Inference for an exponentiated Pareto record values based on the pivotal quantity †
Jung-In Seo 1
1 Department of Big Data, Daejeon University
Received 10 May 2019, revised 9 June 2019, accepted 14 June 2019
Abstract
This article provides estimation methods based on the pivotal quantity to construct exact confidence intervals with the equal-tails and shortest-length for unknown param- eters of an exponentiated Pareto distribution based on lower record values and extends the method to prediction for the future lower record values, which not only entails no computational complexity like the maximum likelihood method, but also leads to valid confidence intervals even if the sample size is not large enough. The provided method is evaluated through Monte Carlo simulations and a real data set is analyzed for illustrative purposes.
Keywords: Exponentiated Pareto distribution, lower record value, pivotal quantity, pre- diction.
1. Introduction
Record values that arise from many real-life situations involving weather, sports and eco- nomics have been first examined by Chandler (1952). Since then, many statistical inference studies on recorded values have been made due to its importance. Balakrishnan et al. (1992) derived of single and double moments of lower record values having the Gumbel distribution and established their relationships. Seo et al. (2012) provided an approximate maximum like- lihood method for unknown parameters of a generalized half-logistic distribution (GHLD) based on the record values and proposed an entropy estimation method of the record val- ues from the GHLD. Seo and Kim (2014) proposed a nonparametric Bayesian method for record values from an exponentiated inverse Weibull distribution as an alternative to the Soland’s method (1969) that has a limitation that parameters of interest are restricted to a finite number of values. Wang and Ye (2015) provided a bias-corrected estimation and ex- act interval estimation for unknown parameters of in a two-parameter Weibull distribution based on upper record values. Wang et al. (2015) proposed exact confidence intervals (CIs) for unknown parameters in the family of proportional reversed hazard distributions based
† This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science, ICT & Future Planning) (No. NRF-2017R1C1B1006792).
1 Assistant professor, Department of Big Data, Daejeon University, Daejeon 34520, Korea. E-mail: lee-
[email protected]
on lower record values. Seo and Kim (2016) proposed inference methods for lower record values having the Gumbel distribution based on the pivotal quantity and objective priors from Bayesian and non-Bayesian perspectives, respectively. Seo and Kang (2016) proposed estimation methods for a shape parameter in the presence of a nuisance parameter in the exponentiated half logistic distribution (EHLD) when lower record values have this distribu- tion. Seo and Kim (2017a) provided an objective Bayesian analysis based on a reference prior with partial information for upper record values from two-parameter Rayleigh distribution.
Seo and Kim (2017b) provided a Bayesian entropy inference method for upper record values from a two-parameter logistic distribution. Yoon et al. (2017) provided Bayes estimators under various error loss functions for the exponentiated Pareto distribution (EPD) with the probability density function (PDF) and cumulative distribution function (CDF)
f (x; λ, θ) = θλ 1 − (1 + x) −λ θ−1
(1 + x) −(λ+1) and
F (x; λ, θ) = 1 − (1 + x) −λ θ
, x > 0, λ, θ > 0, (1.1) respectively, when the lower record values have the distribution.
However, to the best of our knowledge, the issues about the CI and prediction in the EPD based on lower record values have not been studied in the literature. This article provides an approach based on the pivotal quantity to construct exact CIs with equal-tails and shortest- length for unknown parameters of the EPD based on lower record values when data size is not large enough, and the corresponding point estimation is provided. In addition, this approach is extended to prediction for the future lower record values.
The rest of this paper is organized as follows. Section 2 provides estimation methods based on the pivotal quantity for unknown parameters of the EPD based on lower record values. Section 3 examines the validity of the provided approach through the Monte Carlo simulations and analyzes a real data set. Section 4 concludes.
2. Inference based on pivotal quantity
The lower record values can be observed in the following situations: Let {X 1 , X 2 , . . .} be a sequence of independent and identically distributed (iid) random variables with a CDF.
Lower records are values in the sequence lower than all preceding ones. That is, X j is a lower record value if X j < X i for every i < j. The indexes for which lower record values occur are given by the record times {L(k), k ≥ 1} , where L(k) = min j|j > L(k − 1), X j < X L(k−1) , , k > 1, with L(1) = 1. Therefore, a sequence of lower record values is denoted by {x L(k) , k = 1, 2, . . .} from the original sequence {X 1 , X 2 , . . .} .
Let X L(1) , . . . , X L(k) be the lower record values from the EPD with the CDF (1.1). Then, the corresponding likelihood function is given by
L(λ, θ) = θ k λ k 1 − (1 + x L(k) ) −λ θ
k
Y
i=1
(1 + x L(i) ) −(λ+1) 1 − (1 + x L(i) ) −λ ,
and the maximum likelihood estimators (MLEs) ˆ λ and ˆ θ can be found simultaneously by
maximizing its natural logarithm log L(λ, θ) for λ and θ. In addition, the approximate CIs
based on the MLEs can be obtained based on the Fisher information matrix for (λ, θ), given by
I(λ, θ) =
E
− ∂λ ∂ 2 2 log L(λ, θ) E
− ∂λ∂θ ∂ 2 log L(λ, θ) E
− ∂θ∂λ ∂ 2 log L(λ, θ) E
− ∂θ ∂ 2 2 log L(λ, θ)
.
However, it is not only a very tedious task, but also record values rarely observed. Instead, we propose an approach to the pivotal quantity, which provides more efficient estimators and exact CIs for unknown parameters without computational hassles.
2.1. Estimation
Based on the ideas of Wang et al. (2015), the pivotal quantities that corresponds to the lower record values from the EPD is derived here. In addition, we use the following notations X ∼ Exp(1), X ∼ U(0,1), X ∼ χ 2 n , and X ∼ IGam(α, β) to denote that X has a standard exponential distribution with the mean 1, a continuous uniform distribution on the interval (0, 1), a χ 2 distribution with n degrees of freedom, and an inverse gamma distribution with the parameters (k, k − 1), respectively.
Let
Z i = − log F (x L(i) ), i = 1, . . . , k.
Then, S i = Z i − Z i−1 , i = 1, . . . , k(Z 0 ≡ 0) iid ∼ Exp(1) and it leads to the following pivotal quantity
U j =
T j (λ, θ) T j+1 (λ, θ)
j
=
( log[1 − (1 + x L(j) ) −λ ] log[1 − (1 + x L(j+1) ) −λ ]
) j
, j = 1, . . . , k − 1 iid ∼ U(0,1) by letting
T j (λ, θ) = 2
j
X
i=1
S i
= −2θ log 1 − (1 + x L(j) ) −λ , j = 1, . . . , k. (2.1) Note that T k (λ, θ) ∼ χ 2 2k in (2.1). In addition, the pivotal quantity U j leads to the following pivotal quantity
W (λ) = −2
k−1
X
j=1
log U j ∼ χ 2 2(k−1) .
Then, for any 0 < α < 1, an exact 100(1 − α)% equal-tails CI for λ can be constructed based on the pivotal quantity W (λ) as
ϕ(X, χ 2 1−α/2, 2(k−1) ), ϕ(X, χ 2 α/2, 2(k−1) )
because
1 − α = P h
χ 2 1−α/2 < W (λ) < χ 2 α/2 i ,
where ϕ(X, t) is the solution of λ for the equation W (λ) = t, X = {X L(1) , · · · , X L(k) }, and χ 2 α,n is the upper α percentile of the χ 2 n distribution.
Note that an exact 100(1 − α)% CI with the shortest-length based on the pivotal quantity W (λ) not only requires explicit forms for the lower and upper bounds in a CI for W (λ), but also entails computational complexity. Instead, the ideas of Chen and Shao (1999) is used to construct a 100(1 − α)% shortest CI based on the pivotal quantity W (λ) for λ.
Let {w i , i = 1, 2, , . . . , N } be an ergodic MCMC sample from a χ 2 2(k−1) distribution.
According to Chen and Shao (1999), for any 0 < α < 1, a 100(1 − α)% shortest CI for W (λ) is given by
w (l ∗ ) , w (l ∗ +[(1−α)N ]) ,
where w ([αN ]) is the [αN ]th smallest of {w l } and l ∗ is chosen so that w (l ∗ +[(1−α)N ]) − w (l ∗ ) = min
1≤j≤N −[(1−α)N ] w (l+[(1−α)N ]) − w (l) . Then, the 100(1 − α)% shortest CI for λ can be constructed as
ϕ(X, w (l ∗ ) ), ϕ(X, w (l ∗ +[(1−α)N ]) ) .
Similarly, for any 0 < α < 1, an exact 100(1 − α)% equal-tails CI for θ can be constructed based on the pivotal quantity T k (λ, θ) as
χ 2 1−α/2, 2k
2 log[1 − (1 + x L(k) ) −λ ] −1 < θ < χ 2 α/2, 2k
2 log[1 − (1 + x L(k) ) −λ ] −1
! ,
which has a nuisance parameter λ. To overcome it, we provide a generalized pivotal quantity
θ ∗ = T ∗
2 log[1 − (1 + x L(k) ) −ϕ(X, w) ] −1 ,
where T ∗ can be generated from a χ 2 2k distribution. The percentiles of θ ∗ can be obtained through Monte Carlo simulations. Then, the equal-tails and shortest CIs for θ are given by
θ ∗ ([(α/2)N ]) , θ ([(1−α/2)N ]) ∗
and
θ ∗ (l ∗ ) , θ ∗ (l ∗ +[(1−α)N ])
, respectively, where l ∗ is chosen so that
θ ∗ (l ∗ +[(1−α)N ]) − θ ∗ (l ∗ ) = min
1≤j≤N −[(1−α)N ]
h
θ (l+[(1−α)N ]) ∗ − θ ∗ (l) i
.
The validity of the provided CIs is examined in terms of their coverage probabilities (CPs) in subsection 3.1.
Note that both T k (λ, θ)/{2(k−1)} converges with probability one to 1 since T k (λ, θ) ∼ χ 2 2k , and it leads to an unbiased estimator of θ as
θ(λ) = ˆ k − 1
log[1 − (1 + x L(k) ) −λ ] −1
by solving the equation T k (λ, θ) = 2(k − 1) for θ because the estimator ˆ θ(λ) ∼ IGam(k, k − 1) for known λ. Similarly, the nuisance parameter λ in ˆ θ(λ) can be estimated as ˆ λ p,c = ϕ(X, 2(k − c)) for c(∈ Z) < k by solving the equation W (λ) = 2(k − c) for λ. A close examination is made for different c values through real data analysis in subsection 3.2.
2.2. Prediction
This subsection extends the approach provided in the above subsection to the prediction problem for the future record values X L(r) (r = k + 1, k + 2, . . .).
The conditional density function of X L(r) given x L(k) is defined by Ahsanullah (1995) as
f X L(r) |x L(k) (x) = log F (x L(k) ) − log F (x L(r) ) r−k−1
f (x L(r) )
Γ(r − k)F (x L(k) ) , x L(r) < x L(k) . (2.2) In the conditional densty function (2.2), let
Y = θ log 1 − (1 + x L(k) ) −λ − log 1 − (1 + x L(r) ) −λ .
Then, it has a gamma distribution with the parameters (r − k, 1) by the transformation of random variables. Therefore, a random variable from (2.2) can be generated as
X L(r) ∗ | x L(k) = n
1 − [1 − (1 + x L(k) ) −ϕ(X, w) ]e −y/θ ∗ o −1/ϕ(X, w)
− 1,
where y can be generated from the gamma distribution with the parameters (r − k, 1). The equal-tails and shortest predictive intervals (PIs) for the future record value X L(r) can be constructed in the same way as θ.
3. Applications
3.1. Simulation
The provided estimators are evaluated in terms of their relative root mean squared errors
(RMSEs) and biases. In addition, the provided CIs are evaluated in terms of their CPs and
average lengths (ALs). To conduct the performance, we first generate 10,000 lower record
values from the EPD with λ = 2.5 and θ = 3 as in the case of Yoon et al. (2017). Then,
the provided estimators and 95% CIs for each lower record value are computed, and the
corresponding RMSE, bias, CP, and AL are obtained based on 10,000 simulations. Note that
c = −2 in the estimator ˆ λ p,c is only reported here because it has shown the best performance
in terms of the bias for various values of c. Instead, an analysis based on different c values is
performed in the next subsection, Real Data. The results for the Monte Carlo simulations are reported in Table 3.1.
Table 3.1 shows that the estimators based on the pivotal quantity are more efficient than the MLEs in terms of the RMSE and bias. It is noteworthy that the bias of ˆ θ p ˆ λ p,−2
is nearly zero for the moderate sample sizes. For CIs, CPs of the equal-tails CIs are well matched to their corresponding nominal levels, especially for small sample sizes and CPs of the shortest CIs reasonably is very close to the corresponding nominal levels. As expected, the RMSEs and ALs not only decrease with an increase in size of record values k, but also the shortest CIs have shorter ALs than the equal-tails CIs.
3.2. Real Data
Seo and Kang (2014) analyzed a lower record data set obtained from annual rainfall (in inches) recorded at the Los Angeles Civic Center (1877–2012) under the assumption that the lower record data have an EHLD. For illustrative purposes, the lower record values (see Table 3.3) are analyzed in this subsection.
Note that c = −2 in the estimator ˆ λ p,c does not always yield optimal fit for a real data set.
Therefore, we consider different c = −2(1)3 in the estimator ˆ λ p,c to examine which c value shows the best fit for the observed lower record values. The goodness of fit test is conducted based on two types of statistical measures: one is the correlation coefficient (r) for pairs of observed and expected lower record values, the other is the quantile residuals in proposed Dunn and Smyth (1996). The quantile residuals are defined as
z q = Φ −1 n F
y i ; ˆ φ o ,
where Φ(·) is the CDF of the standard normal distribution and F (·; φ) is the CDF of a continuous probability distribution with a parameter φ. Because the CDF corresponding to the lower record value X L(i) (Ahsanullah, 1995; Arnold et al., 1998) is
F X L(i) (x; φ) = e −H(x)
i−1
X
j=0
[H(x)] j
j! , (3.1)
where H(x) = − log F (x; φ), the approach can be applied by substituting F (x; φ) with (1.1) in (3.1). Then, it is possible to the goodness of fit test through the quantile-quantile (Q-Q) plot. These results are provided in Tables 3.2, 3.3, and Figure 3.1. For prediction, we plot the estimated kernel density functions of X L(r) | x L(10) , r = 11(1)13 in Figure 3.2.
Table 3.2 shows that the estimators ˆ λ p,c and ˆ θ p ˆ λ p,c
increase as the value of c in- creases. In addition, Table 3.3 and Figure 3.1 show that the EPD with the estimators
Table 3.1 RMSEs(biases) of the provided estimators, and CPs(ALs) of exact CIs for λ and θ.
RMSEs(biases) CPs(ALs) for λ CPs(ALs) for θ
k ˆ λ λ ˆ p,−2 θ ˆ θ ˆ p ˆ λ p,−2
Equal-tails Shortest Equal-tails Shortest
9 3.400(0.877) 2.732(0.199) 1.797(0.409) 1.015(0.021) 0.950(9.608) 0.939(8.760) 0.952(7.076) 0.957(6.488)
11 3.278(0.790) 2.543(0.215) 1.221(0.278) 0.811(0.013) 0.950(9.266) 0.943(8.492) 0.950(5.385) 0.957(5.107)
13 3.001(0.724) 2.521(0.220) 0.933(0.205) 0.680(0.006) 0.950(9.038) 0.944(8.311) 0.956(4.635) 0.958(4.459)
15 2.871(0.680) 2.211(0.221) 0.761(0.160) 0.589(0.000) 0.950(8.911) 0.943(8.202) 0.951(3.995) 0.955(3.894)
Table 3.2 Estimates and CIs for λ and θ
Estimates CIs
λ ˆ ˆ λ p,−2 λ ˆ p,−1 ˆ λ p,0 λ ˆ p,1 ˆ λ p,2 λ ˆ p,3 Equal-tails Shortest 1.461 1.113 1.275 1.436 1.595 1.752 1.907 (0.463, 2.333) (0.387, 2.223)
θ ˆ θ ˆ p ˆ λ p,−2
θ ˆ p ˆ λ p,−1
θ ˆ p ˆ λ p,0
θ ˆ p ˆ λ p,1
θ ˆ p ˆ λ p,2
θ ˆ p ˆ λ p,3
Equal-tails Shortest 76.570 39.887 51.664 66.305 84.492 107.069 135.085 (11.142, 301.492) (4.144 221.589)
Table 3.3 Observed and expected lower record values, and their corresponding r values i
c 1 2 3 4 5 6 7 8 9 10 r
x L(i) 21.26 11.35 10.40 9.21 6.73 5.59 5.58 4.85 4.42 3.21
E X L(i)
λ, ˆ ˆ θ
54.340 16.540 10.589 7.985 6.481 5.485 4.771 4.230 3.803 3.457 0.954 E
X L(i)
λ ˆ p,c , ˆ θ p ˆ λ p,c
-2 256.975 25.432 13.720 9.426 7.174 5.780 4.828 4.136 3.609 3.194 0.898 -1 92.453 19.334 11.457 8.271 6.510 5.380 4.588 3.999 3.542 3.177 0.931 0 53.912 15.759 9.981 7.476 6.037 5.088 4.410 3.897 3.494 3.167 0.951 1 37.572 13.437 8.947 6.897 5.683 4.867 4.273 3.818 3.457 3.161 0.964 2 28.790 11.818 8.183 6.456 5.409 4.692 4.164 3.755 3.428 3.158 0.973 3 23.398 10.629 7.597 6.109 5.189 4.551 4.075 3.704 3.404 3.156 0.978
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Figure 3.1 Q-Q plots for quantile residuals with the estimators (a) ˆ λ, ˆ θ
; (b) ˆ λ p,−2 , ˆ θ p ˆ λ p,−2
; (c) ˆ λ p,−1 , ˆ θ p ˆ λ p,−1
; (d) ˆ λ p,0 , ˆ θ p ˆ λ p,0
; (e) ˆ λ p,1 , ˆ θ p ˆ λ p,1
; (f) ˆ λ p,2 , ˆ θ p ˆ λ p,2
; (g)
ˆ λ p,3 , ˆ θ p ˆ λ p,3
ˆ λ p,3 , ˆ θ p ˆ λ p,3
has the best fit for the observed lower record values. Finally, Figure 3.2
shows that the estimated kernel density functions of future lower record values all are left-
skewed and the more distant future record value is predicted, the greater the width of the
corresponding PI.
0.5 1.0 1.5 2.0 2.5 3.0
0.00.51.01.52.02.5
XL(11) | xL(10)
Density
95% equal−tails PI 95% shortest PI
(a)
0.5 1.0 1.5 2.0 2.5 3.0
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95% equal−tails PI 95% shortest PI
(b)
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XL(13) | xL(10)
Density
95% equal−tails PI 95% shortest PI
(c)
