Confidence intervals for functions of signal-to-noise ratio with application to economics and finance

Warisa Thangjai; Sa-Aat Niwitpong Article Confidence intervals for functions of signal-to-noise ratio with application to economics and finance Asian Journal of Economics and Banking (AJEB) Provided in Cooperation with: Ho Chi Minh University of Banking (HUB), Ho Chi Minh City Suggested Citation: Warisa Thangjai; Sa-Aat Niwitpong (2024) : Confidence intervals for functions of signal-to-noise ratio with application to economics and finance, Asian Journal of Economics and Banking (AJEB), ISSN 2633-7991, Emerald, Leeds, Vol. 8, Iss. 2, pp. 199-218, https://doi.org/10.1108/AJEB-12-2023-0129 This Version is available at: https://hdl.handle.net/10419/334121 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Confidence intervals for functions of signal-to-noise ratio with application to economics and finance Warisa Thangjai Department of Statistics, Ramkhamhaeng University, Bangkok, Thailand, and Sa-Aat Niwitpong Department of Applied Statistics, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand Abstract Purpose –Confidence intervals play a crucial role in economics and finance, providing a credible range of values for an unknown parameter along with a corresponding level of certainty. Their applications encompass economic forecasting, market research, financial forecasting, econometric analysis, policy analysis, financial reporting, investment decision-making, credit risk assessment and consumer confidence surveys. Signal-to-noise ratio (SNR) finds applications in economics and finance across various domains such as economic forecasting, financial modeling, market analysis and risk assessment. A high SNR indicates a robust and dependable signal, simplifying the process of making well-informed decisions. On the other hand, a low SNR indicates a weak signal that could be obscured by noise, so decision-making procedures need to take this into serious consideration. This research focuses on the development of confidence intervals for functions derived from the SNR and explores their application in the fields of economics and finance. Design/methodology/approach –The construction of the confidence intervals involved the application of various methodologies. For the SNR, confidence intervals were formed using the generalized confidence interval (GCI), large sample and Bayesian approaches. The difference between SNRs was estimated through the GCI, large sample, method of variance estimates recovery (MOVER), parametric bootstrap and Bayesian approaches. Additionally, confidence intervals for the common SNR were constructed using the GCI, adjusted MOVER, computational and Bayesian approaches. The performance of these confidence intervals was assessed using coverage probability and average length, evaluated through Monte Carlo simulation. Findings –The GCI approach demonstrated superior performance over other approaches in terms of both coverage probability and average length for the SNR and the difference between SNRs. Hence, employing the GCI approach is advised for constructing confidence intervals for these parameters. As for the common SNR, the Bayesian approach exhibited the shortest average length. Consequently, the Bayesian approach is recommended for constructing confidence intervals for the common SNR. Originality/value –This research presents confidence intervals for functions of the SNR to assess SNR estimation in the fields of economics and finance. Keywords Average length, Confidence interval, Coverage probability, Monte Carlo simulation, Signal-to-noise ratio Paper type Research paper 1. Introduction Confidence intervals play a crucial role in economics and finance, providing a reliable range of values for an unknown parameter along with a specified level of certainty. Here are diverse Confidence intervals for functions 199 © Warisa Thangjai and Sa-Aat Niwitpong. Published in Asian Journal of Economics and Banking. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/2615-9821.htm Received 18 December 2023 Revised 8 January 2024 Accepted 29 January 2024 Asian Journal of Economics and Banking Vol. 8 No. 2, 2024 pp. 199-218 Emerald Publishing Limited e-ISSN: 2633-7991 p-ISSN: 2615-9821 DOI 10.1108/AJEB-12-2023-0129 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 applications of confidence intervals in these fields. Economic forecasting: confidence intervals are essential for projecting economic metrics like gross domestic product (GDP) growth, inflation rates, and unemployment rates, providing a spectrum of values for the likely actual figures. Market research: in finance, confidence intervals help gauge the potential range of returns on investments. Analysts use them to convey confidence regarding future stock prices or returns on financial instruments. Risk management: confidence intervals play a pivotal role in evaluating and managing financial risk. They assist in approximating potential losses in investment portfolios, enabling informed decisions by investors and financial institutions. Financial forecasting: Confidence intervals are incorporated in financial modeling to project future cash flows, interest rates, and other financial parameters, enhancing the accuracy of predictions about the prospective financial performance of companies. Econometric analysis: in econometrics, confidence intervals gauge the precision of regression coefficients and other model parameters, which is crucial for determining the statistical significance of relationships between economic variables. Policy analysis: Economists use confidence intervals when scrutinizing the repercussions of policy changes, estimating the impact of a tax policy on consumer spending, and providing a confidence interval to convey associated uncertainty. Financial reporting: confidence intervals find application in financial statement analysis to estimate the precision of financial ratios, contributing to the assessment of the financial health and performance of companies. Investment decision-making: Investors rely on confidence intervals to assess potential returns and risks linked to diverse investment opportunities, aiding in making well-informed decisions concerning asset allocation and portfolio management. Credit risk assessment: in banking and finance, confidence intervals are used to evaluate credit risk, estimating the potential range of default probabilities, and establishing suitable interest rates for loans. Consumer confidence surveys: Confidence intervals are employed in the analysis and interpretation of survey data, such as consumer confidence surveys, providing a measure of uncertainty around reported confidence levels. In conclusion, confidence intervals serve as a valuable tool in economics and finance, offering a method to quantify and convey uncertainty in various analyses and decision-making processes. The signal-to-noise ratio (SNR) in the realms of economics and finance originates from signal processing, representing the proportion of valuable information, termed the signal to irrelevant or random background noise. In the context of economic and financial analysis, this concept is commonly utilized to evaluate the information’s quality and the signal’sstrengthcomparedto the surrounding noise. Within economic and financial analysis, the term signal denotes meaningful and pertinent data or patterns, while noise pertains to random fluctuations or inconsequential information. The SNR functions as a metric for assessing the clarity and dependability of a signal amidst background noise. The applications of SNR in economics and finance extend across various domains, encompassing economicforecasting, financial modeling, market analysis, and risk assessment. A high SNR suggests a robust and dependable signal, facilitating more straightforward decision-making. Conversely, a low SNR implies a weak signal, potentially obscured by noise, necessitating careful consideration in decision-making processes. In essence, comprehending and managing the SNR is paramount for extracting meaningful insights and making informed decisions within economic and financial contexts. Point estimation involves providing a single, specific value as an estimate for an unknown parameter in a population. For example, estimate the population mean based on a sample mean. Interval estimation, on the other hand, provides a range of values (an interval) within which the true parameter is likely to lie. This is typically expressed as a confidence interval. Interval estimation is often considered better than point estimation. This is because it incorporates uncertainty, confidence level, decision-making, and robustness. To incorporate uncertainty, interval estimation explicitly acknowledges the uncertainty inherent in estimating population parameters from a sample. It provides a sense of the range of AJEB 8,2 200 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 plausible values. For confidence level, confidence intervals come with a specified confidence level (e.g., 95%). This indicates the proportion of intervals from repeated sampling that would include the true parameter. It offers a clear indication of the reliability of the estimate. For decision-making purposes, having a range of values is often more informative than a single point. It allows decision-makers to consider a spectrum of possibilities. For robustness, point estimates can be sensitive to outliers or extreme values in the data. Confidence intervals, especially those based on robust methods, may be less affected by extreme observations. A confidence interval for a parameter of interest is a statistical range that provides an estimated range of values that is likely to include the true value of the parameter. It is constructed based on the sample data and is associated with a certain level of confidence. The confidence interval is a measure of the precision or uncertainty of the estimated parameter. For example, if you are estimating the SNR of a population, a 95% confidence interval would imply that if you were to take many samples and construct a confidence interval from each, about 95% of those intervals would contain the true population SNR. Moreover, the confidence interval for the difference between parameters of interest is a range of values that is likely to contain the true difference between two population parameters. This type of interval estimation is commonly used in statistical analysis, especially when comparing two groups or assessing the impact of an intervention. In addition, the confidence interval for a common parameter of interest is an interval estimate that provides a range of plausible values for the true value of a parameter. This type of interval estimation is commonly used in statistical analysis when dealing with a single population parameter. 2. Literature review The Generalized Confidence Interval (GCI) approach is designed to be versatile across diverse data types and statistical scenarios. It is not constrained by specific distributional assumptions, making it applicable in situations where classical methods may not be appropriate. The GCI has diverse applications in fields such as economics, finance, biology, and any domain requiring statistical inference. This methodology utilizes the generalized pivotal quantity (GPQ) to construct the confidence interval, enabling the estimation of confidence intervals for complex parameters. However, it’s important to note that the numerical simulation of the GCI approach relies solely on the maximum likelihood estimate. Many researchers have undertaken comparisons between the GCI approach and alternative methods for constructing confidence intervals, as evidenced in studies by Weerahandi (1993), Krishnamoorthy and Lu (2003),Krishnamoorthy and Mathew (2003),Tian (2005),Chen and Zhou (2006),Tian and Wu (2007),Ye et al. (2010),Saothayanun and Thangjai (2018),Thangjai and Niwitpong (2019),Thangjai and Niwitpong (2020a), and Thangjai and Niwitpong (2020b). Constructing confidence intervals using the large sample approach involves exploiting asymptotic properties, particularly when dealing with a substantial volume of data. This method relies on the Central Limit Theorem, which posits that the distribution of sample means converges to a normal distribution as the sample size increases. Utilizing this principle allows for the estimation of confidence intervals under the assumption of normality, enhancing their applicability in extensive datasets. The large sample approach is advantageous due to its simplicity in constructing the confidence interval using the exact formula. However, a limitation is that it requires a large sample size for estimating the confidence interval. Several scholars have evaluated the large sample approach in comparison to alternative methods for constructing confidence intervals, as demonstrated in the research conducted by Tian and Wu (2007),Saothayanun and Thangjai (2018), Thangjai and Niwitpong (2019), and Thangjai and Niwitpong (2020b). The MOVER approach relies on the original confidence interval for a specific parameter of interest to derive the final confidence interval. An advantage of the MOVER approach is its Confidence intervals for functions 201 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 ease of computation using the exact formula. However, a drawback is that it can be constructed with or without the initial confidence interval for a single parameter of interest. Several researchers, including Zou and Donner (2008),Zou et al. (2009),Saothayanun and Thangjai (2018),Thangjai and Niwitpong (2019), and Thangjai and Niwitpong (2020b), have recommended the utilization of the MOVER approach in constructing confidence intervals. The adjusted MOVER approach is inspired by the principles of both the large sample and MOVER approaches. Its advantage lies in the straightforward application of the exact formula for confidence interval computation, although a drawback is that it relies on the initial confidence interval for a single parameter. Thangjai and Niwitpong (2020a) has delved into the investigation of the adjusted MOVER approach. The bootstrap approach involves approximating the sampling distribution of statistics by iteratively resampling with replacements from the population. These multiple bootstrap samples, drawn from the population on numerous occasions, function as representative samples of the entire population. The bootstrap approach provides a simple and reasonably accurate technique for constructing confidence intervals. However, a drawback is the requirement for knowledge regarding the distribution of estimates around the true values because the sampling distribution aligns with the data distribution, considering that the estimates are derived from the data. Various researchers, including Chachi (2017) and Thangjai and Niwitpong (2020b), have advocated for the utilization of the bootstrap approach. The computational approach is employed to formulate confidence intervals for intricate parameters. This technique involves simulations and numerical computations utilizing the maximum likelihood estimate. Scholars have introduced the computational approach for confidence intervals, as demonstrated in works by Pal et al. (2007) and Thangjai and Niwitpong (2020a). The Bayesian approach employs posterior probability and facilitates comparison with alternative methods for constructing credible intervals. The primary motivation for opting for the Bayesian approach is the complexity of models that traditional methods may struggle to address. It is essential to emphasize that, irrespective of the rationale behind adopting the Bayesian approach, conducting a sensitivity analysis of priors is always crucial and should be included. This comparison is substantiated by studies such as Rao and D’Cunha (2016) and Ma and Chen (2018). 3. Methodology The SNR can be described as the reciprocal of the coefficient of variation. The SNR is calculated as the ratio of the mean to the standard deviation. This paper discussed three parts as follow: The SNR, the difference between SNRs, and the common SNR. 3.1 Confidence intervals for the SNR Suppose that random sample X ¼ðX1;X2;...;XnÞfollows any distribution. Suppose that μ and σ are population mean and population standard deviation of the distribution, respectively. The SNR is defined as θ¼ μ σ :(1) Let X and S are sample mean and sample standard deviation of the distribution, respectively. The estimator of the SNR is defined as bθ¼X S:(2) AJEB 8,2 202 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 3.1.1 GCI approach for SNR. The concept of GCI was introduced by Weerahandi (1993). Let X ¼ðX1;X2; :::; XnÞbe a random sample having a density function fðXjθ; υ Þ, where θis the parameter of interest and υ is a nuisance parameter. Let x be the observed sample of X. A generalized pivotal quantity RðX;x;θ; υ Þis considered and satisfies the following conditions: (i) The distribution of RðX;x;θ; υ Þis free of all unknown parameters. (ii) The observed value of RðX;x;θ; υ Þis the parameter of interest. Condition (i) is imposed to guarantee that a subset of the sample space of the possible values of RðX;x;θ; υ Þcan be found at a given value of the confidence coefficient with no knowledge of the parameters. Condition (ii) is imposed to ensure that such probability statements based on the GPQ lead to confidence regions involving observed data x only. The GCI for θis computed using the percentiles of the GPQ. Let ½Rð α =2Þ;Rð1− α =2Þ be a 100ð1− α Þ% twosided GCI for the parameter of interest, where Rð α =2Þand Rð1− α =2Þdenote the 100ð α =2Þ-th and the 100ð1− α =2Þ-th percentiles of RðX;x;θ; υ Þ, respectively. Following Saothayanun and Thangjai (2018). Let R μ be the GPQ of μ and let R σ be the GPQ of σ . The GPQ of θis defined as Rθ¼R μ R σ :(3) The 100ð1− α Þ% two-sided confidence interval for the SNR based on the GCI approach is given by CIθ:GCI ¼½Lθ:GCI;Uθ:GCI¼½Rθð α =2Þ;Rθð1 α =2Þ;(4) where Rθð α =2Þand Rθð1− α =2Þdenote the ð α =2Þ-th and ð1− α =2Þ-th quantiles of Rθ, respectively. The following algorithm is used to construct the GCI for the SNR. Algorithm 1. For a given x and s For g ¼1tom Compute R μ Compute R σ Compute Rθ End g loop Compute the ð α =2Þ-th quantiles of Rθdefined by Rθð α =2Þ Compute the ð1− α =2Þ-th quantiles of Rθdefined by Rθð1− α =2Þ 3.1.2 Large sample approach for SNR. According to Saothayanun and Thangjai (2018) the 100ð1− α Þ% two-sided confidence interval for the SNR based on the large sample approach is given by CIθ:LS ¼½Lθ:LS;Uθ:LS¼ bθz1− α =2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var bθ  r;bθþz1− α =2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var bθ  r  ;(5) where z1− α =2denotes the ð1− α =2Þ-th quantile of a standard normal distribution and VarðbθÞis the variance of the estimator of SNR. 3.1.3 Bayesian approach for SNR. Bayes’rule is utilized to revise the prior distribution, resulting in the posterior distribution, which encompasses all relevant information regarding the unknown parameters inferred from the observed data. The Bayesian approach provides a framework for adjusting beliefs and making predictions based on new evidence or data. It is grounded in Bayes’theorem, which integrates prior probability and likelihood to compute the Confidence intervals for functions 203 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 posterior probability. The prior distribution reflects uncertainty about parameters before observing the data. In this study, we utilized Jeffreys’independence prior. Let σ jx be the posterior distribution of σ . And let μ j σ ;x be the posterior distribution of μ given σ . Let θBS be the posterior distribution using σ jx and μ j σ ;x. The 100ð1− α Þ% two-sided confidence interval for the SNR based on the Bayesian approach is given by CIθ:BS ¼½Lθ:BS;Uθ:BS;(6) where Lθ:BS and Uθ:BS are the lower and upper limits of the shortest 100ð1− α Þ% highest posterior density interval of θBS, respectively. The following algorithm is used to construct the Bayesian credible interval for the SNR. Algorithm 2. For a given x and s For g ¼1tom Compute σ jx Compute μ j σ ;x Compute θBS End g loop Compute the shortest 100ð1− α Þ% highest posterior density interval of θBS The following algorithm is used to evaluate the coverage probabilities and average lengths of the confidence intervals for SNR. Algorithm 3. For a given μ , σ , and θ For h ¼1toM Generate x Calculate x and s Construct the confidence interval ½Lθ:GCI;Uθ:GCI Construct the confidence interval ½Lθ:LS;Uθ:LS Construct the confidence interval ½Lθ:BS;Uθ:BS If L ≤θ≤U, set p ¼1; else set p ¼0 Compute U −L End h loop Compute mean of p defined by the coverage probability Compute mean of U −L defined by the average length 3.2 Confidence intervals for the difference between SNRs Suppose that X ¼ðX1;X2;:::;XnÞfollows any distribution with mean μ Xand standard deviation σ X. Similarly, let Y ¼ðY1;Y2; :::; YmÞbe any distribution with mean μ Yand standard deviation σ Y. Moreover, X and Y are independent. The single SNRs of X and Y are given by θX¼ μ X σ X and θY¼ μ Y σ Y :(7) The difference between of SNRs is defined as δ¼θXθY:(8) Let Xand SXare sample mean and sample standard deviation of X, respectively. Moreover, let Y and SYare sample mean and sample standard deviation of Y, respectively. Suppose that bθX and bθYare the estimators of θXand θY, respectively, which are given AJEB 8,2 204 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 bθX¼X SX and bθY¼Y SY :(9) The difference between of SNRs is defined as bδ¼bθXbθY:(10) Suppose that VarðbθXÞand VarðbθYÞare the variances of bθXand bθY, respectively. The variance of bδ¼bθX−bθYis Var bδ ¼Var bθXbθY  ¼Var bθX  þVar bθY  :(11) 3.2.1 GCI approach for the difference between SNRs. According to Thangjai and Niwitpong (2019) and Thangjai and Niwitpong (2020b). Let R μ Xbe the GPQ of μ Xand let R σ Xbe the GPQ of σ X. The GPQ of θXis defined as RθX¼R μ X R σ X :(12) Moreover, let R μ Ybe the GPQ of μ Yand let R σ Ybe the GPQ of σ Y. The GPQ of θYis defined as RθY¼R μ Y R σ Y :(13) Therefore, the difference between the GPQs of SNRs is Rδ¼RθXRθY:(14) The 100ð1− α Þ%two-sided confidence interval for the difference between SNRs based on the GCI approach is given by CIδ:GCI ¼½Lδ:GCI;Uδ:GCI¼½Rδð α =2Þ;Rδð1 α =2Þ;(15) where Rδð α =2Þand Rδð1− α =2Þdenote the ð α =2Þ-th and ð1− α =2Þ-th quantiles of Rδ, respectively. The following algorithm is used to construct the GCI for the difference between SNRs. Algorithm 4. For a given x, y, sXand sY For g ¼1tom Compute R μ X,R σ X, and RθX Compute R μ Y,R σ Y, and RθY Compute Rδ End g loop Compute the ð α =2Þ-th quantiles of Rδdefined by Rδð α =2Þ Compute the ð1− α =2Þ-th quantiles of Rδdefined by Rδð1− α =2Þ 3.2.2 Large sample approach for the difference between SNRs. Following Thangjai and Niwitpong (2019) and Thangjai and Niwitpong (2020b) using the central limit theorem, the 100ð1− α Þ% two-sided confidence interval for the difference between SNRs based on the large sample approach is given by CIδ:LS ¼½Lδ:LS;Uδ:LS¼ bδz1− α =2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var bδ  r;bδþz1− α =2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var bδ  r  ;(16) where z1− α =2is the ð1− α =2Þ-th quantile of the standard normal distribution and VarðbδÞis the variance of the estimator of difference between SNRs. Confidence intervals for functions 205 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 3.2.3 MOVER for the difference between SNRs. Let lXand uXbe the lower and upper limits of the confidence interval for SNR of X, respectively. Similarly, let lYand uYbe the lower and upper limits of the confidence interval for SNR of Y, respectively. Following Zou and Donner (2008),Zou et al. (2009),Thangjai and Niwitpong (2019) and Thangjai and Niwitpong (2020b), the 100ð1− α Þ% two-sided confidence interval for the difference between the SNRs based on the MOVER approach is given by CIδ:MOVER ¼½Lδ:MOVER;Uδ:MOVER ¼bθXbθYffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bθXlX  2 þuYbθY  2 r;bθXbθYþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uXbθX  2 þbθYlY  2 r " #: (17) 3.2.4 Parametric bootstrap approach for the difference between SNRs. The parametric bootstrap approach is a resampling approach based on independently sampling with a replacement from existing sample data of the same sample size. Let X*¼ðX* 1;X* 2; :::; X* nÞbe sample with replacement from X ¼ðX1;X2; :::; XnÞwith sample size n and let x*¼ðx* 1;x* 2; :::; x* nÞbe the observed values of X*¼ðX* 1;X* 2; :::; X* nÞ. Similarly, let Y*¼ðY* 1;Y* 2; :::; Y* mÞbe the sample from Y ¼ðY1;Y2; :::; YmÞwith replacement sample size m and let y*¼ðy* 1;y* 2; :::; y* mÞbe the observed values of Y*¼ðY* 1;Y* 2; :::; Y* mÞ. The re-sampled sample is called a bootstrap sample. The difference in SNRs from the bootstrap sample is obtained by δ*¼θ* Xθ* Y:(18) An estimator of the difference of SNRs is bδ *¼bθ * Xbθ * Y:(19) For replicate B times, there are totally B estimates of the difference of SNRs δ*from B bootstrap sample. The sampling distribution is constructed with these B bootstrap statistics. The confidence interval for the difference in SNRs is calculated using the distribution. Therefore, the 100ð1− α Þ% two-sided confidence interval for the difference between SNRs based on the parametric bootstrap approach is given by CIδ:PB ¼½Lδ:PB;Uδ:PB¼hbδz1− α =2S*;bδþz1− α =2S*i;(20) where z1− α =2is the ð1− α =2Þ-th quantile of the standard normal distribution and S*is the standard deviation of bδ * . The following algorithm is used to construct the parametric bootstrap confidence interval for the difference between SNRs. Algorithm 5. For a given x*, y*,s * X, and s* Y For g ¼1tom Compute bθ * X Compute bθ * Y Compute bδ * End g loop Compute S* Compute Lδ:PB and Uδ:PB AJEB 8,2 206 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 ðn;mÞðθX;θYÞ CP (AL) CIδ:GCI CIδ:LS CIδ:MOVER CIδ:PB CIδ:BS (30,30) (10,1) 0.9506 (5.3335) 0.9518 (5.3605) 0.9602 (5.5937) 0.9392 (5.6483) 0.9492 (5.2891) (10,2) 0.9516 (5.4272) 0.9528 (5.4499) 0.9608 (5.6870) 0.9410 (5.7394) 0.9478 (5.3764) (10,5) 0.9512 (5.9269) 0.9530 (5.9553) 0.9618 (6.2144) 0.9474 (6.3042) 0.9482 (5.8786) (10,10) 0.9548 (7.5176) 0.9578 (7.5454) 0.9660 (7.8737) 0.9470 (8.0065) 0.9562 (7.4591) (30,50) (10,1) 0.9484 (5.2930) 0.9504 (5.3196) 0.9588 (5.5500) 0.9400 (5.6141) 0.9452 (5.2461) (10,2) 0.9474 (5.3725) 0.9488 (5.3991) 0.9588 (5.6312) 0.9396 (5.6993) 0.9460 (5.3268) (10,5) 0.9484 (5.6744) 0.9500 (5.6973) 0.9584 (5.9316) 0.9410 (6.0113) 0.9472 (5.6214) (10,10) 0.9568 (6.6862) 0.9572 (6.7086) 0.9642 (6.9558) 0.9500 (7.0183) 0.9554 (6.6342) (50,50) (10,1) 0.9448 (4.0614) 0.9456 (4.0721) 0.9518 (4.1751) 0.9340 (4.1708) 0.9410 (4.0268) (10,2) 0.9488 (4.1327) 0.9498 (4.1439) 0.9550 (4.2488) 0.9376 (4.2406) 0.9460 (4.0970) (10,5) 0.9472 (4.5328) 0.9474 (4.5449) 0.9530 (4.6600) 0.9452 (4.6654) 0.9430 (4.4986) (10,10) 0.9498 (5.7174) 0.9508 (5.7315) 0.9560 (5.8766) 0.9454 (5.8767) 0.9486 (5.6727) (50,100) (10,1) 0.9488 (4.0481) 0.9506 (4.0596) 0.9576 (4.1619) 0.9392 (4.1476) 0.9466 (4.0145) (10,2) 0.9466 (4.0751) 0.9480 (4.0896) 0.9536 (4.1919) 0.9428 (4.1858) 0.9426 (4.0444) (10,5) 0.9534 (4.2789) 0.9520 (4.2896) 0.9600 (4.3915) 0.9424 (4.3751) 0.9514 (4.2441) (10,10) 0.9488 (4.9270) 0.9504 (4.9364) 0.9542 (5.0405) 0.9426 (5.0405) 0.9482 (4.8904) (100,100) (10,1) 0.9504 (2.8447) 0.9508 (2.8476) 0.9542 (2.8828) 0.9418 (2.8633) 0.9460 (2.8196) (10,2) 0.9462 (2.8867) 0.9488 (2.8885) 0.9514 (2.9242) 0.9432 (2.9072) 0.9452 (2.8609) (10,5) 0.9484 (3.1648) 0.9470 (3.1688) 0.9500 (3.2080) 0.9460 (3.1970) 0.9464 (3.1406) (10,10) 0.9516 (3.9844) 0.9532 (3.9880) 0.9558 (4.0373) 0.9494 (4.0215) 0.9494 (3.9529) Source(s): Authors’calculation Table 2. The CPs and ALs of 95% two-sided confidence intervals for the difference between SNRs of log-normal distributions Confidence intervals for functions 213 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 Thailand. Similarly, the SET100 Index encompasses the price movements of 100 largecapitalization securities with notable trading liquidity on the same exchange. On the other hand, the sSET Index captures the price changes of common stocks beyond those included in the SET50 and SET100 indices. These stocks exhibit consistent liquidity and adhere to specified requirements related to share distribution among minor shareholders. Price-earnings ratios for the SET50, SET100, and sSET indexes are computed from monthly index data provided by the Stock Exchange of Thailand. This study focuses on monthly index data spanning from January to November 2023, as detailed in Table 4. The histograms depicting daily rainfall data can be found in Figure 1, and Table 5 presents sample sizes, means, standard deviations, and SNRs for the three indexes. Before applying our methods to real data, it is crucial to assess the assumption that the logarithms of the data are drawn from a normal distribution. Traditionally, the Shapiro–Wilk normality test was employed, yielding p-values of 0.3922, 0.1467, and 0.08508 for SET50, SET100, and sSET indexes, respectively. Recognizing the limitations of p-values in testing, alternative methods for checking normality include graphical tools such as QQ-plots or Bayesian tests. Analysis of Table 6 reveals the minimum Akaike Information Criterion (AIC) values from Bayesian tests across five regions, indicating that SET50, SET100, and sSET indexes follow lognormal distributions. Furthermore, the normal QQ-plots of log-data in Figure 2 affirm the results of the Bayesian test. For illustrative purposes, we exclusively select data from lognormal distributions to showcase our estimation approaches. For SET50 index, the 95% confidence intervals for the SNR based on the GCI, large sample, and Bayesian approaches are CIθ:GCI ¼[11.4901,28.5148] with an interval length of 17.0247, CIθ:LS ¼[11.2126,28.7475] with an interval length of 17.5349, and CIθ:BS ¼ [10.8550,28.0229] with an interval length of 17.1679, respectively. For SET100 index, the 95% confidence intervals for the SNR based on the GCI, large sample, and Bayesian approaches are CIθ:GCI ¼[8.6529,21.7304] with an interval length of 13.0775, CIθ:LS ¼ [8.4493,21.6850] with an interval length of 13.2357, and CIθ:BS ¼[8.0436,21.1895] with an interval length of 13.1459, respectively. For sSET index, the 95% confidence intervals for the SNR based on the GCI, large sample, and Bayesian approaches are CIθ:GCI ¼[7.9635,20.0325] with an interval length of 12.0690, CIθ:LS ¼[7.8989,20.2794] with an interval length of 12.3805, and CIθ:BS ¼[7.9815,20.1948] with an interval length of 12.2133, respectively. Notably, the confidence intervals for the SNR based on the GCI, large sample, and Bayesian approaches encompass the true value of the SNR. However, the GCI approach has a shorter length than the large sample and Bayesian approaches. For difference between SET50 index and SET100 index, the true difference between the SNRs is 4.9129. The 95% confidence intervals for the difference between SNRs based on the GCI, large sample, MOVER, parametric bootstrap, Bayesian approaches are CIδ:GCI ¼ ðn1;n2;n3Þð σ 1; σ 2; σ 3Þ CP (AL) CIγ:GCI CIγ:AM CIγ:CA CIγ:BS (30,30,30) (0.10,0.29,0.47) 0.9530 (1.2606) 0.8846 (1.0129) 0.9394 (1.2970) 0.9478 (1.2424) (0.29,0.47,0.83) 0.9484 (0.7425) 0.8908 (0.6071) 0.9386 (0.7596) 0.9444 (0.7317) (50,50,50) (0.10,0.29,0.47) 0.9512 (0.9656) 0.8814 (0.7612) 0.9456 (0.9823) 0.9504 (0.9523) (0.29,0.47,0.83) 0.9550 (0.5699) 0.8952 (0.4578) 0.9498 (0.5776) 0.9528 (0.5614) (30,50,100) (0.10,0.29,0.47) 0.9482 (0.6554) 0.9100 (0.5783) 0.9426 (0.6650) 0.9426 (0.6465) (0.29,0.47,0.83) 0.9484 (0.4010) 0.9170 (0.3554) 0.9508 (0.4065) 0.9444 (0.3954) (100,100,100) (0.10,0.29,0.47) 0.9544 (0.6780) 0.8736 (0.5262) 0.9458 (0.6842) 0.9494 (0.6689) (0.29,0.47,0.83) 0.9448 (0.4000) 0.8792 (0.3169) 0.9454 (0.4032) 0.9414 (0.3944) Source(s): Authors’calculation Table 3. The CPs and ALs of 95% two-sided confidence intervals for the common SNR of several log-normal distributions AJEB 8,2 214 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 Sample statistics Index SET50 SET100 sSET ni11 11 11 yi19.57 18.83 17.21 sYi1.00 1.29 1.26 xi2.97 2.93 2.84 sXi0.05 0.07 0.07 bθi19.98 15.07 14.09 Source(s): Authors’calculation Index Price-earnings ratios SET50 18.90 20.09 20.07 18.71 18.90 18.79 19.69 21.77 20.45 19.58 18.33 SET100 17.54 18.79 18.80 17.52 18.24 18.07 18.66 21.73 20.44 19.43 17.87 sSET 16.00 17.52 17.49 16.45 17.15 16.07 16.15 20.05 18.84 17.20 16.44 Source(s): Stock Exchange of Thailand (https://www.set.or.th/th/market/statistics/market-statistics/main) Authors’calculation Figure 1. Histogram plots of monthly price-earnings ratios of three indexes Table 5. Sample statistics of price-earnings ratios of three indexes Table 4. Price-earnings ratios of three indexes Confidence intervals for functions 215 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 [5.9406,15.6047] with a length of interval of 21.5453, CIδ:LS ¼[6.0718,15.8977] with a length of interval of 21.9695, CIδ:MOVER ¼[7.5748,17.4007] with a length of interval of 24.9755, CIδ:PB ¼[11.6552,21.9753] with a length of interval of 33.6305, and CIδ:BS ¼ [6.2144,15.6602] with a length of interval of 21.8746, respectively. For difference between SET50 index and sSET index, the true difference between the SNRs is 5.8909. The 95% confidence intervals for the difference between SNRs based on the GCI, large sample, MOVER, parametric bootstrap, Bayesian approaches are CIδ:GCI ¼[4.1340,16.4281] with a length of interval of 20.5621, CIδ:LS ¼[4.8416,16.6235] with a length of interval of 21.4651, CIδ:MOVER ¼[6.3101,18.0920] with a length of interval of 24.4021, CIδ:PB ¼[9.7428,21.7973] with a length of interval of 31.5401, and CIδ:BS ¼[5.2779,15.4926] with a length of interval of 20.7705. For difference between SET100 index and sSET index, the true difference between the SNRs is 0.9780. The 95% confidence intervals for the difference between SNRs based on the GCI, large sample, MOVER, parametric bootstrap, Bayesian approaches are CIδ:GCI ¼ [7.7565,9.7044] with a length of interval of 17.4609, CIδ:LS ¼[8.0837,10.0398] with a length of interval of 18.1235, CIδ:MOVER ¼[9.3236,11.2796] with a length of interval of 20.6032, CIδ:PB ¼[14.3997,16.7647] with a length of interval of 31.1644, and CIδ:BS ¼[8.0435,9.8604] with a length of interval of 17.9039. The results indicate that all confidence intervals contain the true difference between the SNRs. However, the GCI approach stands out by providing the shortest length, making it the most preferable among the alternatives. The true common SNRs is 15.6870. The 95% confidence intervals for the common SNR based on GCI, adjusted MOVER, computational, and Bayesian approaches are CIγ:GCI ¼ Distribution AIC SET50 index SET100 index sSET index Normal 34.09 39.76 39.31 Log-normal 33.67 39.00 38.50 Gamma 33.88 39.34 38.86 Exponential 88.43 87.58 85.61 Source(s): Authors’calculation Figure 2. The normal QQ-plots of log-monthly priceearnings ratios of three indexes Table 6. The AIC values of price-earnings ratios of three indexes AJEB 8,2 216 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/2/199/9598675/ajeb-12-2023-0129.pdf by ZBW German National Library of Economics user on 16 December 2025 [9.7447,18.5943] with a length of interval of 8.8496, CIγ:AM ¼[11.1191,20.2548] with a length of interval of 9.1357, CIγ:CA ¼[12.0582,21.0089] with a length of interval of 8.9507, and CIγ:BS ¼ [9.7514,18.7842] with a length of interval of 9.0328. 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