Pith. sign in

REVIEW 2 major objections 5 minor 82 references

Minimum Density Power Divergence Estimation for the Gamma Distribution with Applications to Robust Rainfall Modeling

T0 review · 2 major / 5 minor · reviewed 2026-07-11 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A tunable robust estimator for the gamma distribution keeps rainfall fits stable when outliers appear while staying nearly as efficient as maximum likelihood on clean data.

desk verdict Solid, usable MDPDE specialization for gamma: closed-form sandwich matrices, IF/ARE results, and a clean rainfall application; incremental but referee-ready. read the letter →

arxiv 2607.04226 v1 pith:IR4JIFG5 submitted 2026-07-05 stat.ME stat.AP

Arnab Hazra This is my paper · ORCID
classification stat.MEstat.AP MSC 62F3562F1262P12
keywords densitypowerdivergencegammadistributionrobustestimationinfluencefunctionrainfallmodelingasymptoticrelativeefficiencyIndianmeteorologicalsubdivisions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Rainfall amounts are routinely modeled with the two-parameter gamma distribution, but maximum likelihood is easily pulled off course by a few extreme or erroneous observations. This paper builds the minimum density power divergence estimator (MDPDE) for that distribution: a single tuning parameter α continuously moves the estimator from ordinary maximum likelihood (α = 0) to progressively more outlier-resistant versions. Closed-form estimating equations, an explicit sandwich covariance matrix, and influence-function proofs show that every positive α yields bounded influence (when the shape exceeds 1) while asymptotic relative efficiency stays high for moderate α. Simulations under pure and contaminated gamma samples confirm that bias and mean-squared error remain controlled once contamination reaches 5–10 percent, outperforming moments, L-moments, and least-squares competitors. Applied to 64 years of detrended monsoon totals from India’s 36 meteorological subdivisions, the data-driven choice of α produces stable shape and rate maps and reliable 30/50/70 percent rainfall quantiles without sacrificing much precision relative to maximum likelihood.

What carries the argument

The density-power-divergence objective whose gradient yields weighted score equations: each observation is weighted by the α-power of its model density, automatically down-weighting points that lie far from the fitted gamma density.

What would settle it

Re-estimate the same Indian subdivision series with a model that keeps residual temporal dependence (or re-introduces the raw undetrended series); if the MDPDE shape and rate maps, their standard errors, or the CVM-chosen α change substantially, the i.i.d. gamma claim fails.

Watch

Extended reading notes

Core claim

The MDPDE family for the gamma distribution supplies explicit estimating equations and a closed-form asymptotic covariance matrix; for every tuning parameter α > 0 the influence function is bounded (when true shape exceeds 1), asymptotic relative efficiency declines only gradually with α, and finite-sample bias and mean-squared error under 5–10 percent contamination are markedly smaller than those of maximum likelihood and several classical alternatives, while efficiency on clean data remains competitive.

Load-bearing premise

After a simple log-scale trend is removed, the 64 annual monsoon totals inside each subdivision can be treated as independent and identically distributed gamma draws.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a complete minimum density power divergence estimation (MDPDE) framework for the two-parameter gamma distribution, motivated by rainfall modeling under contamination. It derives the closed-form objective (7), the weighted score estimating equations (11), and explicit sandwich asymptotic covariance expressions involving digamma/trigamma functions (13). Consistency and asymptotic normality follow from Basu et al. (1998) under the regularity a > 2α/(1+2α). Robustness is established via influence-function analysis (Theorem 3: bounded IF for every α>0 when a*>1) and asymptotic relative efficiency (Table 1, Theorems 1–2). Finite-sample performance is assessed by Monte Carlo under pure and two contamination schemes against ML, MM, percentile, LS, WLS and L-moments (Tables 2–3). A data-driven CVM choice of α is proposed. The method is applied to detrended 1951–2014 monsoon totals for India’s 36 meteorological subdivisions, producing parameter maps, rainfall quantiles and standard errors.

Significance. The contribution is a clean, self-contained specialization of the general MDPDE theory to the gamma model that is widely used in hydrology and agro-meteorology. Closed-form estimating equations, sandwich covariance and the bounded-IF result (Theorem 3) are load-bearing and immediately usable; the ARE table and the contamination simulations against six competitors give concrete guidance on the efficiency–robustness trade-off. The Indian rainfall application, with public code and data, demonstrates practical value. The work is incremental relative to the authors’ earlier generalized-exponential and multi-distribution papers, yet the gamma-specific algebra and the real-data analysis make it a useful reference for practitioners who need robust gamma fits.

major comments (2)
  1. Section 5 (paragraph after Cox–Stuart tests): after L1 log-detrending the 64 annual totals are treated as i.i.d. gamma. Residual serial dependence or unmodeled non-stationarity would invalidate both the asymptotic standard errors derived from (12)–(13) and the leave-one-out CVM selector (16). A short residual-diagnostics check (e.g., lag-1 ACF or Ljung–Box on the fitted residuals) or a brief sensitivity statement would strengthen the applied claims without altering the methodological core.
  2. Tables 2–3 and Section 4.2: all Monte Carlo designs fix the true model at Gamma(2,1). Table 1 already shows that ARE collapses for a=0.5 and the regularity a>2α/(1+2α) becomes restrictive. One additional simulation block with a smaller shape (e.g., a=0.8 or a=1) would confirm that the reported superiority under 5–10 % contamination is not an artifact of the chosen shape.
minor comments (5)
  1. Figure 5 caption: the third panel is labelled R_50 instead of R_70; correct the typo.
  2. Equation (7) and the definition of r_{a,α}: the domain restriction a>α/(1+α) is stated later; it would help the reader if it appeared immediately after (7).
  3. Section 3.1: the phrase “Proof of Theorem 2 is provided in the Appendix” is redundant because the short proof already appears in the main text.
  4. References: several self-citations (Hazra 2025, Hazra & Ghosh 2024, Bhowmik & Hazra 2026) are appropriate but could be condensed slightly to keep the focus on the gamma contribution.
  5. Data/code availability: the GitHub link is welcome; a short README note on the R packages required (lmom, univOutl, trend, numDeriv) would improve reproducibility.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: closed-form estimating equations, sandwich covariance, IF boundedness and ARE results are direct specializations of Basu et al. (1998) MDPDE theory to the gamma density; self-citations supply only parallel applications and a methodological template.

  1. self citation load bearing [Introduction, final paragraph before Section 2]
    "Overall, in this paper, we mostly follow the steps in the exploration of MDPDE for a generalized exponential distribution in Hazra (2025), while keeping the main focus on the gamma distribution."

    The paper explicitly anchors its organizational template and several robustness/efficiency checks to the author’s own prior MDPDE paper on a different distribution. While the gamma-specific algebra (eqs. 7–13, Theorems 1–3) is re-derived and self-contained, the citation supplies the only external warrant for the particular sequence of analyses (ARE table, IF plots, CVM selector, contamination designs). This is a minor, non-load-bearing self-citation rather than a definitional loop.

full rationale

The load-bearing derivation chain (objective V_α in (7), estimating equations (11), sensitivity/variability matrices and explicit J_α elements in (13), Theorems 1–3 on covariance scaling/ARE independence/IF boundedness for α>0 when a*>1, and the CVM selector (16)) follows the general M-estimator asymptotics and influence-function formulae of Basu et al. (1998) by substituting the gamma score (10) and the closed integral M_θ,α. These reductions are algebraic specializations, not tautologies that redefine the target quantities in terms of themselves. Simulation bias/MSE comparisons (Tables 2–3) and the Indian rainfall application are external Monte-Carlo and data exercises that do not feed back into the asymptotic claims. Self-citations (Hazra & Ghosh 2024; Hazra 2025) are used only to note prior rainfall applications and to indicate that the present paper follows the same exploratory outline previously used for the generalized exponential; they do not supply uniqueness theorems, ansätze, or fitted constants that force the gamma-specific closed forms. Consequently the central claims remain independently derived and falsifiable. A single minor self-citation of methodological template is recorded for completeness but is not load-bearing, yielding score 1.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper rests on the general MDPDE theory of Basu et al. (1998), the classical gamma density, and the modeling assumption that detrended monsoon totals are i.i.d. gamma. The single free tuning parameter α is selected by a data-driven CVM criterion rather than fixed a priori. No new physical entities are postulated.

free parameters (2)
  • α (MDPDE tuning parameter) = data-driven, 0.127–0.499 across subdivisions
    Controls the robustness–efficiency trade-off; selected for each data set by minimizing the leave-one-out Cramér–von Mises distance (eq. 16). In the rainfall application the selected values range from 0.127 to 0.499.
  • simulation contamination design = 0/1/5/10 % at fixed extreme quantiles
    Outlier percentages (0/1/5/10 %) and locations ((1–10^{-3}) and (1–10^{-5}) quantiles of Gamma(2,1)) are chosen by the author; reported bias/MSE rankings depend on these choices.
assumptions (3)
  • standard math Regularity conditions of Basu et al. (1998) hold for the gamma family whenever a > 2α/(1+2α), yielding consistency and asymptotic normality of the MDPDE.
    Invoked without re-proof in §3.1 to obtain the sandwich covariance Σ̃θ = Jα^{-1} Kα Jα^{-1}.
  • domain assumption After L1 detrending of log-rainfall, the 64 annual monsoon totals within each meteorological subdivision are independent and identically distributed gamma random variables.
    Stated in §5 after Cox–Stuart tests; independence is justified by the claim that weather conditions in different monsoon seasons are ‘likely independent’.
  • domain assumption The bulk of the data-generating distribution is gamma (or a contaminated gamma), so that the MDPDE targets the gamma parameters of interest.
    Maintained throughout the methodology, simulations, and application; supported by the extensive rainfall-modeling literature cited in the introduction.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Minimum Density Power Divergence Estimation for the Gamma Distribution with Applications to Robust Rainfall Modeling." pith.science (2026). https://pith.science/paper/IR4JIFG5

@misc{pith2026260704226,
  author       = {Pith},
  title        = {Pith review of: Minimum Density Power Divergence Estimation for the Gamma Distribution with Applications to Robust Rainfall Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IR4JIFG5}},
  note         = {Machine review of arXiv:2607.04226}
}
read the original abstract

Statistical modeling of rainfall amounts is of considerable importance in meteorology, hydrology, and agriculture. The gamma distribution remains one of the most popular choices for modeling rainfall data due to its flexibility and ability to capture the skewness of rainfall observations. Rainfall datasets often contain atypical observations due to measurement errors and extreme weather events, making maximum likelihood estimation (MLE) highly sensitive to contamination. In this paper, we develop a robust estimation framework for the two-parameter gamma distribution based on the minimum density power divergence estimator (MDPDE). Explicit estimating equations are derived, and several theoretical properties of the proposed estimators are established. In particular, closed-form expressions for the asymptotic covariance matrix are obtained, and robustness is investigated through influence function analysis and asymptotic relative efficiency. The finite-sample performance of the estimators is examined through simulation studies under both pure and contaminated gamma models. The proposed methodology is further implemented to analyze detrended areally weighted monsoon rainfall data from the 36 meteorological subdivisions of India for the period 1951--2014. The results demonstrate that the MDPDE provides a useful compromise between robustness and efficiency, yielding more stable inference than MLE in the presence of outliers while maintaining high efficiency for uncontaminated data.

Figures

Figures reproduced from arXiv: 2607.04226 by the authors.

Figure 1
Figure 1. Variation of the elements of the asymptotic covariance matrix [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Components of the influence function IF ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Meteorological subdivision-wise rainfall data for 36 subdivisions of India during [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Meteorological subdivision-wise minimum density power divergence estimates of [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Meteorological subdivision-wise estimates (in the left column) for 30%, 50%, and [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

82 extracted references · 3 linked inside Pith

  1. [1]

    Water , volume=

    Flood frequency analysis using the gamma family probability distributions , author=. Water , volume=. 2023 , publisher=

    2023

  2. [2]

    Entropy , volume=

    An entropy-based generalized gamma distribution for flood frequency analysis , author=. Entropy , volume=. 2017 , publisher=

    2017

  3. [3]

    2026 , publisher=

    Basu, Arghadeep and Hazra, Arnab , booktitle=. 2026 , publisher=

    2026

  4. [4]

    Computational Statistics & Data Analysis , volume=

    An adjusted boxplot for skewed distributions , author=. Computational Statistics & Data Analysis , volume=. 2008 , publisher=

    2008

  5. [5]

    2022 , note =

    univOutl: Detection of Univariate Outliers , author =. 2022 , note =

    2022

  6. [6]

    2023 , note =

    trend: Non-Parametric Trend Tests and Change-Point Detection , author =. 2023 , note =

    2023

  7. [7]

    New rainfall series for the districts, meteorological sub-divisions and country as whole of

    Guhathakurta, P and Koppar, AL and Krishan, U and Menon, Preetha , journal=. New rainfall series for the districts, meteorological sub-divisions and country as whole of

  8. [8]

    Trends in the rainfall pattern over

    Guhathakurta, P and Rajeevan, M , journal=. Trends in the rainfall pattern over. 2008 , publisher=

    2008

Show all 82 references
  1. [9]

    Biometrika , volume=

    Some quick sign tests for trend in location and dispersion , author=. Biometrika , volume=. 1955 , publisher=

    1955

  2. [10]

    2026 , note =

    L-Moments , author =. 2026 , note =

    2026

  3. [11]

    1994 , publisher=

    Continuous univariate distributions, volume 1 , author=. 1994 , publisher=

    1994

  4. [12]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    L-moments: analysis and estimation of distributions using linear combinations of order statistics , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 1990 , publisher=

    1990

  5. [13]

    2010 , publisher=

    Kendall's advanced theory of statistics, distribution theory , author=. 2010 , publisher=

    2010

  6. [14]

    Journal of the Atmospheric Sciences , volume=

    Why do precipitation intensities tend to follow gamma distributions? , author=. Journal of the Atmospheric Sciences , volume=. 2019 , publisher=

    2019

  7. [15]

    Journal of Hydrology , volume=

    The suitability assessment of a generalized exponential distribution for the description of maximum precipitation amounts , author=. Journal of Hydrology , volume=. 2015 , publisher=

    2015

  8. [16]

    Communications in Statistics-Theory and Methods , volume=

    Minimum density power divergence estimation for the generalized exponential distribution , author=. Communications in Statistics-Theory and Methods , volume=. 2025 , publisher=

    2025

  9. [17]

    Selection of the best-fit probability distribution in rainfall frequency analysis for

    Mamoon, Abdullah Al and Rahman, Ataur , journal=. Selection of the best-fit probability distribution in rainfall frequency analysis for. 2017 , publisher=

    2017

  10. [18]

    Estimation and prediction of maximum daily rainfall at

    Mandal, S and Choudhury, BU , journal=. Estimation and prediction of maximum daily rainfall at. 2015 , publisher=

    2015

  11. [19]

    A water harvesting model for optimizing rainwater harvesting in the wadi

    Adham, Ammar and Wesseling, Jan G and Riksen, Michel and Ouessar, Mohamed and Ritsema, Coen J , journal=. A water harvesting model for optimizing rainwater harvesting in the wadi. 2016 , publisher=

    2016

  12. [20]

    Japanese Journal of Statistics and Data Science , volume=

    A Bayesian latent Gaussian conditional autoregressive copula model for analyzing spatially-varying trends in rainfall , author=. Japanese Journal of Statistics and Data Science , volume=. 2026 , publisher=

    2026

  13. [21]

    Sankhya B , volume=

    Robust statistical modeling of monthly rainfall: The minimum density power divergence approach , author=. Sankhya B , volume=. 2024 , publisher=

    2024

  14. [22]

    Environmetrics , volume=

    A multivariate spatial skew-t process for joint modeling of extreme precipitation indexes , author=. Environmetrics , volume=. 2020 , publisher=

    2020

  15. [23]

    The Annals of Applied Statistics , volume=

    Estimating high-resolution Red Sea surface temperature hotspots, using a low-rank semiparametric spatial model , author=. The Annals of Applied Statistics , volume=. 2021 , publisher=

    2021

  16. [24]

    Sankhya B , volume=

    Absolute continuous multivariate generalized exponential distribution , author=. Sankhya B , volume=. 2015 , publisher=

    2015

  17. [25]

    Sankhya B , pages=

    Stationary GE-Process and its Application in Analyzing Gold Price Data , author=. Sankhya B , pages=. 2021 , publisher=

    2021

  18. [26]

    Journal of Statistical Computation and Simulation , volume=

    Choosing a robustness tuning parameter , author=. Journal of Statistical Computation and Simulation , volume=. 2005 , publisher=

    2005

  19. [27]

    Journal of the Korean Statistical Society , volume=

    Automatic Selection of the Turning Parametter in the Minimum Density Power Divergence Estimation , author=. Journal of the Korean Statistical Society , volume=. 2001 , publisher=

    2001

  20. [28]

    Journal of Applied Statistics , volume=

    On the ‘optimal’ density power divergence tuning parameter , author=. Journal of Applied Statistics , volume=. 2021 , publisher=

    2021

  21. [29]

    Journal of Statistical Computation and Simulation , volume=

    Generalized exponential distribution: different method of estimations , author=. Journal of Statistical Computation and Simulation , volume=. 2001 , publisher=

    2001

  22. [30]

    Environmetrics: The official journal of the International Environmetrics Society , volume=

    Bayesian prediction of rainfall records using the generalized exponential distribution , author=. Environmetrics: The official journal of the International Environmetrics Society , volume=. 2007 , publisher=

    2007

  23. [31]

    Journal of Statistical planning and inference , volume=

    Generalized exponential distribution: Existing results and some recent developments , author=. Journal of Statistical planning and inference , volume=. 2007 , publisher=

    2007

  24. [32]

    arXiv preprint arXiv:1909.08035 , year=

    Robust statistical modeling of monthly rainfall: The minimum density power divergence approach , author=. arXiv preprint arXiv:1909.08035 , year=

    1909 arXiv

  25. [33]

    Exponentiated exponential family: an alternative to gamma and

    Gupta, Rameshwar D and Kundu, Debasis , journal=. Exponentiated exponential family: an alternative to gamma and. 2001 , publisher=

    2001

  26. [34]

    Australian & New Zealand Journal of Statistics , volume=

    Theory & methods: Generalized exponential distributions , author=. Australian & New Zealand Journal of Statistics , volume=. 1999 , publisher=

    1999

  27. [35]

    Journal of the Royal Statistical Society: Series C (Applied Statistics) , volume=

    Fitting a Gamma or Log-Normal Distribution to Fibre-Diameter Measurements on Wool Tops , author=. Journal of the Royal Statistical Society: Series C (Applied Statistics) , volume=. 1969 , publisher=

    1969

  28. [36]

    Quarterly Journal of the Royal Meteorological Society , volume=

    Monsoon seasonal forecasting , author=. Quarterly Journal of the Royal Meteorological Society , volume=. 1953 , publisher=

    1953

  29. [37]

    Environmetrics , volume=

    Correspondence estimation of the source profiles in receptor modeling , author=. Environmetrics , volume=. 2004 , publisher=

    2004

  30. [38]

    BMC Bioinformatics , volume=

    Partial mixture model for tight clustering of gene expression time-course , author=. BMC Bioinformatics , volume=. 2008 , publisher=

    2008

  31. [39]

    Beran, Rudolf , journal=. Minimum. 1977 , publisher=

    1977

  32. [40]

    Annals of the Institute of Statistical Mathematics , volume=

    Minimum disparity estimation for continuous models: efficiency, distributions and robustness , author=. Annals of the Institute of Statistical Mathematics , volume=. 1994 , publisher=

    1994

  33. [41]

    Water , volume=

    Extreme Precipitation Frequency Analysis Using a Minimum Density Power Divergence Estimator , author=. Water , volume=. 2017 , publisher=

    2017

  34. [42]

    On robustness of large quantile estimates of log-

    Strupczewski, WG and Kochanek, K and Weglarczyk, S and Singh, VP , journal=. On robustness of large quantile estimates of log-. 2005 , publisher=

    2005

  35. [43]

    Computational Statistics & Data Analysis , volume=

    Robust fitting of mixtures using the trimmed likelihood estimator , author=. Computational Statistics & Data Analysis , volume=. 2007 , publisher=

    2007

  36. [44]

    Hydrological Processes: An International Journal , volume=

    On robustness of large quantile estimates to largest elements of the observation series , author=. Hydrological Processes: An International Journal , volume=. 2007 , publisher=

    2007

  37. [45]

    International Journal of Climatology , volume=

    Development of statistical models for at-site probabilistic seasonal rainfall forecast , author=. International Journal of Climatology , volume=. 2012 , publisher=

    2012

  38. [46]

    Frequency analysis of the monthly rainfall data at

    Al-Suhili, Rafa H and Khanbilvardi, Reza , journal=. Frequency analysis of the monthly rainfall data at

  39. [47]

    Journal of Statistical Planning and Inference , volume=

    Robust estimation in the normal mixture model , author=. Journal of Statistical Planning and Inference , volume=. 2006 , publisher=

    2006

  40. [48]

    Biometrika , volume=

    Robust and efficient estimation by minimising a density power divergence , author=. Biometrika , volume=. 1998 , publisher=

    1998

  41. [49]

    Rainfall intensity patterns derived from the urban network of

    Lana, Xavier and Serra, C and Casas-Castillo, MC and Rodr. Rainfall intensity patterns derived from the urban network of. Theoretical and Applied Climatology , pages=. 2017 , publisher=

    2017

  42. [50]

    Environmetrics , volume=

    Small sample inference for gamma parameters: one-sample and two-sample problems , author=. Environmetrics , volume=. 2014 , publisher=

    2014

  43. [51]

    Use of the gamma distribution to represent monthly rainfall in

    Husak, Gregory J and Michaelsen, Joel and Funk, Chris , journal=. Use of the gamma distribution to represent monthly rainfall in. 2007 , publisher=

    2007

  44. [52]

    Meteorological Applications , volume=

    Optimal stochastic multi-states first-order Markov chain parameters for synthesizing daily rainfall data using multi-objective differential evolution in Thailand , author=. Meteorological Applications , volume=. 2013 , publisher=

    2013

  45. [53]

    Statistical distributions of daily rainfall regime in

    Burgue. Statistical distributions of daily rainfall regime in. Theoretical and Applied Climatology , volume=. 2010 , publisher=

    2010

  46. [54]

    Modelling weekly rainfall data for crop planning in a sub-humid climate of

    Sharda, VN and Das, PK , journal=. Modelling weekly rainfall data for crop planning in a sub-humid climate of. 2005 , publisher=

    2005

  47. [55]

    Stochastic Environmental Research and Risk Assessment , volume=

    Analysis of extreme rainfall using the log-logistic distribution , author=. Stochastic Environmental Research and Risk Assessment , volume=. 2005 , publisher=

    2005

  48. [56]

    The statistical distribution of annual maximum rainfall in

    Mayooran, T and Laheetharan, A , journal=. The statistical distribution of annual maximum rainfall in. 2014 , publisher=

    2014

  49. [57]

    Determination of the best fit probability distribution for annual extreme precipitation in

    Khudri, Md Mohsan and Sadia, Farhana , journal=. Determination of the best fit probability distribution for annual extreme precipitation in

  50. [58]

    The probability distribution of daily rainfall in the

    Hanson, Lars S and Vogel, Richard , booktitle=. The probability distribution of daily rainfall in the

  51. [59]

    Meteorology and Atmospheric Physics , volume=

    A note on the misuses of the variance test in meteorological studies , author=. Meteorology and Atmospheric Physics , volume=. 2017 , publisher=

    2017

  52. [60]

    Modelling

    Hazra, Arnab and Bhattacharya, Sabyasachi and Banik, Pabitra , journal=. Modelling. 2014 , publisher=

    2014

  53. [61]

    Open Journal of Modern Hydrology , volume=

    Introducing the mixed distribution in fitting rainfall data , author=. Open Journal of Modern Hydrology , volume=. 2011 , publisher=

    2011

  54. [62]

    Hazra, Arnab and Bhattacharya, Sourabh and Banik, Pabitra , journal=. A. 2018 , publisher=

    2018

  55. [63]

    Characterization and frequency analysis of one day annual maximum and two to five consecutive days maximum rainfall of

    Kwaku, Xeflide Seth and Duke, Ophori , journal=. Characterization and frequency analysis of one day annual maximum and two to five consecutive days maximum rainfall of

  56. [64]

    New York Science Journal , volume=

    Use of probability distribution in rainfall analysis , author=. New York Science Journal , volume=

  57. [65]

    Gamma distribution probability model for

    Mooley, Diwakar A , journal=. Gamma distribution probability model for

  58. [66]

    Hydrological sciences journal , volume=

    Rainfall distribution function for Libya and rainfall prediction , author=. Hydrological sciences journal , volume=. 1999 , publisher=

    1999

  59. [67]

    Journal of Applied Meteorology , volume=

    A stochastic model of n -day precipitation , author=. Journal of Applied Meteorology , volume=

  60. [68]

    Statistical distributions of the daily rainfall regime in

    Burgue. Statistical distributions of the daily rainfall regime in. International Journal of Climatology , volume=. 2005 , publisher=

    2005

  61. [69]

    A comparison of stochastic models for generating daily precipitation at the

    Duan, Jinfan and Sikka, Alok K and Grant, Gordon E , year=. A comparison of stochastic models for generating daily precipitation at the. Northwestern Science , volume=

  62. [70]

    Assessment of interstate virtual water flows embedded in agriculture to mitigate water scarcity in

    Katyaini, Suparana and Barua, Anamika , journal=. Assessment of interstate virtual water flows embedded in agriculture to mitigate water scarcity in. 2017 , publisher=

    2017

  63. [71]

    Global warming and its possible impact on agriculture in

    Chauhan, Bhagirath Singh and Mahajan, Gulshan and Randhawa, Ramanjit Kaur and Singh, Harpreet and Kang, Manjit S and others , booktitle=. Global warming and its possible impact on agriculture in

  64. [72]

    Hazra, Arnab and Huser, Rapha. Latent. arXiv preprint arXiv:2110.02680 , year=

    arXiv

  65. [73]

    Rainfed agriculture in

    Venkateswarlu, B , journal=. Rainfed agriculture in

  66. [74]

    1993 , publisher=

    Statistics for spatial data , author=. 1993 , publisher=

    1993

  67. [75]

    Arjun, Kekane M , journal=

  68. [76]

    Agronomy Journal , pages=

    Evaluation of drought hazard , author=. Agronomy Journal , pages=

  69. [77]

    An application of the gamma distribution function to

    Mooley, Diwakar A and Crutcher, Harold L , volume=. An application of the gamma distribution function to

  70. [78]

    2019 , address =

    Statistical inference: the minimum distance approach , author=. 2019 , address =

    2019

  71. [79]

    1986 , publisher=

    Robust Statistics: The Approach Based on Influence Functions , author=. 1986 , publisher=

    1986

  72. [80]

    and Karagrigoriou, A

    Mattheou, K., and Lee, S. and Karagrigoriou, A. , journal=. A model selection criterion based on the

  73. [81]

    arXiv preprint arXiv:2009.04710 , year=

    Robust Clustering with Normal Mixture Models: A Pseudo -Likelihood Approach , author=. arXiv preprint arXiv:2009.04710 , year=

    2009 arXiv

  74. [82]

    The Annals of Applied Statistics , year=

    Approximate Bayesian inference for analysis of spatio-temporal flood frequency data , author=. The Annals of Applied Statistics , year=

Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.