OPEN ACCESS 
PEER-REVIEWED
Inference on P(X<Y) for Extreme Values
Sudhansu S. Maiti1,
, Sudhir Murmu2
1Department of Statistics, Visva-Bharati University Santiniketan, India
2District Rural Development Agency Khunti, Jharkhand, India
Abstract
The article considers the problem of 
, when X and Y belong to independently distributed two extreme value distributions. Maximum likelihood estimate of R has been found out and the estimates assuming different distributions have been compared for complete samples. Lower confidence limits of R have been found out by Delta method and bootstrap method. The Bayes estimate of R has also been calculated using MCMC approach.
Keywords: Bayes estimate, delta method, Lower Confidence Limit, Metropolis-Hastings algorithm
American Journal of Applied Mathematics and Statistics, 2014 2 (3),
pp 121-128.
DOI: 10.12691/ajams-2-3-6
Received April 12, 2014; Revised April 24, 2014; Accepted May 03, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.Cite this article:
- Maiti, Sudhansu S., and Sudhir Murmu. "Inference on P(X<Y) for Extreme Values." American Journal of Applied Mathematics and Statistics 2.3 (2014): 121-128.
- Maiti, S. S. , & Murmu, S. (2014). Inference on P(X<Y) for Extreme Values. American Journal of Applied Mathematics and Statistics, 2(3), 121-128.
- Maiti, Sudhansu S., and Sudhir Murmu. "Inference on P(X<Y) for Extreme Values." American Journal of Applied Mathematics and Statistics 2, no. 3 (2014): 121-128.
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1. Introduction
Inference of 
is used in various applications e.g. stress-strength reliability, statistical tolerancing, measuring demand-supply system performance, measuring heritability of a genetic trait, bio-equivalence study etc. It is observed especially in military and medical sciences that the system designers, reliability practitioners and experts in medical field seek to assign high probability to the event that the system/unit remains operable at its minimum strength encountering maximum stress at that time epoch. To meet this objective, it seems reasonable to define 
with 
and 
.
Now the cumulative distribution function of 
is given by
 |
if 
are independent and identical, and the cumulative distribution function of Y is given by
 |
if 
are independent and identical.
Here we assume that 
and 
follow independent Weibull distributions with common shape parameter and the probability density functions are given by
 |
and
 |
respectively.
Then
 |
and the probability density function is
 |
and
 |
and the probability density function is
 |
Hence
 |
 | (1.1) |
If k=1 i.e. in case of stress-strength reliability for component only, 
and its inferential aspects have been studied in McCool (1991) and Mukherjee and Maiti (1998), if 
also, then 
. It is just exponential case and the case is studied by a host of authors. If 
only, then the situation reduces to exponential case of a system with 
identical components and then
 |
In this article, we have attempted estimation problem of 
for Weibull family of distributions. We have found out maximum likelihood estimate (MLE) of 
for complete samples. An emphasis has been given for finding out lower confidence limits (lcls) as this is the one of practical importance-practioners want to assert that the system is at least attained this limit. We use delta method and bootstrap method to find out lcls. We also derive Bayes estimate of 
using MCMC approach.
The paper is organized as follows. Section 2 is devoted for finding out MLE and lcls of 
. Bayes estimation of 
has been discussed in section 3. Simulation results have been discussed in section 4. Data analysis has been presented in Section 5 and section 6 concludes.
2. Inference about R
2.1. Maximum Likelihood Estimation of RTo compute the MLE of 
, we have to obtain the MLEs of 
and 
. Suppose 
is a random sample from 
and 
is a random sample from 
. Hence, the underlying log-likelihood function is
 |
Then the MLE of 
is to be obtained from the relation
 |
and that of 
is from
 |
and the MLE of 
is to be obtained by solving the equation
 |
An estimate 
of 
is to be obtained from 
replacing 
and 
by 
and 
respectively.
We have already mentioned that when 
, it reduces to exponential case. We will concentrate further inference in this situation only. Under such situation, the estimates of 
and 
are of the form
 |
 |
respectively.
Let us write
 |
Where
 |
 |
Now, the asymptotic variance-covariance matrix of 
is given by
 |
Let 
, with 
yield the asymptotic variance of 
as 
Here 
and
 |
Assuming 
as a standard normal variate a lower confidence bound to 
can be constructed.
Remark 2.1 
is to be obtained by replacing the parameters by their ML estimators.
In this subsection, we propose to use two lower confidence limits based on the parametric bootstrap methods; (i) percentile bootstrap method (we call it from now on as Boot-p) based on the idea of Efron (1982, 1988), (ii) bootstrap-t method (we refer it as Boot-t from now on) based on the idea of Hall (1988). We illustrate briefly how to estimate lower confidence limits of 
using both methods.
Boot-p Methods:
Step 1: From the sample 
and 
compute 
and 
.
Step 2: Using 
generate a bootstrap sample 
and similarly using 
generate a bootstrap sample 
. Based on 
and 
compute the bootstrap estimate of 
using (1), say
Step 3 : Repeat step 2, NBOOT times.
Step 4: Let 
be the cumulative distribution function of 
Define 
for a given 
. The approximate 
lower confidence limits of
is given by 
.
Bootstrap-t Methods:
Step 1: From the sample 
and 
compute 
and 
.
Step 2: Using 
generate a bootstrap sample 
and similarly using 
generate a bootstrap sample 
. Based on 
and 
compute the bootstrap estimate of 
using (1), say
and the following statistic:
 |
Compute 
using Remark 2.1.
Step 3 : Repeat step 2, NBOOT times.
Step 4: From the NBOOT 
values obtained, determine the lower bound of the 
confidence limits of 
as follows: Let 
be the cumulative distribution function of 
. For a given 
define
 |
Here also, 
can be computed as mentioned in Remark 2.1. The approximate 
lower confidence limit of 
is given by 
.
3. Bayes Estimation of R
In this section, we obtain the Bayes estimation of 
under assumption that the shape parameters 
and 
are random variables. We mainly obtain the Bayes estimate of 
under the squared error loss by Gibbs sampling technique. It is assumed that 
and 
have independent gamma priors with the parameter 
~ 
, 
~ 
and 
~ 
. Based on the above assumptions, we have the likelihood function of the observed data as
 |
Therefore, the joint density of the data, 
and 
can be obtained as
 |
where 
is the prior distribution. Therefore, the joint posterior density of 
and 
given data is
 |
Since these equations cannot be obtained analytically, we adopt the Gibbs sampling technique to compute the Bayes estimate of 
The posterior pdfs of 
and 
are as follows:
 |
and
 |
To generate random numbers from these distributions, we use the Metropolis-Hastings method with appropriate proposal distributions. Therefore, the algorithm of Gibbs sampling is as follows:
Step 1: Start with an initial guess
.
Step 2: Set
.
Step 3: Using Metropolis-Hastings, generate 
from 
with appropriate proposal distribution.
Step 4: Using Metropolis-Hastings, generate 
from 
with appropriate proposal distribution.
Step 5: Using Metropolis-Hastings, 
from 
with appropriate proposal distribution.
Step 6: Compute 
from the expression of
.
Step 7: Set 
Step 8: Repeat step 3-6, 
times.
Note that in steps 3-5, we use the Metropolis-Hastings algorithm with 
proposal distribution as follows:
Let 
.
Generate 
from proposal distribution 
Let 
Accept 
with probability 
or accept 
with probability 
In case of exponential distributions (i.e.
), we have the posterior pdfs of 
and 
are as follows:
 |
 |
Now the appropriate posterior mean and posterior variance of 
become
 |
and
respectively.
4. Simulation and discussion
In this section we present some results based on Monte Carlo simulations to compare the performance of 
for different values of 
Also the values of 
and 
are mentioned under each table. All computations were performed using R-software and these are available on request from the corresponding author. We consider to draw inference on 
when the baseline distribution of extended distribution is known. All the results are based on 1000 replications.
We report the average biases and mean squared errors (MSEs) over 1000 replications. We also compute the 95% lower confidence limit (lcl) of 
based on asymptotic distribution of
, using Boot-p and Boot-t methods. The bootstrap lcls are obtained using 1000 bootstrap replications in both cases. All the results are reported in Table 7, Table 8, Table 9.
Some of the points are quite clear from this experiment. The performances of the MLEs are quite satisfactory in terms of biases and MSEs. It is observed that when 
increases then MSEs decreases for low value of
but increase for high value of 
High value of 
is underestimated slightly (i.e. bias is negative), where as low value of 
is overestimated generally. All lower confidence bounds are estimated satisfactorily. Particularly, Boot-t lcls perform very well. Based on all these, we recommend using the parametric percentile bootstrap lcls, particularly Boot-t lcls.
We do not have any prior information on
, therefore, we prefer to use the non-informative prior to compute different Bayes estimates. Since the non-informative prior, i.e. 
provides prior distributions which are not proper, we adopt the suggestion of Congdon (2001, pp 20), i.e. choose 
which are almost like Jeffrey’s prior, but they are proper. Under the same prior distributions, we compute Bayes estimate of 
and 
and have approximate Bayes estimates of 
under squared error loss function. To generate random observations from the posterior distributions of 
and 
we use the Metropolis-Hastings method with proposal distributions 
and 
respectively. The algorithms of Gibbs sampling is described in section 3. The burn in sample in each case is taken 5000. The results are reported in Table 10, Table 11, Table 12 with the change of 
the averages biases and the MSEs do not show clear picture. Therefore, if we do not have prior information about 
and 
then using Bayes estimates we may not gain much. Since the MLE is consistent and it can be used for constructing lower confidence limits also, we recommend using MLEs in this case.
5. Simulated Data Analysis
In this section we present the analysis of simulated data. The data set are presented in Table 1, Table 3 and Table 5. The results are summarized in Table 2, Table 4 and Table 6.
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View current table in a new windowThe true values of R for the simulated data sets in Table 1, Table 3 and Table 5 are 0.8, 0.5333333 and 0.3324675 respectively (see Table 7, Table 8 and Table 9 for
, 
). We observe that, in all the cases, the MLE of R is very close to the true value. One should note that in real life data situation, the true value of R is not possible to get and hence comparison of biases and MSEs are not possible. However, in the present scenario, one can get almost the true picture from the simulation results presented in section 4. From Table 7, Table 8 and Table 9 , it is ensured that the MLEs of R has minimum biases and MSEs comparing the values corresponding to 
, 
. It is evident from the analysis of data sets and the results presented in Table 2, Table 4 and Table 6 that the MLE of R is fairly good compared to the Bayes estimate – the fact was also reported in simulation study. Some improvements in the case of Bayes estimate of R may be expected if the appropriate prior distributions are selected when it is available besides non-informative prior. In all the data sets, lcl’s in Bootstrap-t are better from maximum coverage probability point of view.
6. Concluding Remark
In this article, we have discussed inference problem of 
with 
and 
. The 
and 
distributions have been considered Weibull. We have considered maximum likelihood estimate and Bayes estimate of 
Comparing these two, we recommend to use MLE for 
An emphasis has been given on lower confidence limits as this is the one of practical importance-practitioners want to assert that the system is at least attained this limit. To construct lcls, we consider Delta method and two bootstrap methods -percentile (Boot-p) and bootstrap-t (Boot-t). We recommend using the parametric bootstrap lcls, particularly Boot-t lcls.
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View current table in a new windowAcknowledgement
The authors are thankful to the referee for valuable comments, which led to an improvement in the presentation of this paper.
References
| [1] | Congdon, P., Bayesian Statistical Modeling, John Wiley, , 2001. | ||
 In article | |||
| [2] | Efrom, B., The jackknife, the bootstrap and other resampling plans, In CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Phiadelphia, PA, 3, 1982. | ||
| In article | |||
| [3] | Efron, B., Discussion: Theoritical comparison of bootstrap intervals, The Annals of Statistics, 16, 969-972, 1988. | ||
| In article | CrossRef | ||
| [4] | Hall, P. Theoritical comparison of bootstrap confidence intervals, Annals of Statistics, 16, 927-953, 1988. | ||
| In article | CrossRef | ||
| [5] | McCool, I.J., Inference o P(X<Y) in the Weibull case, Commun. Statist.-Simula. and Comp., 20, 129-148, 1991. | ||
| In article | CrossRef | ||
| [6] | Mukherjee, S.P. and Maiti, S.S., Stress-Strength reliability in the Weibull case, Frontiers in reliability, World Scientific, 4, 231-248, 1998. | ||
| In article | |||
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