arxiv: 2605.03044 · v1 · submitted 2026-05-04 · 📊 stat.ME · math.ST· stat.AP· stat.TH

Recognition: 2 theorem links

· Lean Theorem

Tweedie-based nonparametric estimation for semicontinuous mixed densities

Guanjie Lyu , Fr\'ed\'eric Ouimet , Cindy Feng

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:05 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.APstat.TH

keywords nonparametric density estimationsemicontinuous dataTweedie distributionasymmetric kernelsboundary effectszero-inflated densitiesmixed distributionscross-validation

0 comments

The pith

Tweedie asymmetric kernels unify nonparametric estimation of semicontinuous mixed densities on [0, ∞).

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

This paper develops a nonparametric estimator for densities that combine a point mass at zero with a continuous positive component, a structure common in health costs, insurance claims, and lengths of stay. Standard kernel methods distort the boundary at zero or require separate handling of the discrete and continuous parts. The proposed approach uses the Tweedie distribution with power parameter between 1 and 2 as an asymmetric kernel whose own form already includes a mass at zero and a density on the positives. This single construction smooths the positive data while leaving the zero atom intact and draws on every observation. Asymptotic expansions, optimal bandwidth rates, normality results, and a joint cross-validation selector for bandwidth and power parameter are derived, with simulations showing gains in boundary-spike and heavy-tailed regimes.

Core claim

We introduce an asymmetric kernel estimator for mixed densities on [0,∞) based on the Tweedie distribution. For a power parameter p∈(1,2), the Tweedie kernel itself has a point mass at zero and an absolutely continuous component on (0,∞), yielding a unified smoothing construction that preserves the atom at zero and smooths the positive component using the full semicontinuous sample. We establish pointwise bias and variance expansions, derive asymptotic formulae for the mean squared error and mean integrated squared error, obtain optimal bandwidth rates, and prove asymptotic normality. We propose a profile least-squares cross-validation procedure to jointly select the bandwidth and the power.

What carries the argument

The Tweedie kernel with power p in (1,2), an asymmetric kernel that carries both a built-in point mass at zero and a continuous density on (0,∞) to weight the entire semicontinuous sample in one step.

If this is right

The estimator attains standard optimal convergence rates for density estimation while respecting the mixed structure.
Asymptotic normality at interior points supports construction of pointwise confidence bands.
Profile least-squares cross-validation jointly tunes bandwidth and power parameter in finite samples.
Performance advantages appear most clearly in boundary-spike and heavy-tailed positive components.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction may extend directly to regression settings for conditional semicontinuous densities.
Similar Tweedie-based kernels could handle other nonnegative supports with atoms, such as count-inflated continuous data.
Multivariate versions would allow joint modeling of several semicontinuous variables without separate marginal steps.
The automatic mass preservation suggests the method can serve as a building block for more complex zero-inflated models in applied statistics.

Load-bearing premise

The Tweedie distribution with power p in (1,2) provides a valid kernel that correctly separates the discrete mass at zero from the continuous component on (0,∞) for the target density class.

What would settle it

A simulation or real-data comparison in which the Tweedie estimator yields higher mean integrated squared error than a two-part estimator (separate zero-probability estimate plus positive kernel) under heavy tails or strong boundary spikes would refute the unified advantage.

Figures

Figures reproduced from arXiv: 2605.03044 by Cindy Feng, Fr\'ed\'eric Ouimet, Guanjie Lyu.

**Figure 2.1.** Figure 2.1: Illustration of the estimator construction under a semicontinuous sample with positive observations and an atom at zero. The left panel illustrates the proposed Tweedie kernel estimator as the average of Tweedie kernels centered at each observation, including both zero and positive values, whereas the right panel shows a conventional two-part approach based only on the positive observations. The blue das… view at source ↗

**Figure 4.1.** Figure 4.1: displays the true positive-component target functions g+(x) under the four simulation scenarios, with the structural-zero probability fixed at p0 = 0.3. Scenario M.1 corresponds to the Tweedie case and exhibits a unimodal shape with moderate right skewness. Scenario M.2 is a zero-inflated Gamma model that is likewise unimodal, but more sharply concentrated near the origin and supported over a shorter ran… view at source ↗

**Figure 4.2.** Figure 4.2: Monte Carlo boxplots (500 replicates) of the integrated squared error (ISE+) of the density estimators on the positive half-line (0, ∞) under the four simulation scenarios. Each panel corresponds to a data-generating mechanism (M.1– M.4), with columns showing different values of the zero mass p0 ∈ {0.15, 0.30, 0.45} and sample sizes n ∈ {100, 200, 500}. The proposed Tweedie kernel estimator is compared w… view at source ↗

**Figure 4.3.** Figure 4.3: Monte Carlo boxplots (500 replicates) of the integrated absolute error (IAE+) of the density estimators on the positive half-line (0, ∞) under the four simulation scenarios. Each panel corresponds to a data-generating mechanism (M.1– M.4), with columns showing different values of the zero mass p0 ∈ {0.15, 0.30, 0.45} and sample sizes n ∈ {100, 200, 500}. The proposed Tweedie kernel estimator is compared … view at source ↗

**Figure 4.** Figure 4: reports the resulting rejection rates under the M.1 data-generating process. In the left panel, view at source ↗

**Figure 4.4.** Figure 4.4: reports the resulting rejection rates under the M.1 data-generating process. In the left panel, the alternative is indexed by the Tweedie power parameter p, with the null corresponding to p = 1.1. In the right panel, the alternative is indexed by the mean parameter µ, with the null corresponding to µ = 2. In both panels, the rejection rates under the null configuration are close to the nominal 5% level. … view at source ↗

**Figure 5.** Figure 5: highlights two salient features of the excess length-of-stay distribution for mental health view at source ↗

**Figure 5.1.** Figure 5.1: highlights two salient features of the excess length-of-stay distribution for mental health visits. The left panel shows a substantial point mass at zero together with a strongly right-skewed positive component, indicating that many visits do not exceed the 4-hour benchmark, while positive exceedances are concentrated near zero and taper off gradually over a long right tail. In the right panel, the densi… view at source ↗

read the original abstract

Semicontinuous outcomes occur frequently in health services, insurance, and cost studies. Standard nonparametric density estimators are not well suited to such data because they do not naturally accommodate the mixed structure, the nonnegative support, or the pronounced boundary effects near zero. To address these limitations, we introduce an asymmetric kernel estimator for mixed densities on $[0,\infty)$ based on the Tweedie distribution. For a power parameter $p\in(1,2)$, the Tweedie kernel itself has a point mass at zero and an absolutely continuous component on $(0,\infty)$, yielding a unified smoothing construction that preserves the atom at zero and smooths the positive component using the full semicontinuous sample. We establish pointwise bias and variance expansions, derive asymptotic formulae for the mean squared error and mean integrated squared error, obtain optimal bandwidth rates, and prove asymptotic normality. We propose a profile least-squares cross-validation procedure to jointly select the bandwidth and the power parameter. Simulation results show competitive performance, particularly in challenging boundary-spike and heavy-tailed settings, and an application to emergency department length-of-stay data illustrates the practical value of the method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Tweedie kernels give a clean way to estimate semicontinuous densities while keeping the zero mass intact, with standard asymptotics attached.

read the letter

The main takeaway is that this paper uses the Tweedie distribution with p in (1,2) to build an asymmetric kernel that has a built-in point mass at zero plus a density on the positive side. That structure lets the estimator smooth the full sample without splitting the data or patching boundary effects separately. They work out the usual pointwise bias and variance, the MISE formula, optimal bandwidth rates, and asymptotic normality, then add a profile least-squares CV that picks both h and p at once. Simulations are reported to perform competitively on boundary spikes and heavy tails, with a real-data illustration on emergency department lengths of stay. The construction itself appears new for this mixed-density setting and rests on known Tweedie properties plus ordinary kernel arguments, so there is no obvious circularity or unsupported regularity condition. The stress-test note aligns with that reading. Soft spots are limited. The extra tuning parameter p is handled by the CV, but its practical sensitivity is only shown through simulations rather than deeper analysis. The method stays specialized to nonnegative semicontinuous data, so it will not displace general-purpose estimators. Computational scaling for very large samples is not discussed. This is aimed at statisticians who routinely work with health, insurance, or cost data that mix a zero atom with a positive continuous part. A reader who needs a nonparametric density that respects that structure without ad-hoc fixes will get a usable tool with supporting theory. It is worth sending to referees because the central idea is coherent, the derivations follow standard lines, and the simulations provide some empirical grounding.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a nonparametric estimator for semicontinuous mixed densities supported on [0, ∞) that employs the Tweedie distribution with power parameter p ∈ (1,2) as an asymmetric kernel. This kernel inherently carries a point mass at zero together with a density on (0, ∞), allowing a single smoothing construction that preserves the atom while using the full sample for the continuous component. The authors derive pointwise bias and variance expansions, asymptotic MSE and MISE formulae, optimal bandwidth rates, and asymptotic normality; they also introduce a profile least-squares cross-validation procedure for joint selection of the bandwidth h and power p. Simulation experiments and an application to emergency-department length-of-stay data are reported.

Significance. If the stated derivations hold, the work supplies a theoretically grounded, unified kernel method for a data type that appears routinely in health-services, insurance, and cost studies. The approach exploits the compound-Poisson-gamma structure of the Tweedie family to avoid ad-hoc separation of the discrete and continuous parts and to mitigate boundary effects near zero. The claimed optimal rates, normality result, and joint CV selector constitute concrete methodological advances. The reported simulations indicate competitive performance in boundary-spike and heavy-tailed regimes, and the real-data illustration demonstrates practical utility.

major comments (2)

[§3] §3 (asymptotic theory): the bias and variance expansions are stated to follow from standard kernel arguments applied to the Tweedie kernel, yet the precise regularity conditions on the target density (e.g., smoothness at zero, behavior of the continuous component) that justify interchanging limits and integrals are not spelled out; these conditions are load-bearing for the subsequent MISE and normality claims.
[§4] §4 (profile least-squares CV): the procedure is defined and implemented, but no consistency or rate result for the selected (ĥ, p̂) is provided; without such a result the practical estimator lacks the theoretical backing given to the oracle bandwidth, which weakens the overall contribution.

minor comments (3)

[§2] The notation for the mixed density (atom plus continuous part) is introduced only in the abstract and should be restated explicitly in §2 with a clear decomposition f = π δ_0 + (1-π) f_c.
[§5] Simulation tables would benefit from reporting the selected p̂ values alongside ĥ; this would illustrate how the joint selector behaves across the boundary-spike and heavy-tailed designs.
[Introduction] A brief comparison with the existing literature on asymmetric kernels for nonnegative data (e.g., gamma or inverse-Gaussian kernels) would help situate the Tweedie choice.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and the recommendation for minor revision. The comments highlight important points regarding the presentation of assumptions and the theoretical properties of the cross-validation procedure. We address each comment below and indicate the changes we will implement in the revised manuscript.

read point-by-point responses

Referee: [§3] §3 (asymptotic theory): the bias and variance expansions are stated to follow from standard kernel arguments applied to the Tweedie kernel, yet the precise regularity conditions on the target density (e.g., smoothness at zero, behavior of the continuous component) that justify interchanging limits and integrals are not spelled out; these conditions are load-bearing for the subsequent MISE and normality claims.

Authors: We concur that explicitly stating the regularity conditions is necessary for rigor. In the revised version, we will insert a new paragraph at the beginning of §3 that specifies the assumptions: the mixed density f is such that its continuous part f_c on (0, ∞) belongs to the class of twice continuously differentiable functions with bounded second derivative, f is continuous from the right at 0, and suitable moment conditions hold to allow differentiation under the integral sign in the bias and variance calculations. These conditions ensure the validity of the expansions and the subsequent results on MISE and asymptotic normality. We believe this addition will address the concern without altering the main results. revision: yes
Referee: [§4] §4 (profile least-squares CV): the procedure is defined and implemented, but no consistency or rate result for the selected (ĥ, p̂) is provided; without such a result the practical estimator lacks the theoretical backing given to the oracle bandwidth, which weakens the overall contribution.

Authors: This is a fair observation. Deriving consistency and rates for the joint selector (ĥ, p̂) would require proving uniform convergence of the CV criterion to the MISE over a suitable range of h and p, which involves more advanced empirical process techniques and is not included in the present work. We will revise §4 to include a brief discussion acknowledging this gap and noting that the selector is supported by the simulation studies showing good finite-sample performance. The primary theoretical contribution remains the asymptotic properties of the Tweedie kernel estimator itself, with the CV serving as a data-driven implementation. If the editor deems it necessary, we can explore adding a heuristic argument, but a full proof is left for future research. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard Tweedie kernel properties and kernel asymptotics

full rationale

The paper constructs the estimator directly from the known mixed structure of the Tweedie distribution (point mass at zero plus continuous component on (0,∞) for p ∈ (1,2)), then applies standard bias/variance expansions, MISE derivations, optimal bandwidth rates, asymptotic normality, and profile least-squares CV. No step reduces a claimed prediction or result to a fitted quantity defined by the same data, nor relies on self-citation chains for uniqueness or ansatz smuggling. The central construction and asymptotics remain independent of the target estimates.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central construction rests on the existence and properties of Tweedie distributions for p in (1,2) that produce the required mixed kernel, plus standard regularity conditions for kernel asymptotics that are not detailed in the abstract.

free parameters (1)

power parameter p
Chosen jointly with bandwidth via profile least-squares cross-validation; remains a tunable parameter even after selection.

axioms (1)

domain assumption Tweedie distributions with p ∈ (1,2) possess a point mass at zero and an absolutely continuous component on (0,∞) that can serve as a valid kernel.
Invoked to justify the unified smoothing construction for semicontinuous data.

pith-pipeline@v0.9.0 · 5506 in / 1350 out tokens · 82102 ms · 2026-05-08T18:05:15.731245+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

For a power parameter p in (1,2), the Tweedie kernel itself has a point mass at zero and an absolutely continuous component on (0,infty) ... We propose a profile least-squares cross-validation procedure to jointly select the bandwidth and the power parameter.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 22 canonical work pages · 1 internal anchor

[1]

Tweedie-based nonparametric estimation for semicontinuous mixed densities

Introduction Semicontinuous outcomes, characterized by a point mass at zero together with a continuously dis- tributed positive component, arise naturally in many applied settings where no-event or nonuse observa- tions coexist with positive, often strongly right-skewed realizations. Such data have long been recognized in statistics; an early treatment is...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

Equivalently, the cumulative distribution functionGofXhas the form G(x) =p 01(x≥0) + (1−p 0)F(x), x∈R,(2.1) where1(·) is the indicator function

Semicontinuous data and Tweedie kernel estimator Formally, a random variableXissemicontinuousif it admits the representation X d= ( 0,with probabilityp 0, Y,with probability 1−p 0, wherep 0 ∈(0,1) andYhas distribution functionFsupported on (0,∞) with densityfwith respect to the Lebesgue measure. Equivalently, the cumulative distribution functionGofXhas th...
[3]

Main results Becausebgh(0) =bp0, the only nontrivial asymptotic analysis concerns the casex >0. The results in this section are therefore pointwise at a fixed interior locationx, but they must still account for the influence of the boundary at zero through the support of the Tweedie kernel. The assumptions below separate these two aspects: local smoothnes...
[4]

coarse”, “medium

Simulation study This section investigates the finite-sample performance of the proposed Tweedie kernel estimator for semicontinuous data. The simulation study is designed to assess how well the estimator recovers the positive component of the target density across a range of settings, including the Tweedie case and several misspecified zero-inflated alte...
[5]

The data were obtained from Nova Scotia Health’s centralized Emergency Department Information System (EDIS), which uses standardized reporting procedures across hospitals

Real-data application For the empirical illustration, we analyze 732 mental health-related emergency department visits recorded in March 2023. The data were obtained from Nova Scotia Health’s centralized Emergency Department Information System (EDIS), which uses standardized reporting procedures across hospitals. The data come from four principal emergenc...

2023
[6]

Concluding remarks In this paper, we studied nonparametric estimation for semicontinuous distributions by introducing a Tweedie kernel estimator for the full mixed density on [0,∞). The proposed construction exploits the compound Poisson–Gamma structure of Tweedie laws with power parameterp∈(1,2), so that the point mass at zero and the continuous positive...

1984
[7]

Aitchison

J. Aitchison. On the distribution of a positive random variable having a discrete probability mass at the origin.J. Amer. Statist. Assoc., 50(271):901–908, 1955.DOI:10.1080/01621459.1955.1050 1976

work page doi:10.1080/01621459.1955.1050 1955
[8]

Billingsley.Probability and Measure

P. Billingsley.Probability and Measure. Wiley Series in Probability and Mathematical Statistics. Wiley, New York, 3rd edition, 1995. ISBN 978-0-471-00710-4

1995
[9]

Bouezmarni and J.-M

T. Bouezmarni and J.-M. Rolin. Consistency of the beta kernel density function estimator.Canad. J. Statist., 31(1):89–98, 2003.DOI:10.2307/3315905

work page doi:10.2307/3315905 2003
[10]

Bouezmarni and J.-M

T. Bouezmarni and J.-M. Rolin. Bernstein estimator for unbounded density function.J. Non- parametr. Stat., 19(3):145–161, 2007.DOI:10.1080/10485250701441218

work page doi:10.1080/10485250701441218 2007
[11]

Bouezmarni and O

T. Bouezmarni and O. Scaillet. Consistency of asymmetric kernel density estimators and smoothed histograms with application to income data.Econometric Theory, 21(2):390–412, 2005.DOI:10.1 017/S0266466605050218

2005
[12]

S. X. Chen. Beta kernel estimators for density functions.Comput. Statist. Data Anal., 31(2):131–145, 1999.DOI:10.1016/S0167-9473(99)00010-9

work page doi:10.1016/s0167-9473(99)00010-9 1999
[13]

S. X. Chen. Probability density function estimation using gamma kernels.Ann. Inst. Statist. Math., 52(3):471–480, 2000.DOI:10.1023/A:1004165218295

work page doi:10.1023/a:1004165218295 2000
[14]

Cowling and P

A. Cowling and P. Hall. On pseudodata methods for removing boundary effects in kernel density estimation.J. R. Stat. Soc. Ser. B. Stat. Methodol., 58(3):551–563, 1996.DOI:10.1111/j.2517-6 161.1996.tb02100.x

work page doi:10.1111/j.2517-6 1996
[15]

Duong, M

T. Duong, M. Wand, J. Chac´ on, and A. Gramacki.ks: Kernel Smoothing, 2025. URLhttps: //CRAN.R-project.org/package=ks. R package version 1.15.1

2025
[16]

Funke and M

B. Funke and M. Hirukawa. Nonparametric estimation and testing on discontinuity of positive supported densities: a kernel truncation approach.Econom. Stat., 9:156–170, 2019.DOI:10.1016/ j.ecosta.2017.07.006

2019
[17]

Funke and M

B. Funke and M. Hirukawa. Density derivative estimation using asymmetric kernels.J. Nonparametr. Stat., 36(4):994–1017, 2024.DOI:10.1080/10485252.2023.2291430

work page doi:10.1080/10485252.2023.2291430 2024
[18]

Funke and M

B. Funke and M. Hirukawa. On uniform consistency of nonparametric estimators smoothed by the gamma kernel.Ann. Inst. Statist. Math., 77(3):459–489, 2025.DOI:10.1007/s10463-024-00923-8

work page doi:10.1007/s10463-024-00923-8 2025
[19]

Funke and M

B. Funke and M. Hirukawa. Nonparametric estimation of splicing points in skewed cost distributions: a kernel-based approach.J. Nonparametr. Stat., pages 1–40, 2025.DOI:10.1080/10485252.2025. 2505639. 29

work page doi:10.1080/10485252.2025 2025
[20]

Funke and R

B. Funke and R. Kawka. Nonparametric density estimation for multivariate bounded data using two non-negative multiplicative bias correction methods.Comput. Statist. Data Anal., 92:148–162, 2015. DOI:10.1016/j.csda.2015.07.006

work page doi:10.1016/j.csda.2015.07.006 2015
[21]

Gawronski and U

W. Gawronski and U. Stadtm¨ uller. On density estimation by means of Poisson’s distribution.Scand. J. Statist., 7(2):90–94, 1980

1980
[22]

Gawronski and U

W. Gawronski and U. Stadtm¨ uller. Smoothing histograms by means of lattice- and continuous distributions.Metrika, 28:155–164, 1981.DOI:10.1007/BF01902889

work page doi:10.1007/bf01902889 1981
[23]

Hirukawa, I

M. Hirukawa, I. Murtazashvili, and A. Prokhorov. Uniform convergence rates for nonparametric estimators smoothed by the beta kernel.Scand. J. Statist., 49(3):1353–1382, 2022.DOI:10.1111/ sjos.12573

2022
[24]

M. C. Jones. Simple boundary correction for kernel density estimation.Stat. Comput., 3(3):135–146, 1993.DOI:10.1007/BF00147776

work page doi:10.1007/bf00147776 1993
[25]

Jørgensen

B. Jørgensen. Exponential dispersion models.J. R. Stat. Soc. Ser. B. Stat. Methodol., 49(2):127–145, 1987.DOI:10.1111/j.2517-6161.1987.tb01685.x

work page doi:10.1111/j.2517-6161.1987.tb01685.x 1987
[26]

Jørgensen.The Theory of Dispersion Models, volume 76 ofMonographs on Statistics and Applied Probability

B. Jørgensen.The Theory of Dispersion Models, volume 76 ofMonographs on Statistics and Applied Probability. Chapman & Hall, London, 1997. ISBN 978-0-412-99711-2

1997
[27]

R. J. Karunamuni and T. Alberts. On boundary correction in kernel density estimation.Stat. Methodol., 2(3):191–212, 2005.DOI:10.1016/j.stamet.2005.04.001

work page doi:10.1016/j.stamet.2005.04.001 2005
[28]

J. S. Marron and D. Ruppert. Transformations to reduce boundary bias in kernel density estimation. J. R. Stat. Soc. Ser. B. Methodol., 56(4):653–671, 1994.DOI:10.1111/j.2517-6161.1994.tb020 06.x

work page doi:10.1111/j.2517-6161.1994.tb020 1994
[29]

Min and A

Y. Min and A. Agresti. Modeling nonnegative data with clumping at zero: a survey.J. Iran. Stat. Soc., 1(1):7–33, 2002

2002
[30]

Moss and M

J. Moss and M. Tveten.kdensity: Kernel Density Estimation with Parametric Starts and Asymmetric Kernels, 2025. URLhttps://CRAN.R-project.org/package=kdensity. R package version 1.1.1

2025
[31]

M. K. Olsen and J. L. Schafer. A two-part random-effects model for semicontinuous longitudinal data.J. Amer. Statist. Assoc., 96(454):730–745, 2001.DOI:10.1198/016214501753168389

work page doi:10.1198/016214501753168389 2001
[32]

Charlie Frogner, Chiyuan Zhang, Hossein Mobahi, Mauricio Araya, and Tomaso A Poggio

M. Rosenblatt. Remarks on some nonparametric estimates of a density function.Ann. Math. Statist., 27(3):832–837, 1956.DOI:10.1214/aoms/1177728190

work page doi:10.1214/aoms/1177728190 1956
[33]

E. F. Schuster. Incorporating support constraints into nonparametric estimators of densities.Comm. Statist. Theory Methods, 14(5):1123–1136, 1985.DOI:10.1080/03610928508828965

work page doi:10.1080/03610928508828965 1985
[34]

V. A. Smith, J. S. Preisser, B. Neelon, and M. L. Maciejewski. A marginalized two-part model for semicontinuous data.Stat. Med., 33(28):4891–4903, 2014.DOI:10.1002/sim.6263

work page doi:10.1002/sim.6263 2014
[35]

V. A. Smith, B. Neelon, M. L. Maciejewski, and J. S. Preisser. Two parts are better than one: modeling marginal means of semicontinuous data.Health Serv. Outcomes Res. Methodol., 17(3–4): 198–218, 2017.DOI:10.1007/s10742-017-0169-9

work page doi:10.1007/s10742-017-0169-9 2017
[36]

Census profile, 2021 census of population: Zone 4 - Central, Nova Scotia [Health region], 2023

Statistics Canada. Census profile, 2021 census of population: Zone 4 - Central, Nova Scotia [Health region], 2023. URLhttps://www12.statcan.gc.ca/census-recensement/2021/dp-pd/prof/i ndex.cfm?Lang=E. Statistics Canada Catalogue no. 98-316-X2021001; released November 15, 2023; accessed April 27, 2026

2021
[37]

M. C. K. Tweedie. An index which distinguishes between some important exponential families. In J. K. Ghosh and J. Roy, editors,Statistics: Applications and New Directions, pages 579–604. Indian Statistical Institute, Calcutta, 1984. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference

1984
[38]

Yiu and B

S. Yiu and B. D. M. Tom. Two-part models with stochastic processes for modelling longitudinal semicontinuous data: computationally efficient inference and modelling the overall marginal mean. Stat. Methods Med. Res., 27(12):3679–3695, 2018.DOI:10.1177/0962280217710573. 30

work page doi:10.1177/0962280217710573 2018