Literature DB >> 35421161

Bayesian interval estimations for the mean of delta-three parameter lognormal distribution with application to heavy rainfall data.

Patcharee Maneerat1, Pisit Nakjai2, Sa-Aat Niwitpong3.   

Abstract

Flash flooding is caused by heavy rainfall that frequently occurs during a tropical storm, and the Thai population has been subjected to this problem for a long time. The key to solving this problem by planning and taking action to protect the population and infrastructure is the motivation behind this study. The average weekly rainfall in northern Thailand during Tropical Storm Wipha are approximated using interval estimations for the mean of a delta-three parameter lognormal distribution. Our proposed methods are Bayesian confidence intervals-based noninformative (NI) priors (equal-tailed and highest posterior density (HPD) intervals based on NI1 and NI2 priors). Our numerical evaluation shows that the HPD-NI1 prior was closer to the nominal confidence level and possessed the narrowest expected length when the variance was small-to-medium for a large threshold. The efficacy of the methods was illustrated by applying them to weekly natural rainfall data in northern Thailand to examine their abilities to indicate flooding occurrence.

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Year:  2022        PMID： 35421161      PMCID： PMC9009634          DOI： 10.1371/journal.pone.0266455

Source DB:  PubMed          Journal:  PLoS One        ISSN： 1932-6203            Impact factor:   3.240


Introduction

Human beings and all living things need water to survive, so life would not exist without it. The amount of water that is usually available depends on the amounts of rain and snow that fall. Unfortunately, some areas barely see rain while others get more than their fair share. These situations can cause natural disasters such as floods and droughts, which are dramatic changes that occur when most people least expect them. In Thailand, long periods of rain caused by tropical storms have triggered significant flooding. In July-August 2019, Tropical Storm Wipha crossed the North Vietnam coast and headed westward toward upper Thailand (mainly the northern and northeastern regions). Heavy rain passing through northern Thailand caused flash flooding in Phayao province and also triggered landslides in Nan province in northern Thailand [1]. As a consequence, these extreme events caused loss of life and significantly damaged assets and transport infrastructure in these at-risk areas. One of the most important factors in solving this problem is how to use the historical rainfall data to plan and prevent more flooding in the future by taking direct action accordingly. These reasons have led to our interest in the estimation of the mean rainfall amount using the observed data from extreme rainfall events. Importantly, the weekly natural rainfall amounts in the week 29 July to 4 August 2019 follow the assumptions of a delta-three parameter lognormal distribution. A three parameter lognormal distribution is considered to be suitable for the observed data that are highly skewed and cannot be modeled using a two-parameter lognormal distribution [2]. A delta-three parameter lognormal distribution is a combination of zero and non-zero values following the three-parameter lognormal distribution first introduced by Aitchison and Brown [3]. The three-parameter lognormal distribution has threshold parameter a, an unknown parameter that makes it differ from a two-parameter lognormal distribution in that a > 0. Thus, a two-parameter lognormal distribution is a special case of a three-parameter lognormal distribution when a = 0. In application involving real-world data, the three-parameter lognormal distribution has been used in Hydrology [4-6], rainfall network [7, 8], and flood frequency analysis [9, 10]. In probability and statistical inference, there are two types of estimation: point and interval, the latter also being known as the confidence interval (CI). Point estimations for the parameters of a three-parameter lognormal distribution have been developed and discussed by many researchers. For example, Cohen and Whitten [11] proposed modifications of the moment and maximum likelihood estimates (MLEs). After that Cohen et al. [12] modified the moment estimates by replacing the function of the first-order statistic in the third moment. Singh et al. [2] conducted a performance evaluation of the estimates through Monte Carlo simulation. Later, interval estimations were constructed for the parameters of a three-parameter lognormal distribution. Royston [13] constructed CIs for the reference range of random samples from a three-parameter lognormal distribution. Pang et al. [14] used a simulation-based approach to assess the Bayesian CIs for the coefficient of variation of a three-parameter lognormal distribution as one of their proposed distributions. Next, Basak et al. [15] evaluated a maximum likelihood estimate created from an expectation-maximization algorithm when progressive Type-II censored samples were drawn from a three-parameter lognormal distribution while providing interval estimations for the mean, variance and threshold in a three-parameter lognormal distribution using large-sample theory. Finally, Chen and Miao [16] studied the exact CIs and exact upper CIs for the location parameter (also known as the threshold parameter) of a three-parameter lognormal distribution. Recently, Maneerat et al. [17] revealed that the highest posterior density (HPD) interval-based beta prior was the best-performing method for estimating a single mean and the difference between two delta-lognormal means. After that the HPD-based normal gamma prior was developed in the comparison of the estimated rainfall dispersion between northern and northeastern regions in Thailand proposed by Maneerat et al. [18], while the HPD-based probability matching prior was recommended to construct the CIs for the difference between two delta-lognormal variances [19]. Unfortunately, the CI for the mean of a delta-three parameter lognormal distribution, especially when the highly skewed non-zero values are present with the zero observations, has not yet been established. Therefore, the first goal in the study was to propose a Bayesian CI (BCI) estimation-based approach (equal-tailed (ET) and HPD intervals based on different noninformative (NI) priors), generalized CI (GCI), and the method of variance estimates recovery (MOVER) for the mean of a delta-three parameter lognormal distribution. Using this as a starting point, the other goal was to estimate the weekly natural rainfall amounts during Tropical Storm Wipha in northern Thailand using our proposed methods since the observed rainfall data can be used to indicate an extreme event that can cause flooding. The article is outlined as follows. The notation and the proposed methods in the construction of the CIs of the mean of a delta-three parameter lognormal distribution are elaborated in Section. In Sections -, details of the simulation studies, including the parameter settings together with the criteria for assessing the proposed methods and numerical computation to identify the best-performing method, are presented. The efficacies of the proposed methods using weekly natural rainfall amounts during the period of Tropical Storm Wipha are examined in Section. Finally, this article is ended with a brief discussion and conclusions in Sections -.

Notation and methods

Let X = (X1, X2, …, X) be a random sample follows a delta-three parameter lognormal distribution (DTPLN), denoted as where δ is the probability of having zero, μ is the scale parameter, is the shape parameter, and a is the threshold parameter or a quantity defined as a lower bound of X. These parameters δ, μ, , a satisfy 0 < δ < 1, μ > 0, and a ≥ 0, respectively. The distribution function of X is where is the three-parameter lognormal (TPLN) distribution, introduced by Aitchison and Brown [3], Cohen and Whitten [11] so that the probability density function of X is where μ = E[ln(x − a)] and . For x > a, it has the relation between random variables X and Y = ln(X − a), that is, if . The mean of X is defined as which is log-transformed a The likelihood of X is where n(0) = #{i:x = 0} and n(1) = n − n(0). The likelihood (4) leads to obtain the log-likelihood is The first and second derivatives are and From , we obtain that The estimate of threshold () was obtained by the modified method of moments estimation, proved by Cohen and Whitten [11]. Thus, the maximum likelihood estimates (MLEs) of δ, μ and are For formulating the CIs for θ, the methods are detailed as follows:

Bayesian confidence intervals

The BCI for a parameter of interest is constructed from the posterior distribution, introduced by Gelman et al. [20]. Based on Bayesian approach, the 100(1 − α)% equal-tailed CI or central interval for the parameter of interest can be computed the lower and upper limits from the 100(α/2)% and 100(1 − α/2)% quantiles of the posterior probability, respectively. Box and Tiao [21] defined the HPD region (Definition 1) can be led to construct the HPD interval which is different from the equal-tailed CI. Definition 1. Let p(θ|y) be a posterior density function. A region R in the parameter space of θ is called a HPD region of content (1 − α) if i) Pr(θ ∈ R|y) = 1 − α, ii) For θ1 ∈ R and θ2 ∉ R, p(θ1|y)≥p(θ2|y). The HPD region is defined as the value set that contains the 100(1 − α)% of the posterior probability, importantly its density within the region is never less than that outside. In the present study, the equal-tailed CIs and HPD intervals for the log-transformed mean of a delta-three parameter lognormal distribution are proposed and constructed using the noninformative (NI) priors as follows:

NI1 prior

The NI1 prior is derived from the square root of the Fisher information matrix of , given by Recall that the likelihood is given by where . The likelihood (11) is updated with the NI1 prior (10) to obtain the posterior of , that is The posterior of δ is It can be implied that . From Eq (12), the posterior of a|data becomes where which can be obtained its random samples using Metropolis algorithm. Let , such that leads to obtain . The posterior of μ is Thus, ; df = n(1) − 1. The posterior of is where ; α = n(1)/2 and . The posterior of θ based on NI1 can be expressed as Finally, the 100(1 − α)%BCIs for θ based on NI1 prior are where denotes the αth quantile of θ and .

NI2 prior

This prior belief is obtained from the δ, a, μ and are treated as random variables of the beta, uniform, normal and gamma distributions, denoted as beta(c, c), U(a′ = 0, b′ = 1), N(μ, 1/kη) and G(a, b), respectively, where and . The prior of δ was derived by Jin et al. [22]. The prior of is When k0 = 0, a0 = −1/2 and b0 = 0, the NI2 prior of (μ, σ2, δ) is derived from , and δ ∼ beta(c, c) as which is combined with the likelihood (11) in term of η as Meanwhile, the posterior of θ is This leads to obtain the posterior of δ is This is δ ∼ beta(n(0)+ c, n(1)+ c). The posterior of a|data becomes which can be written as . Let and T = ηW such that ; . The posterior of μ|data is which is the Student’ t distribution with n(1) − 1 the degrees of freedom (df), denoted as . From Eq (23), the posterior of η|data is It can be concluded that . Thus, the posterior of is The posterior of θ based on NI2 can be written as Hence, the 100(1 − α)%BCIs for θ based on NI2 prior are where denotes the αth quantile of θ and . Algorithm 1 describes the steps for constructing BCIs for the delta-three parameter lognormal means. Algorithm 1: BCIs 1: Compute the unbiased estimates , , , and . 2: For NI1 prior, generate the posterior distributions of δ, a, μ, and , denoted by δ, a , and in Eqs (13), (14), (15) and (16), respectively. 3: For NI2 prior, generate the posterior distributions of δ, a, μ, and , denoted by δ, a , and in Eqs (24), (25), (26) and (27), respectively. 4: Compute θ and θ based on NI1 and NI2 priors, respectively. 5: Repeat 2–4 a number of times, say, m. 6: For m times, compute the 100(1 − α)% ET and HPD intervals for θ based on priors: BCIET-NI1, BCIHPD-NI1, BCIET-NI2 and BCIHPD-NI2.

Generalized confidence interval

The GCI is established based on the concept of generalized pivotal quantity (GPQ), defined by Weerahandi [23]. The CI of θ can be constructed using GCI. Recall that if Y = ln(X − a) be a random variable of normal distribution with mean μ and variance . Cohen [24], Cohen et al. [12] and Cohen and Whitten [11] derived the MLE of threshold (6), and the asymptotic variance of is based on the Fisher information matrix, given by By replacing the estimates and , the is estimated and denoted by . Let be a random variable. From the approximation result, it is transformed as which has a Student’s t distribution with n(1) − 2 df, where and are independent random variables of standard normal and chi-square distribution with n(1) − 2 df, denoted by Z ∼ N(0, 1) and , respectively. By the information of pivotal quantity T, the GPQ of a is Furthermore, Wu and Hsieh [25] proposed the GPQs of δ and (), defined as where , and . The GPQ of θ is which satisfies the conditions of Weerahandi [23], i.e., the distribution of R is free from all unknown parameters, and the observed value of R depends only on the parameters of interest. Therefore, the 100(1 − α)% GCI for θ is given by where R(α) denotes the αth quantile of R. The steps for computing GCI for θ are detailed in Algorithm 2. Algorithm 2: GCI 1: Generate T ∼ t, K, W ∼ N(0, 1) and . 2: Compute the GPQs of a, δ, μ and , denoted as R, R, , and , respectively. 3: Compute the GPQ of θ, denoted as R. 4: Repeat 1–3 a number of times, say, m. 5: For m times, compute the 100(1 − α)%GCI for θ in Eq (38).

Method of variance estimates recovery

Let λ be the parameter of interest for the population i; i = 1, 2, …, p. Also, let be the point estimate of λ. The MOVER interval for the function of parameters λ is a closed-form CI constructed by obtaining the variance estimates at the neighborhood of the lower and upper limits separately (to recover from confidence limits), given in Zou and Donner [26] and Zou et al. [27]. Thus, 100(1 − α)% MOVER interval for λ is where be the 100(1 − α)% CIs for λ. Approximate closed-form CI for the logarithm of delta-three parameter lognormal mean is considered and developed using the MOVER. Recall that where δ′ = 1 − δ. Let be the point estimate of θ; and . The MOVER intervals for ln θ1 and ln θ2 are described as follows. First, the 100(1 − α)% Wilson interval for ln θ1 is proposed by Zou et al. [27], given by where k denotes the αth quantile of standard normal distribution. Next, the 100(1 − α)% MOVER interval for lnθ2 where Note that t and w denote the αth quantile of Student’s t distribution with n(1) − 2 df and standard normal distributions, respectively. The is given in Zou et al. [27]. Applying Eq (39), the 100(1 − α)% MOVER interval for θ is The MOVER for θ can be computed in Algorithm 3. Algorithm 3: MOVER 1) Compute the CIs for a and in Eqs (43) and (44), respectively. 2) Compute and . 3) Compute the 100(1 − α)%MOVER for θ in Eq (45).

Simulation studies

Simulation studies were conducted to calculate the performances of the methods: the coverage probabilities (CPs) and expected lengths (ELs) of BCIs (HPD and ET intervals)-based NI1 and NI2 priors, GCI, and MOVER for the logarithm of the delta-three parameter lognormal mean. Both performances are defined as follows: CP: the proportion of intervals in which the true parameter falls within the intervals. EL: the average lengths of simulated intervals. Monte Carlo simulation studies were undertaken to compare the performances of our proposed methods and provide insight into their sampling behavior. In the comparison of the methods, a CI with a CP close to the nominal level 0.95 and the narrowest EL are the criteria for the best performance. In the simulation studies, the values of the threshold parameter were chosen as a = 1, 5, 15. For each threshold value, the parameter combinations were sample sizes n = 30, 50, 100; proportion of zero δ = 10%, 30%, 50%; mean μ = 2 and variance σ2 = 0.3, 0.5, 0.8, 1.0, 2.0. For each set of parameter settings, 5,000 simulation runs were generated and 5,000 GPQs were fixed for the GCI. The steps of the simulation study were executed as shown in Algorithm 4. Algorithm 4 1: Generate . 2: Compute the unbiased estimates , , , and . 3: Compute the CIs: BCIs, GCI and MOVER in Algorithms 1, 2 and 3, respectively. 4: For the 5000 generated values, the CIs in Step (3) are computed. 5: Computed the estimated CPs and ELs of the CIs in Step (4).

Monte Carlo simulation results

The simulation results for threshold a = 1 (Table 1 and Fig 1) show that GCI and MOVER generated CPs close to nominal level 0.95 when the variance was small for all of the proportions of zero observations, although those of the BCIs (HPD and ET intervals based on the NI1 and NI2 priors) were under it. For threshold a = 5 (Table 2 and Fig 2),the CP and EL performances of HPD-NI1, ET-NI1, GCI, and MOVER were better than a = 1, while HPD-NI1 performed the best in terms of EL for small-to-large sample sizes except for large variance. For a large threshold a = 15 (Table 3 and Fig 3), HPD-NI1 performed better than the other methods with a CP close to the nominal level and the narrowest EL when the variance was small-to-medium.
Table 1

CP and EL performances of 95% CI for θ: a = 1.

a = 1CPEL
n δ σ 2 HPD-NI1HPD-NI2ET-NI1ET-NI2GCIMOVERHPD-NI1HPD-NI2ET-NI1ET-NI2GCIMOVER
3010%0.30.93580.94200.92560.93220.99660.99680.51810.52570.49220.4994 0.7926 0.8028
0.50.91860.92580.90600.91360.99160.99140.64490.65090.61260.6184 0.8484 0.8548
0.80.92660.93040.91240.91560.97380.97240.79150.79660.75190.7568 0.9320 0.9342
1.00.93100.93300.91600.91860.96420.96060.88580.89050.84150.8460 0.9958 0.9963
2.00.91960.91600.90880.90240.92380.92221.31631.31961.25051.25361.37101.3653
30%0.30.95520.96160.94460.95340.99280.99380.71720.72900.68140.69250.9426 0.9395
0.50.93740.94460.92540.93160.98400.98300.85270.86290.81010.81981.0288 1.0232
0.80.93040.93700.91860.92380.97460.97261.01371.02200.96300.97091.1550 1.1451
1.00.93480.93520.92100.92380.96600.96321.11941.12781.06341.07141.2412 1.2292
2.00.92020.91440.90560.90300.92220.91741.55051.55681.47291.47891.63591.6182
50%0.30.94900.96200.93980.95100.98460.98501.00571.02480.95540.97351.1897 1.1661
0.50.94200.95400.92940.94360.97880.97981.18431.20101.12511.14101.3465 1.3199
0.80.94460.95380.93100.94100.97900.97701.38011.39451.31111.32481.5224 1.4924
1.00.94680.95280.93440.94200.97320.96981.49451.50761.41981.43231.6316 1.5976
2.00.94400.94500.93120.93340.95660.95041.95941.97131.86141.87272.0981 2.0567
5010%0.30.85180.85980.83760.84700.99720.99720.37210.37570.35350.3569 0.6789 0.6841
0.50.86920.87560.85120.86120.99040.99120.46010.46310.43710.4399 0.6570 0.6609
0.80.92620.93020.91200.91380.97400.97260.59180.59430.56220.5645 0.6850 0.6869
1.00.93000.93240.91720.92060.95140.94940.67830.68060.64440.6466 0.7358 0.7371
2.00.91440.91380.90080.90120.91960.91761.04441.04550.99220.99321.06271.0611
30%0.30.90720.91420.89260.90180.99360.99320.52070.52630.49460.50000.7935 0.7927
0.50.90240.90980.88640.89580.98320.98380.60990.61510.57940.58430.8062 0.8039
0.80.93100.93280.91580.92100.97220.97120.74140.74560.70440.70830.8528 0.8493
1.00.93880.93840.92220.92200.95940.95420.82640.83000.78500.78850.9044 0.8997
2.00.92520.92080.90820.90800.92500.92301.20951.21321.14901.15261.24331.2362
50%0.30.92900.94080.91600.92560.98900.98940.72540.73550.68910.69870.9587 0.9496
0.50.92500.93440.91160.92320.98560.98580.83170.84040.79020.79841.0115 1.0004
0.80.92580.93500.91440.92020.96960.96700.97370.98100.92500.93191.1021 1.0884
1.00.93980.94480.92680.93260.97420.97181.06241.06991.00931.01641.1687 1.1537
2.00.92360.92240.91340.90980.92420.92061.46471.46931.39141.39581.52911.5113
10010%0.30.73820.74280.72160.72900.99840.99840.24960.25100.23710.2385 0.5141 0.5161
0.50.88000.88500.86280.86880.98640.98680.32220.32350.30610.3073 0.4356 0.4373
0.80.94440.94220.93180.93040.96200.96340.41940.42020.39840.3992 0.4589 0.4599
1.00.94100.94040.92880.92980.94880.94740.48360.48450.45940.4603 0.5046 0.5052
2.00.92120.91920.90740.90740.92560.92560.75990.76050.72190.72240.76650.7661
30%0.30.79560.80520.77860.78640.99480.99460.35200.35400.33440.33630.6199 0.6202
0.50.88500.89200.86880.87520.98480.98540.41970.42170.39870.40060.5532 0.5528
0.80.93780.93940.92180.92560.96060.96060.51950.52130.49350.49530.5697 0.5686
1.00.94200.94260.93020.92920.94900.94780.58400.58530.55480.55600.6135 0.6120
2.00.92160.91860.91040.90740.92400.92460.87900.87970.83510.83570.88830.8866
50%0.30.85940.86620.84280.85320.99020.99060.48980.49360.46530.46890.7544 0.7522
0.50.89260.90000.87360.88280.98400.98260.55600.55920.52820.53120.7139 0.7106
0.80.93340.93400.91780.92280.96060.95840.66110.66410.62810.63090.7347 0.7303
1.00.94400.94560.93260.93260.95700.95620.73550.73820.69870.70130.7822 0.7772
2.00.92280.91780.90660.90720.92180.91941.05361.05581.00101.00301.07151.0662

Remark: Boldface indicates the recommended method for each case.

Fig 1

Performance measures of 95%CIs for θ: a = 1 (A) Coverage probabilities and (B) Expected lengths.

Table 2

CP and EL performances of 95% CI for θ: a = 5.

a = 5CPEL
n δ σ 2 HPD-NI1HPD-NI2ET-NI1ET-NI2GCIMOVERHPD-NI1HPD-NI2ET-NI1ET-NI2GCIMOVER
3010%0.30.96220.96340.95360.95440.99360.9942 0.4043 0.41380.38410.39310.64840.6627
0.50.95400.95460.94340.94240.99000.9894 0.4996 0.50770.47460.48230.67330.6857
0.80.94820.94560.93600.93480.97320.9726 0.6316 0.63790.60000.60600.73360.7436
1.00.94220.93900.93100.92860.96280.9610 0.7196 0.72560.68360.68930.79200.8005
2.00.90260.89960.88880.88360.91540.91361.12001.12401.06401.06781.15471.1576
30%0.30.95560.96260.94580.95340.98960.9904 0.6013 0.61530.57130.58450.80730.8056
0.50.95640.96080.94620.95160.98020.9812 0.6989 0.71070.66390.67520.85780.8554
0.80.95760.95820.94660.94660.97220.9714 0.8309 0.84150.78940.79950.94300.9393
1.00.95580.95380.94440.94340.96600.9646 0.9294 0.93970.88300.89281.01881.0139
2.00.91360.90940.89900.89460.92220.91481.34221.35031.27511.28271.39951.3906
50%0.30.94120.95580.93000.94400.98280.9850 0.8606 0.88250.81760.83841.04401.0217
0.50.95500.96340.94260.95660.97880.9792 0.9784 0.99870.92950.94871.12861.1065
0.80.96440.96800.95320.95920.97600.9738 1.1517 1.16951.09411.11111.27711.2531
1.00.95960.96140.94640.95240.96940.9662 1.2491 1.26581.18671.20251.36681.3415
2.00.94000.93800.92620.92400.94560.93821.72401.73861.63781.65161.82711.7974
5010%0.30.95840.95760.94820.94600.99720.9982 0.2984 0.30300.28350.28790.54770.5545
0.50.94880.94800.93460.93780.98880.9886 0.3665 0.37050.34820.35190.50660.5131
0.80.94380.94200.92920.93040.97000.9692 0.4735 0.47670.44980.45290.53170.5374
1.00.93400.93220.91980.91560.95000.94740.54410.54670.51690.5193 0.5791 0.5842
2.00.90980.90680.89600.89240.92080.92060.88220.88400.83810.83980.89780.8995
30%0.30.94660.95100.93560.93900.99020.9906 0.4478 0.45440.42540.43170.67230.6721
0.50.94980.95260.94000.94140.98000.9820 0.5117 0.51800.48610.49210.66180.6613
0.80.94880.94820.93720.93560.97080.9686 0.6172 0.62250.58640.59140.69180.6907
1.00.94180.93900.92980.92700.95380.9508 0.6874 0.69180.65300.65720.73900.7376
2.00.91100.90960.89760.89300.91840.91461.04211.04540.99000.99321.06641.0633
50%0.30.94620.95220.93260.94140.98380.9854 0.6370 0.64800.60520.61560.83950.8305
0.50.94660.95240.93640.94280.97720.9774 0.7110 0.72120.67550.68510.86250.8531
0.80.95880.95960.94680.94780.97300.9706 0.8252 0.83450.78390.79270.92410.9133
1.00.95240.95420.94240.94040.96360.9612 0.9074 0.91570.86200.86990.98570.9742
2.00.91520.90900.89900.89620.91800.91201.28121.28781.21711.22341.32641.3136
10010%0.30.94480.94620.93280.93320.99700.9972 0.2046 0.20620.19440.19590.38030.3829
0.50.95000.95220.93620.93820.98740.9884 0.2546 0.25590.24190.24310.32890.3317
0.80.94040.93900.92620.92700.95960.9582 0.3310 0.33210.31440.31550.35640.3588
1.00.94000.93900.92940.92400.94940.94900.38140.38230.36230.3632 0.3958 0.3980
2.00.91340.91280.89820.89960.92140.92100.64000.64080.60800.60870.64680.6477
30%0.30.93840.94060.92720.92780.99360.9932 0.3106 0.31300.29510.29730.49040.4907
0.50.94640.94580.93480.93540.97600.9762 0.3566 0.35890.33880.34100.43960.4397
0.80.94580.94800.93380.93360.95860.9582 0.4313 0.43300.40980.41130.46160.4615
1.00.94340.94360.92740.92720.94980.9498 0.4813 0.48310.45730.45890.50030.5002
2.00.92580.92560.91000.91240.92800.92720.74820.74940.71080.71190.75740.7571
50%0.30.94100.94400.93020.93260.98400.9842 0.4431 0.44720.42090.42490.63150.6291
0.50.94520.94640.93440.93320.97240.9734 0.4891 0.49270.46460.46800.59700.5939
0.80.94320.94580.92740.93060.96000.9580 0.5692 0.57240.54080.54370.61430.6106
1.00.94920.94880.93540.93700.95420.9516 0.6217 0.62460.59060.59340.65180.6481
2.00.92200.92080.91200.90760.92300.92260.91090.91310.86540.86750.92580.9222

Remark: Boldface indicates the recommended method for each case.

Fig 2

Performance measures of 95%CIs for θ: a = 5 (A) Coverage probabilities and (B) Expected lengths.

Table 3

CP and EL performances of 95% CI for θ: a = 15.

a = 15CPEL
n δ σ 2 HPD-NI1HPD-NI2ET-NI1ET-NI2GCIMOVERHPD-NI1HPD-NI2ET-NI1ET-NI2GCIMOVER
3010%0.30.95960.96700.94800.95860.99080.9936 0.3033 0.31510.28810.29940.48810.5065
0.50.96100.96200.94860.95100.98220.9876 0.3554 0.36600.33760.34770.47860.4978
0.80.95300.95240.94100.93940.96900.9708 0.4430 0.45240.42090.42980.50510.5248
1.00.95040.94600.93780.93440.96220.9614 0.5077 0.51620.48230.49040.55120.5704
2.00.91020.90200.89340.88920.92060.91680.84170.84770.79960.80530.87380.8878
30%0.30.93900.94880.93000.93880.97520.9818 0.4983 0.51400.47340.48830.66230.6609
0.50.95160.95940.93880.94980.97420.9806 0.5538 0.56880.52610.54030.67900.6782
0.80.95740.96040.94400.94920.96880.9698 0.6388 0.65230.60680.61970.72190.7228
1.00.95900.95780.94760.94660.96380.9634 0.7044 0.71740.66920.68160.76880.7698
2.00.92480.91900.91080.90200.92700.92181.04011.04990.98810.99741.09071.0926
50%0.30.94100.95240.92820.94060.97540.9768 0.7395 0.76420.70260.72600.90090.8773
0.50.94300.95880.93360.94820.97520.9774 0.8024 0.82540.76230.78410.93800.9159
0.80.95340.96100.94320.95020.96960.9706 0.9092 0.93110.86380.88461.02020.9993
1.00.96260.96840.95100.95820.97140.9710 0.9787 1.00000.92970.95001.08011.0607
2.00.94640.94220.93580.93160.94240.9390 1.3712 1.38931.30261.31981.47201.4523
5010%0.30.95040.95860.93880.94800.99360.9946 0.2266 0.23230.21520.22070.37670.3859
0.50.95180.95440.93980.94060.97920.9826 0.2666 0.27170.25330.25810.34430.3544
0.80.94980.94760.93800.93720.96560.9668 0.3342 0.33860.31750.32160.36500.3753
1.00.94560.94500.93460.93360.95540.9542 0.3832 0.38710.36410.36780.40220.4125
2.00.91040.90780.89960.89360.91940.91900.64500.64780.61270.61540.66010.6673
30%0.30.94440.94940.93260.93760.98180.9844 0.3856 0.39330.36630.37360.53260.5322
0.50.95720.95740.94680.94780.97520.9766 0.4211 0.42840.40010.40690.50770.5075
0.80.94580.94640.93100.93520.95720.9556 0.4804 0.48690.45630.46260.52370.5245
1.00.94920.94960.93740.93680.95100.9516 0.5267 0.53290.50040.50620.55670.5581
2.00.92640.92320.91280.91040.92760.92540.79730.80210.75740.76200.81920.8213
50%0.30.94300.95220.93320.94220.97420.9760 0.5699 0.58200.54140.55290.72100.7107
0.50.94920.95520.94000.94700.97280.9738 0.6060 0.61760.57570.58670.71150.7010
0.80.94580.95100.93340.93800.95920.9598 0.6728 0.68360.63920.64940.73960.7305
1.00.95240.95600.94120.94240.95940.9598 0.7215 0.73220.68540.69560.77530.7667
2.00.93700.93220.92360.91600.93220.9294 1.0160 1.02420.96520.97301.05581.0490
10010%0.30.94620.95160.93440.94060.99220.9920 0.1587 0.16090.15080.15280.24480.2483
0.50.94940.94940.93500.93600.97700.9776 0.1856 0.18750.17640.17810.22190.2259
0.80.94660.94440.93380.93240.95520.9554 0.2314 0.23280.21980.22120.24440.2486
1.00.94480.94360.93040.92900.94880.9482 0.2649 0.26610.25160.25280.27260.2766
2.00.92280.92160.90800.90780.92880.92840.46530.46610.44200.44280.47170.4746
30%0.30.94680.94880.93460.93700.98120.9840 0.2732 0.27610.25960.26230.36000.3600
0.50.94720.95200.93180.93560.97020.9702 0.2945 0.29730.27980.28240.33390.3340
0.80.95400.95580.94240.94460.95940.9608 0.3348 0.33720.31800.32030.35140.3520
1.00.94680.94740.93380.93400.95020.9484 0.3674 0.36950.34900.35100.37830.3792
2.00.92140.92040.90620.90760.92340.92220.56880.57070.54040.54210.57760.5788
50%0.30.94460.94800.93320.93700.97640.9758 0.4052 0.40950.38490.38900.51170.5081
0.50.94720.94900.93240.93780.96360.9650 0.4265 0.43070.40520.40920.47900.4755
0.80.94040.94300.92940.93180.94980.9498 0.4670 0.47070.44360.44710.49260.4894
1.00.94620.94680.93300.93180.94920.9482 0.5015 0.50500.47640.47970.52020.5173
2.00.93780.93460.92240.92260.93600.9338 0.7187 0.72110.68280.68500.73230.7306

Remark: Boldface indicates the recommended method for each case.

Fig 3

Performance measures of 95%CIs for θ: a = 15 (A) Coverage probabilities and (B) Expected lengths.

Remark: Boldface indicates the recommended method for each case. Remark: Boldface indicates the recommended method for each case. Remark: Boldface indicates the recommended method for each case.

An illustrative example

We applied the CIs constructed with the proposed methods to real-world data. In the week 29 July to 4 August 2019, Tropical Storm Wipha moved from Vietnam to northern Thailand, thereby putting the area at high risk of flash floods and landslides caused by heavy rain [1]. Thus, predicting the weekly natural rainfall data in the above-mentioned period is of interest. Data on the weekly rainfall during this period was collected by the Thailand Meteorological Department (TMD) (Table 4). The northern station includes 62 substations: 55 with positive rainfall records (88.71%) and the rest with no recorded rainfall.
Table 4

Data on weekly natural rainfall in northern Thailand in the week 29 July to 4 August 2019.

Weekly natural rainfall data
125.3160.1118.5148.85066.752.6131.145.2025.2106.50
050.176.871.831.4032.934.526.883.4189.1179.3309.7
206.6114.9283.161.525181516.64614.51524.723
8.120.8122.8228.610.2107.4026.926.217.715.622.934.1
279.146.134.6025.818.215.18.50

Source: Thailand Meteorogical Deparment

URL: https://www.tmd.go.th/services/weekly_report.php

Source: Thailand Meteorogical Deparment URL: https://www.tmd.go.th/services/weekly_report.php By applying the theory in Section and with known , the weekly positive rainfall data follow a normal distribution when they are log-transformed as . It is possible that this positive rainfall data have a lognormal distribution (the histogram and the empirical cumulative distribution function (CDF) plots in Fig 4). To determine which model fits the positive rainfall data, Nguyen [28] suggested that it might be insufficient to use the probability value (p-value) for decision-making alone in statistical testing of hypotheses. Thus, the Akaike information criterion (AIC) and Bayesian information criterion (BIC) are used to avoid using the p-value for model evaluation. Akaike [29] and Stone [30] defined the AIC and BIC which are the methods for scoring and selecting a suitable model derived from frequentist and Bayesian probabilities, respectively. Let θ be the set (vector) of model parameters and be the likelihood of the candidate model when evaluated at the MLE of θ. The AIC and BIC of a model are expressed as where k stands for the number of estimated parameters in the candidate model and n stands for the number of recorded measurements. The AIC and BIC results, in Table 5, reveal that the reduced rainfall data fit a lognormal distribution. When factoring in the empty rainfall records, the weekly positive rainfall data in the week 29 July to 4 August 2019 follow a delta-three parameter lognormal distribution. The descriptive statistics for the data are as follows: n = 62, , δ = 11.29%, and .
Fig 4

Histogram and empirical CDF plots of weekly rainfall records in northern Thailand in the week 29 July to 4 August 2019.

Table 5

Results of AIC and BIC for weekly positive rainfall data.

DistributionsCriteria
AICsBICs
Cauchy610.1506614.1653
Exponential575.2178577.2251
Gamma576.7389580.7536
Logistic622.7518626.7665
Lonormal 569.3073 573.3219
Normal628.9230632.9376
T-distribution612.0378618.0598
Weibull577.1703581.1850
The mean of the weekly positive rainfall records is 62.5183 mm/wk. Computations of the 95% CIs for the BCIs, GCI, and MOVER for the estimated mean are reported in Table 6. The weekly rainfall amounts infer heavy rain (35.1–90.0 millimetre), as per the criteria of the TMD [31]. Importantly, it is in line with the TMD warnings of heavy downpours and flash floods to the population living in the at-risk areas. For a = 1, the evidence in support of the estimated CIs can be found in the simulation results in Section.
Table 6

95%CIs for the weekly average natural rainfall in northern Thailand.

Methods95% CIs for θLengths
LowerUpper
HPD-NI144.929090.939946.0109
HPD-NI244.843990.683245.8393
ET-NI144.748790.661145.9124
ET-NI243.389484.189340.7999
GCI45.361891.503246.1414
MOVER44.793391.674346.8810

Discussion

Random samples were drawn from data following a delta-three parameter lognormal distribution including zero observations of proportion δ and highly skewed non-zero values in the remaining proportion following a three-parameter lognormal distribution. This distribution offers a solution for how to handle highly skewed observed data that cannot be modeled using a two-parameter lognormal distribution. In this study, CI estimates for the mean of a delta-three parameter lognormal distribution were developed based on BCIs (HPD and ET-based NI intervals), GCI, and MOVER. Applying our proposed methods to predict the weekly natural rainfall amount was the motivation for this study. When the threshold was large, HPD-NI1 provided better performance than the other methods in the extreme situation where the variance was small-to-medium, although it did not deal well with a large variance. The first reason is that the ET interval can substantially differ from the HPD region if the posterior density is highly skewed, as noted by Gelman et al. [20]. The next reason is that the NI1 prior was obtained from the prior of using its Fisher information matrix, which might make it stronger than the NI2 prior (the normal-gamma prior of ). However, it is important to note the limitation of HPD-NI1 when dealing with large . Likewise, the current study has a research gap in the perspectives on spatial information because it could be useful for statistical estimation if this study has been enabled by considerable insights associated with modeling framework using rainfall spatial analysis. See details in Banerjee et al. [32]. These need further research in the future.

Conclusions

The present study aimed to propose BCIs-based NI1 and NI2 priors, GCI, and MOVER for the logarithm of the mean of delta-three parameter lognormal model. Our numerical evaluation shows that in situations of small threshold, MOVER maintained a good performance and obtained the recommended CIs for large proportion of zeros except for large variance, while the next recommended CIs were obtained by apply GCI. On the other hand, HPD-NI1 performed quite well in situations of small-to-medium variance and a large threshold. Therefore, the HPD-NI1 is recommended for constructing CI estimation for the mean of a delta-three parameter lognormal distribution under these conditions. Furthermore, the GCI and MOVER are considered as the alternative methods.

Abbreviations commonly used throughout this article.

(PDF) Click here for additional data file.

Data on weekly natural rainfall in northern Thailand in the week 29 July to 4 August 2019.

(XLSX) Click here for additional data file.

Performance measures of 95%CIs for θ: a = 1.

(A) Coverage probabilities and (B) Expected lengths. (PDF) Click here for additional data file.

Performance measures of 95%CIs for θ: a = 5.

(A) Coverage probabilities and (B) Expected lengths. (PDF) Click here for additional data file.

Performance measures of 95%CIs for θ: a = 15.

(A) Coverage probabilities and (B) Expected lengths. (PDF) Click here for additional data file.

Histogram and empirical CDF plots of weekly rainfall records in northern Thailand in the week 29 July to 4 August 2019.

(PDF) Click here for additional data file.

CP and EL performances of 95% CI for θ: a = 1.

(PDF) Click here for additional data file.

CP and EL performances of 95% CI for θ: a = 5.

(PDF) Click here for additional data file.

CP and EL performances of 95% CI for θ: a = 15.

(PDF) Click here for additional data file. (PDF) Click here for additional data file.

Results of AIC and BIC for weekly positive rainfall data.

(PDF) Click here for additional data file.

95%CIs for the weekly average natural rainfall in northern Thailand.

(PDF) Click here for additional data file. 12 Jan 2022
PONE-D-21-29828
Bayesian interval estimations for the mean of delta-three parameter lognormal distribution with application to heavy rainfall data
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17 Mar 2022 Response to Reviewers Journal: PLOS ONE Manuscript ID: PONE-D-21-29828 Title name: Bayesian interval estimations for the mean of delta-three parameter lognormal distribution with application to heavy rainfall data Authors: Patcharee Maneerat, Pisit Nakjai and Sa-Aat Niwitpong Dear Academic Editor and Reviewers: We would like to thank you for your time and patience in reviewing our manuscript and making insightful comments toward improving it for resubmission. The following are our responses to each of your comments. Academic Editor” comments 1. We note that your manuscript is not formatted using one of PLOS ONE’s accepted file types. Please reattach your manuscript as one of the following file types: .doc, .docx, .rtf, or .tex (accompanied by a .pdf). If your submission was prepared in LaTex, please submit your manuscript file in PDF format and attach your .tex file as “other.” Response: The revised submission is prepared in LaTex and attached .tex file as “other.” 2. Please upload a Response to Reviewers letter which should include a point by point response to each of the points made by the Editor and / or Reviewers. (This should be uploaded as a 'Response to Reviewers' file type.) Please follow this link for more information: http://blogs.PLOS.org/everyone/2011/05/10/how-to-submit-your-revised-manuscript/ Response: Yes, we have uploaded a Response to Reviewers letter which should include a point by point response to each of the points made by the Editor and / or Reviewers. 3. Please state what role the funders took in the study. If the funders had no role, please state: "The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript." If this statement is not correct you must amend it as needed. Please include this amended Role of Funder statement in your cover letter; we will change the online submission form on your behalf. Response: We have stated "The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript." In the cover letter. Reviewers' comments: Reviewer #1: I enjoyed reading the manuscript. Bayesian inference certainly has several advantages over classical inference and the use of Bayesian confidence intervals to analyze rainfall is appealing. The manuscript is overall well written, but I have some suggestions and recommendations that should situate the paper better in terms of its scientific merits. 1. While the authors have painstakingly derived analytical formulas for the Bayesian credible intervals, these could be derived using sampling-based inference by drawing posterior samples using MCMC. Software frameworks such as WinBUGS and STAN and RJAGS can easily handle such methods. Response: Thank you for your valuable comments. We are looking forward to use these software frameworks in the future study. In this studies, we have used only package R version 4.1.3, in Algorithm 1 : BCIs (page 8), drawing posterior samples from parameters of the mean of delta-three parameter lognormal distribution can be approximately derived analytically, see pages 5-9. So we understand that MCMC is required when the posterior cannot be computed analytically. Moreover, it is also known form https://www.quora.com/Why-do-we-use-MCMC-algorithm-in-Bayesian-estimation that “Regarding Markov Chain, one of its bottleneck is the slower mixing time (or too many steps) to reach stationary distribution which usually worsen with an increasing number of dimensionality of the problem. Moreover, even when MCMC reaches stationary distribution, it still can only approximate the target distribution (to a normalising constant). In other words, the samples are not perfect.” So, drawing posterior samples using MCMC may be take longer times than the posterior samples in Algorithm 1: BCIs (page 8). 2. The one aspect of this research that has a gap is the lack of use of spatial information. A modeling framework that uses spatial analysis of rainfall will be very useful. There are a number of resources for Bayesian spatial analysis. The book, Banerjee, S., Carlin, B.P. and Gelfand, A.E. (2014). Hierarchical Modeling and Analysis for Spatial Data. Second Edition. Taylor and Francis CRC. This book should be referred. Response: Thank you for your suggestions. This book is referred our manuscript in the Discussion section. We look forward to hearing from you in the near future. Sincerely yours, Patcharee Maneerat, Pisit Nakjai, and Sa-Aat Niwitpong The authors Submitted filename: Response to Reviewers 17-3-2022.pdf Click here for additional data file. 22 Mar 2022 Bayesian interval estimations for the mean of delta-three parameter lognormal distribution with application to heavy rainfall data PONE-D-21-29828R1 Dear Dr. Niwitpong, We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements. Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication. An invoice for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at http://www.editorialmanager.com/pone/, click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. If you have any billing related questions, please contact our Author Billing department directly at authorbilling@plos.org. If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. Kind regards, Inés P. Mariño, Ph.D. Academic Editor PLOS ONE Additional Editor Comments (optional): Reviewers' comments: 24 Mar 2022 PONE-D-21-29828R1 Bayesian interval estimations for the mean of delta-three parameter lognormal distribution with application to heavy rainfall data Dear Dr. Niwitpong: I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department. If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org. If we can help with anything else, please email us at plosone@plos.org. Thank you for submitting your work to PLOS ONE and supporting open access. Kind regards, PLOS ONE Editorial Office Staff on behalf of Dr. Inés P. Mariño Academic Editor PLOS ONE
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Authors:  Patcharee Maneerat; Sa-Aat Niwitpong; Suparat Niwitpong
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4.  A Bayesian approach to construct confidence intervals for comparing the rainfall dispersion in Thailand.

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1.  Estimation methods for the ratio of medians of three-parameter lognormal distributions containing zero values and their application to wind speed data from northern Thailand.

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