Unbiased Estimates Research Articles

Early versus Delayed Cholecystectomy in Mild Gallstone Pancreatitis (Prospective Comparative Study)

Abstract Background Gallstone pancreatitis is the most common etiology of acute pancreatitis accounting for up to sixty percent of all cases. In patients with mild gallstone pancreatitis – characterized by the absence of organ failure, peripancreatic fluid collections or necrosis, and typical resolution within one week – cholecystectomy is the definitive treatment. Aim of the Work To determine the feasibility of early cholecystectomy during index admission as long as the symptoms and laboratory values are improving for patients with gallstone pancreatitis predicted to be mild on admission and to obtain unbiased estimates of the effect of early cholecystectomy on hospital LOS, complications, and patient-reported outcomes (PROs) in order to determine the need for further evaluation. Patients and Methods This Prospective comparative study was conducted at tertiary care hospitals at Ain Shams University hospitals and Nasser institute for medical research and treatment from May 2022 till December 2022 and performed on a total of 20 adult patients who aged 18 years old and over with mild pancreatitis and without cholangitis. Results Patients treated with early or late procedure experienced approximately the same intra operative bleeding (30 vs 50%), conversion to open (10 vs 10%), biliary injury (0 vs 0%), and operation time (50 vs 50 minutes) with no statistically significant differences between them (p value= 1.00, 0.65, 0.912) respectively. Conclusion Early cholecystectomy after gallstone pancreatitis is more advantageous than late cholecystectomy after gallstone pancreatitis because it is associated with less rate of recurrence of biliary pancreatitis, less complication of pancreatitis, less perioperative complication (adhesions, blood loss, biliary events, infection, postoperative pain) and shorter duration of postoperative hospital stay.

Phenological Adaptation Is Insufficient to Offset Climate Change-Induced Yield Losses in US Hybrid Maize.

Climate change is projected to decrease maize yields due to warmer temperatures and their consequences. Studies using crop growth models (CGMs), however, have predicted that, through a combination of alterations to planting date, flowering time, and maturity, these yield losses can be mitigated or even reversed. Here, we examine three assumptions of such studies: (1) that climate has driven historical phenological trends, (2) that CGM ensembles provide unbiased estimates of yields under high temperatures, and (3) that the effects of temperature on yields are an emergent property of interactions between phenology and environment. We used data on maize phenology from the United States Department of Agriculture, a statistical model of maize hybrid heat tolerance derived from 80 years of public yield trial records across four US states, and outputs of an ensemble of CMIP6 climate models. While planting dates have advanced historically, we found a trend toward later planting dates after 2005 and no trend for silking or maturity, shifting more time into the reproductive period. We then projected maize yields using the historical model and crop calendars devised using three previously proposed adaptation strategies. In contrast to studies using CGMs, our statistical yield model projected severe yield losses under all three strategies. Finally, we projected maize yields accounting for historical genetic variability for heat tolerance, discovering that it was insufficient to overcome the negative effects of projected warming. These projections are driven by greater heat stress exposure under all crop calendars and climate scenarios. Combined with analysis of the internal sensitivities of CGMs to temperature, our results suggest that current projections do not adequately account for the effects of increasing temperatures on maize yields. Climate adaptation in the US Midwest must utilize a richer set of strategies than phenological adaptation, including improvements to heat tolerance and crop diversification.

GWAS and Meta-QTL Analysis of Kernel Quality-Related Traits in Maize.

The quality of corn kernels is crucial for their nutritional value, making the enhancement of kernel quality a primary objective of contemporary corn breeding efforts. This study utilized 260 corn inbred lines as research materials and assessed three traits associated with grain quality. A genome-wide association study (GWAS) was conducted using the best linear unbiased estimator (BLUE) for quality traits, resulting in the identification of 23 significant single nucleotide polymorphisms (SNPs). Additionally, nine genes associated with grain quality traits were identified through gene function annotation and prediction. Furthermore, a total of 697 quantitative trait loci (QTL) related to quality traits were compiled from 27 documents, followed by a meta-QTL analysis that revealed 40 meta-QTL associated with these traits. Among these, 19 functional genes and reported candidate genes related to quality traits were detected. Three significant SNPs identified by GWAS were located within the intervals of these QTL, while the remaining eight significant SNPs were situated within 2 Mb of the QTL. In summary, the findings of this study provide a theoretical framework for analyzing the genetic basis of corn grain quality-related traits and for enhancing corn quality.

Adaptive radar target detection in nonzero-mean compound Gaussian sea clutter with random texture

This paper deals with the radar target detecting problem in nonzero-mean compound Gaussian sea clutter with random texture. The texture is considered to be an inverse Gamma, Gamma, or inverse Gaussian variable. Three novel adaptive detectors using the two-step maximum a posteriori (MAP) generalized likelihood ratio test (GLRT) are proposed. More precisely, we derive the test statistics of the proposed detectors for known mean vector (MV) and speckle covariance matrix (CM) in the first step. In the second step, unbiased and consistent estimators are proposed to estimate the MV and CM in nonzero-mean compound Gaussian circumstances. We acquire the fully adaptive nonzero-mean GLRT detectors by substituting the estimates into the test statistics. Then, the constant false alarm rate (CFAR) properties of the proposed detectors with respect to (w.r.t.) the speckle CM are proved. Finally, the performance of three proposed detectors is verified by simulation experiments using the synthetic and real sea clutter data.

Bayesian approach for assessment of co-variances and genetic parameters of production and reproduction traits in Sahiwal cattle.

Precise as well as unbiased estimates of genetic parameters of economically important traits is important for framing breeding programmes meant for genetic progress of dairy animals. Present work was carried out to assess the co-variances along with genetic parameters of production and reproduction traits in Sahiwal cattle with Bayesian approach. Records of Sahiwal cattle kept at ICAR-National Dairy Research Institute, Karnal, Haryana for the period 1988-2016 were used. Heritability estimates ± posterior standard deviation for 305 days' milk yield (305DMY), lactation milk yield (LMY), lactation length (LL), peak yield (PY), days to attain peak yield (DPY), dry period (DP), gestation period (GP), calving to first insemination interval (CFI), days open (DO) and calving interval (CI) as obtained were 0.23 ± 0.03, 0.26 ± 0.03, 0.18 ± 0.03, 0.18 ± 0.03, 0.07 ± 0.02, 0.08 ± 0.02, 0.08 ± 0.02, 0.07 ± 0.02, 0.09 ± 0.02 and 0.02 ± 0.02, respectively and estimates of repeatability for the corresponding traits were 0.23 ± 0.03, 0.26 ± 0.03, 0.21 ± 0.03, 0.21 ± 0.03, 0.12 ± 0.02, 0.14 ± 0.03, 0.16 ± 0.03, 0.16 ± 0.03, 0.17 ± 0.03 and 0.17 ± 0.03. Moderate heritability values for 305DMY and LMY implies that sufficient additive genetic variability is available to obtain selection response. Repeatability estimates for 305DMY, LMY and PY were moderate, pointing towards the possibility of early appraisal of Sahiwal cattle for selection. Antagonistic genetic associations were observed between reproduction and production traits suggesting that reproduction traits need due attention while selecting high producing Sahiwal animals.

The Impact of Environmental Social Governance (ESG) on Firm Performance with the Moderating Role of CEO Turnover

This research aims to investigate the influence of Environmental, Social, and Governance (ESG) on firm performance, measured by Return on Equity (ROE). Additionally, the study explores the moderating role of CEO turnover in the relationship between ESG and firm performance in publicly listed companies on the Indonesia Stock Exchange, with a sample of 33 companies from 2016 to 2022. The GMM estimator method is employed to address endogeneity issues, ensuring consistent and unbiased estimation results. Findings indicate that overall ESG performance significantly affects firm performance, emphasizing the importance of implementing ESG principles in Indonesian businesses to enhance ROE. Other results reveal the pure moderating role of CEO turnover, weakening the relationship between ESG and firm performance. Therefore, proper planning and guidelines for CEO succession are crucial for improving firm performance and sustainability. The research results carry important implications for Indonesian companies to consider ESG implementation in their business strategies. Furthermore, CEO succession planning management is key to maximizing the positive impact of ESG on firm performance. This study contributes to a deeper understanding of how ESG factors influence firm performance in the Indonesian business context, highlighting the significance of executive leadership in managing these impacts.

Implementing multiple imputations for addressing missing data in multireader multicase design studies

BackgroundIn computer-aided diagnosis (CAD) studies utilizing multireader multicase (MRMC) designs, missing data might occur when there are instances of misinterpretation or oversight by the reader or problems with measurement techniques. Improper handling of these missing data can lead to bias. However, little research has been conducted on addressing the missing data issue within the MRMC framework.MethodsWe introduced a novel approach that integrates multiple imputation with MRMC analysis (MI-MRMC). An elaborate simulation study was conducted to compare the efficacy of our proposed approach with that of the traditional complete case analysis strategy within the MRMC design. Furthermore, we applied these approaches to a real MRMC design CAD study on aneurysm detection via head and neck CT angiograms to further validate their practicality.ResultsCompared with traditional complete case analysis, the simulation study demonstrated the MI-MRMC approach provides an almost unbiased estimate of diagnostic capability, alongside satisfactory performance in terms of statistical power and the type I error rate within the MRMC framework, even in small sample scenarios. In the real CAD study, the proposed MI-MRMC method further demonstrated strong performance in terms of both point estimates and confidence intervals compared with traditional complete case analysis.ConclusionWithin MRMC design settings, the adoption of an MI-MRMC approach in the face of missing data can facilitate the attainment of unbiased and robust estimates of diagnostic capability.

Improved weak lensing photometric redshift calibration via StratLearn and hierarchical modelling

ABSTRACT Discrepancies between cosmological parameter estimates from cosmic shear surveys and from recent Planck cosmic microwave background measurements challenge the ability of the highly successful $\Lambda$CDM model to describe the nature of the Universe. To rule out systematic biases in cosmic shear survey analyses, accurate redshift calibration within tomographic bins is key. In this paper, we improve photo-z calibration via Bayesian hierarchical modeling of full galaxy photo-z conditional densities, by employing ${\it StratLearn}$, a recently developed statistical methodology, which accounts for systematic differences in the distribution of the spectroscopic training/source set and the photometric target set. Using realistic simulations that were designed to resemble the KiDS + VIKING-450 data set, we show that ${\it StratLearn}$-estimated conditional densities improve the galaxy tomographic bin assignment, and that our ${\it StratLearn}$-Bayesian framework leads to nearly unbiased estimates of the target population means. This leads to a factor of $\sim 2$ improvement upon often used and state-of-the-art photo-z calibration methods. Our approach delivers a maximum bias per tomographic bin of $\Delta \langle z \rangle = 0.0095 \pm 0.0089$, with an average absolute bias of $0.0052 \pm 0.0067$ across the five tomographic bins.

Estimation of within-gap regeneration height growth in managed temperate deciduous forests using bi-temporal airborne laser scanning data

Key messageMulti-temporal airborne laser scanning (ALS) data were used to estimate regeneration stem height growth within gaps in uneven-aged deciduous forests. The height and height growth measured in the field were used to calibrate and validate ALS estimates. This method provided highly precise estimates of height and unbiased height increment estimates of regeneration at stem level.ContextAssessing regeneration height growth is essential for evaluating forest dynamics and optimizing silvicultural operations. However, regeneration description at high spatiotemporal resolution has remained limited to restricted areas by the limiting cost constraints of field measurements. Highly precise airborne laser scanning (ALS) data are currently acquired over wide areas. Such datasets are promising for characterizing regeneration dynamics.AimsWe aimed to estimate height and height growth within regenerating areas at the stem level using multi-temporal ALS data.MethodsALS data were acquired from 56,150 ha of uneven-aged deciduous forest in Belgium in 2014 and 2021. Stem tops were detected using local maxima (LM) within regenerating areas in both ALS datasets and matched. Field data were collected in 2021 and used to calibrate the ALS-estimated heights using linear and non-linear models at stem level. Height growth estimation was then validated using field-measured increments.ResultsWithout height calibration, the 2021 ALS-estimated height had a − 1.06 m bias and 1.39 m root-mean-squared error (RMSE). Likewise, the 2014 ALS-estimated height had a − 0.58 m bias and 1.14 m RMSE. The non-linear calibration seemed more appropriate for small regeneration stems (height < 4 m). Using height calibration, the 2021 ALS-estimated height had a − 0.01 m bias and 0.84 m RMSE. In 2014, the bias and RMSE were 0.02 and 0.91 m, respectively. ALS-estimated height growth was unbiased and had an RMSE of 0.10 m·year−1.ConclusionsThis original method is based on the bi-temporal ALS datasets calibrated by limited field measurements. The proposed method is the first to provide unbiased regeneration height growth of regeneration stems in uneven-aged forests and new perspectives for studying and managing forest regeneration.

Local differential privacy federated learning based on heterogeneous data multi-privacy mechanism

Federated learning enables the development of robust models without accessing users data directly. However, recent studies indicate that federated learning remains vulnerable to privacy leakage. To address this issue, local differential privacy mechanisms have been incorporated into federated learning. Nevertheless, local differential privacy will reduce the availability of data. To explore the balance between privacy budgets and data availability in federated learning, we propose federated learning for clustering hierarchical aggregation with adaptive piecewise mechanisms under multiple privacy-FedAPCA as a way to balance the relationship between privacy preservation and model accuracy. First, we introduce an adaptive piecewise mechanism that dynamically adjusts perturbation intervals based on the data ranges across different layers of the model, ensuring minimized perturbation variance while maintaining the same level of privacy. Second, we propose two dynamic privacy budget allocation methods, which are allocating the privacy budget based on global accuracy and global loss, and allocating the privacy budget based on local accuracy and loss, to ensure that better model accuracy can be achieved under the same privacy budget. Finally, we propose a clustering hierarchical aggregation method in the model aggregation stage, and the model is updated and aggregated after the unbiased estimation of the disturbance in each cluster according to the variance of each layer. FedAPCA improves the balance between privacy preservation and model accuracy. Our experimental results, comparing FedAPCA with the SOTA multi-privacy local differential privacy federated learning frameworks on the MNIST and CIFAR-10 datasets, demonstrate that FedAPCA improves model accuracy by 1%–2%.

A class of infinite number of unbiased estimators using weighted squared distance for two-deck randomized response model

ABSTRACT We develop a collection of unbiased estimators for the proportion of a population bearing a sensitive characteristic using a randomized response technique with two decks of cards for any choice of weights. The efficiency of the estimator depends on the weights, and we demonstrate how to find an optimal choice. The coefficients of skewness and kurtosis are introduced. We support our findings with a simulation study that models a real survey dataset. We suggest that a careful choice of such weights can also lead to all estimates of proportion lying between [0, 1]. In addition, we illustrate the use of the estimators in a recent study that estimates the proportion of students, 18 years and over, who had returned to the campus and tested positive for COVID-19.

Statistical inference of Poisson censored δ-shock model

In this article, the sufficient statistics, complete statistics, and a uniformly minimum variance unbiased estimate (UMVUE) of the failure parameters of the Poisson censored δ -shock model are obtained, which are based on the sample data of shock arrival time. The maximum likelihood estimates of the shock parameters are also obtained by using the EM algorithm, which is based on the total number of observed shock samples. Finally, the estimation errors and convergence rate of the EM iteration are simulated numerically using Matlab R2016b.

Bayesian estimation of spatially varying mortality risk using tagged animal data

Abstract The survival of animals is impacted by landscapes of spatially varying mortality factors including habitat type, predation risk or harvest risk, among others. Characterization of these spatial mortality processes is important for managing animal populations and their habitats, yet this information has proved challenging to capture. Advances in telemetry now make it possible to monitor tagged individuals' locations through time for a wide range of taxa, providing opportunity to assess movement and mortality simultaneously through spatial time‐to‐event data. Existing time‐to‐event modelling frameworks have largely ignored spatially varying mortality processes or have treated an animal's location as fixed at a regional level. Here we present a general spatial survival model for analysing time‐to‐event data arising from animal telemetry. Our model has a flexible Cox regression form and can estimate the effects of conventional non‐spatial risk factors (size, seasonality, etc.), spatial covariates (e.g. habitat type) and spatial variation in risk not explained by covariates on mortality. We show how to fit our model using Bayesian methods and demonstrate applications of our model with several simulated examples and two animal telemetry case studies. Our model produced consistent and unbiased parameter estimates throughout simulations with a variety of spatial and non‐spatial hazards. In the first case study, data from 147 tagged caribou in British Columbia, Canada, revealed a spatially heterogeneous mortality landscape with caribou survival varying by elevation, likely in response to space use by predators. Our second case study involved a dataset of 390 acoustically tagged Atlantic cod in a southern Norwegian fjord where a marine protected area (MPA) was established during the study. We found that the MPA led to a shift from mostly fishing mortality to mostly natural mortality within the fjord and that these risks had markedly different spatial hazard patterns. Our spatial time‐to‐event model can make use of data from a variety of widely used telemetry technologies to characterize landscapes of mortality risks for different taxa. Our work provides new opportunities to inform the spatial ecology and management of fish and wildlife populations.

Three-dimensional characteristics of the alveolar capillary network in infant and adult human lungs.

A comprehensive understanding of vascular development in the human lung is still missing. Therefore, samples of infant (n = 5, 26 days to 18 months postnatally) and adult (n = 5, 20 to 40 years) human lungs were subjected to unbiased stereological estimation of the total number of capillary loops. Serial sections were segmented to visualize the alveolar capillary network (ACN) in 3D. The number of capillary loops increased in parallel to lung volume from 26 days to 18 months, while in adults, it was not correlated to lung volume. In infant lungs, two capillary layers were separated by a connective tissue sheet with a growing number of interconnections. In adults, the mature ACN was almost, but not completely, single-layered. Here, the connective tissue was thinner but still centrally positioned, suggesting the persistence of interconnected parts of both layers of the previously double-layered ACN. Small parts of the capillaries remain double-layered and seem to be grouped around the thin connective tissue sheet, suggesting a different mechanism of microvascular maturation than simple fusion of the two layers. These spots are a potential basis for further alveolarization after completion of bulk formation. The 3D data offer a new conceptual approach to microvascular maturation of the lung. Microvascular maturation rather results from reduction than simple fusion of capillary fragments. Adult lungs maintain small double-layered capillary spots. These could offer a potential source of regeneration. The data are important to better understand normal and pathological lung development.

AIGen: an artificial intelligence software for complex genetic data analysis.

The recent development of artificial intelligence (AI) technology, especially the advance of deep neural network (DNN) technology, has revolutionized many fields. While DNN plays a central role in modern AI technology, it has rarely been used in genetic data analysis due to analytical and computational challenges brought by high-dimensional genetic data and an increasing number of samples. To facilitate the use of AI in genetic data analysis, we developed a C++ package, AIGen, based on two newly developed neural networks (i.e. kernel neural networks and functional neural networks) that are capable of modeling complex genotype-phenotype relationships (e.g. interactions) while providing robust performance against high-dimensional genetic data. Moreover, computationally efficient algorithms (e.g. a minimum norm quadratic unbiased estimation approach and batch training) are implemented in the package to accelerate the computation, making them computationally efficient for analyzing large-scale datasets with thousands or even millions of samples. By applying AIGen to the UK Biobank dataset, we demonstrate that it can efficiently analyze large-scale genetic data, attain improved accuracy, and maintain robust performance. Availability: AIGen is developed in C++ and its source code, along with reference libraries, is publicly accessible on GitHub at https://github.com/TingtHou/AIGen.

UIR-ES: An unsupervised underwater image restoration framework with equivariance and stein unbiased risk estimator

Underwater imaging faces challenges for enhancing object visibility and restoring true colors due to the absorptive and scattering characteristics of water. Underwater image restoration (UIR) seeks solutions to restore clean images from degraded ones, providing significant utility in downstream tasks. Recently, data-driven UIR has garnered much attention due to the potent expressive capabilities of deep neural networks (DNNs). These DNNs are supervised, relying on a large amount of labeled training samples. However, acquiring such data is expensive or even impossible in real-world underwater scenarios. While recent researches suggest that unsupervised learning is effective in UIR, none of these frameworks consider signal physical priors. In this work, we present a novel physics-inspired unsupervised UIR framework empowered by equivariance and unbiased estimation techniques. Specifically, equivariance stems from the invariance, inherent in natural signals to enhance data-efficient learning. Given that degraded images invariably contain noise, we propose a noise-tolerant loss for unsupervised UIR based on the Stein unbiased risk estimator to achieve an accurate estimation of the data consistency. Extensive experiments on the benchmark UIR datasets, including the UIEB and RUIE datasets, validate the superiority of the proposed method in terms of quantitative scores, visual outcomes, and generalization ability, compared to state-of-the-art counterparts. Moreover, our method demonstrates even comparable performance with the supervised model.

Pretest Estimation for the Common Mean of Several Normal Distributions: In Meta-Analysis Context

The estimation of unknown quantities from multiple independent yet non-homogeneous samples has garnered increasing attention in various fields over the past decade. This interest is evidenced by the wide range of applications discussed in recent literature. In this study, we propose a preliminary test estimator for the common mean (μ) with unknown and unequal variances. When there exists prior information regarding the population mean with consideration that μ might be equal to the reference value for the population mean, a hypothesis test can be conducted: H0:μ=μ0 versus H1:μ≠μ0. The initial sample is used to test H0, and if H0 is not rejected, we become more confident in using our prior information (after the test) to estimate μ. However, if H0 is rejected, the prior information is discarded. Our simulations indicate that the proposed preliminary test estimator significantly decreases the mean squared error (MSE) values compared to unbiased estimators such as the Garybill-Deal (GD) estimator, particularly when μ closely aligns with the hypothesized mean (μ0). Furthermore, our analysis indicates that the proposed test estimator outperforms the existing method, particularly in cases with minimal sample sizes. We advocate for its adoption to improve the accuracy of common mean estimation. Our findings suggest that through careful application to real meta-analyses, the proposed test estimator shows promising potential.

Forced Response in Dichotomous Randomised Response Technique

In sample survey research when sensitive information such as illegal drug usage, rape, examination malpractices among students, induced abortion and the likes are to be obtained from individuals or respondents, false or no response is imminent. To correct this bias, this paper proposes a new randomized response model by incorporating the unrelated question in force-response technique. The unbiased estimate and variance of the proposed model were also obtained. The proposed randomized response model is then compared with the Alternative Unbiased Estimator in Unrelated Dichotomous Randomized Response Model, (AEUDRRT) and Alternative Unbiased Estimator in Dichotomous Randomized Response Model, (AEDRRT). The numerical investigation shows that the variance of the proposed model reduces while the V(AEUDRRT) and V(AEDRRT) increases as the sensitive issues increases. Hence, the proposed model has percentage relative efficiency (PRE) as high as 1195.45% and 1977.20% over AEUDRRT and AEDRRT, respectively. Also, the application of the proposed method revealed that it is practically more efficient in estimating the proportion of people involved in sensitive character (cyber threat) than the direct method of data collection. Therefore, the proposed technique is shown to be more efficient than conventional ones.

Structured orthogonal random features based on DCT for kernel approximation

Random Fourier features are a popular technique used to improve nonlinear kernel methods in large-scale problems. Recent studies have shown that replacing random Gaussian matrices of random feature maps with appropriately scaled random orthogonal matrices, such as SORF, can significantly improve kernel approximation performance. However, the SORF method theoretically requires zero-padding for datasets whose input feature dimension does not meet d=2p for some p∈N+, resulting in increased memory and computation time. To address this limitation, we propose a new structured orthogonal random features method based on discrete cosine transform (SORF-DCT) to approximate Gaussian kernel functions. The SORF-DCT method does not require the input feature dimension to be d=2p. The key to SORF-DCT is that we combine the DCT matrix with diagonal ”sign-flipping” matrices and scale them appropriately using a diagonal matrix whose diagonal elements follow the chi distribution, resulting in a structured orthogonal matrix whose elements follow the standard Gaussian distribution instead of a random Gaussian matrix. SORF-DCT generates D-dimensional structured orthogonal random features in O(Dlogd) time and O(D) memory by employing fast discrete cosine transform, where d and D represent the input feature dimension and expansion dimension, respectively. We prove that SORF-DCT is an unbiased estimator of the Gaussian kernel function and analyze the concentration error bounds. Experimental results on eleven benchmark datasets show that SORF-DCT achieves lower kernel approximation errors, requires less time, and has comparable nonlinear learning capabilities.

Existence, uniqueness, and efficiency of numerically unbiased attenuation pathlength estimators for photon counting detectors at low count rates.

The first step in computed tomography (CT) reconstruction is to estimate attenuation pathlength. Usually, this is done with a logarithm transformation, which is the direct solution to the Beer-Lambert Law. At low signals, however, the logarithm estimator is biased. Bias arises both from the curvature of the logarithm and from the possibility of detecting zero counts, so a data substitution strategy may be employed to avoid the singularity of the logarithm. Recent progress has been made by Li etal. [IEEE Trans Med Img 42:6, 2023] to modify the logarithm estimator to eliminate curvature bias, but the optimal strategy for mitigating bias from the singularity remains unknown. The purpose of this study was to use numerical techniques to construct unbiased attenuation pathlength estimators that are alternatives to the logarithm estimator, and to study the uniqueness and optimality of possible solutions, assuming a photon counting detector. Formally, an attenuation pathlength estimator is a mapping from integer detector counts to real pathlength values. We constrain our focus to only the small signal inputs that are problematic for the logarithm estimator, which we define as inputs of<100 counts, and we consider estimators that use only a single input and that are not informed by adjacent measurements (e.g., adaptive smoothing). The set of all possible pathlength estimators can then be represented as points in a 100-dimensional vector space. Within this vector space, we use optimization to select the estimator that (1) minimizes mean squared error and (2) is unbiased. We define "unbiased" as satisfying the numerical condition that the maximum bias be less than 0.001 across a continuum of 1000 object thicknesses that span the desired operating range. Because the objective function is convex and the constraints are affine, optimization is tractable and guaranteed to converge to the global minimum. We further examine the nullspace of the constraint matrix to understand the uniqueness of possible solutions, and we compare the results to the Cramér-Rao bound of the variance. We first show that an unbiased attenuation pathlength estimator does not exist if very low mean detector signals (equivalently, very thick objects) are permitted. It is necessary to select a minimum mean detector signal for which unbiased behavior is desired. If we select two counts, the optimal estimator is similar to Li's estimator. If we select one count, the optimal estimator becomes non-monotonic. The oscillations cause the unbiased estimator to be noise amplifying. The nullspace of the constraint matrix is high-dimensional, so that unbiased solutions are not unique. The Cramér-Rao bound of the variance matches well with the expected scaling law and cannot be attained. If arbitrarily thick objects are permitted, an unbiased attenuation pathlength estimator does not exist. If the maximum thickness is restricted, an unbiased estimator exists but is not unique. An optimal estimator can be selected that minimizes variance, but a bias-variance tradeoff exists where a larger domain of unbiased behavior requires increased variance.