Modelling income data using two extensions of the exponential distribution

Earlier comparative work on income distribution has tended to suggest that Australia is characterised by less income inequality than other industrialised economies. Concerns about the quality of the Australian data used in such comparisons have led to the need for more detailed assessment of the situation. The Luxembourg Income Study has been a focus for this work by bringing together microdata sets for a range of countries and reorganising them to conform to standardised concepts and definitions. This paper builds on earlier work undertaken as part of the Luxembourg Income Study by including Australia in an international comparative analysis of income distribution and redistribution. The Australian data are those from the 1981–82 Income and Housing Survey, with income tax imputed onto the data file. Results are presented for the gross and net income distributions between both families and individuals in seven countries. A common set of equivalence scales is also used to adjust for differing family needs. The results indicate that, using several summary measures of inequality, the distribution of income in Australia is less equal than in four of the other six countries studied. Earlier research which placed Australia high on the international league table of income equality is thus not confirmed by the results.

The last two strong earthquakes, 21 July 2017 Bodrum-Kos (Mw = 6.5) and 12 June 2017 Karaburun-Aegean Sea (Mw = 6.2), indicate that western Anatolia (Aegean region) is seismo-tectonically active and has experienced several destructive earthquakes. Therefore, we aim to evaluate the seismic hazard of two study regions in western Anatolia by calculating conditional probabilities of earthquake (M ≥ 5.5) recurrences according to exponential, Weibull, gamma, log-normal, and Rayleigh distribution models. The study data include earthquakes (M ≥ 5.5) that occurred in the study regions from 1900 to 2018. We calculate the distribution model parameters with maximum likelihood estimates by using the inter-event time of earthquake recurrences. The fittings of the models with data are tested with the selection criteria, which are the lnL, the Akaike information criterion, Bayesian information criteria, and the Kolmogorov–Smirnov minimum distance criterion. After evaluating these selection criteria, the best fit is provided by the log-normal distribution model and the poorest fit is from the Rayleigh distribution model in both regions. The exponential, Weibull, and log-normal model conditional probability results (> 0.50) are close in the two regions (2017, te = 0 and 2020–2042, 3 ≤ t ≤ 25). The gamma distribution conditional probability gives the highest values. The minimum conditional probability value begins at 0.10 for the Rayleigh distribution model. According to the different distribution models in the study, the conditional probability results contribute to seismic hazard assessment in the regions.

The Distribution of Top Incomes in Australia

294 Background: Economic evaluations in oncology aim to assess the value of new therapies in the long term based on clinical trial data that often have restricted follow-up times (< 5 years) and small sample sizes (< 500 patients). This requires the use of extrapolation assumptions on long-term survival that go beyond the observed data. In this analysis, differences in survival extrapolation methods are tested in samples of sizes and follow-up reflecting typical clinical trials against a background of known survival in prostate cancer from a US based cancer registry. Methods: Data from the National Cancer Institute's Surveillance Epidemiology and End Results (SEER) registry on long-term survival in patients with stage IV prostate cancer were employed. The data set comprised those patients diagnosed between 1988 and 2003, with follow-up data available until 2012. Additional survival for those who received surgery (compared to those who did not), was estimated based on extrapolations using standard parametric statistical models (exponential, Weibull, log-logistic, log-normal, Gamma) fitted to the observed data. Survival analyses were run for 5 sample size scenarios (n = 27,670, 1000, 500, 200, 50) and 6 follow-up scenarios (follow-up years = 25, 20, 10, 5, 2, 1) yielding 30 combination scenarios. Performance of the methods was tested relative to the maximum follow-up, maximum sample size scenario (i.e. reference case) from the SEER registry. Results: Log-logistic and log-normal models were associated with flat tails which led to inflated survival estimations. For scenarios with smaller sizes, gamma models often did not converge. Exponential models were the most frequently reported as best model fit (in approximately 50% of scenarios). Also, gains in OS were consistent when exponential models were selected, and closely matched gain in OS from the reference case. Conclusions: Since clinical trials in oncology are often associated with small patient sample sizes and restricted follow-up, selecting an exponential model may lead to the most consistent and stable results based on the experiment constructed here. Further research should confirm these results for other types of cancer.

Hospital length of stay (LOS) is a key indicator of healthcare efficiency and resource utilization, particularly for chronic infectious diseases such as hepatitis B virus (HBV) infection. Prolonged hospitalization increases costs and strains limited hospital capacity in low-resource settings. Accurate statistical modelling of LOS is therefore essential for planning bed utilization and improving patient management. This study compares Exponential and Log-Normal accelerated failure time (AFT) survival regression models for predicting LOS among patients living with HBV in Maiduguri, Nigeria. A retrospective cohort of 60 HBV admissions at a tertiary hospital in Maiduguri was analyzed. LOS (days) was defined as the time from admission to discharge, with deaths and transfers treated as non-discharge outcomes in sensitivity analyses. Covariates included age, gender, marital status, diagnostic method, disease stage, comorbidity, antiviral treatment, and admission type. Exponential and Log-Normal AFT models were fitted and compared using log-likelihood, Akaike and Bayesian information criteria (AIC, BIC), likelihood ratio tests, graphical diagnostics (Q–Q and residual plots), and prediction accuracy metrics (mean absolute error (MAE) and root mean squared error (RMSE)). Of the 60 patients, 38 were discharged alive, 6 died, and 16 were transferred. LOS was positively skewed. The Log-Normal AFT model outperformed the Exponential model with higher log-likelihood (−139.56 vs −194.48), lower AIC (161.87 vs 402.97) and BIC (178.62 vs 417.63), and substantially improved prediction accuracy (MAE = 3.66 vs 12.97 days; RMSE = 8.50 vs 28.20 days). In the Log-Normal model, age significantly prolonged LOS (TR ≈ 1.06 per year), implying about a 5–6% increase in expected stay per additional year of age. Antiviral treatment markedly reduced hospitalization time (TR ≈ 0.55), resulting in a roughly 45% shorter LOS. Gender, disease stage, and comorbidity also showed significant associations with LOS. Diagnostic plots indicated better conformity to Log-Normal assumptions than to the Exponential specification. The log-normal AFT model offers better fit, prediction, and interpretability than the exponential model for HBV-related LOS. Effects on time ratios may be translated to indicate that timely use of antiviral therapy may save 4–5 bed-days per patient, whereas older age alone significantly increases demand for bed types. These results justify the application of Log-Normal AFT models to predict LOS and hospital resource planning in resource-constrained environments.

This study introduces and analyses existing models of wind speed frequency distribution in wind farms, such as the Weibull distribution model, the Rayleigh distribution model, and the lognormal distribution model. Inspired by the shortcomings of these models, we propose a distribution model based on an exponential polynomial, which can describe the actual wind speed frequency distribution. The fitting error of other common distribution models is too large at zero or low wind speeds. The proposed model can solve this problem. The exponential polynomial distribution model can fit multimodal distribution wind speed data as well as unimodal distribution wind speed data. We used the linear-least-squares method to acquire the parameters for the distribution model. Finally, we carried out contrast simulation experiments to validate the effectiveness and advantages of the proposed distribution model.

This study introduces the Lindley Stochastic Frontier Analysis—LSFA model, a novel approach that incorporates the Lindley distribution to enhance the flexibility and accuracy of efficiency estimation. The LSFA model is compared against traditional SFA models, including the half-normal, exponential, and gamma models, using advanced numerical methods such as the Gauss–Hermite Quadrature, Monte Carlo Integration, and Simulated Maximum Likelihood Estimation for parameter estimation. Simulation studies revealed that the LSFA model outperforms in scenarios involving small sample sizes and complex, skewed distributions, particularly those characterized by gamma distributions. In contrast, traditional models such as the half-normal model perform better in larger samples and simpler settings, while the gamma model is particularly effective under exponential inefficiency distributions. Among the numerical techniques, the Gauss–Hermite Quadrature demonstrates a strong performance for half-normal distributions, the Monte Carlo Integration offers consistent results across models, and the Simulated Maximum Likelihood Estimation shows robustness in handling gamma and Lindley distributions despite higher errors in simpler cases. The application to a banking dataset assessed the performance of 12 commercial banks pre-COVID-19 and during COVID-19, demonstrating LSFA’s superior ability to handle skewed and intricate data structures. LSFA achieved the best overall reliability in terms of the root mean square error and bias, while the gamma model emerged as the most accurate for minimizing absolute and percentage errors. These results highlight LSFA’s potential for evaluating efficiency during economic shocks, such as the COVID-19 pandemic, where data patterns may deviate from standard assumptions. This study highlights the advantages of the Lindley distribution in capturing non-standard inefficiency patterns, offering a valuable alternative to simpler distributions like the exponential and half-normal models. However, the LSFA model’s increased computational complexity highlights the need for advanced numerical techniques. Future research may explore the integration of generalized Lindley distributions to enhance model adaptability with enriched numerical optimization to establish its effectiveness across diverse datasets.

Bu çalışmada dört farklı istatistiksel dağılım modeli (Weibull, Rayleigh, üstel ve log-normal dağılım modelleri) uygulanarak 24 Ocak 2020 Sivrice (Elazığ) depreminin şartlı olasılığı hesaplanmıştır. Bu hesaplamalar için, bölgede 1900 yılı ile 2019 yılları arasında meydana gelmiş M≥4,7 depremlerin tekrarlama periyotları kullanılmıştır. Modeller ile kullanılan veri arasındaki uyum ilişkisi üç farklı test kriteri (Olabilirlik değeri (lnL), Akaike ve Bayesian bilgi kriteri) ile değerlendirilirmiştir. Log-normal model, çalışma verisini en iyi temsil eden model olarak belirlenmiştir. Çalışma sonuçlarına göre Sivrice (Elazığ) depreminin şartlı olasılık değerleri (t=0 ve te=2 için); Log-normal modele göre %80, üstel modele göre %72, Weibull modele göre %70, Rayleigh modele göre %36 olarak hesaplanmıştır.

The International Committee for Standardization in Hematology recommended three methods for calculating objectively the mean platelet survival time, ie, weighted mean of linear estimate and logarithmic estimate, truncated exponential model, and gamma model. For determining which method is better, data of the survival study were fitted to each of three methods, as well as the Paulus polynomial and exponential polynomial models, and the mean survival time and the mean square error were compared. Mean survival obtained by each method was always longest in healthy controls and shortest in ITP. In 3rd degree of polynomial estimate and 2nd and 3rd degrees of logarithmic estimate, several cases showed practically outlying mean survival time of more than 12 days. In healthy controls, highest mean square error was observed in 1st degree logarithmic estimate, and in cases with short mean survival time the highest one was obtained in 1st degree polynomial estimate. Mean square error in ITP pre- and postsplenectomy and in aplastic anemia was smallest in the gamma model. Among nine methods recommended by the International Committee for Standardization in Hematology and by Dr. Paulus, the gamma model so far provides the best method for calculating the mean survival time of 51Cr-platelets.

BackgroundWhen conducting a survival analysis, researchers might consider two broad classes of models: nonparametric models and parametric models. While nonparametric models are more flexible because they make few assumptions regarding the shape of the data distribution, parametric models are more efficient. Here we sought to make concrete the difference in efficiency between these two model types using effective sample size.MethodsWe compared cumulative risk of AIDS or death estimated using four survival models – nonparametric, generalized gamma, Weibull, and exponential – and data from 1164 HIV patients who were alive and AIDS-free in 1995. We added pseudo-observations to the sample until the spread of the 95% confidence limits for the nonparametric model became less than that for the parametric models.ResultsWe found the 3-parameter generalized gamma to be a good fit to the nonparametric risk curve, but the 1-parameter exponential both underestimated and overestimated the risk at different times. Using two year-risk as an example, we had to add 354, 593, and 3960 observations for the nonparametric model to be as efficient as the generalized gamma, Weibull, and exponential models, respectively.ConclusionsThese added observations represent the hidden observations underlying the efficiency gained through parametric model form assumptions. If the model is correctly specified, the efficiency gain may be justified, as appeared to be the case for the generalized gamma model. Otherwise, precision will be improved, but at the cost of specification bias, as was the case for the exponential model.

A number of empirical models are used in literature to simulate the response of soil respiration (Rs) to soil temperature (Ts). The most widely used ones are the exponential Q10 model and the sigmoid‐shaped Lloyd‐Taylor and logistic models. None of these models are applicable across a wide range of ecosystems or climates, and none allow Rs to decrease at high Ts values. Here we present a new, more flexible, empirical model, the so‐called Gamma model, which can take on the shapes of the three models mentioned above and is mathematically flexible enough to allow for Rs to decrease at high Ts values, as dictated by data. We compared the Gamma model fits to the Q10, Lloyd‐Taylor, and logistic models, using coefficient of determination (R2), residual sum of squares, and Akaike's Information Criterion. The models were tested across a wide Ts range (−18 to 35°C), in five forest ecosystems, spanning three different climate zones: boreal, temperate, and Mediterranean. Compared to the other three models, the Gamma model performed either better or as good as the other models in simulating the Rs‐Ts relationship at all sites. Simulations were carried out using models parameterized by the ordinary least squares and weighted absolute deviation estimation methods. Rs values derived from the two estimation methods were comparable once the proper functional form for the Rs‐Ts model was chosen. We also show how the Gamma model can be expanded, using simple mathematics to help researchers analyze the Rs‐Ts relationship in the context of other environmental factors, such as soil moisture and nutrients.

A new parametric model is proposed for modeling the density function of perpendicular distances in line transects sampling. The model can be considered a weighted exponential model in the sense that it combines two exponential models with different weights. The proposed model is appealing because it is monotone decreasing with distance from transect line; in contrast to the classical exponential model, it satisfies the shoulder condition at the origin. Simulation results for a wide range of target densities show reasonable and good performances of the weighted exponential model in most considered cases compared to the classical exponential and the half-normal models.

BackgroundMulti-state models are being increasingly used to capture complex disease pathways. The convenient formula of the exponential multi-state model can facilitate a quick and accessible understanding of the data. However, assuming time constant transition rates is not always plausible. On the other hand, obtaining predictions from a fitted model with time-dependent transitions can be challenging. One proposed solution is to utilise a general simulation algorithm to calculate predictions from a fitted multi-state model.MethodsPredictions obtained from an exponential multi-state model were compared to those obtained from two different parametric models and to non-parametric Aalen-Johansen estimates. The first comparative approach fitted a multi-state model with transition-specific distributions, chosen separately based on the Akaike Information Criterion. The second approach was a Royston-Parmar multi-state model with 4 degrees of freedom, which was chosen as a reference model flexible enough to capture complex hazard shapes. All quantities were obtained analytically for the exponential and Aalen-Johansen approaches. The transition rates for the two comparative approaches were also obtained analytically, while all other quantities were obtained from the fitted models via a general simulation algorithm. Metrics investigated were: transition probabilities, attributable mortality (AM), population attributable fraction (PAF) and expected length of stay. This work was performed on previously analysed hospital acquired infection (HAI) data. By definition, a HAI takes three days to develop and therefore selected metrics were also predicted from time 3 (delayed entry).ResultsDespite clear deviations from the constant transition rates assumption, the empirical estimates of the transition probabilities were approximated reasonably well by the exponential model. However, functions of the transition probabilities, e.g. AM and PAF, were not well approximated and the comparative models offered considerable improvements for these metrics. They also provided consistent predictions with the empirical estimates in the case of delayed entry time, unlike the exponential model.ConclusionWe conclude that methods and software are readily available for obtaining predictions from multi-state models that do not assume constant transition rates. The multistate package in Stata facilitates a range of predictions with confidence intervals, which can provide a more comprehensive understanding of the data. User-friendly code is provided.

Wei et al. [B.C. Wei, J.Q. Shi, W.K. Fung, and Y.Q. Hu, Testing for varying dispersion in exponential family nonlinear models, Ann. Inst. Statist. Math. 50 (1998), pp. 277–294.] developed the score diagnostics for varying dispersion in exponential family nonlinear models, such as the normal, inverse Gaussian, and gamma models, and investigated the powers of these tests through Monte Carlo simulations. In this paper, the asymptotic behaviours, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied and examined by Monte Carlo simulations. The methods to estimate local powers of the score tests are illustrated with Grass yield data [P. McCullagh, and J.A. Nelder, Generalized Linear Models, Chapman and Hall, London (1989).].

Research in event cognition has shown that how people segment events plays an important role in how they are perceived and remembered. One potential source of information for anticipating upcoming event boundaries is the typical duration of the event. The small amount of prior work that exists suggests that the durations of daily events have normal distributions and that people have accurate prior beliefs about these durations (Griffiths & Tenenbaum, 2006). This research study aims to further explore the temporal distribution of self-reported events from daily life. Forty-eight participants provided information regarding the duration of events from their daily life for 14 days. Event durations for various activity types were characterised using duration modelling with truncated normal, exponential, and gamma models. Predictions based on mean and median values, including ± 5% and ± 10% error margins, were obtained, and the cumulative hazard functions were derived. Exponential and gamma distributions fit most event categories better than the truncated normal distribution. Mean-based predictions of boundaries based on duration showed only 4–5% accuracy, indicating limited usefulness. This aligns with the cumulative hazard results, where most events followed the exponential distribution, suggesting no optimal time for predicting the end of the event. This suggests that when daily events are studied in an ecological context, most of them have little sign of a typical duration. Consequently, it is unlikely that duration estimation plays a large role in the anticipation of event boundaries.