Bayesian Quantiles of Extremes

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Bayesian Quantiles of Extremes a

Branko Miladinovic & Chris P. Tsokos

b

a

Center for Evidence Based Medicine and Health Outcomes Research, University of South Florida, Tampa, Florida, USA b

Department of Mathematics and Statistics, University of South Florida, Tampa, Florida, USA Version of record first published: 10 Aug 2012

To cite this article: Branko Miladinovic & Chris P. Tsokos (2012): Bayesian Quantiles of Extremes, Journal of Statistical Theory and Practice, 6:3, 566-579 To link to this article: http://dx.doi.org/10.1080/15598608.2012.698206

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Journal of Statistical Theory and Practice, 6:566–579, 2012 Copyright © Grace Scientific Publishing, LLC ISSN: 1559-8608 print / 1559-8616 online DOI: 10.1080/15598608.2012.698206

Bayesian Quantiles of Extremes BRANKO MILADINOVIC1 AND CHRIS P. TSOKOS2 Downloaded by [University of South Florida], [Branko Miladinovic] at 07:41 10 August 2012

1

Center for Evidence Based Medicine and Health Outcomes Research, University of South Florida, Tampa, Florida, USA 2 Department of Mathematics and Statistics, University of South Florida, Tampa, Florida, USA Extreme value distributions are increasingly being applied in biomedical literature to model unusual behavior or rare events. Two popular methods that are used to estimate the location and scale parameters of the type I extreme value (or Gumbel) distribution, namely, the empirical distribution function and the method of moments, are not optimal, especially for small samples. Additionally, even with the more robust maximum likelihood method, it is difficult to make inferences regarding outcomes based on estimates of location and scale parameters alone. Quantile modeling has been advocated in statistical literature as an intuitive and comprehensive approach to inferential statistics. We derive Bayesian estimates of the Gumbel quantile function by utilizing the Jeffreys noninformative prior and Lindley approximation procedure. The advantage of this approach is that it utilizes information on the prior distribution of parameters, while making minimal impact on the estimated posterior distribution. The Bayesian and maximum likelihood estimates are compared using numerical simulation. Numerical results indicate that Bayesian quantile estimates are closer to the true quantiles than their maximum likelihood counterparts. We illustrate the method by applying the estimates to published extreme data from the analysis of streak artifacts on computed tomography (CT) images. AMS Subject Classification: 62F15. Keywords: Bayesian inference; Extreme value distribution; Lindley procedure; noninformative prior; Quantiles.

1. Introduction Extreme value distributions (EVDs) have been used to model various problems in engineering and bioinformatics. The applications range from alignment of biological sequences (Bastien et al. 2008), postoperative breathing patterns (Leong et al. 2002), streak artifacts on computed tomography (CT) images (Imai et al. 2002; 2007; 2009), differential expression in microarrays (Ivanek et al. 2009), record linkage (Sariyar et al. 2011), probabilistic patient monitoring (Hugueny et al. 2010), and spatial scan statistics (Abrams et al. 2010), among others. A comprehensive theory of EVDs was developed by Gumbel (1958), following the pioneering works by Fisher and Tippet (1928). Briefly, given Received July 28, 2011; accepted March 15, 2012. Address correspondence to: Branko Miladinovic, Center for Evidence Based Medicine and Health Outcomes Research, University of South Florida, Tampa, FL 33612, USA. Email: bmiladin@ health.usf.edu

566

Bayesian Quantiles of Extremes

567

Mn = max/min(T1 , ..., Tn ) for a sequence of independent and identically distributed random variables T1 , ..., Tn , classical EVD theory is concerned with the limiting distribution of Mn as n approaches infinity. Its normalized version states that there exist sequences of constants {an } > 0 and {bn } such that

Downloaded by [University of South Florida], [Branko Miladinovic] at 07:41 10 August 2012

P

Mn − bn ≤ t → F(t) as n → ∞ an

and that F(x) must belong to one of three families of distributions, namely, Gumbel, Frechet, or Weibull. The Gumbel cumulative distribution function has the form t−μ F(t) = exp −exp − σ

(1)

where μ and σ are the location and scale parameters, respectively. Haan and Ferreira (2006) provide an excellent introductory text to the subject area. Based on record values, Ahsanullah (1990; 1991) obtained the maximum likelihood (ML), best linear invariant (BLI), and minimum variance unbiased (MVU) estimators of the Gumbel location and scale parameters. Ali Mousa et al. (2001) obtained the Bayesian estimators of the same under the Jeffreys noninformative prior and found them to be superior to the MVU and BLI estimates. Following a brief introduction to the Gumbel distribution, in this article we derive Bayesian estimates of the Gumbel quantile function under the Jeffreys noninformative prior, which has the advantage of freeing the researcher from the burden of having to specify a subjective prior. Our approach is motivated in part by works of Parzen, who has advocated using quantiles to integrate statistical approaches to data modeling (Parzen 1979; 2004; 2009).

2. The Gumbel model For the Gumbel model, the probability distribution function (PDF) of the extrema (minima or maxima) T is given by f (t) =

1 dF(t) = e− dt σ

t−μ t−μ − σ σ −e

, −∞ < t < ∞,

−∞ < μ < ∞,

σ >0

(2)

Two popular but inefficient methods of estimating parameters of the Gumbel distribution have been applied in practice, namely, the empirical distribution plot and estimation of moments of the Gumbel distribution. In the first method, the Gumbel distribution location ˆ parameter μ and scale parameter σ are estimated from the empirical function F(t) by fitting a line ˆ −ln(−ln(F(t))) =

μˆ 1 t− σˆ σˆ

to the ordered data (Kimball 1960). The second method calculates the moments of the Gumbel distribution to obtain the following parameter estimates: √ s 6 σˆ = π

568

B. Miladinovic and C. P. Tsokos μˆ = ¯t − 0.5772σˆ

where ¯t is the sample mean and s is the sample standard deviation. The asymptotically efficient and generally robust is the method of maximum likelihood (ML), which maximizes the Gumbel likelihood function L(μ, σ ) given by


L(μ, σ ) = σ

−n

exp −

n ti − μ

σ

i=1

−

n i=1

ti − μ exp − σ

and its logarithmic form is n ti − μ

LogL(μ, σ ) = −nlnσ −

σ

i=1

−

n

e−(

ti −μ σ )

(3)

i=1

The maximum likelihood (ML) estimates for μ and σ can be obtained by solving the following equations: n

σˆ +

ti e−ti /σˆ

i=1 n

e−ti /σˆ

= ¯t

(4)

i=1

and

1 −ti /σˆ e μˆ = −σˆ ln n i=1 n

(5)

Equations (4) and (5) are not analytically tractable and must be solved numerically. The corresponding (1–α) 100% confidence intervals for μ and σ are given by μˆ ± z1− α2

Var(μ); ˆ σˆ ± z1− α2 Var(σˆ )

where Var(μ) ˆ =

σˆ 2 (γ 2 − 2γ + 2 + η(2, 2)) n((1 + η(2, 2))

Var(σˆ ) =

σˆ 2 n(1 + η(2, 2))

γ = 0.57722 is the Euler’s constant and 1 η(p, q) = (p)

∞ 0

tp−1 e−qt dt 1 − e−t


569

Furthermore, the Anderson–Darling (AD) goodness-of-fit test can be used for the Gumbel distribution for a pair of ML parameter estimates (Shorak and Wellner 1986) and is superior to using the line of best fit and linear correlation coefficient r under the empirical distribution method. The AD test is a modification of the Kolmogorov–Smirnov test that gives more weight to the tails. However, unlike the Kolmogorov–Smirnov test, which is distribution free, it makes use of the specific distribution being tested and allows for a more sensitive test. The p-value is obtained using the test statistic 1 (2i − 1)(ln(F(ti )) + ln(1 − F(tn−i+1 ))) n i=1


n

A2 = −n −

where F(t) is the CDF of the Gumbel distribution and ti are the ordered data. A small sample adjustment to A2 is given by A2ADJ

0.2 =A 1+ √ n 2

A test statistic of 0.461 corresponds to the significance level of α = 0.05, and the lower the test statistic, the less likely we are to reject the null hypothesis that the Gumbel distribution provides a good fit.

The Quantile Function In general, for a continuous random variable T with a distribution function FT (t) = P(T ≤ t) the αth quantile of T is defined to be QT (α) = FT−1 (t) = inf(t|FT (t) ≥ α, 0 ≤ α ≤ 1)

(6)

By taking the natural logarithm of both sides of Eq. (1) and solving for t we obtain the expression for the quantile function Q(α) Q(α) = μ − σ (ln(− ln(α))

(7)

The Fisher information matrix, which is defined as

∂2

2 ln L I(μ, σ ) = −E

∂μ∂ 2 ln L ∂μ∂σ

∂ 2 ln L

∂μ∂σ

∂2 ln L

∂σ 2

can be used to derive the ML estimates of the quantile function Q(α). Using log L in Eq. (3), we obtain

n

− σn2 (1 − γ )

I(μ, σ ) =

−n σ 2 (8) n 2 (1 − γ ) σ 2 {γ − 2γ + 2 + η(2, 2)}

σ2 By the invariance property, the ML estimate of the quantile function Q(α) and the corresponding (1 – α)100% confidence interval are given by

570

B. Miladinovic and C. P. Tsokos ˆ ML = μˆ − σˆ (ln(− ln(α)) Q ˆ ML ) ˆ ML ± z1− α Var(Q Q 2

(9)


where ˆ ML ) = Var(Q

σˆ 2 (γ 2 − 2γ + 2 + η(2, 2)) n((1 + η(2, 2))

+ (−ln(−ln(α)))2

σˆ 2 (1 − γ ) σˆ 2 − 2ln(−ln(α)) n(1 + η(2, 2)) n(1 + η(2, 2))

and γ and η(p, q) as previously defined. Our objective is to model the quantile function for the Gumbel model in both the frequentist and Bayesian settings. We develop the Bayesian estimate of the quantile function based on the Jeffreys noninformative prior in order to circumvent the issue of having to specify a subjective prior and introduce Lindley’s procedure to obtain closed-form approximations. We perform a numerical study to compare the estimates using the mean integrated ˆ squared error (MISE), which for the quantile function Q(α) and its estimate Q(α) is defined as ˆ MISE(Q(α)) =

α

2 ˆ E(Q(α) − Q(α)) dα =

α

ˆ Var(Q(α)) dα +

2 ˆ bias(Q(α)) dα (10) α

An advantage of MISE is that it can be split into the integrated variance and integrated square bias components, so that the change in each can be studied given a particular estimate.

3. Bayesian Approach to the Gumbel Model In the Bayesian approach we regard μ and σ behaving as random variables with a joint PDF π (μ, σ ). We investigate the Bayesian point estimator of Q(α) for √ the Jeffreys noninformative prior, which chooses the prior π (μ, σ ) to be proportional to det(I(μ, σ )), where I(μ, σ ) is the Fisher information matrix defined in (8). The result det(I(μ, σ )) =

K σ4

implies that the Jeffreys noninformative prior π (μ, σ ) is proportional to

1 . σ2

Posterior Distribution The posterior probability density function of (μ, σ ) given t1 , ..., tn is given by L(μ, σ |t1 , ..., tn )π (μ, σ ) −∞ L(μ, σ |t1 , t2 , ..., tn )π (μ, σ ) dμdσ

π (μ, σ |t1 , t2 , ..., tn ) = ∞ ∞ 0


571

where L(μ, σ |t1 , ..., tn ) is given by Eq. (3). We first compute the marginal probability density function, that is,

∞


∞

∞

∞

= μ

n

ti , we obtain

i=1

L(μ, σ |t1 , ..., tn )π (μ, σ ) dμdσ

σ −n−2

0

Let u = e σ , and a =

n

σ > 0, and letting x =

1 , σ2

−∞

0

L(μ, σ |t1 , ..., tn )π (μ, σ ) dμ dσ

−∞

0

Using the prior π (μ, σ ) =

∞

+∞

−ti /σ

e−x/σ e(nμ/σ ) e−e(μ/θ) i=1 e n

dμdσ

(11)

−∞

e−(ti /σ ) ; then expression (11) can be written as

i=1

∞

∞

−∞

0

L(μ, σ |t1 , ..., tn )π (μ, σ ) dμdσ

∞ =

σ

−n−1 −x/σ

∞

e

un−1 e−au dudσ

0

0

∞

= (n)

σ −n−1 e−x/σ a−n dσ

0

∞

= (n)

vn−1 e−xv

n

0

where x =

n

n−1

v 0

e−ti v

dv

i=1 ∞

= (n)

n

n

n −v(ti +¯t)

e

dv

i=1

ti = n¯t.

i=1

Bayesian Estimation Under Jeffreys Noninformative Prior The Bayes estimate of Q(α) under squared error loss is given by ˆ B = E(Q(α)|t1 , t2 , ..., tn ) = Q

∞ 0

∞

−∞

[μ − σ ln(− ln α)]L(μ, σ |t1 , t2 , ..., tn )π (μ, σ ) dμdσ

or ˆB = Q

∞ ∞ 0

−∞ [μ

− σ ln(− ln α)]L(μ, σ |t1 , t2 , ..., tn )π (μ, σ ) dμdσ n ∞ −(t +¯t)v n (n) 0 vn−1 e i dv i=1

(12)

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B. Miladinovic and C. P. Tsokos

Proceeding as we did before for obtaining the marginal probability distribution, we can write E(μ|t) =

∞

vn−2 e−xv

0

∞

(ln u)un−1 eau dudv

(13)

0


where a=

n

e−ti v

i=1

and E(σ |t) = (n)

∞

n−2

v

n

0

n −v(ti +¯t)

e

dv

(14)

i=1

Hence, ∞ vn−2 e−xv 0 (ln u)un−1 e−au dudv ∞ n n (n) 0 vn−1 i=1 e−v(ti +¯t) dv

∞ E(Q(α)|t1 , ..., tn ) =

0

∞

−

n n−2 e−v(ti +¯t) dv 0 v ln(− ln α) ∞ n−1 n e−v(ti +¯t) n dv i=1 0 v

where a=

n

e−ti v

i=1

and n

¯t =

ti

i=1

n

Lindley’s approximation method is used to evaluate the expectation just given. ˆ ML is the ML estimate of the true quantile function Q, then the Bayesian If Q approximate estimate is given by ˆB = Q ˆ ML + P Q

(15)

1 2 2 + L03 u2 σ22 + L21 u2 σ11 σ22 + L12 σ22 σ11 ) P = p2 u2 σ22 + (L30 u1 σ11 2

(16)

where


573

reflects the Bayesian “correction” of the ML estimate. The full details of the derivations are described in the Appendix.


Numerical Analysis In this section we present a numerical comparison between the ML and Bayesian estimates. We use MISE as the measure of performance of each estimate. All the analysis was performed using the R evd package and Maple computational software. Our numerical simulation was conducted in the following manner: 1. We assumed that the location parameter followed N(25,1) and simulated data from the Gumbel distribution with the scale parameter σ equal to 1, 2, and 4, respectively, in order to see what effects the increase in variance has on our estimates. 2. We generated n (n = 10, 20, 30, 50) observations from the Gumbel PDF and calculated both the ML and Bayes estimates of the quantile function. 3. For comparison purposes, we calculated the difference of MISE between the true quantile function and the corresponding ML and Bayes estimates. The numerical results are given in Table 1. By varying the sample size of the quantile model ˆ B ) decrease. The Bayesian estimate is ˆ ML ) and MISE(Q from n = 10 to n = 50, both MISE(Q consistently lower for both MISE and integrated square bias. We note that for a number of ˆ B ) and that the results in Table 1 ˆ ML ) was lower than MISE(Q individual simulations MISE(Q are the average of 500 simulations, with standard errors (SE) provided. Table 1 shows that the percent of times ML estimates have lower MISE increases as sample size increases from n = 10 to n = 50. This behavior is consistent as we increase σ and n, except that we notice a significant improvement in the ML estimate. Also, the average difference between the ˆ ML ), decreases, indicating that the two estimates ˆ B ) – MISE(Q two MISE, namely, MISE(Q are asymptotically equivalent. The percent of lower individual MISE for ML estimates is higher than for Bayes estimates with the increase in sample size. However, as an average of 500 simulations the Bayesian estimate is closer to the true quantile function than its ML counterpart, with both less integrated variance and integrated square bias.

Table 1 MISE(SE) and Bias2 under ML and Bayesian estimates of the quantile function Q(α) n

μ, σ

10 20 30 50 10 20 30 50 10 20 30 50

N(25,1), 1 N(25,1), 1 N(25,1), 1 N(25,1), 1 N(25,1), 2 N(25,1), 2 N(25,1), 2 N(25,1), 2 N(25,1), 4 N(25,1), 4 N(25,1), 4 N(25,1), 4

MISE(ML)(SE) Bias2 (ML) MISE(Bay)(SE) Bias2 (Bay) % ML < Bay 0.434 (0.002) 0.143 (0.003) 0.061 (0.001) 0.032 (0.0005) 1.072 (0.003) 0.832 (0.005) 0.112 (0.001) 0.076 (0.0009) 1.282 (0.004) 0.952 (0.004) 0.461 (0.003) 0.172 (0.001)

0.116 0.084 0.052 0.032 0.232 0.192 0.104 0.048 0.68 0.432 0.36 0.224

0.348 (0.001) 0.081 (0.001) 0.024 (0.007) 0.018 (0.0006) 0.95 (0.001) 0.742 (0.008) 0.07 (0.001) 0.068 (0.007) 1.129 (0.003) 0.817 (0.003) 0.375 (0.003) 0.135 (0.002)

0.108 0.069 0.048 0.024 0.216 0.168 0.088 0.04 0.664 0.416 0.34 0.192

4.4 7.2 10.7 13.3 5.1 7.1 11.2 14.2 4.9 8.8 11.1 15.8

574

B. Miladinovic and C. P. Tsokos Table 2 Largest differences between adjacent CT values


Order

Largest difference

Order

Largest difference

143 151 155 164 167 168 176 184 184 185 186 187 187 187 188 197 204 206 212 214

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

218 218 221 224 235 236 238 239 244 245 256 263 275 278 286 301 303 308 363 382

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Table 3 Twelve randomly selected largest distances between adjacent CT values Order

Largest difference

Order

Largest difference

151 164 168 184 185 187

7 8 9 10 11 12

206 224 238 275 278 382

1 2 3 4 5 6

Table 4 Parameter estimates under three estimation methods Dataset Full (n = 40) Subsample (n = 12)

μED , σED

μMM , σMM

μML , σML

185.32, 67.28 186.23, 66.65

199.63, 43.12 190.63, 51.17

199.78, 41.46 193.17, 42.17

575



Figure 1. Maximum likelihood and Bayesian estimates of the quantile function for the full (Table 2) and subsample (Table 3) data sets (color figure available online).

Example Computed tomography (CT) scans use x-rays to make detailed pictures of structures inside of the body. Since the radiation doses of CT examinations are relatively high compared with


576

B. Miladinovic and C. P. Tsokos

those of ordinary x-rays, it is of interest to find the optimal dose while preserving the quality of the image. The quality is measured by the amount of noise (density of streak artifacts) on the CT image. Imai et al. (2007; 2009) studied the largest differences between 40 adjacent CT values as part of the imaging quality study in order to establish which dose corresponds to the best image quality. Table 2 presents the full ordered data set. The adjusted AD goodness-of-fit test statistics for the Gumbel ML estimate was A2ADJ = 0.22, suggesting a good fit at 0.05 level of significance. Since the simulation showed that the Bayesian estimate performed better for small samples and to illustrate our method, we randomly selected a small subsample of n = 12 data points from the reported data set (Table 3). The small sample adjusted AD goodness-of-fit test statistics for the Gumbel ML estimate was A2ADJ = 0.34, suggesting a good fit at 0.05 level of significance. In Table 4 we present parameter estimates based on three estimation methods (empirical CDF, method of moments, and maximum likelihood) for both the full and subsample data sets. The location estimates are relatively close to each other; however, there is significant variation in the scale parameter estimates, with the smallest being the ML estimate. Figure 1a presents the estimates of quantile functions under the two methods and the full data set, and suggests that the Bayesian quantile estimates are more conservative for the lower quantiles and less conservative for the upper quantiles. As suggested by the simulation, the difference is more pronounced for the small sample size case, as shown in Figure 1b.

4. Conclusion The numerical simulation indicates that the Bayes estimate under the noninformative prior is closer to the true quantile function than its ML counterpart over 500 simulations. We note that in some individual instances ML estimates produced smaller MISE values than Bayesian and that an increase in the sample size of the simulated Gumbel data resulted in the improvement of both. When we increased the variance of the simulated Gumbel data from 1 to 2 to 4, we noticed an increase in MISE, but also improvement in the ML estimate. A larger percentage of individual MISE for ML estimates was lower than for Bayes with the increase in sample size. We conclude that the ML and Bayes estimates of the Gumbel quantile function get asymptotically closer as sample size increases. However, because the Bayesian approach incorporates the Gumbel parameter uncertainty and performs better for small samples, we recommend it should be used in practice.

References Abrams, A. M., K. Kleinman, and M. Kulldorff. 2010. Gumbel based p-value approximations for spatial scan statistics. Int. J. Health Geogr. 9; 61. Ahsanullah, M. 1990. Estimation of the parameters of the Gumbel distribution based on the m record values. Comput. Stat. Qu., 6, 231–239. Ahsanullah, M. 1991. Inference and prediction of the Gumbel distribution based on record values. Pakistan. J. Stat., 7(3)B, 53–62. Ali Mousa, M. A., Z. F. Jaheen, and A. A. Ahmad. 2001. Bayesian estimation, prediction, and characterization for the Gumbel model based on records. Statistics, 36(1), 65–74. Bastien, O., and E. Marechel. 2008. Evolution of biological sequences implies and extreme value distribution of type I for both global and local pairwise alignment scores. BMC Bioinformatics, 9, 332. Fisher, R. A., and L. H. C. Tippett. 1928. Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Cambridge Philosophical Society W24, 180–290.



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Gumbel, E. J. 1958. Statistics of extremes. New York, Columbia University Press. Haan, L. D., and A. Ferreira. 2006. Extreme value theory: An introduction. Springer Series in Operations Research. New York, Springer. Hugueny, S., D. A. Clifton, and L., Tarassenko. 2010. Probabilistic patient monitoring with multivariate, multimodal extreme value theory. Commun. Comput. and Information Sci. Invited article, from IEEE Biomedical Engineering Systems and Technologies. 127, 199–211. Imai, K., et al. 2002. A detection method for streak artifacts and radiological noise in a non-uniform region in a CT image. Phys. Med., 26(3), 157–165. Imai, K., et al. 2007. Analysis of streak artifacts on CT images using statistics of extremes. Br. J. Radiol., 80(959), 911–918. Imai, K., et al. 2009. Statistical characteristics of streak artifacts on CT images: Relationship between streak artifacts and mA s values. Med. Phys., 36(2), 492–499. Ivanek, R., et al., 2008. Extreme value theory in analysis of differential expression in microarrays where either only up- or down-regulated genes are relevant or expected. Genet. Res. (Cambr.) 90(4), 347–361. Kimball B. F., 1960. On the choice plotting positions on probability paper. JASA, 55(291), 546–560. Leong, Y. P., J. W. Sleigh, and J. M. Torrance. 2002. Extreme value theory applied to postoperative breathing patterns. Br. J. Anaesth., 88(1), 61–64. Parzen, E., 1979. Nonparametric statistical data modeling. JASA, 74, 105–131. Parzen, E. 2004. Quantile probability and statistical data modeling. Stat. Sci., 19, 652–662. Parzen, E. 2009. Quantiles, conditional quantiles, confidence quantiles for p, logodds(p). Commun. Stat. Theory Methods, 38, 3048–3058. Sariyar, M., A. Borg, and K. Pommerening. 2011. Controlling false match rates in record linkage using extreme value theory. J. Biomed. Inform. 44(4), 648–654. Shorak, G. R. and J. A. Wellner. 1986. Empirical processes with applications to statistics. New York, Wiley.

Appendix The Lindley Approximation Lindley’s approximation method evaluates ratios of integrals of the type I=

L(θ) dθ θ u(θ )v(θ )e L(θ) dθ θ v(θ )e

where θ = (θ1 , θ2 , ..., θk ) is a vector of parameters. Note that I is the posterior expectation given the extrema, for a prior v(θ ). Denote by of u(θ) u1 =

∂u ∂u u2 = ∂θ1 ∂θ2

u11 =

∂ 2u ∂ 2u u22 = 2 2 ∂θ1 ∂θ2

p = π (θ1 , θ2 ) p1 =

∂p , ∂θ1

p2 =

∂p ∂θ2

578

B. Miladinovic and C. P. Tsokos L20 =

∂ 2L , ∂θ12

L02 =

∂ 2L ∂θ22

L30 =

∂ 3L , ∂θ13

L03 =

∂ 3L ∂θ23


and σ11 = −

1 1 and σ22 = − L20 L02

the quantile function for the Gumbel model given by Q(α) = μ − bσ = u(μ, σ ) where θ1 = μ and θ2 = σ . Also, u1 = 1 and u2 = −b where b = ln(− ln α) where u11 = 0 and u22 = 0. Thus, we can write 1 σ2

p(θ1 , θ2 ) = π (μ, σ ) = and p1 = 0 and p2 = −

2 σ3

Given the Gumbel log-likelihood function ln L = −n ln σ −

n ti − μ i=1

σ

−

n

e−

ti −μ σ

i=1

then, n n 1 − ∂ ln L = − e ∂μ σ σ i=1

ti −μ σ

and L2,0 =

n ∂ 2 ln L 1 − = − e ∂μ2 σ 2 i=1

ti −μ σ

Also, n n 1 − ( tiσ−μ) ∂ ln L n (ti − μ) − e (ti − μ) =− + ∂σ σ σ2 σ 2 i=1 i=1

and


L0,2

n

∂ 2 ln L n 1 = = 2− 3 2 ∂σ σ σ

−

2(ti − μ) 1 − e

(ti −μ) σ

i=1

579

1 + 4 σ

n

−(

e

ti −μ σ )

(ti − μ)

2

i=1

We proceed to find L3,0 and L0,3 , that is,


L3,0 =

n ∂ 3 ln L 1 − = − e ∂μ3 σ 3 i=1

ti −μ σ

and L0,3

n ∂ 3 ln L −2n = = + 6 (t − μ) 1 − e− i ∂σ 3 σ3 i=1 +6

n

−

e

(ti −μ) σ

(ti − μ)

2

i=1

(ti −μ) σ

n 1 −1 e− 5 σ i=1

1 σ4

(ti −μ) σ

(ti − μ)

3

1 σ6

Also, L21

∂ = ∂σ

∂ 2 ln L ∂μ2

=

n

−(

e

ti −μ σ )

i=1

n t −μ 2 1 − ( iσ ) − (ti − μ)e 3 σ σ4 i=1

and L12

∂ = ∂μ

∂ 2 ln L ∂σ 2

or L12

2 = 3 σ

n−

n i=1

−

e

ti −μ σ

+

n 4 (ti − μ)e− σ 4 i=1

ti −μ σ

−

n 1 (ti − μ)2 e− σ 5 i=1

ti −μ σ

For the pair of parameter estimates (μ, ˆ σˆ ) = (θˆ1 , θˆ2 ), 1 E(u(θ )|t) = u(θˆ1 , θˆ2 ) + (u11 σ11 + u22 σ22 ) + p1 u1 σ11 + p2 u2 σ22 2 1 2 2 + (L30 u1 σ11 + L03 u2 σ22 + L21 u2 σ11 σ22 + L12 u1 σ22 σ11 ) 2 evaluated at (θˆ1 , θˆ2 ), where θˆ1 and θˆ2 are the MLEs of θ1 and θ2 . Thus, the Bayesian approximate estimate of Q(α) is given by 2 2 ˆB = Q ˆ ML + p2 u2 σ22 + 1 (L30 u1 σ11 Q + L03 u2 σ22 + L21 u2 σ11 σ22 + L12 σ22 σ11 ) 2