0.0135
2.50     0.2266     0.00116556     0.0009073532
0.1045          0.1075
94.7371         94.7634

n = 50

t         R(t)        [??](t)         [??](t)

0.9656           0.964
0.0014          -1 - 04
0.40     0.9642    0.0001467748    0.0001442984
0.0477          0.0476
94.0056         94.1243

0.9399          0.9386
8e-04          -5 - 04
0.50     0.9391    0.0005898798    0.0005745804
0.0958          0.0947
94.5567         94.6561

0.8319          0.8325
-0.0011         -5 - 04
0.80     0.8330     0.001419582     0.001364689
0.1474          0.1446
94.952          94.9534

0.7548          0.7568
0.0074          0.0093
1.00     0.7474     0.002428885     0.002383391
0.1915          0.1884
95.0586         95.0578

0.5256           0.53
-0.0055         -0.0011
1.50     0.5311     0.001922681     0.001875105
0.1747          0.1738
95.0374         95.0517

0.3485          0.3529
-0.005         -6e - 04
2.00     0.3535     0.001298761     0.001284863
0.1442          0.1448
95.0453         95.0583

0.226          0.2295
-6e - 04         0.0029
2.50     0.2266    0.0004148542    0.0004315035
0.0809          0.0817
95.0835         95.0883

TABLE 2: Simulation results for "P"

([[alpha].sub.1],              (1, 0.50)
[[alpha].sub.2])                0.6667
P
(n, m)                   [??]            [??]

0.4160          0.4071
-0.2507         -0.2595
(5,5)                 0.08305567      0.09161194
0.5225          0.5686
92.2467         91.7696

0.4219          0.4179
-0.2448         -0.2487
(10,10)               0.06910139      0.07194417
0.3560          0.3727
93.4513         93.3550

0.4507          0.4491
-0.2159         -0.2176
(15, 15)              0.05200412      0.05309595
0.2739          0.2826
93.2657         93.2813

0.4676          0.4670
-0.1990         -0.1997
(25, 25)              0.04415965       0.0446212
0.2529          0.2578
93.8839        93.87700

0.4869          0.4867
-0.1798         -0.1800
(30, 30)              0.03621819      0.03642065
0.2470          0.2511
95.2267         95.2211

([[alpha].sub.1],             (1, 0.75)
[[alpha].sub.2])              0.5714286
P
(n, m)                   [??]            [??]

0.5196          0.5198
-0.0518         -0.0516
(5,5)                 0.04060813      0.04711423
0.6963          0.7375
92.6155         91.9685

0.5324          0.5340
-0.0391         -0.0374
(10,10)               0.01062704      0.01145625
0.3524          0.3698
93.5220         93.4724

0.5601          0.5620
-0.0113         -0.0095
(15, 15)              0.01157199       0.0122409
0.4016          0.4130
93.8776         93.8404

0.5625          0.5637
-0.0090         -0.0077
(25, 25)              0.002974049     0.003065954
0.2048          0.2088
94.2551         94.2501

0.5740          0.5753
0.0026          0.0038
(30, 30)              0.003972225     0.004105375
0.243          0.2468
94.5145         94.5023

([[alpha].sub.1],             (1, 1.00)
[[alpha].sub.2])               0.500000
P
(n, m)                   [??]            [??]

0.4937          0.4925
-0.0063         -0.0075
(5,5)                 0.01738566      0.02128741
0.4996          0.5469
93.9423         93.6839

0.5028          0.5029
0.0028          0.0029
(10,10)               0.01023848      0.01128198
0.3863          0.4050
94.1638         94.1400

0.5015          0.5016
0.0015          0.0016
(15, 15)              0.00717329      0.007664479
0.3234          0.3340
94.2206         94.1996

0.4994          0.4994
-6e - 04        -6e - 04
(25, 25)              0.004487199     0.004669894
0.2579          0.2630
94.3152         94.3066

0.4999          0.4999
-1e - 04         -1e-04
(30, 30)              0.003633788     0.003756843
0.2391           0.243
95.1531         95.1467

([[alpha].sub.1],             (1, 1.25)
[[alpha].sub.2])              0.4444000
P
(n, m)                   [??]            [??]

0.4503          0.4445
0.0058           1e-04
(5,5)                 0.03447488      0.04041304
0.7179          0.7577
93.2283         92.5227

0.4477          0.4451
0.0033           6e-04
(10,10)               0.01168588      0.01284482
0.4024          0.4213
93.2875         93.2019

0.4499          0.4481
0.0054          0.0037
(15, 15)              0.002471958     0.002628193
0.1911          0.1976
94.6752         94.6696

0.4466          0.4455
0.0021           0.001
(25, 25)              0.004399884     0.004568917
0.266           0.271
95.5079         95.4975

0.447          0.4461
0.0025          0.0017
(30, 30)              0.00493606      0.005091428
0.2797          0.2841
95.3624         95.3519
```