Exploring the Lindley Distribution in Stochastic Frontier Analysis: Numerical Methods and Applications

In this paper, we introduce a new distribution named as the Pseudo Lindley Distribution (PsLD) as a generalization of the Lindley distribution (LD). A full and detailed description are provided in terms of moments, cumulates, characteristic function, failure, rate function, stochastic ordering, distributions of sums, and parameters estimation. Simulations studies and data driven applications are also reported. Keywords: Lindley distribution; Exponential distribution; Gamma distribution; Stochastic ordering; Maximum-likelihood estimation.

To assess the adequacy of a statistical model based on the gamma distribution for diffusion signal decays of prostate cancer (PCa) using b-values ranging up to 2000 sec/mm(2) , and to evaluate the differences in gamma model parameters for PCa, benign prostatic hyperplasia (BPH), and peripheral zone (PZ). Twenty-six patients with histologically proven PCa underwent diffusion-weighted magnetic resonance imaging using five b-values (0, 500, 1000, 1500, 2000 sec/mm(2) ). The acquired signal decay curves were fit with both gamma and truncated Gaussian models and a statistical comparison between the two fits was performed. The acquired parameters using the gamma model (mean, standard deviation, the area fraction for D < 1.0 mm(2) /s [Frac<1.0], the area fraction of D > 3.0 mm(2) /s [Frac>3.0]) were compared between PCa, BPH, and PZ. The gamma model provided a statistically improved fit over the truncated Gaussian model in PCa. The mean and the standard deviation were significantly lower in PCa than in BPH and PZ (P < 0.01). Frac<1.0 was significantly higher in PCa than in BPH and PZ, and Frac>3.0 was significantly lower in PCa than in BPH and PZ (P < 0.01). A statistical model based on the gamma distribution proved suitable for describing diffusion signal decay curves of PCa. This approach may provide better correlation between diffusion signal decay and histological information in the prostate gland.

The probabilistic models achieved considerable attention in the field of seismology due to their dynamic properties. In this study, an attempt has been made to quantify the uncertainty observed in the earthquake data of the Hindu Kush region by using probabilistic models. The conditional probabilities of major earthquake (Mw ≥ 5.5) occurrence in the Hindu Kush region are estimated by using the Weibull, lognormal, gamma, and log-logistic probability distributions. The maximum likelihood estimation method has been used to estimate the model parameters. Results based on the three model selection criteria suggest that the Weibull and log-logistic distributions provide the most appropriate fit, while the lognormal and gamma distributions appeared as the second and third appropriate fitted distributions to the elapsed time data of the Hindu Kush region. Using the most appropriate fitted Weibull and log-logistic distributions, it has been observed that the estimated earthquake occurrence probability of magnitude 5.5 or greater in the Hindu Kush region reaches 0.94–0.99 within next 3 to 6 (2021–2024) years since the last event occurred in 2018 in the study region. The conditional probability values are continuously increasing with increasing time interval and it approximately reaches to one in next 6 to 12 (2024–2030) years without any elapsed time, i.e., (T = 0) years. The estimated earthquake conditional probability curves for different elapsed time and time interval values are also presented to understand the earthquake genesis in the study region. The earthquake mean recurrence interval is 0.68, 0.81, 0.68, and 1.11 years by using the Weibull, lognormal, gamma, and log-logistic models respectively. According to the obtained results, the earthquake of magnitude Mw ≥ 5.5 occurrence probability in the study region is significantly high. The obtained results provide important information’s regarding the underlying physical mechanism of earthquake generation process in the Hindu Kush region.

Purpose:Prostate cancer management includes identification of clinically significant cancers that may require curative treatment. Statistical models based on gamma distribution can describe diffusion signal decay curves of prostate cancer. The purpose of this study was to evaluate the ability of parameters obtained with the gamma model in differentiating prostate cancers with different Gleason score values.Methods:This study included 155 patients with prostate cancer who underwent multiparametric magnetic resonance imaging prior to prostate biopsy (127 patients) or radical prostatectomy (28 patients) between January 2015 and June 2017; 159 foci of prostate cancer were included in our study. We compared cases scored as Gleason score (GS) 3 + 3 and GS ≥ 3 + 4, and analyzed cases scored as GS ≤ 3+ 4 and GS ≥ 4 + 3 based on the gamma model (Frac < 1.0, Frac < 0.8, Frac < 0.5, Frac < 0.3, and Frac > 3.0), and apparent diffusion coefficient (ADC).Results:Among 159 cancerous lesions in 155 patients, 13 (8.2%) were GS 3 + 3 prostate cancers, 51 (32.0%) were GS 3 + 4 prostate cancers, 30 (18.2%) were GS 4 + 3 cancers, and 65 (40.9%) were GS ≥ 4 + 4 cancers. Frac < 0.3, Frac < 0.5, Frac < 0.8, and Frac < 1.0 were significantly higher and ADC values were significantly lower in GS ≥ 4 + 3 cancers than in GS ≤ 3 + 4 cancers (P < 0.01, P < 0.01, P < 0.01, P = 0.01, and P < 0.01, respectively). With receiver operating characteristic (ROC) analysis, Frac < 0.3 and Frac < 0.5 had significantly greater area under the ROC curve for discriminating GS ≥ 4 + 3 cancers from GS ≤ 3 + 4 cancers than ADC (P = 0.03, P < 0.01, respectively).Conclusion:Frac < 0.3 and Frac < 0.5 showed higher diagnostic performance than ADC for differentiating GS ≥ 4 + 3 from GS ≤ 3 + 4 cancers. The gamma model may add additional value in discrimination of tumor grades.

In this article, we introduce a notable bounded distribution based on a modification of the epsilon function that creates an upper bound on the domain of a distribution. Further, a key feature of the distribution is to have asymptotic connections with the famous Lindley distribution, which is a weighted variant of the exponential distribution and also a mixture of exponential and gamma distributions. In some ways, the proposed distribution provides a flexible solution to the modeling of bounded characteristics that can be almost well-fitted by the Lindley distribution if the domain is restricted. Moreover, we have also explored its application, particularly with reference to lifetime and environmental points of view, and found that the proposed model exhibits a better fit among the competing models. Namely, we demonstrate the practical applicability of the new distribution on two data sets containing lifetime data, as well as on two other data sets of rainfall data. Further, from the annual rainfall analysis, the proposed model exhibits a realistic return period of the rainfall.

Although many models have been proposed to interpret non-Gaussian diffusion MRI data in biological tissues, it is often difficult to see the correlation between the MRI data and the histological changes in the tissue. Among these models, so called statistical models, which assume the diffusion coefficient D is distributed continuously within a voxel, are more suitable for interpreting the data in a histological context than others. In this work, we examined a statistical model based on the gamma distribution. First, the proposed gamma model, the bi-exponential model, and the truncated Gaussian model were compared for goodness of fit. To evaluate diagnostic capability, area fractions of certain D ranges were evaluated. The area fraction for D < 1.0 mm2/s (frac < 1) was attributed to small cancer cells with restricted diffusion, and the area fraction for D > 3.0 mm2/s (frac > 3) was considered to reflect perfusion component. A clinical data set of histologically proven prostate cancer cases from previous study was used. For the cancer tissue, the gamma model was better fit than the truncated Gaussian model, and there was no significant difference between the gamma model and the bi-exponential model. For the normal peripheral zone tissue, there was no significant differences among all models. In the 2D scatter plot of frac < 1 vs. frac > 3, Cancer and non-cancer tissues were clearly separated. Using the proposed model, the diffusion MR data was well fit, and histological interpretation of the data appears possible.

Purpose:To determine the appropriateness of statistical models using the truncated Gaussian distribution and gamma distribution for diffusion signal decay, and to assess the correlation between the parameters obtained from the statistical models and estimated glomerular filtration rate (eGFR).Methods:Twenty-nine patients with chronic kidney disease and 21 healthy volunteers were included and classified in four groups according to eGFR (ml/min/1.73 m2): group 1 (90 ≤ eGFR, n = 10), group 2 (eGFR 60–90, n = 15), group 3 (eGFR 30–60, n = 17), and group 4 (eGFR < 30, n = 8). Diffusion-weighted imaging using five b-values (0, 500, 1000, 1500, and 2000 s/mm2) was performed. Truncated Gaussian and gamma models were compared for goodness of fit. Area fractions for the diffusion coefficient D < 1.0 × 10−3 mm2/s (Frac < 1.0) and D > 3.0 × 10−3 mm2/s (Frac > 3.0) obtained from the statistical model were compared among four groups. Correlation between proposed parameters and conventional apparent diffusion coefficient (ADC) with eGFR was calculated.Results:There was no significant difference in goodness of fit between the truncated Gaussian and gamma models. Frac < 1.0 and Frac > 3.0 showed good correlation with eGFR, as did ADC. Comparison between groups 1 and 2 showed that only Frac < 1.0 in both distribution models had significant differences.Conclusion:Statistical models yield robust interpretation of diffusion magnetic resonance (MR) signals with relevance to histological changes in the kidney. The parameters of the statistical models, particularly Frac < 1.0, strongly correlated with eGFR.

Statistical Downscaling (SDS) models might involve ill-conditioned covariates (large dimension and high correlation/multicollinear). This problem could be solved by a variable selection technique using L1 regularization/LASSO or a dimension reduction approach using principal component analysis (PCA). In this paper, both methods were applied to generalized linear modeling with gamma distribution and compared to predict rainfall models at 11 rain posts in Indramayu. More over, generalized linear model with gamma distribution was used to obtain non-negative rainfall prediction and compared with principal component regression (PCR). Two types of ill-conditioned data with different characteristics (CMIP5 and GPCP version 2.2) were used as covariates in SDS modeling. The results show that three methods (PCR, Gamma-PC, and Gamma-L1) did not demonstrate significant difference in term of Root Mean Square Error (RMSE) after addition of dummy variables (month) in the models. However, a generalized linear modeling with gamma distribution could be considered as the best methods since it provided non-negative rainfall predictions.

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.

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.

This paper examines the relation between black zone and number of feeder roads joining the four lane, undivided, 8.5 km long National Highway. Primary and secondary data were collected about geometrical features, number of intersection created due to feeder road per kilometer, accident number/frequency and locations. To indentify the black zone on the study stretch, three methods are used such as an accident rate method, accident frequency method and accident severity. After ascertaining a black zone, a technique of negative binomial distribution with gamma distribution and prediction error minimization method was applied for identifying minimum distances in between two feeder road and/or intersection to avoid creation of black zones. The results of this study suggest that feeder network intersection characterized by left turning do in fact play a role in creating black zones other than the traffic flow and geometry. The prediction model suggests that the number of accidents reduces gradually, if feeder road traffic merges with national highway in between minimum spacing should be more than 750 m to 1 km. The performance of the model was successfully tested on the basis of three statistical measures, mean- absolute error (MAE), root mean square error (RMSE) and normalized error (NER). This work suggests that black zone can be prevented through controlling merging of left turning feeder road traffic to national highway traffic by providing minimum in-between safe spacing. Language: en

Prior research on structured products has demonstrated that equity-linked notes (ELNs) sold to retail investors in initial public offerings are typically issued at above their fair market value. A particular type of ELN – reverse convertibles – embed down-and-in put options and offer investors relatively high coupon payments in exchange for bearing some of the downside risk of the equity underlying the note. We analytically study the magnitude of the overpricing of reverse convertibles – one of the most popular structured products on the market today – within a stochastic volatility model.We extend the current literature to include analytical valuation formulas within a model of stochastic volatility – the variance gamma (VG) model. We show that these complex notes are even more overpriced than previously estimated when stochastic volatility is taken into account. As a result of their complex payoffs and the lack of a secondary market to correct the mispricing, reverse convertible notes continue to be sold at prices substantially in excess of their fair market value.

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.

SYNOPTIC ABSTRACTFluctuating asymmetry (FA), is assumed to be the joint outcome of random developmental noise and stabilizing mechanisms. Because development is never perfect, a realized phenotype will always deviate from its expected value conditional on the current environmental and genetic conditions. Unfortunately, each trait develops only once, such that it is impossible to calculate the deviations from its expectation. However, in bilaterally symmetric traits, the left and right side develop under exactly the same conditions. Therefore, the expected phenotypes are identical on both sides and the expectation of left minus right trait value equals zero. Any deviation from perfect symmetry reflects the joint outcome of developmental noise and/or stability, i.e., of developmental instability (DI). While the concept of FA is relatively simple, its statistical analysis is not. FA is a reflection of variation in development, so in other words, at the individual level we attempt to estimate a variance with two data points, which is bound to be prone to high sampling variation. Theoretical models of morphological development predict either log-normal or γ-distributions, while traditional analysis tools of FA and DI have assumed normality. We present the analysis of simulated datasets to explore the robustness of results under the different alternative distributions. Results from normal and gamma models appeared comparable, while data generated assuming log-normality of developmental noise yielded biased results if analysed assuming a normal or gamma distribution.

We analyze the model used to assess most major commercial marine fish populations, namely, sequential population analysis (SPA). This model estimates population abundance by combining catch-at-age data with research surveys or commercial catch per unit effort indices of abundance. We examine two maximum likelihood estimators of SPA parameters. These estimators are based on assuming that the stock-size indices are from lognormal or gamma distributions. Using simulations, we find that both types of estimators can have significant biases; however, our results indicate that it is preferable to use the gamma model, because it tends to have lower bias and variability, even when the true distribution of the stock-size indices is lognormal.