The Bayesian approach is a suitable alternative in constructing appropriate models for observed record values because the number of these values is small. This paper provides an objective Bayesian analysis method for upper record values arising from the Rayleigh distribution. For the objective Bayesian analysis, the Fisher information matrix for unknown parameters is derived in terms of the second derivative of the log-likelihood function by using Leibniz’s rule; subsequently, objective priors are provided, resulting in proper posterior distributions. We examine if these priors are the PMPs. In a simulation study, inference results under the provided priors are compared through Monte Carlo simulations. Through real data analysis, we reveal a limitation of the appropriate confidence interval based on the maximum likelihood estimator for the scale parameter and evaluate the models under the provided priors.
Observations of survival times of objects, precipitation levels, Olympic records, or daily stock prices greater than the existing respective records, are called the upper record values. This concept was introduced by Chandler (1952). Let {
The statistical inference based on the record values has limitations due to small sample sizes, although the modelling for small samples is an important issue in statistical application. In addition, the likelihood function for unknown parameters and the predictive likelihood function respectively provided by Arnold
However, subjective Bayesian approaches cannot be properly used in situations in which little or no prior information is available. In this case, the Bayesian inference can rely on the noninformative or objective priors. The most widely used noninformative priors are the Jeffreys prior (Jeffreys, 1961) and the reference prior (Bernardo, 1979; Berger and Bernardo, 1989, 1992). In addition, the probability matching prior (PMP) introduced by Welch and Peers (1963) has gained recent popularity due to its frequentist properties.
This article provides an objective Bayesian approach based on noninformative priors to estimate the unknown parameters of the two-parameter Rayleigh distribution with the CDF
and the PDF
where
This article focuses on inference based on the objective priors to avoid the risk from inappropriate prior information and reduce the effort in obtaining sufficient prior information. To develop the method based on the objective priors, it needs to obtain a closed form of the Fisher information matrix for unknown parameters (
The rest of this paper is organized as follows. Section 2 provides the Fisher information matrix in terms of the second derivative of the log-likelihood and preferred objective priors (the Jeffreys and reference priors and the second-order PMP) for unknown parameters (
This section provides the Fisher information matrix for unknown parameters (
Let
where
is the marginal density function of
which has computational convenience compared with (
Suppose that the marginal density function (
By Leibniz’s rule, the right-hand side in (
where
Let
Then, we have
Suppose that the function (
For
because
This completes the proof.
By (
In addition, in
by taking the expectation, we have
Therefore,
by (
Let
Then, we have
By taking the expectation,
Then, the first term is zero by Remark 2 and the second term is also zero because − log (1 −
Similarly, we can obtain the following result:
The result (
Let
and
where
Therefore, the expectation of the partial derivative of the log-likelihood function is
by the relationship
By Remark 2, the Fisher information matrix for (
where
Then, the Fisher information (
from the expectations (
for
the Fisher information matrix (
where
and
where
Based on the Fisher information (
The Jeffreys prior is proportional to the square root of the determinant of the Fisher information. Therefore, the Jeffreys prior for (
Note that the Jeffreys prior may lead to some undesirable frequentist properties in the presence of nuisance parameters (Bernardo and Smith, 1994). The following theorem provides a reference prior for (
This is proved by using the algorithm provided by Berger and Bernardo (1989). We first give a proof procedure when
where
and the following proper prior is obtained:
where
and
where
When
From the above sequence of compact sets, it follows that
and
Then, the marginal reference prior for
and the reference prior for (
where
The formula for finding the second-order PMP for the multi-parameter case is provided in Peers (1965). When
where
When
where
Since the Fisher information (
Therefore, the prior distribution
The reference prior (
The following subsection investigates the property of posteriors under the proposed priors.
The posterior distribution under the Jeffreys prior (
where
By Remark 2, the resulting posterior is the same as that of Seo and Kim (2017). For comparison, we re-write the results with those based on the reference prior (
where
Seo and Kim (2017) proved that the posterior distribution (
By integrating out
In addition, because the inequality
holds, we can obtain the following result:
Therefore, the posterior distribution (
However, a Markov chain Monte Carlo (MCMC) technique should be applied to generate the MCMC samples from the marginal posterior distributions since marginal posterior distributions (
The conditional posterior density function (
By Remark 3, the MCMC samples
and
where
This section assesses how the proposed analysis method is valid through Monte Carlo simulations and real data analysis.
This subsection reports the mean squared errors (MSE) and biases of the proposed estimators, and the coverage probabilities (CPs) and average lengths (ALs) for the proposed intervals at the 0.95 level to assess their validity. The upper record values are generated from two-parameter Rayleigh distribution with
From Figures 1 and 2, we can see that the Bayes estimators under the reference prior (
In this subsection, we analyze a real data set that represents survival times in days for a group of lung cancer patients, as provided in Lawless (1982):
We can observe the following upper record values:
which have been analyzed by some authors. Soliman and Al-Aboud (2008) showed that the Rayleigh distribution with the scale parameter fits in the analysis of the observed record data. Seo and Kim (2017) applied an objective Bayesian method under the reference prior with partial information to the observed record data and showed that the proposed Bayesian model fits the observed record data well. We focus on comparing the Bayesian models under the Jeffreys prior (
Tables 1 and 2 show that the Bayes estimates based on the generated MCMC samples and the numerical results are very close to each other. In addition, the 95% HPD CrIs satisfy their PPs well. It is worth noting that the lower bound of the approximate 95% CI based on the MLE
The quality of models under the derived priors can be evaluated through posterior predictive checking. The data drawn from the fitted model, namely replications, should look similar to observed data if the model is adequate. Let
where
The replications from the model under the reference prior (
Under the provided priors (
Table 3 shows that the replications under the Jeffreys prior (
This paper provides an objective Bayesian analysis method based on the objective priors (the Jeffreys and reference priors, and the second-order PMP) for unknown parameters of the two-parameter Rayleigh distribution when the upper record values are observed. To obtain the objective priors, we derived the Fisher information matrix for unknown parameters in terms of the second derivative of the log-likelihood function using Leibniz’s rule. In the simulation study, we showed that the model under the reference prior (
Estimates and the corresponding 95% CIs and HPD CrIs for
Estimate | 5.205 | Numerical | 4.196 | 3.867 |
MCMC | 4.205 | 3.863 | ||
CI | (−7.651, 18.062) | HPD CrI | (0.906, 6.872) | (0.514, 6.684) |
PP | 0.949 | 0.951 |
CI = confidence interval; HPD = highest posterior density; CrIs = credible intervals; MCMC = Markov chain Monte Carlo; PP = posterior probability; JB = Jeffreys prior; RB = reference prior.
Estimates and the corresponding 95% CIs and HPD CrIs for
Estimate | 2.446 | Numerical | 2.835 | 3.109 |
MCMC | 2.831 | 3.113 | ||
CI | (−1.662, 6.554) | HPD CrI | (1.451, 4.584) | (1.663, 5.409) |
PP | 0.949 | 0.951 |
CI = confidence interval; HPD = highest posterior density; CrIs = credible intervals; MCMC = Markov chain Monte Carlo; PP = posterior probability; JB = Jeffreys prior; RB = reference prior.
Replications of the observed upper record values under the provided priors
1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|
7.75 | 9.52 | 10.85 | 11.96 | 12.93 | |
7.76 | 9.71 | 11.17 | 12.39 | 13.46 |