Unimodal Probability Distribution Research Articles

Application of unimodal probability distribution models for morphological phenotyping of budding yeast.

Morphological phenotyping of the budding yeast Saccharomyces cerevisiae has helped to greatly clarify the functions of genes and increase our understanding of cellular functional networks. It is necessary to understand cell morphology and perform quantitative morphological analysis (QMA) but assigning precise values to morphological phenotypes has been challenging. We recently developed the Unimodal Morphological Data image analysis pipeline for this purpose. All true values can be estimated theoretically by applying an appropriate probability distribution if the distribution of experimental values follows a unimodal pattern. This reliable pipeline allows several downstream analyses, including detection of subtle morphological differences, selection of mutant strains with similar morphology, clustering based on morphology, and study of morphological diversity. In addition to basic research, morphological analyses of yeast cells can also be used in applied research to monitor breeding and fermentation processes and control the fermentation activity of yeast cells.

Discretizing Continuous Action Space With Unimodal Probability Distributions for On-Policy Reinforcement Learning.

For on-policy reinforcement learning (RL), discretizing action space for continuous control can easily express multiple modes and is straightforward to optimize. However, without considering the inherent ordering between the discrete atomic actions, the explosion in the number of discrete actions can possess undesired properties and induce a higher variance for the policy gradient (PG) estimator. In this article, we introduce a straightforward architecture that addresses this issue by constraining the discrete policy to be unimodal using Poisson probability distributions. This unimodal architecture can better leverage the continuity in the underlying continuous action space using explicit unimodal probability distributions. We conduct extensive experiments to show that the discrete policy with the unimodal probability distribution provides significantly faster convergence and higher performance for on-policy RL algorithms in challenging control tasks, especially in highly complex tasks such as Humanoid. We provide theoretical analysis on the variance of the PG estimator, which suggests that our attentively designed unimodal discrete policy can retain a lower variance and yield a stable learning process.

Self-excited intermittent thermoacoustic fluctuations in an annular combustor exhibiting flame transient phenomena: Physical mechanisms and modeling

We experimentally examine the physical mechanisms causing intermittent thermoacoustic fluctuations in a turbulent annular combustor exhibiting flame transient phenomena (FTP). The combustor consists of 12 burners. Flames are stabilized by conical bluff bodies, resembling the afterburner and ramjet burner configurations. The combustor exhibits a dominant 1A–1L (first azimuthal–first longitudinal, ∼630 Hz) thermoacoustic mode. Instability manifests as intermittent fluctuations in acoustic pressure. From the heat release and acoustic pressure measurements, FTPs are found to create large amplitude heat release fluctuations occurring on a slow timescale compared with 1A–1L thermoacoustic mode. The heat release, in turn, leads to the observed intermittent acoustic pressure fluctuations. Four FTPs are found to occur in the combustor: (1) near blow-off, (2) flame extinction, (3) successful reignition, and (4) unsuccessful reignition. Their occurrences and the associated time spans are found to be random. The time span follows an unimodal probability distribution, peaking around 100 ms. A low-order model is developed by incorporating the distribution for the FTP time span in the heat release. An additive Gaussian white noise is added to represent background turbulent fluctuations. The model qualitatively reproduces the experimentally observed probability density function of the acoustic pressure fluctuations. This indicates that the stochastic FTP time span and turbulence are essential for reproducing intermittent thermoacoustic fluctuations.

Hydraulic characterization and modeling of hydrophobic substrates

The hydraulic characterization of hydrophobic substrates, which strongly depends on the water content, is an understudied topic in the field of soil physics. Based on laboratory experiments, this work presents a physically-based model, denoted as Soil Water Repellency Infiltration Model (SWRIM), to describe the double-slope cumulative infiltration curves commonly observed in hydrophobic substrates, whose behavior is conditioned by the dual hydraulic behavior (under dry and saturated conditions) of the water repellency materials. The experimental setup consists of 1D infiltration tests on hydrophobic substrates and tension of 0 cm, following by a free drainage. The SWRIM model assumes that water flow in the hydrophobic substrate is gravity driven, representing the pore system as a set of disconnected pathways or capillaries with dual behavior. Each capillary is governed by two hydraulic conductivity values corresponding to dry or partially saturated and saturated conditions, Kd and Ks respectively. During infiltration, water flow within each pathway is regulated by Kd, but once the pathways are filled and the substrate is fully saturated, the water flow is governed by Ks, whose value is directly measured from the slope of the drainage curve. In order to incorporate the variability of the pore system, unimodal and bimodal probability distributions of Kd values were considered. The application of the SWRIM model allowed determining the distribution of Kd and the percentage of water-conducting pores, δ, of the hydrophobic layer. Additional determinations of the soil hydraulic conductivity, K, and sorptivity under water repellent conditions were also obtained from the inverse analysis of the initial times of the infiltration test. The proposed model was validated on experimental soil columns of 5 cm internal diameter and 1.0 and 2.5 cm height filled with pine forest organic material, Pine, rosemary leaf litter, RS, and, blond peat, Peat. Overall, convex-to-linear curves with a monotonously increasing infiltration, typical of hydrophobic substrates, were obtained. SWRIM model allowed robust fits of the infiltration curves, showing in all cases a distribution of Kd with a bimodal shape. Pine and Peat presented the largest values of δ. The estimated average Kd was within the same order of magnitude as K values. Overall, the results showed that the proposed physically-based model allows to satisfactory describe and characterize the dynamic hydraulic behavior of hydrophobic soil substrates.

Evaluation of the Pacific Decadal Oscillation from 1901 to 2014 in CMIP6 models

The simulation of internal climate variability is key to understanding climate change. Internal fluctuations can fully obscure or amplify the underlying climate-change signal in many fields over years to decades. The Pacific Decadal Oscillation (PDO) is an important mechanism that dominates interdecadal climate variations. Here, the capacity of 36 Coupled Model Intercomparison Project Phase 6 (CMIP6) models for reproducing the PDO during the period 1901-2014 was evaluated across spatial pattern, amplitude, periodicity and phase aspects. The results suggested that approximately 71% of the models were capable of capturing the spatial pattern of the PDO. However, most of the other models underestimated the percentage of the variance explained and the pattern amplitude. CMIP6 models can capture the periodicity over shorter periods (<20 yr), while most CMIP6 models underestimate the periodicity of the PDO on the interdecadal scale (60 yr periods). In particular, the probability distribution functions (PDFs) of the raw PDO index showed that most CMIP6 models could capture the unimodal probability distribution. Some of the models were able to reproduce the bimodal probability distribution of the PDO index filtered by the ensemble empirical mode decomposition method. Moreover, the findings also showed that the multimodel ensemble (MME) for 36 CMIP6 models could effectively capture the spatial pattern and periodicity over a shorter period (<20 yr) of the PDO. Moreover, the MME exhibited higher fidelity to the observation in the PDF of the PDO index, although the positive area was larger than the observation. These findings are crucial because they may help detect simulation gaps and provide valuable references for the design and applicability of future models worldwide.

Assignment of unimodal probability distribution models for quantitative morphological phenotyping

BackgroundCell morphology is a complex and integrative readout, and therefore, an attractive measurement for assessing the effects of genetic and chemical perturbations to cells. Microscopic images provide rich information on cell morphology; therefore, subjective morphological features are frequently extracted from digital images. However, measured datasets are fundamentally noisy; thus, estimation of the true values is an ultimate goal in quantitative morphological phenotyping. Ideal image analyses require precision, such as proper probability distribution analyses to detect subtle morphological changes, recall to minimize artifacts due to experimental error, and reproducibility to confirm the results.ResultsHere, we present UNIMO (UNImodal MOrphological data), a reliable pipeline for precise detection of subtle morphological changes by assigning unimodal probability distributions to morphological features of the budding yeast cells. By defining the data type, followed by validation using the model selection method, examination of 33 probability distributions revealed nine best-fitting probability distributions. The modality of the distribution was then clarified for each morphological feature using a probabilistic mixture model. Using a reliable and detailed set of experimental log data of wild-type morphological replicates, we considered the effects of confounding factors. As a result, most of the yeast morphological parameters exhibited unimodal distributions that can be used as basic tools for powerful downstream parametric analyses. The power of the proposed pipeline was confirmed by reanalyzing morphological changes in non-essential yeast mutants and detecting 1284 more mutants with morphological defects compared with a conventional approach (Box–Cox transformation). Furthermore, the combined use of canonical correlation analysis permitted global views on the cellular network as well as new insights into possible gene functions.ConclusionsBased on statistical principles, we showed that UNIMO offers better predictions of the true values of morphological measurements. We also demonstrated how these concepts can provide biologically important information. This study draws attention to the necessity of employing a proper approach to do more with less.

Probabilistic parameter estimation using a Gaussian mixture density network: application to X-ray reflectivity data curve fitting

X-ray reflectivity (XRR) is widely used for thin-film structure analysis, and XRR data analysis involves minimizing the difference between experimental data and an XRR curve calculated from model parameters describing the thin-film structure. This analysis takes a certain amount of time because it involves many unavoidable iterations. However, the recently introduced artificial neural network (ANN) method can dramatically reduce the analysis time in the case of repeated analyses of similar samples. Here, the analysis of XRR data using a mixture density network (MDN) is demonstrated, which enables probabilistic prediction while maintaining the advantages of an ANN. First, under the assumption of a unimodal probability distribution of the output parameter, the trained MDN can estimate the best-fit parameter and, at the same time, estimate the confidence interval (CI) corresponding to the error bar of the best-fit parameter. The CI obtained in this manner is similar to that obtained using the Neumann process, a well known statistical method. Next, the MDN method provides several possible solutions for each parameter in the case of a multimodal distribution of the output parameters. An unsupervised machine learning method is used to cluster possible parameter sets in order of probability. Determining the true value by examining the candidates of the parameter sets obtained in this manner can help solve the inherent inverse problem associated with scattering data.

Temperature and friction-induced tunable current reversal, anomalous mobility and diffusions

Abstract We numerically investigate the transport properties and dynamic relaxation of a classical inertia Brownian particle in a biased periodic potential under the action of friction and thermal fluctuations. The both fluctuations may be related to friction, thus they may be correlated with each other. We mainly focus on how particle’s transport and dynamic relaxation are affected by fluctuations and the correlation between them. It is unexpected that the combinations of them can tune transport of the particle, including current reversal, and anomalous mobility and diffusions. These results are explained via the stationary probability distributions, which suggest that the bias of the potential can lead to a transition between unimodal and bimodal stationary probability distributions of particle’s velocity. To understand the physical mechanism of these findings, we present dynamic relaxations, namely, the transient probability distributions of velocity and position. These distributions suggest that the motion of the particle is not only related to systematic parameters, but also sensitive to the initial conditions. Our findings may be potentially helpful for research on various transport of the particle and on similar circuits, such as particle separation, Josephson junctions, the phase-locked loops, etc.

Stochastic parametrization with VARX processes

In this study we investigate a data-driven stochastic methodology to parameterize small-scale features in a prototype multiscale dynamical system, the Lorenz '96 (L96) model. We propose to model the small-scale features using a vector autoregressive process with exogenous variable (VARX), estimated from given sample data. To reduce the number of parameters of the VARX we impose a diagonal structure on its coefficient matrices. We apply the VARX to two different configurations of the 2-layer L96 model, one with common parameter choices giving unimodal invariant probability distributions for the L96 model variables, and one with non-standard parameters giving trimodal distributions. We show through various statistical criteria that the proposed VARX performs very well for the unimodal configuration, while keeping the number of parameters linear in the number of model variables. We also show that the parameterization performs accurately for the very challenging trimodal L96 configuration by allowing for a dense (non-diagonal) VARX covariance matrix.

A new transformation of continuous unimodal asymmetric probability distributions into possibility distributions

This paper deals with a new transformation, so-called two-sided normalized (TSN), of continuous unimodal asymmetric probability distributions into possibility distributions. Many properties are derived and interpretations are discussed. A comparison with the optimal transformation is provided. In particular, the respective positions of right or left branches relative to the resulting optimal and TSN possibility distributions are given. It is also shown that the TSN transformation is the optimal transformation for the particular family of two-piece skewed distributions. The preservation of the asymmetry property is then analyzed and illustrated for several conventional asymmetric distributions and counter-examples of asymmetry preservation are provided. A multilinear approximation of the TSN transformation is finally proposed.

A general frame for uncertainty propagation under multimodally distributed random variables

Uncertainty propagation under multimodally distributed random variables, called multimodal distribution propagation for short, is a challenging problem due to the complicate probability density function of the random variables. In this paper, a general frame based on a new finite mixture model (λMM) constructed by derivative lambda probability density function and polynomial chaos expansion method is put forward to efficiently solve the multimodal distribution propagation problem. First, λMM with high accuracy and extensive applicability is proposed to represent the arbitrary unimodal distributions and multimodal distributions. Second, a pseudo EM method proved strictly is proposed to estimate the parameters of this λMM. Third, the statistical moments of the response by polynomial chaos expansion method is derived mathematically. Since the new mixture model can be decomposed into several unimodal probability distributions, the multimodal distribution propagation can be successfully converted into several unimodal distribution propagations, which further can be easily solved by polynomial chaos expansion method. Finally, the maximum entropy principle is adopted to evaluate the probability density function of the result of every unimodal distribution propagation, which is then superimposed into the final probability density function of the system response. The proposed frame only requires the first four statistical moments to evaluate the probability density function and avoid calculating the higher statistical moments, especially for the situation that the probability distribution of the system response is multimodal. Four examples are presented to verify the high accuracy and efficiency of the proposed general frame for uncertainty propagation.

Multimodal water age distributions and the challenge of complex hydrological landscapes

AbstractTravel time distributions (TTDs) are concise descriptions of transport processes in catchments based on water ages, and they are particularly efficient as lumped hydrological models to simulate tracers in outflows. Past studies have approximated catchment TTDs with unimodal probability distribution functions (pdf) and have successfully simulated tracers in outflows with those. However, intricate flow paths and contrasting water velocities observed in complex hydrological systems may generate multimodal age distributions. This study explores the occurrence of multimodal age distributions in hydrological systems and investigates the consequences of multimodality for tracer transport. Lumped models based on TTDs of varying complexity (unimodal and multimodal) are used to simulate tracers in the discharge of hydrological systems under well‐known conditions. Specifically, we simulate tracer data from a controlled lysimeter irrigation experiment showing a multimodal response and we provide results from a virtual catchment‐scale experiment testing the ability of a unimodal age distribution to simulate a known and more complex multimodal age system. Models are based on composite StorAge Selection functions, defined as weighted sums of pdfs, which allow a straightforward implementation of uni‐ or multi‐modal age distributions while accounting for unsteady conditions. These two experiments show that simple unimodal models provide satisfactory simulations of a given tracer, but they fail in reproducing processes occurring at different temporal scales. Multimodal distributions, instead, can better capture the detailed dynamics embedded in the observations. We conclude that experimental knowledge of flow paths and the systematic use of data from multiple, independent tracers can be used to validate the assumption of water age unimodality. Multimodal age distributions are more likely to emerge in landscapes where the distributions of flow path lengths and/or water velocities are themselves multimodal. In general, age multimodality may not be particularly pronounced and detectable, unless comparable amounts of water with contrasting ages reach a given outflow.

Candidate massive galaxies atz ∼ 4 in the Dark Energy Survey

Using stellar population models, we predicted that the Dark Energy Survey (DES) - due to its special combination of area (5000 deg. sq.) and depth ($i = 24.3$) - would be in the position to detect massive ($\gtrsim 10^{11}$ M$_{\odot}$) galaxies at $z \sim 4$. We confront those theoretical calculations with the first $\sim 150$ deg. sq. of DES data reaching nominal depth. From a catalogue containing $\sim 5$ million sources, $\sim26000$ were found to have observed-frame $g-r$ vs $r-i$ colours within the locus predicted for $z \sim 4$ massive galaxies. We further removed contamination by stars and artefacts, obtaining 606 galaxies lining up by the model selection box. We obtained their photometric redshifts and physical properties by fitting model templates spanning a wide range of star formation histories, reddening and redshift. Key to constrain the models is the addition, to the optical DES bands $g$, $r$, $i$, $z$, and $Y$, of near-IR $J$, $H$, $K_{s}$ data from the Vista Hemisphere Survey. We further applied several quality cuts to the fitting results, including goodness of fit and a unimodal redshift probability distribution. We finally select 233 candidates whose photometric redshift probability distribution function peaks around $z\sim4$, have high stellar masses ($\log($M$^{*}$/M$_{\odot})\sim 11.7$ for a Salpeter IMF) and ages around 0.1 Gyr, i.e. formation redshift around 5. These properties match those of the progenitors of the most massive galaxies in the local universe. This is an ideal sample for spectroscopic follow-up to select the fraction of galaxies which is truly at high redshift. These initial results and those at the survey completion, which we shall push to higher redshifts, will set unprecedented constraints on galaxy formation, evolution, and the re-ionisation epoch.

How much data do you need? An operational, pre-asymptotic metric for fat-tailedness

This paper presents an operational metric for univariate unimodal probability distributions with finite first moments in [0,1], where 0 is maximally thin-tailed (Gaussian) and 1 is maximally fat-tailed. It is based on the question, “how much data does one need to make meaningful statements about a given dataset?”Applications: Among others, it •helps to determine the sample size n needed for statistical significance outside the Gaussian distribution,•helps to measure the speed of convergence to the Gaussian (or stable basin),•allows practical comparisons across classes of fat-tailed distributions,•allows the number of securities needed in portfolio construction to achieve a certain level of risk-reduction from diversification to be assessed,•helps in assessing risks under various settings, and•helps to explain some inconsistent attributes of the lognormal, depending on the parametrization of its variance.The literature in regard to asymptotic behavior is rich, but there is a large void regarding finite values of n, which are those that are needed for operational purposes.Background: Conventional measures of fat-tailedness, namely (1) the tail index for the power law class, and (2) the kurtosis for finite moment distributions, fail to apply to some distributions, and do not allow comparisons across classes and parametrization; that is, between power laws outside the Levy-stable basin, or power laws and distributions in other classes, or power laws for different numbers of summands. How can one compare a sum of 100 Student t distributed random variables with three degrees of freedom to one in a Levy-stable or a Lognormal class? How can one compare a sum of 100 Student t with three degrees of freedom to a single Student t with two degrees of freedom?We propose an operational and heuristic metric that allows us to compare n-summed independent variables under all distributions with finite first moments. The method is based on the rate of convergence of the law of large numbers for finite sums, specifically n-summands.We get either explicit expressions or simulation results and bounds for the lognormal, exponential, Pareto, and Student t distributions in their various calibrations, in addition to the general Pearson classes.

A closed-form estimation of the travel time percentile function for characterizing travel time reliability

Travel time reliability (TTR) has received great attention in the past decades. The majority of TTR measures rely on the travel time percentile function as a basic element for performance evaluation. There are two main approaches for deriving the travel time percentile function: simple unimodal probability distribution models and mixture/nonparametric models. Despite the tractability of the former approach, they cannot sufficiently capture the travel time distributions (TTDs) due to their heterogeneity, and also often encounters many issues such as the failure of significance tests and the indecisiveness among multiple fitted distributions. On the other hand, the latter approach possesses greater flexibility for capturing diverse TTDs, but it does not have a simple and closed-form travel time percentile function. Motivated by the above drawbacks, this paper proposes a closed-form and flexible approach for estimating the travel time percentile function of diverse TTDs based on the Cornish–Fisher expansion without the need to assume/fit a certain distribution type. To ensure a high-quality estimation, we introduce and integrate two improvements with theoretically proven foundation into the Cornish–Fisher expansion while guaranteeing a closed-form expression of the travel time percentile function. Specifically, the first improvement, logarithm transformation, increases the probability of satisfying the validity domain of the Cornish–Fisher expansion; while the second improvement, rearrangement, guarantees a monotone travel time percentile function when travel time datasets cannot satisfy the validity domain after the logarithm transformation. Realistic travel time datasets are used to examine the accuracy and robustness of the proposed method. Compared to five widely-used probability distributions, the proposed method is sufficiently adaptable to estimating percentile function of diverse TTDs with lower estimation error. More importantly, it has a closed-form expression of the travel time percentile function, which would facilitate characterizing TTR in large-scale network applications.

A Quantal Response Statistical Equilibrium Model of Induced Technical Change in an Interactive Factor Market: Firm-Level Evidence in the EU Economies.

This paper studies the pattern of technical change at the firm level by applying and extending the Quantal Response Statistical Equilibrium model (QRSE). The model assumes that a large number of cost minimizing firms decide whether to adopt a new technology based on the potential rate of cost reduction. The firm in the model is assumed to have a limited capacity to process market signals so there is a positive degree of uncertainty in adopting a new technology. The adoption decision by the firm, in turn, makes an impact on the whole market through changes in the factor-price ratio. The equilibrium distribution of the model is a unimodal probability distribution with four parameters, which is qualitatively different from the Walrasian notion of equilibrium in so far as the state of equilibrium is not a single state but a probability distribution of multiple states. This paper applies Bayesian inference to estimate the unknown parameters of the model using the firm-level data of seven advanced OECD countries over eight years and shows that the mentioned equilibrium distribution from the model can satisfactorily recover the observed pattern of technical change.

Does prediction error drive one-shot declarative learning?

The role of prediction error (PE) in driving learning is well-established in fields such as classical and instrumental conditioning, reward learning and procedural memory; however, its role in human one-shot declarative encoding is less clear. According to one recent hypothesis, PE reflects the divergence between two probability distributions: one reflecting the prior probability (from previous experiences) and the other reflecting the sensory evidence (from the current experience). Assuming unimodal probability distributions, PE can be manipulated in three ways: (1) the distance between the mode of the prior and evidence, (2) the precision of the prior, and (3) the precision of the evidence. We tested these three manipulations across five experiments, in terms of peoples' ability to encode a single presentation of a scene-item pairing as a function of previous exposures to that scene and/or item. Memory was probed by presenting the scene together with three choices for the previously paired item, in which the two foil items were from other pairings within the same condition as the target item. In Experiment 1, we manipulated the evidence to be either consistent or inconsistent with prior expectations, predicting PE to be larger, and hence memory better, when the new pairing was inconsistent. In Experiments 2a-c, we manipulated the precision of the priors, predicting better memory for a new pairing when the (inconsistent) priors were more precise. In Experiment 3, we manipulated both visual noise and prior exposure for unfamiliar faces, before pairing them with scenes, predicting better memory when the sensory evidence was more precise. In all experiments, the PE hypotheses were supported. We discuss alternative explanations of individual experiments, and conclude the Predictive Interactive Multiple Memory Signals (PIMMS) framework provides the most parsimonious account of the full pattern of results.

Adaptive robust regression with continuous Gaussian scale mixture errors

Model based regression analysis always requires a certain choice of models which typically specifies the behavior of regression errors. The normal distribution is the most common choice for this purpose, but the estimator under normality is known to be too sensitive to outliers. As an alternative, heavy tailed distributions such as t distributions have been suggested. Though this choice can reduce the sensitivity to outliers, it also requires the choice of distributions and tuning parameters for practical use. In this paper, we propose a class of continuous Gaussian scale mixtures for the error distribution that contains most symmetric unimodal probability distributions including normal, t, Laplace, and stable distributions. With this quite flexible class of error distributions, we provide the asymptotic property and robust property of the proposed method, and show its successes along with numerical examples.

Goals as Reference Points: Empirical Evidence from a Virtual Reward System

Heath et al. [Heath C, Larrick RP, Wu G (1999) Goals as reference points. Cognitive Psych. 38(1):79–109] propose a prospect theory model for goal behavior. Their analytical model is based on the assumption that goals inherit the main properties of the prospect theory value function, i.e., reference point dependence, loss aversion, and diminishing sensitivity. We investigate whether these main properties transfer to goal behavior in the field. We take user activity data from a gamified question and answer (Q&A) community and analyze how users adjust their contribution behavior in the days surrounding goal achievement, where goals are represented by badges. We find that users gradually increase their performance in the days prior to earning a badge, with performance peaking on the day of the promotion. In subsequent days, user performance gradually diminishes again, with the decline being strongest on the day immediately following the badge achievement. These findings reflect the characteristic S-shape of the prospect theory value function which is convex below the reference point and concave above it. Employing the target-based approach, we can interpret the value function as a cumulative density function of a unimodal probability distribution. Our results suggest that it is more likely that active members of the community focus on the next badge relative to the status already achieved, as their next goal and are less likely to focus on more remote (higher-ranked) badges. Our results thus support the transferability of the main properties of the prospect theory value function to goal behavior in the field and suggest a distinct shape of the value function around goals.

Score function of distribution and revival of the moment method

The article deals with the scalar-valued score function SF defined for a regular unimodal and continuous probability distribution F with arbitrary interval support , recently introduced by the author. The concept of the central characteristic that describes a relative influence of is described in a much more general way in order to improve its usefulness in parametric estimation by means of the general moment method. In particular, the whole approach is elucidated by describing it in a more suitable framework than before. Further, we show that the inference function to be used in the moment estimating equations is either the score function of distribution (bounded for heavy-tailed parametric distribution models) or its modification based on the Huber’s approach.