#  Statistical Inference for Scale Mixture Models via Mellin Transform
Approach111The article was prepared in the framework of a research grant
funded by the Ministry of Science and Higher Education of the Russian
Federation (grant ID: 075-15-2022-325).

Denis Belomestny<EMAIL_ADDRESS>University of Duisburg-Essen,
Thea-Leymann-Str. 9, 45127 Essen, Germany  Ekaterina Morozova
<EMAIL_ADDRESS>Vladimir Panov<EMAIL_ADDRESS>
###### Abstract

This paper deals with statistical inference for the scale mixture models. We
study an estimation approach based on the Mellin – Stieltjes transform that
can be applied to both discrete and absolute continuous mixing distributions.
The accuracy of the corresponding estimate is analysed in terms of its
expected pointwise error. As an important technical result, we prove the
analogue of the Berry – Esseen inequality for the Mellin transforms. The
proposed statistical approach is illustrated by numerical examples.

##  1 Introduction

In this paper we consider the problem of statistical inference for
multiplicative mixture models. More precisely, given a sample of i.i.d. random
variables $X_{1},\dots,X_{n}$, $n\in{\mathbb{N}}$, from the multiplicative
mixture model of the form

$X=Y\eta,$ (1)

where $Y$ and $\eta$ are independent random variables, we aim at estimating
the distribution of one of these variables (say, $Y$) assuming that the law of
another random variable ($\eta$) is known. For simplicity, we assume that both
$Y$ and $\eta$ are almost surely positive, so that $\eta$ can be viewed as the
(stochastic) scaling parameter.

The aforementioned problem can be viewed as the problem of reconstructing the
original signal from the contaminated sample and naturally arises in many
applications. For instance, the case when $\eta$ has a standard uniform
distribution is known as the multiplicative censoring model and is widely
employed in survival analysis (Vardi, 1989). In this context, $X$ corresponds
to the time elapsed since the beginning of the disease, whereas $Y$ represents
the true survival time (Van Es et al., 2000). The precise estimation of the
distribution of $Y$ in this case would help the development of treatment
programmes, as well as the assessment of their performance. Another example
comes from finance, where the model (1) with normally distributed $\eta$ and
positive $Y$ corresponds to the stochastic volatility model for describing the
log-returns of an asset (Van Es et al., 2003, Belomestny and Schoenmakers,
2015).

While certain methods for estimation of the distribution of $Y$ in model (1)
already exist, they mostly assume some specific form of the distribution of
$\eta$. For instance, for the case when $\eta$ follows a standard uniform
distribution, some nonparametric estimation techniques are proposed by Vardi
(1989), Asgharian et al. (2012), Brunel et al. (2016). Later, Comte and Dion
(2016) and Belomestny et al. (2016) generalise the setting and develop the
estimators based on the projection techniques for the case when $\eta$ follows
the uniform distribution symmetric about one and the beta distribution,
respectively.

The problem of statistical inference for the multiplicative mixture models can
be reduced to the additive deconvolution problem by taking logarithms of both
parts in (1) or by taking logarithms of the squares in the alternating case.
For the additive models, a wide range of estimation methods is available; see,
e.g., Zhang (1990), Meister (2009), Belomestny and Goldenschluger (2021), and
numereous references therein. However, as was pointed out by Brunel et al.
(2016) and Belomestny and Goldenschluger (2020), this idea leads to several
undesired consequences such as inability of estimation at zero and the loss of
the information about the sign of the random variable.

For almost all papers mentioned above, the parametric assumption on the
distribution of $\eta$ is essential. The nonparametric case was considered by
Belomestny and Goldenschluger (2020) and Miguel, Comte and Johannes (2021),
where the kernel-type and series estimators based on the Mellin transform are
introduced. However, these papers significantly employ the assumption of
absolute continuity of the distributions of $Y$ and $\eta$.

In the current paper, we do not impose any restrictions on the class of
distributions of $\eta$. The only assumption is that the set $\mathcal{H}_{G}$
determined by (3) from the distribution of $\eta$, is non-empty. This
assumption holds for a wide class of discrete distributions (see Section 6 for
the detailed discussion). Moreover, as we also show in the article, this
assumption yields the optimal rate of convergence without any specific
restrictions on the class of the probability density functions (p.d.f.s) of
$Y.$

In order to avoid the assumption of absolute continuity of $\eta,$ we
formulate all results in terms of the distribution functions, assuming for
simplicity that both random variables $Y$ and $\eta$ are a.s. positive. Note
that the distribution function of $X$ is equal to

$F_{\rm{mix}}(x)=\int_{{\mathbb{R}}^{+}}F\left(\frac{x}{\theta}\right)\,dG(\theta),$

where $F$ and $G$ are the cumulative distribution functions (c.d.f.s) of $Y$
and $\eta,$ respectively. Given observations $X_{1},\ldots,X_{n}$ from
$F_{\rm{mix}}$, we aim to estimate the function $F(x),x\in{\mathbb{R}}_{+}$,
provided that $G(x),x\in{\mathbb{R}}_{+},$ is known.

Our estimation method is based on the Mellin – Stieltjes transform, defined
for a function $\varphi:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$, which is assumed
to be a function of bounded variation over any bounded interval, as

${\mathcal{M}}[\varphi](z):=\int\limits_{0}^{\infty}x^{z-1}\,d\varphi(x),\quad
z\in{\mathbb{C}}.$ (2)

The integral on the right-hand side is known to converge in a vertical strip
$\\{z\in{\mathbb{C}}:\operatorname{Re}(z)\in[\alpha_{\varphi},\beta_{\varphi}]=:\mathcal{C}_{\varphi}\\}$,
with some $\alpha_{\varphi},\beta_{\varphi}>0$ (the degenerate case
$\alpha_{\varphi}=\beta_{\varphi}$ is also possible). Using the properties of
the Mellin – Stieltjes transform, we construct an estimator $\widehat{F}$ of
$F$ (to be defined in Section 2) and study the accuracy of this estimator at a
fixed point $x\in{\mathbb{R}}_{+}$ in terms of the expected pointwise error

$\mathcal{R}^{*}(\widehat{F})=\mathcal{R}^{*}(\widehat{F};u^{\circ},x):={\mathbb{E}}\Bigl{[}\bigl{(}\mathcal{R}(\widehat{F};u^{\circ},x)\bigr{)}^{2}\Bigr{]},$

where

$\mathcal{R}(\widehat{F})=\mathcal{R}(\widehat{F};u^{\circ},x):=x^{u^{\circ}-1}\bigl{|}F(x)-\widehat{F}(x)\bigr{|}$

and $u^{\circ}$ is a technical parameter. We show that under rather mild
assumptions on $G$ and $F$, the estimate $\widehat{F}$ has $1/n$ rate of
convergence to $F$ as measured in terms of $\mathcal{R}^{*}(\widehat{F})$ with
$n$ being the sample size.

The paper is organised as follows. In the following section (Section 2) we
recall the most important properties of the Mellin – Stieltjes transform and
introduce the estimator $\widehat{F}$ for $F$. In Section 3 we prove the
analogue of the Berry – Esseen inequality for the Mellin – Stieltjes
transforms (Lemma 3.1), which plays an essential role for establishing the
upper bounds for $\mathcal{R}(\widehat{F})$ and
$\mathcal{R}^{*}(\widehat{F})$. The exact statements are given in Section 4,
see Theorems 4.1 and 4.2. Next, Section 5 contains the detailed discussion on
the subclasses of distribution functions $F$ and $G$, for which the rate of
convergence of the proposed estimators is polynomial. Section 6 is devoted to
one of the key assumptions of our estimation procedure, namely, that there
exists a line parallel to the imaginary axis such that
${\mathcal{M}}[G](z)\neq 0$ for any $z$ on this line. It is shown that the
aforementioned assumption is fulfilled, in particular, for any discrete
positive distribution separated from zero. Finally, Section 7 contains a
numerical example demonstrating the performance of our estimator via a
simulation study. Appendix A contains a numerical example of the application
of the Berry – Esseen inequality. All proofs are collected in Appendix B.

##  2 Estimation procedure

Note that the Mellin – Stieltjes transform of $F_{\rm{mix}}$ is equal to

${\mathcal{M}}[F_{\rm{mix}}](z):={\mathbb{E}}[X^{z-1}]={\mathcal{M}}[F](z)\cdot{\mathcal{M}}[G](z)$

for any $z\in{\mathbb{C}}$ such that both Mellin transforms on the right-hand
side are well defined. Since we assume that the distribution of $\eta$ is
known, we can estimate the Mellin transform of $F$ by

$\widehat{M[F]}(z):=\frac{1}{n}\sum\limits_{i=1}^{n}X_{i}^{z-1}/{\mathcal{M}}[G](z),$

at any point $z\in{\mathbb{C}}$ such that ${\mathcal{M}}[G](z)\neq 0.$ Let us
introduce the notation

$\displaystyle\mathcal{H}_{G}:=\bigl{\\{}u\in\mathcal{C}_{G}:\;{\mathcal{M}}[G](u+\mathrm{i}v)\neq
0\quad\forall\;v\in{\mathbb{R}}\bigr{\\}}.$ (3)

The set $\mathcal{H}_{G}$ is non-empty in most cases; for instance, as we show
in Section 6, for positive discrete distributions, there exists some
$\tilde{u}<1$ such that $(-\infty,\tilde{u})\subset\mathcal{H}_{G},$ provided
that the distribution is separated from 0.

The estimator of $F$ is based on the inversion formula for the Mellin
transform. Some versions of this formula are known in the literature, see,
e.g., Section 7 from [11], but here we need a slightly different form.

######  Lemma 2.1.

Let $\varphi:{\mathbb{R}}_{+}\to{\mathbb{R}}$ be a non-decreasing function.
Then

  1. 1.

if there exists some $u^{\circ}\in(-\infty,1)\cap\mathcal{C}_{\varphi}$, then
$\varphi(0)<\infty$ and

$\frac{1}{2\pi
i}\int\limits_{u^{\circ}-i\infty}^{u^{\circ}+i\infty}x^{-z+1}\frac{{\mathcal{M}}[\varphi](z)}{-(z-1)}\,dz=\frac{1}{2}\Bigl{(}\varphi(x+0)+\varphi(x-0)\Bigr{)}-\varphi(0);$

  2. 2.

if there exists some $u^{\circ}\in(1,\infty)\cap\mathcal{C}_{\varphi}$, then
$\varphi(\infty)<\infty$ and

$\frac{1}{2\pi
i}\int\limits_{u^{\circ}-i\infty}^{u^{\circ}+i\infty}x^{-z+1}\frac{{\mathcal{M}}[\varphi](z)}{-(z-1)}\,dz=\frac{1}{2}\Bigl{(}\varphi(x+0)+\varphi(x-0)\Bigr{)}-\varphi(\infty).$

###### Proof.

The proof is given in Appendix B.1.∎

######  Remark 2.2.

In what follows, for any non-decreasing function
$\varphi:{\mathbb{R}}_{+}\to{\mathbb{R}}$, we will use the same notation
$\varphi(x),x\in{\mathbb{R}}_{+},$ for the standardised version of the
function, that is, for $\bigl{(}\varphi(x+0)+\varphi(x-0)\bigr{)}/2.$

Motivated by Lemma 2.1, we define the estimator $\widehat{F}(x)$ of $F(x)$ as

$\widehat{F}(x):=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}x^{-u^{\circ}-\mathrm{i}v+1}\frac{\widehat{{\mathcal{M}}[F]}(u^{\circ}+\mathrm{i}v)}{-(u^{\circ}+\mathrm{i}v-1)}K(v)\,dv$
(4)

for some $u^{\circ}\in(-\infty,1)\cap\mathcal{C}_{F}\cap\mathcal{H}_{G}$, and
the kernel function $K$ of the form

$\displaystyle K(x):=\left(1-\frac{|x|}{T}\right)\mathbb{I}\\{|x|\leq T\\},$
(5)

with a positive number $T$. This choice of the kernel function is inspired by
the Berry – Esseen inequality for the Mellin transforms. In the next section,
we discuss this inequality.

##  3 Berry – Esseen inequality for the Mellin transforms

The following lemma is motivated by a similar result for the Fourier
transform, see Section 4.1 from [12].

######  Lemma 3.1.

Let $\varphi,\psi:{\mathbb{R}}_{+}\to{\mathbb{R}}$ be two non-decreasing left
continuous functions such that $\varphi(0)=\psi(0)=0$. Let
$u^{\circ}\in(-\infty,1)\cap\mathcal{C}_{\varphi}\cap\mathcal{C}_{\psi}$.
Denote

$\displaystyle\rho_{u^{\circ}}(\varphi,\psi):=\sup\limits_{x\geq
0}\bigl{|}x^{u^{\circ}-1}\bigl{(}\varphi(x)-\psi(x)\bigr{)}\bigr{|}.$

Assume that the supremum in the definition of $\rho_{u^{\circ}}(\varphi,\psi)$
is attained at some point $x_{0}>0$. Then for any $b>2/\pi$, it holds

$\rho_{u^{\circ}}(\varphi,\psi)\leq\frac{b}{2}\int\limits_{-T}^{T}\frac{|{\mathcal{M}}[\varphi](u^{\circ}+\mathrm{i}v)-{\mathcal{M}}[\psi](u^{\circ}+\mathrm{i}v)|}{|v|}\,dv\\\
+bTx_{0}^{u^{\circ}-1}\int_{0}^{2c(b)/T}\left|\psi(x_{0})-\psi\left(x_{0}e^{r}\right)\right|\,dr,$
(6)

where $c(b)$ can be found as a unique root of the equation

$\int\limits_{|r|\leq c(b)}\frac{\sin^{2}r}{\pi
r^{2}}\,dr=\frac{2}{3}\left(1+\frac{1}{\pi b}\right),$ (7)

and $T$ is an arbitrary positive number such that

$\displaystyle T>2c(b)(1-u^{\circ})/\log 2.$ (8)

###### Proof.

The proof is given in Appendix B.2. ∎

######  Remark 3.2.

Since the left-hand side of the equation (7) tends to one as $c(b)\to\infty$,
and to zero as $c(b)\to 0$, it can take any values from the interval $(0,1).$
This is the reason for the restriction $b>2/\pi$.

######  Remark 3.3.

It is a worth mentioning that the second term in the upper bound (6) tends to
zero as $T\to\infty$ at a polynomial rate in the case when $\psi$ is uniformly
$\alpha$-Hölder continuous with $\alpha\in(0,1]$.222The asymptotic behaviour
of the first term in (6) will be discussed later. Definitely, it depends on
the closeness between the functions $\varphi$ and $\psi.$ Indeed, we have

$\int\limits_{0}^{2c(b)/T}|\psi(x_{0})-\psi(x_{0}e^{r})|\,dr\leq
L_{\psi}x_{0}^{\alpha}\int\limits_{0}^{2c(b)/T}|1-e^{r}|^{\alpha}\,dr$

for some $L_{\psi}\in(0,\infty)$. Using the inequality $e^{x}-1\leq xe^{x}$
which holds for any $x>0,$ we get

$\displaystyle\int\limits_{0}^{2c(b)/T}|1-e^{r}|^{\alpha}\,dr\leq e^{2\alpha
c(b)/T}\frac{(2c(b))^{1+\alpha}}{(1+\alpha)T^{1+\alpha}}.$

Hence,

$bTx_{0}^{u^{\circ}-1}\int_{0}^{2c(b)/T}\left|\psi(x_{0})-\psi\left(x_{0}e^{r}\right)\right|\,dr\leq
L_{\psi}x_{0}^{\alpha+u^{\circ}-1}e^{2\alpha
c(b)/T}\frac{b(2c(b))^{1+\alpha}}{(1+\alpha)T^{\alpha}},$

yielding the polynomial rate of decay.

The numerical example of the use of the Berry – Esseen inequality is given in
Appendix A.

##  4 Main results

In this section, we provide the rates of convergence of the estimate (4) with
the kernel (5). The following theorem holds.

######  Theorem 4.1.

Let $u^{\circ}\in(-\infty,1)\cap\mathcal{C}_{F}\cap\mathcal{H}_{G}.$ Assume
that the distribution function $F$ is continuous at least in a small vicinity
of $0$, and

$\displaystyle x^{u^{\circ}-1}F(x)\to 0\qquad\mbox{as}\quad x\to 0.$ (9)

Let $b>2/\pi$ be an arbitrary number, $c(b)$ be the solution of the equation
(7) and $T$ satisfy (8). Then there exists some $x_{0}>0$ not depending on $T$
and $n$ such that for any fixed $x>0,$ it holds with probability $1$

$\displaystyle\mathcal{R}(\widehat{F};u^{\circ},x)$ $\displaystyle\leq$
$\displaystyle\frac{b}{2T}\int\limits_{-T}^{T}\left|{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v)\right|\,dv$
$\displaystyle\hskip
28.45274pt+bTx_{0}^{u^{\circ}-1}\int\limits_{0}^{2c(b)/T}\left|F(x_{0})-F\left(x_{0}e^{r}\right)\right|\,dr$
$\displaystyle\hskip 28.45274pt+\frac{x^{u^{\circ}-1}}{2\pi
n}\left|\sum\limits_{k=1}^{n}\Lambda(X_{k},x)\right|,$

where

$\displaystyle\Lambda(X_{k},x)=\int\limits_{-\infty}^{\infty}x^{-(u^{\circ}-1)-\mathrm{i}v}\frac{{\mathcal{M}}[F_{\rm{mix}}](u^{\circ}+\mathrm{i}v)-X_{k}^{(u^{\circ}-1)+\mathrm{i}v}}{-(u^{\circ}+\mathrm{i}v-1){\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)}K(v)\,dv.$

###### Proof.

The proof is given in Appendix B.3. ∎

The preceding theorem allows us to further analyze the convergence of the
estimator $\widehat{F}$ to the true c.d.f. $F$ in the $\mathcal{L}^{2}$-sense.

######  Theorem 4.2.

Under the same notations and assumptions as in Theorem 4.1, we get for any
fixed $x>0,$

$\displaystyle\mathcal{R}^{*}(\widehat{F};u^{\circ},x)$ $\displaystyle\leq$
$\displaystyle\frac{3b^{2}}{4T^{2}}\left(\int\limits_{-T}^{T}|{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v)|\,dv\right)^{2}$
(10) $\displaystyle\hskip
28.45274pt+3b^{2}T^{2}x_{0}^{2(u^{\circ}-1)}\left(\int\limits_{0}^{2c(b)/T}\left|F(x_{0})-F\left(x_{0}e^{r}\right)\right|\,dr\right)^{2}$
$\displaystyle\hskip
28.45274pt+\frac{3x^{2(u^{\circ}-1)}}{4\pi^{2}n}\int\limits_{-T}^{T}\frac{1}{((u^{\circ}-1)^{2}+v^{2})|{\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)|^{2}}\,dv\hskip
8.5359pt$ $\displaystyle\hskip
71.13188pt\times\int\limits_{{\mathbb{R}}}|{\mathcal{M}}[F_{\rm{mix}}](2u^{\circ}-1+\mathrm{i}w)|\,dw.$

###### Proof.

The proof is given in Appendix B.4. ∎

##  5 Convergence rates

Denote the summands in (10) by $I_{1},I_{2},I_{3},$ respectively. Let us
consider these summands separately.

1. Let us first note that the condition

$\displaystyle\int\limits_{{\mathbb{R}}}|{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v)|\,dv<\infty$
(11)

is fulfilled for many distributions, including the distributions with
exponential and polynomial decay of the Mellin transform, see [4]. The
condition (11) yields that the first summand is of order $I_{1}=O(T^{-2})$ as
$T\to\infty.$ Nevertheless, it can be shown that any distribution satisfying
(11) is absolutely continuous. This fact can be proved in various ways, e.g.,
via the convergence in $\mathcal{L}^{2}({\mathbb{R}})$, along the same lines
as the proof of the similar fact for the Fourier transform, see Lemma 1.1 from
[10]. It can be also derived from the properties of the Fourier transform (see
Theorem 3.2.2 from [13]). In fact,

$\displaystyle\int\limits_{{\mathbb{R}}}\Bigl{|}{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v)\Bigr{|}\,dv$
$\displaystyle=$
$\displaystyle\int\limits_{{\mathbb{R}}}\Bigl{|}\int_{{\mathbb{R}}+}x^{u^{\circ}+\mathrm{i}v-1}dF(x)\Bigr{|}\,dv$
$\displaystyle=$
$\displaystyle\int\limits_{{\mathbb{R}}}\Bigl{|}\int_{{\mathbb{R}}}e^{\mathrm{i}yv}\cdot
e^{y(u^{\circ}-1)}dF(e^{y})\Bigr{|}\,dv$ $\displaystyle=$
$\displaystyle{\mathcal{M}}[F](u^{\circ})\int\limits_{{\mathbb{R}}}\Bigl{|}\int_{{\mathbb{R}}}e^{\mathrm{i}yv}dH(y)\Bigr{|}\,dv,$

where

$\displaystyle H(y)$ $\displaystyle:=$
$\displaystyle\frac{1}{{\mathcal{M}}[F](u^{\circ})}\int_{-\infty}^{y}e^{s(u^{\circ}-1)}dF(e^{s})$
$\displaystyle=$
$\displaystyle\frac{1}{{\mathcal{M}}[F](u^{\circ})}\int_{0}^{e^{y}}x^{u^{\circ}-1}dF(x),\qquad\qquad
y\in{\mathbb{R}},$

is a distribution function (here we use that
${\mathcal{M}}[F](u^{\circ})<\infty$ since $u^{\circ}\in\mathcal{C}_{F}$).
Therefore, the condition (11) is equivalent to
$\mathcal{F}[H]\in\mathcal{L}^{1}(\mathbb{R})$, which yields that the $H$ is a
c.d.f. of an absolutely continuous distribution with bounded and continuous
density function. Then the distribution of $F$ is absolutely continuous as
well.

2. The typical order for the second summand is also $O(T^{-2}).$ In fact, if $F$ is Lipschitz continuous, we get from Remark 3.3

$\displaystyle I_{2}\leq
L_{F}^{2}x_{0}^{2u^{\circ}}e^{4c(b)/T}\frac{12b^{2}(c(b))^{4}}{T^{2}}$

where $L_{F}\in(0,\infty)$ is the Lipschitz constant of $F$.

3. To establish the asymptotic order of the third summand $I_{3}$, we observe that, since $u^{\circ}\in\mathcal{H}_{G}\subset\mathcal{C}_{G}$, for any $T>0,$ there exist positive constants $C_{u^{\circ},T}^{(1)},C_{u^{\circ},T}^{(2)}$ such that

$C_{u^{\circ},T}^{(1)}\leq|{\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)|\leq
C_{u^{\circ},T}^{(2)},\quad\forall v\in[-T,T].$ (12)

Assuming that $2u^{\circ}-1\in\mathcal{H}_{G},$ we similarly get with some
positive constants $C_{2u^{\circ}-1,T}^{(1)},C_{2u^{\circ}-1,T}^{(2)}$

$C_{2u^{\circ}-1,T}^{(1)}\leq|{\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)|\leq
C_{2u^{\circ}-1,T}^{(2)},\quad\forall v\in[-T,T].$ (13)

Let us assume additionally that as $T\to\infty$,

$C_{u^{\circ},T}^{(1)}\to C^{(1)}_{u^{\circ}}>0\quad\mbox{ and }\quad
C_{2u^{\circ}-1,T}^{(2)}\to C_{2u^{\circ}-1}^{(2)}<\infty.$

We refer to Section 6 for examples of mixing distributions which satisfy this
assumption. Finally, assume that the analogue of (11) holds along the line
$\operatorname{Re}(z)=2u^{\circ}-1$,

$\displaystyle\int\limits_{{\mathbb{R}}}|{\mathcal{M}}[F](2u^{\circ}-1+\mathrm{i}v)|\,dv\leq
C_{1}<\infty.$ (14)

Then we get

$I_{3}\leq\frac{3x^{2(u^{\circ}-1)}}{4\pi^{2}n}\frac{C_{1}C_{2u^{\circ}-1}^{(2)}}{\left(C_{u^{\circ}}^{(1)}\right)^{2}}\int\limits_{-T}^{T}\frac{1}{(u^{\circ}-1)^{2}+v^{2}}\,dv=\frac{3x^{2(u^{\circ}-1)}C_{1}C_{2u^{\circ}-1}^{(2)}}{4(1-u^{\circ})\pi\left(C_{u^{\circ}}^{(1)}\right)^{2}n}.$

provided that (11) holds. Finally, let us note that the choice $T=O(n^{1/2})$
leads to the rate of convergence $I_{1}+I_{2}+I_{3}=O(1/n)$ which coincides
with the rate in the case when the direct observations from $F$ are available.

##  6 The set $\mathcal{H}_{G}$

The set $\mathcal{H}_{G}$ defined by (3) plays a crucial role in the analysis.
Let us show that this set is nonempty for positive discrete distributions
separated from $0$. Indeed, consider a random variable $\eta$ taking $K$
positive values $\sigma_{1},\dots,\sigma_{K}$ with probabilities
$p_{1},\dots,p_{K}>0$, $\sum_{k=1}^{K}p_{k}=1$ (the analysis of the case
$K=\infty$ follows the same lines). Without loss of generality, assume that
$0<\sigma_{1}<\sigma_{2}<\dots<\sigma_{K}$. Then it holds

$\displaystyle{\mathcal{M}}[G](u+\mathrm{i}v)=\sum_{k=1}^{K}\sigma_{k}^{u+\mathrm{i}v-1}p_{k}=\sigma_{1}^{u+\mathrm{i}v-1}p_{1}+\sum_{k=2}^{K}\sigma_{k}^{u+\mathrm{i}v-1}p_{k},$
(15)

where the modulus of the second summand can be upper bounded as

$\displaystyle\left|\sum_{k=2}^{K}\sigma_{k}^{u+\mathrm{i}v-1}p_{k}\right|\leq\sum_{k=2}^{K}|\sigma_{k}^{u+\mathrm{i}v-1}|p_{k}=\sum_{k=2}^{K}\sigma_{k}^{u-1}p_{k}<\sigma_{2}^{u-1}(1-p_{1})$

for any $u<1,$ while the modulus of the first summand in (15) is equal to
$\sigma_{1}^{u-1}p_{1}.$ From this it follows that

$\displaystyle\bigl{|}M[G](u+\mathrm{i}v)\bigr{|}\geq\bigl{|}\sigma_{1}^{u+\mathrm{i}v-1}p_{1}\bigr{|}-\Bigl{|}\sum_{k=2}^{K}\sigma_{k}^{u+\mathrm{i}v-1}p_{k}\Bigr{|}>\sigma_{1}^{u-1}p_{1}-\sigma_{2}^{u-1}(1-p_{1}),$

where the expression on the right-hand side is strongly positive for any

$\displaystyle u<\frac{\log(p_{1}/(1-p_{1}))}{\log(\sigma_{2}/\sigma_{1})}+1.$
(16)

Let us provide a couple of examples.

  1. 1.

Consider the geometric distribution defined as a distribution of a r.v. $\eta$
such that

$\displaystyle\mathbb{P}\bigl{\\{}\eta=k\bigr{\\}}=pq^{k-1},\qquad k=1,2,...$

According to (16), the absolute value of the Mellin transform of this
distribution is strongly positive for any

$\displaystyle u<\frac{\log\Bigl{(}p/(1-p)\Bigr{)}}{\log(2)}+1,$

where the expression on the r.h.s. is positive for $p>1/3$ and negative for
$p<1/3.$

  2. 2.

Consider an analogue of the Poisson distribution on $\\{1,2,...\\}:$

$\displaystyle\mathbb{P}\bigl{\\{}\eta=k\bigr{\\}}=\frac{e^{-\lambda}}{1-e^{-\lambda}}\cdot\frac{\lambda^{k}}{k!},\qquad
k=1,2,3,...,$

where $\lambda>0.$ Returning to (16), we get that the absolute value of the
Mellin transform is strongly positive in absolute value for any

$\displaystyle u<\frac{\log\Bigl{(}\lambda
e^{-\lambda}/(1-e^{-\lambda}-\lambda e^{-\lambda})\Bigr{)}}{\log(2)}+1,$

which is positive whenever $\lambda\in(0,\lambda_{0})$ with $\lambda_{0}$
defined as the positive solution of the equation $3\lambda+1=e^{\lambda}.$
Numerically we get $\lambda_{\circ}\approx 1.9$.

Note that in some simple cases the result can be enhanced.

  1. 1.

For instance, consider the zeta distribution

$\mathbb{P}\\{\eta=k\\}=\frac{1}{\zeta(s)}k^{-s},\quad k=1,2,\dots,$ (17)

where $s>1$ and $\zeta(s)$ is the Riemann zeta function defined as

$\zeta(s)=\sum\limits_{k=1}^{\infty}k^{-s}.$

Since ${\mathcal{M}}[G](u+\mathrm{i}v)=\zeta(1+s-u-\mathrm{i}v)/\zeta(s)$, and
the zeta-function has no zeros with real part larger than 1, we get that the
set $\mathcal{H}_{G}$ contains all $u<s.$

  2. 2.

Consider the distribution of the random variable taking values $\sigma_{1}=1$
and $\sigma_{2}>1$ with probabilities $p\in(0,1)$ and $1-p$. Then any

$u\neq u_{\circ}:=1+\log_{\sigma_{2}}(p/(1-p))$ (18)

belong to the set $\mathcal{H}_{G}$. In fact, in this case the Mellin
transform can be lower bounded for any $v\in{\mathbb{R}}$ by

$\displaystyle\left|{\mathcal{M}}[G](u+\mathrm{i}v)\right|\geq\bigl{|}p-(1-p)\sigma_{2}^{u-1}\bigr{|},$

which is strongly larger than $0$ for any $u\neq u_{\circ}.$

##  7 Numerical examples

In this section we use simulated data to illustrate the behaviour of the
proposed estimator for different types of mixing distributions $G$. We also
compare our estimator with another one based on a logarithmic transformation
of both sides in (1) and interpretation of the problem as an additive
deconvolution problem. Since $\phi_{\log X}(t)=\phi_{\log
Y}(t)\phi_{\log\eta}(t)$, $\forall t\in{\mathbb{R}}$ where $\phi$ denotes a
characteristic function, the inversion formula for the Fourier-Stieltjes
transform (see Theorem 4.4.1 in Kawata, 1972) suggests that $F$ can be
estimated using

$\widehat{F}_{\log Y}(x)-\widehat{F}_{\log
Y}(0)=\frac{1}{2\pi}\int\limits_{-R_{n}}^{R_{n}}\frac{e^{-\mathrm{i}tx}-1}{-it}\cdot\frac{\widehat{\phi}_{\log
X}(t)}{\phi_{\log\eta}(t)}\phi_{\kappa}(ht)\,dt,$ (19)

where $\widehat{\phi}_{\log X}(t)=\frac{1}{n}\sum_{j=1}^{n}e^{\mathrm{i}t\log
X_{j}}$, $R_{n}\to\infty$ as $n\to\infty$, $h$ is a bandwidth parameter chosen
by the cross-validation approach, and $\phi_{\kappa}$ is the characteristic
function of some kernel function $\kappa$ which we choose to be density of the
standard normal law; for details see, e.g., Delaigle (2021).

In what follows, we assume that the unknown distribution $F$ is absolutely
continuous. More precisely, consider the cases when $F$ is a c.d.f. of the
beta distribution

$F(x)=\frac{{\mathbb{I}}\\{x\in[0,1]\\}}{B(\alpha_{1},\alpha_{2})}\int\limits_{0}^{x}y^{\alpha_{1}-1}(1-y)^{\alpha_{2}-1}\,dy+{\mathbb{I}}\\{x>1\\}$

and the gamma distribution

$F(x)=\frac{\theta^{k}}{\Gamma(k)}\int\limits_{0}^{x}y^{k-1}e^{-\theta
y}\,dy\cdot{\mathbb{I}}\\{x>0\\}$

with parameters $\alpha_{1}=\alpha_{2}=k=\theta=2$. Note that
$\mathcal{C}_{F}=(1-\alpha_{1},\infty)$ for the beta distribution and
$\mathcal{C}_{F}=(1-k,\infty)$ for the gamma distribution. Also note that the
assumption (9) holds for both examples: in fact, since $F(0)=0,$ we get by the
L’Hôpital rule

$\displaystyle\lim_{x\to 0}x^{u^{\circ}-1}F(x)=\frac{1}{1-u^{\circ}}\lim_{x\to
0}x^{u^{\circ}}F^{\prime}(x),$

where the limit on the right-hand side is finite for any
$u^{\circ}\in(1-\alpha_{1},1)=(-1,1)$ in the case of the beta distribution,
and for any $u^{\circ}\in(1-k,1)=(-1,1)$ in the case of the gamma
distribution. In what follows, we consider the estimate (4) with the kernel
(5).

  1. 1.

Discrete mixing distribution with finite support. Let $G$ be a purely discrete
probability law taking values 1 and 2 with probabilities 1/3 and 2/3,
respectively. As was discussed in Section 6, we have in this case

$\displaystyle|{\mathcal{M}}[G](u+\mathrm{i}v)|\neq 0\quad\mbox{ for any
}\hskip 6.6864ptu\neq 1+\log_{2}\frac{1/3}{2/3}=0\hskip 6.6864pt\mbox{ and
any}\hskip 6.6864ptv\in{\mathbb{R}}.$

Therefore the parameter $u^{\circ}$ can be taken from the set

$\displaystyle(-\infty,1)\cap\mathcal{C}_{F}\cap\mathcal{H}_{G}=(-1,0)\cup(0,1).$

In what follows, we take $u^{\circ}=1/2.$ Based on the discussion in Section
5, we choose $T_{n}=n=1000$. As for the estimator (19), the parameter $R_{n}$
minimises the corresponding mean-square error based on 500 samples of size
$n=1000$. The exact values are 3.5 and 9.7 for the beta and the gamma
distributions, respectively.

Figure 1 shows boxplots of the mean-squared errors of the Mellin-based
estimator (4) and the Fourier-based estimator (19) for different values of
$n\in\\{100,500,1000\\}$ obtained via $100$ simulation runs. The average (over
runs) values of MSE for each $n$ are presented in Table 1(b). Note that for
both beta and gamma distributions the proposed estimator (4) converges to the
true c.d.f. $F$ rather fast with the MSE being less than $0.003$ for $n=1000$.
To the contrary, the accuracy of the estimator (19) does not significantly
increase with the growth of $n$, and the average errors are more than $60$
times greater than those of the estimator (4). This point is also justified by
Figures 2 and 3 showing the estimated and true c.d.f.s.

Table 1: Average MSE for the Mellin-based estimator (4) and the Fourier-based
estimator (19) obtained via 100 simulation runs (a) Beta(2,2) distribution

Mixing distribution | Method | Relative error
---|---|---
Mellin | Fourier | Fourier/Mellin
Two point | $n=100$ | 0.0012 | 0.2128 | 177.33
$n=500$ | 0.0003 | 0.2093 | 697.67
$n=1000$ | 0.0002 | 0.2092 | 1046
Zeta | $n=100$ | 0.000069 | 0.00011 | 1.59
$n=500$ | 0.000074 | 0.00011 | 1.49
$n=1000$ | 0.000068 | 0.0001 | 1.47
Uniform | $n=100$ | 0.0051 | 0.0046 | 0.9
$n=500$ | 0.0014 | 0.0012 | 0.86
$n=1000$ | 0.00071 | 0.00072 | 1.01

(b) Gamma(2,2) distribution

Mixing distribution | Method | Relative error
---|---|---
Mellin | Fourier | Fourier/Mellin
Two point | $n=100$ | 0.0012 | 0.0728 | 60.67
$n=500$ | 0.0003 | 0.0723 | 241
$n=1000$ | 0.0001 | 0.0716 | 716
Zeta | $n=100$ | 0.0008 | 0.0022 | 2.75
$n=500$ | 0.0002 | 0.0004 | 2
$n=1000$ | 0.00008 | 0.0002 | 2.5
Uniform | $n=100$ | 0.004 | 0.0035 | 0.88
$n=500$ | 0.001 | 0.0007 | 0.7
$n=1000$ | 0.0006 | 0.0004 | 0.67

Figure 1: The mean-squared errors of the estimators (4) (top) and (19)
(bottom) for the c.d.f.s of Beta(2,2) (left) and Gamma(2,2) (right)
distributions based on 100 simulation runs for $G$ being a c.d.f. of a
discrete distribution supported on two points. Figure 2: The estimates
$\widehat{F}$ of the c.d.f. of the Beta(2,2) distribution for the estimators
(4) based on the Mellin transform (top) and (19) based on the Fourier
transform (bottom) superimposed with the true distribution function for $G$
being a c.d.f. of a discrete distribution supported on two points. Figure 3:
The estimates $\widehat{F}$ of the c.d.f. of the Gamma(2,2) distribution for
the estimators (4) based on the Mellin transform (top) and (19) based on the
Fourier transform (bottom) superimposed with the true distribution function
for $G$ being a c.d.f. of a discrete distribution supported on two points

Figure 4 represents the dependence of the left-hand side in Theorem 4.1 on $x$
(with fixed $u=0.5$) and $u^{\circ}$ (with fixed $x=0.5$) for $25$ samples of
size $n=1000$. The plots indicate that for both beta and gamma distributions
the accuracy of the estimator increases with growing $x$ which is coherent
with the theoretical findings of Theorem 4.1. As for dependence on
$u^{\circ}$, one can observe that, as suggested by Theorem 4.1, the proposed
estimator becomes more accurate as $u^{\circ}$ increases with an evident
exception for $u^{\circ}=0$. As was mentioned earlier in this section, the
reason for this peculiarity is that $u^{\circ}=0$ turns out to be exactly the
threshold value (18), for which the Mellin transform ${\mathcal{M}}[G](z)$ of
the mixing distribution is not guaranteed to be separated from zero in
absolute value.

Figure 4: The estimates $\widehat{\mathcal{R}}(\widehat{F};u^{\circ},x)$ for
different values of $x$ (left) and $u^{\circ}$ (right) for Beta(2,2) (top) and
Gamma(2,2) (bottom) distributions for $G$ being a c.d.f. of a discrete
distribution supported on two points

  2. 2.

Discrete mixing distribution with infinite support. Now let $G$ be the c.d.f.
of the zeta distribution (17) with parameter $s=5$. As was discussed in
Section 6, in this case $|{\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)|$ is
strictly positive for any

$u^{\circ}<-\frac{\log(\zeta(5))}{\log(2)}+1\approx 0.95.$

Figure 5 shows the mean-square errors of the estimators (4) and (19). As
before, the parameters for (4) are fixed as $T_{n}=1000$ and $u^{\circ}=0.5$.
The cutoff parameter $R_{n}$ for the estimator (19) was chosen by the same
methodology as before resulting in the values $9.6$ for beta distribution and
$45.4$ for gamma law. It can be observed that in case of the latter the errors
for both estimators converge to zero as $n$ grows. However, the proposed
estimator outperforms the one based on the Fourier transform as follows from
the average errors depicted in Table 1(b). In case of beta distribution, the
average errors for both estimators are smaller than $0.0009$ already for
$n=100$, and further convergence to zero appears to be very slow.
Nevertheless, the estimator (4) still has slightly smaller errors and yields a
more accurate fit compared to (19), as follows from Figures 6 and 7.

Figure 5: The mean-squared errors of the estimators (4) (top) and (19)
(bottom) for the c.d.f.s of Beta(2,2) (left) and Gamma(2,2) (right)
distributions based on 100 simulation runs for $G$ being a c.d.f. of the zeta
distribution  Figure 6: The estimates $\widehat{F}$ of the c.d.f. of the
Beta(2,2) distribution for the estimators (4) based on the Mellin transform
(top) and (19) based on the Fourier transform (bottom) superimposed with the
true distribution function for $G$ being a c.d.f. of the zeta distribution
Figure 7: The estimates $\widehat{F}$ of the c.d.f. of the Gamma(2,2)
distribution for the estimators (4) based on the Mellin transform (top) and
(19) based on the Fourier transform (bottom) superimposed with the true
distribution function for $G$ being a c.d.f. of the zeta distribution

Figure 8 illustrates the dependence of the estimated left-hand side in Theorem
4.1 on $x$ and $u^{\circ}$ for 25 samples of size $n=1000$. As before, in the
first case the parameter $u^{\circ}$ was fixed as $u^{\circ}=0.5$. It can be
observed that for both beta and gamma distributions the accuracy of the
proposed estimator increases as $x$ grows, though for the former the
dependence is more unstable. In case of dependence on $u^{\circ},$ the
parameter $x$ was fixed at the same level $x=0.5$. From the plots on the right
of Figure 8 one can observe that the error strictly decreases with growing
$u^{\circ}$. Hence, the estimator becomes more accurate as $x$ and $u^{\circ}$
increase, which supports the theoretical findings of Theorem 4.1.

Figure 8: The estimates $\widehat{\mathcal{R}}(\widehat{F};u^{\circ},x)$ for
different values of $x$ (left) and $u^{\circ}$ (right) for Beta(2,2) (top) and
Gamma(2,2) (bottom) distributions for $G$ being a c.d.f. of the zeta
distribution

  3. 3.

Continuous mixing distribution. Finally, assume that $G$ corresponds to the
c.d.f. of the uniform distribution on $[0,1]$. In this case, we have

$|{\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)|=|u^{\circ}+\mathrm{i}v|^{-1}>0$

for any $u^{\circ},v\in{\mathbb{R}}$. However, since
$|{\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)|$ decreases in $v$ and can become
arbitrarily close to zero as $|v|\to\infty$, the choice of the truncation
level $T_{n}$ for (4) becomes crucial. In what follows, we use the values
$T_{n}=34.6$ and $T_{n}=29.7$ for the cases of beta and gamma distributions,
respectively. These values were obtained by the same methodology as described
above for the estimator (19). As for the latter, the values of $R_{n}$
minimising the MSE turned out to be $R_{n}=9.7$ for the beta distribution and
$R_{n}=3.4$ for the gamma law.

Figure 9 represents the mean-square errors for estimators (4) and (19). It can
be observed that both methods provide very similar results, with the latter
estimator performing a bit better in case of the gamma distribution. In both
cases, the error decreases as the sample size grows, though the rate of
convergence to zero appears to be slower than for discrete mixing
distributions considered above. This observation is further supported by
Figures 10 and 11 showing the fits provided by the above two estimators.

Figure 9: The mean-squared errors of the estimators (4) (top) and (19)
(bottom) for the c.d.f.s of Beta(2,2) (left) and Gamma(2,2) (right)
distributions based on 100 simulation runs for $G$ being a c.d.f. of the
uniform distribution  Figure 10: The estimates $\widehat{F}$ of the c.d.f. of
the Beta(2,2) distribution for the estimators (4) based on the Mellin
transform (top) and (19) based on the Fourier transform (bottom) superimposed
with the true distribution function for $G$ being a c.d.f. of the uniform
distribution Figure 11: The estimates $\widehat{F}$ of the c.d.f. of the
Gamma(2,2) distribution for the estimators (4) based on the Mellin transform
(top) and (19) based on the Fourier transform (bottom) superimposed with the
true distribution function for $G$ being a c.d.f. of the uniform distribution

Finally, Figure 12 represents the boxplots of the estimated
$\widehat{\mathcal{R}}(\widehat{F};u^{\circ},x)$ for the proposed estimator
(4) for different values of $x$ and $u^{\circ}$. The parameters were fixed as
$x=u^{\circ}=0.5$, and the boxplots were obtained based on $25$ samples of
size $n=1000$ as before. It can be seen that the accuracy of the estimator
increases with growing $x$ and $u^{\circ}$.

Figure 12: The estimates $\widehat{\mathcal{R}}(\widehat{F};u^{\circ},x)$ for
different values of $x$ (left) and $u^{\circ}$ (right) for Beta(2,2) (top) and
Gamma(2,2) (bottom) distributions for $G$ being a c.d.f. of the uniform
distribution

##  Appendix A Berry – Esseen inequality in a particular case

Let us consider the case when $\varphi$ and $\psi$ are the distribution
functions of the exponential distribution, that is,

$\displaystyle\varphi(x)=\bigl{(}1-e^{-\lambda_{1}x}\bigr{)}{\mathbb{I}}\bigl{\\{}x\geq
0\bigr{\\}},\qquad\psi(x)=\bigl{(}1-e^{-\lambda_{2}x}\bigr{)}{\mathbb{I}}\bigl{\\{}x\geq
0\bigr{\\}},$

where $\lambda_{1},\lambda_{2}>0.$ Since

$\displaystyle{\mathcal{M}}[\varphi](z)=\lambda_{1}^{1-z}\Gamma(z),\qquad{\mathcal{M}}[\psi](z)=\lambda_{2}^{1-z}\Gamma(z)$

for any $z$ with $\operatorname{Re}(z)>0,$ we can take $u^{\circ}=1/2$. We
have

$\displaystyle\rho_{1/2}(\varphi,\psi):=\sup\limits_{x\geq
0}\Bigl{|}\frac{e^{-\lambda_{1}x}-e^{-\lambda_{2}x}}{\sqrt{x}}\Bigr{|}.$

The maximum is attained at some point $x_{0}\in(0,\infty).$ For this numerical
example we take the values $\lambda_{1}=1,\lambda_{2}=1.5,$ for which
$x_{0}=0.4$.

We also take $b=0.8$, and numerically get $c(0.8)\approx 4.8$ as the solution
of (7). We aim to study the right-hand side of (6) depending on $T.$

Figure 13 represents the plots of the first and the second summands in (6),
namely,

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{b}{2}\int\limits_{-T}^{T}\frac{|\lambda_{1}^{1/2-\mathrm{i}v}-\lambda_{2}^{1/2-\mathrm{i}v}|}{|v|}\bigl{|}\Gamma((1/2)+\mathrm{i}v)\bigr{|}\,dv,$
$\displaystyle I_{2}$ $\displaystyle=$ $\displaystyle
bTx_{0}^{u^{\circ}-1}\int_{0}^{2c(b)/T}\bigl{(}e^{x_{0}e^{r}}-e^{x_{0}}\bigr{)}\,dr.$

An interesting observation is that $I_{1}$ is almost equal to a constant for
large $T$, while $I_{2}$ decays rather fast as $T$ grows.

Figure 13: The graphs of the first and the second summands, and the right-hand
side in (6), depending on the value of $T.$ The blue dotted line represents
the first summand and is for large $T$ is close to 1.1. The maroon dashed line
shows the behaviour of the second summand, which tends to 0 as $T\to\infty.$
Finally, the red solid line shows the total behaviour of the r.h.s. in (6).

##  Appendix B Proofs

###  B.1 Proof of Lemma 2.1

Let us rewrite (2) as

${\mathcal{M}}[\varphi](z)=\int\limits_{0}^{\infty}x^{z-1}\,d\varphi(x)=\int\limits_{0}^{\infty}e^{(z-1)\log
x}\,d\varphi(e^{\log x})\\\
=\int\limits_{-\infty}^{\infty}e^{-sy}\,d\varphi(e^{y})=:\mathcal{L}[\varphi(e^{y})](s),$

where the right-hand side defines the bilateral Laplace transform of
$\varphi(e^{y})$, $y=\log x$, at the point $s=-(z-1)$. Applying Theorem 7.7.5
from [11] for the bilaterial Laplace transform, we arrive at

$\displaystyle\frac{1}{2}\Bigl{(}\varphi(e^{y}+0)+\varphi(e^{y}+0)\Bigr{)}-\varphi(0)$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\mathrm{i}}\int\limits_{\tilde{u}-\mathrm{i}\infty}^{\tilde{u}+\mathrm{i}\infty}e^{sy}\frac{\mathcal{L}[\varphi(e^{y})](s)}{s}\,ds$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\mathrm{i}}\int\limits_{u^{\circ}-\mathrm{i}\infty}^{u^{\circ}+\mathrm{i}\infty}x^{-z+1}\frac{{\mathcal{M}}[\varphi](z)}{-(z-1)}\,dz,$

provided $\tilde{u}:=1-u^{\circ}>0$. The second statement of the theorem
follows from another part of Theorem 7.7.5 from [11]

###  B.2 Proof of Lemma 3.1

Let

$\displaystyle w(x)=\frac{\sin^{2}(T\log x)}{\pi
Tx^{u^{\circ}}\log^{2}x}\qquad\mbox{and}\qquad
W(x)=\int\limits_{0}^{x}w(y)\,dy,x\in{\mathbb{R}}_{+}.$ (20)

Define

$\widetilde{\varphi}(x):=\int\limits_{0}^{\infty}\varphi(x/y)\,dW(y),\qquad\widetilde{\psi}(x):=\int\limits_{0}^{\infty}\psi(x/y)dW(y).$

Due to Theorem 7.8.3 from [11],

${\mathcal{M}}[\widetilde{\varphi}](z)={\mathcal{M}}[\varphi](z){\mathcal{M}}[W](z),\qquad{\mathcal{M}}[\widetilde{\psi}](z)={\mathcal{M}}[\psi](z){\mathcal{M}}[W](z).$

Then, by the inverse formula for the Mellin transform (Lemma 2.1),

$\widetilde{\varphi}(x)-\widetilde{\psi}(x)=\frac{1}{2\pi\mathrm{i}}\int\limits_{u^{\circ}-\mathrm{i}\infty}^{u^{\circ}+\mathrm{i}\infty}\frac{{\mathcal{M}}[\varphi](z)-{\mathcal{M}}[\psi](z)}{-(z-1)}{\mathcal{M}}[W](z)x^{-z+1}\,dz\\\
=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\frac{{\mathcal{M}}[\varphi](u^{\circ}+\mathrm{i}v)-{\mathcal{M}}[\psi](u^{\circ}+\mathrm{i}v)}{-(u^{\circ}+\mathrm{i}v-1)}{\mathcal{M}}[W](u^{\circ}+\mathrm{i}v)x^{-u^{\circ}-\mathrm{i}v+1}\,dv.$

We get

$\displaystyle{\mathcal{M}}[W](u^{\circ}+\mathrm{i}v)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{\infty}x^{(u^{\circ}-1)+\mathrm{i}v}\frac{\sin^{2}(T\log
x)}{\pi Tx^{u^{\circ}}\log^{2}x}\,dx$ (21) $\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{\infty}e^{\mathrm{i}vy}\frac{\sin^{2}(Ty)}{\pi
Ty^{2}}\,dy$ $\displaystyle=$
$\displaystyle\left(1-\left|\frac{v}{T}\right|\right){\mathbb{I}}\\{|v|\leq
T\\},$

hence,

$\displaystyle\rho_{u^{\circ}}(\widetilde{\varphi},\widetilde{\psi})$
$\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\int\limits_{-T}^{T}\frac{|{\mathcal{M}}[\varphi](u^{\circ}+\mathrm{i}v)-{\mathcal{M}}[\psi](u^{\circ}+\mathrm{i}v)|}{\sqrt{(u^{\circ}-1)^{2}+v^{2}}}\,dv$
(22) $\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\int\limits_{-T}^{T}\frac{|{\mathcal{M}}[\varphi](u^{\circ}+\mathrm{i}v)-{\mathcal{M}}[\psi](u^{\circ}+\mathrm{i}v)|}{|v|}\,dv.$

At the same time,

$\widetilde{\varphi}(x)-\widetilde{\psi}(x)=\int\limits_{0}^{\infty}\Bigl{(}\varphi\left(x/y\right)-\psi\left(x/y\right)\Bigr{)}w(y)\,dy\\\
\
=\int\limits_{0}^{\infty}\Bigl{(}\varphi\left(x/y\right)-\psi\left(x/y\right)\Bigr{)}\frac{\sin^{2}(T\log
y)}{\pi Ty^{u^{\circ}}\log^{2}y}\,dy.$

Let us denote $\Delta=\rho_{u^{\circ}}(\varphi,\psi)$. Then

$x_{0}^{u^{\circ}-1}\bigl{(}\varphi(x_{0})-\psi(x_{0}\pm 0)\bigr{)}=\Delta$

or

$x_{0}^{u^{\circ}-1}\bigl{(}\psi(x_{0}\pm 0)-\varphi(x_{0}-0)\bigr{)}=\Delta.$

Since for both cases the proof follows the same lines, we will only consider
the first one. In this case

$x_{0}^{u^{\circ}-1}e^{\frac{c(b)(u^{\circ}-1)}{T}}\Bigl{(}\widetilde{\varphi}\left(x_{0}e^{\frac{c(b)}{T}}\right)-\widetilde{\psi}\left(x_{0}e^{\frac{c(b)}{T}}\right)\Bigr{)}\\\
=x_{0}^{u^{\circ}-1}e^{\frac{c(b)(u^{\circ}-1)}{T}}\int\limits_{0}^{\infty}\Bigl{(}\varphi\bigl{(}x_{0}e^{\frac{c(b)}{T}}/y\bigr{)}-\psi\bigl{(}x_{0}e^{\frac{c(b)}{T}}/y\bigr{)}\Bigr{)}\frac{\sin^{2}(T\log
y)}{\pi Ty^{u^{\circ}}\log^{2}y}\,dy\\\
=x_{0}^{u^{\circ}-1}\int\limits_{-\infty}^{\infty}\Bigl{(}\varphi\bigl{(}x_{0}e^{\frac{c(b)-r}{T}}\bigr{)}-\psi\bigl{(}x_{0}e^{\frac{c(b)-r}{T}}\bigr{)}\Bigr{)}\frac{\sin^{2}r}{\pi
r^{2}}\,e^{(c(b)-r)(u^{\circ}-1)/T}dr\\\
=x_{0}^{u^{\circ}-1}\int\limits_{|r|<c(b)}\Bigl{(}\varphi\bigl{(}x_{0}e^{\frac{c(b)-r}{T}}\bigr{)}-\psi\bigl{(}x_{0}e^{\frac{c(b)-r}{T}}\bigr{)}\Bigr{)}\frac{\sin^{2}r}{\pi
r^{2}}e^{(c(b)-r)(u^{\circ}-1)/T}\,dr\\\ \hskip
28.45274pt+x_{0}^{u^{\circ}-1}\int\limits_{|r|\geq
c(b)}\Bigl{(}\varphi\bigl{(}x_{0}e^{\frac{c(b)-r}{T}}\bigr{)}-\psi\bigl{(}x_{0}e^{\frac{c(b)-r}{T}}\bigr{)}\Bigr{)}\frac{\sin^{2}r}{\pi
r^{2}}e^{(c(b)-r)(u^{\circ}-1)/T}\,dr\\\ \geq
x_{0}^{u^{\circ}-1}\int\limits_{|r|<c(b)}\Bigl{(}\psi\bigl{(}x_{0}\pm
0\bigr{)}-\psi\bigl{(}x_{0}e^{\frac{c(b)-r}{T}}\bigr{)}\Bigr{)}\frac{\sin^{2}r}{\pi
r^{2}}e^{(c(b)-r)(u^{\circ}-1)/T}\,dr\\\ \hskip
28.45274pt+x_{0}^{u^{\circ}-1}\int\limits_{|r|<c(b)}\Bigl{(}\varphi\bigl{(}x_{0}e^{\frac{c(b)-r}{T}}\bigr{)}-\psi\bigl{(}x_{0}\pm
0\bigr{)}\Bigr{)}\frac{\sin^{2}r}{\pi r^{2}}e^{(c(b)-r)(u^{\circ}-1)/T}\,dr\\\
\hskip 142.26378pt-\Delta\int\limits_{|r|\geq c(b)}\frac{\sin^{2}r}{\pi
r^{2}}dr.$

Since for any $r<c(b),$ it holds

$x_{0}^{u^{\circ}-1}\Bigl{(}\varphi\bigl{(}x_{0}e^{\frac{c(b)-r}{T}}\bigr{)}-\psi\bigl{(}x_{0}\pm
0\bigr{)}\Bigr{)}\geq
x_{0}^{u^{\circ}-1}\bigl{(}\varphi\left(x_{0}\right)-\psi\left(x_{0}\pm
0\right)\bigr{)}=\Delta,$

we further have

$x_{0}^{u^{\circ}-1}e^{\frac{c(b)(u^{\circ}-1)}{T}}\Bigl{(}\widetilde{\varphi}\bigl{(}x_{0}e^{\frac{c(b)}{T}}\bigr{)}-\widetilde{\psi}\bigl{(}x_{0}e^{\frac{c(b)}{T}}\bigr{)}\Bigr{)}\\\
\geq-\frac{x_{0}^{u^{\circ}-1}}{\pi}\int\limits_{|r|<c(b)}\left|\psi(x_{0}\pm
0)-\psi\left(x_{0}e^{\frac{c(b)-r}{T}}\right)\right|e^{(c(b)-r)(u^{\circ}-1)/T}\,dr\\\
\hskip 28.45274pt+\Delta\int\limits_{|r|<c(b)}\frac{\sin^{2}r}{\pi
r^{2}}e^{(c(b)-r)(u^{\circ}-1)/T}\,dr-\Delta\int\limits_{|r|\geq
c(b)}\frac{\sin^{2}r}{\pi r^{2}}\,dr\\\
\geq-\frac{T}{\pi}x_{0}^{u^{\circ}-1}\int\limits_{0}^{2c(b)/T}|\psi(x_{0}\pm
0)-\psi(x_{0}e^{r})|\,dr\\\ +\Delta\int\limits_{|r|<c(b)}\frac{\sin^{2}r}{\pi
r^{2}}e^{(c(b)-r)(u^{\circ}-1)/T}\,dr-\Delta\int\limits_{|r|\geq
c(b)}\frac{\sin^{2}r}{\pi r^{2}}\,dr.\hskip 17.07182pt$

Now, noting that

$\displaystyle\int\limits_{|r|<c(b)}\frac{\sin^{2}r}{\pi
r^{2}}e^{(c(b)-r)(u^{\circ}-1)/T}dr$ $\displaystyle\geq$ $\displaystyle
e^{2c(b)(u^{\circ}-1)/T}\int\limits_{|r|<c(b)}\frac{\sin^{2}r}{\pi r^{2}}dr,$
$\displaystyle\int\limits_{|r|\geq c(b)}\frac{\sin^{2}r}{\pi r^{2}}\;dr$
$\displaystyle=$ $\displaystyle 1-\int\limits_{|r|<c(b)}\frac{\sin^{2}r}{\pi
r^{2}}dr$

we obtain

$\rho_{u^{\circ}}(\widetilde{\varphi},\widetilde{\psi})\geq-\frac{T}{\pi}x_{0}^{u^{\circ}-1}\int\limits_{0}^{2c(b)/T}|\psi(x_{0}\pm
0)-\psi(x_{0}e^{r})|\,dr\\\
+\Delta\Bigl{(}-1+\int\limits_{|r|<c(b)}\frac{\sin^{2}r}{\pi
r^{2}}\,dr\cdot\bigl{(}e^{2c(b)(u^{\circ}-1)/T}+1\bigr{)}\Bigr{)},$

where $e^{2c(b)(u^{\circ}-1)/T}+1>3/2$ due to our choice of $T$. Choosing
$c(b)$ as in (7), arrive at the upper bound for $\Delta$

$\Delta\leq\pi
b\cdot\rho_{u^{\circ}}(\widetilde{\varphi},\widetilde{\psi})+Tbx_{0}^{u^{\circ}-1}\int\limits_{0}^{2c(b)/T}|\psi(x_{0}\pm
0)-\psi(x_{0}e^{r})|\,dr.$ (23)

Combining this bound with (22), we obtain the result.

###  B.3 Proof of Theorem 4.1

Let us consider

$x^{u^{\circ}-1}|F(x)-\widehat{F}(x)|\leq
x^{u^{\circ}-1}|F(x)-\widetilde{F}(x)|+x^{u^{\circ}-1}|\widetilde{F}(x)-\widehat{F}(x)|$
(24)

with $\widetilde{F}(x)$ being an auxiliary function defined as the
multiplicative convolution

$\widetilde{F}(x):=F\star
W(x)=\int\limits_{0}^{\infty}F\left(x/y\right)\,dW(y),$

where $W:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ is defined by (20). Since
${\mathcal{M}}[\widetilde{F}](z)={\mathcal{M}}[F](z){\mathcal{M}}[W](z)$ for
any $z$ such that
$\operatorname{Re}(z)\in\mathcal{C}_{\widetilde{F}}=\mathcal{C}_{F}\cap\mathcal{C}_{W}$,
and $u^{\circ}\in\mathcal{C}_{W}$ by (21), we get from Lemma 2.1,

$\widetilde{F}(x)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}x^{-u^{\circ}-\mathrm{i}v+1}\frac{{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v){\mathcal{M}}[W](u^{\circ}+\mathrm{i}v)}{-(u^{\circ}+\mathrm{i}v-1)}\,dv,\quad
v\in{\mathbb{R}}.$

Noticing that ${\mathcal{M}}[W](u^{\circ}+\mathrm{i}v)=K(v),$ see (21), we
have for the second summand in (24)

$x^{u^{\circ}-1}|\widetilde{F}(x)-\widehat{F}(x)|\\\
=x^{u^{\circ}-1}\left|\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}x^{(-u^{\circ}+1)-\mathrm{i}v}\frac{{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v)-\widehat{{\mathcal{M}}[F]}(u^{\circ}+\mathrm{i}v)}{-(u^{\circ}+\mathrm{i}v-1)}K(v)\right|\\\
=x^{u^{\circ}-1}\left|\frac{1}{2\pi
n}\sum\limits_{k=1}^{n}\int\limits_{-\infty}^{\infty}x^{(-u^{\circ}+1)-\mathrm{i}v}\frac{{\mathcal{M}}[F_{\text{mix}}](u^{\circ}+\mathrm{i}v)-X_{k}^{(u^{\circ}-1)+\mathrm{i}v}}{-(u^{\circ}+\mathrm{i}v-1){\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)}K(v)\right|\\\
=\frac{x^{u^{\circ}-1}}{2\pi
n}\left|\sum\limits_{k=1}^{n}\Lambda_{k}(X_{k},x)\right|.\hskip 204.85974pt$

As for the first summand in (24), let us note that

$\sup_{x\geq
0}\Bigl{|}x^{u^{\circ}-1}\bigl{(}F(x)-\widetilde{F}(x)\bigr{)}\Bigr{|}=\sup_{x\geq
0}\Biggl{|}x^{u^{\circ}-1}F(x)-\int\limits_{0}^{\infty}\left(x/y\right)^{u^{\circ}-1}F\left(x/y\right)\widetilde{w}(y)dy\Biggr{|}\\\
=\sup_{x\geq
0}\Biggl{|}\int\limits_{0}^{\infty}\left(x^{u^{\circ}-1}F\bigl{(}x\bigr{)}-\left(x/y\right)^{u^{\circ}-1}F\left(x/y\right)\right)\widetilde{w}(y)dy\Biggr{|},$
(25)

where $\widetilde{w}(y):=w(y)y^{u^{\circ}-1},y\in{\mathbb{R}}_{+}$ is a
density function of some distribution. Since we assume that the distribution
function $F$ is continuous at least in a small vicinity of $0$, and (9) holds,
the supremum in (25) is attained at some point $x_{0}>0.$ Thus, by Lemma 3.1,

$x^{u^{\circ}-1}|F(x)-\widetilde{F}(x)|\leq\frac{b}{2}\int\limits_{-T}^{T}\frac{\left|{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v)-{\mathcal{M}}[\widetilde{F}](u^{\circ}+\mathrm{i}v)\right|}{|v|}\,dv\\\
+bTx_{0}^{u^{\circ}-1}\int_{0}^{2c(b)/T}\left|F(x_{0}\pm
0)-F\left(x_{0}e^{r}\right)\right|\,dr.$

Now it only remains to observe that

$\int\limits_{-T}^{T}\frac{\left|{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v)-{\mathcal{M}}[\widetilde{F}](u^{\circ}+\mathrm{i}v)\right|}{|v|}\,dv\\\
=\int\limits_{-T}^{T}\frac{\left|{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v)(1-{\mathcal{M}}[W](u^{\circ}+\mathrm{i}v))\right|}{|v|}\,dv=\frac{1}{T}\int\limits_{-T}^{T}\left|{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v)\right|\,dv.$

###  B.4 Proof of Theorem 4.2

By the upper bound established in Theorem 4.1,

$\displaystyle\mathcal{R}^{*}(\widehat{F};u^{\circ},x)$ $\displaystyle\leq$
$\displaystyle\frac{3b^{2}}{4T^{2}}\Bigl{(}\int\limits_{-T}^{T}|{\mathcal{M}}[F](u^{\circ}+\mathrm{i}v)|\,dv\Bigr{)}^{2}$
(26) $\displaystyle\hskip
28.45274pt+3b^{2}T^{2}x_{0}^{2(u^{\circ}-1)}\Bigl{(}\int\limits_{0}^{2c(b)/T}|F(x_{0}\pm
0)-F(x_{0}e^{r})|\,dr\Bigr{)}^{2}$ $\displaystyle\hskip
28.45274pt+\frac{3x^{2(u^{\circ}-1)}}{4\pi^{2}n^{2}}{\mathbb{E}}\left[\left|\sum\limits_{k=1}^{n}\Lambda_{k}(X_{k},x)\right|^{2}\right].$

Now, since $\Lambda_{k}(X_{k},x)$ are centred i.i.d. for all $k=1,\dots,n$,

$\displaystyle{\mathbb{E}}\left[\left|\sum\limits_{k=1}^{n}\Lambda_{k}(X_{k},x)\right|^{2}\right]=\operatorname{Var}\left(\sum\limits_{k=1}^{n}\Lambda_{k}(X_{k},x)\right)=n{\mathbb{E}}\left[\Lambda_{1}(X_{1},x)\overline{\Lambda_{1}(X_{1},x)}\right],$

where for the latter quantity we have that

${\mathbb{E}}\left[\Lambda_{1}(X_{1},x)\overline{\Lambda_{1}(X_{1},x)}\right]\\\
={\mathbb{E}}\left[\int\limits_{-T}^{T}\int\limits_{-T}^{T}x^{2(-u^{\circ}+1)-\mathrm{i}(v-w)}\frac{({\mathcal{M}}[F_{\text{mix}}](u^{\circ}+\mathrm{i}v)-X_{1}^{(u^{\circ}-1)+\mathrm{i}v})}{(u^{\circ}-1+\mathrm{i}v)(u^{\circ}-1-\mathrm{i}w)}\left(1-\frac{|v|}{T}\right)\right.\\\
\hskip
99.58464pt\times\left.\frac{(\overline{{\mathcal{M}}[F_{\text{mix}}](u^{\circ}+\mathrm{i}w)}-X_{1}^{(u^{\circ}-1)-\mathrm{i}w})}{{\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)\overline{{\mathcal{M}}[G](u^{\circ}+\mathrm{i}w)}}\left(1-\frac{|w|}{T}\right)\,dv\,dw\right]\\\
\hskip 113.81102pt\times
x^{2(-u^{\circ}+1)-\mathrm{i}(v-w)}\left(1-\frac{|v|}{T}\right)\left(1-\frac{|w|}{T}\right)\,dv\,dw.$

Since

${\mathbb{E}}[({\mathcal{M}}[F_{\text{mix}}](u^{\circ}+\mathrm{i}v)-X_{1}^{(u^{\circ}-1)+\mathrm{i}v})(\overline{{\mathcal{M}}[F_{\text{mix}}](u^{\circ}+\mathrm{i}w)}-X_{1}^{(u^{\circ}-1)-\mathrm{i}w})]\\\
=-{\mathcal{M}}[F_{\text{mix}}](u^{\circ}+\mathrm{i}v){\mathcal{M}}[F_{\text{mix}}](u^{\circ}-\mathrm{i}w)+{\mathcal{M}}[F_{\text{mix}}](2u^{\circ}-1+\mathrm{i}(v-w)),$

we further have

${\mathbb{E}}\left[\Lambda_{1}(X_{1},x)\overline{\Lambda_{1}(X_{1},x)}\right]\\\
\hskip 19.91684pt\\\
=\int\limits_{-T}^{T}\int\limits_{-T}^{T}\frac{{\mathcal{M}}[F_{\text{mix}}](2u^{\circ}-1+\mathrm{i}(v-w))}{(u^{\circ}-1+\mathrm{i}v)(u^{\circ}-1-\mathrm{i}w){\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)\overline{{\mathcal{M}}[G](u^{\circ}+\mathrm{i}w)}}\\\
\hskip 128.0374pt\times
x^{2(-u^{\circ}+1)-\mathrm{i}(v-w)}\left(1-\frac{|v|}{T}\right)\left(1-\frac{|w|}{T}\right)\,dv\,dw\\\
-\int\limits_{-T}^{T}\int\limits_{-T}^{T}\frac{{\mathcal{M}}[F_{\text{mix}}](u^{\circ}+\mathrm{i}v){\mathcal{M}}[F_{\text{mix}}](u^{\circ}-\mathrm{i}w)}{(u^{\circ}-1+\mathrm{i}v)(u^{\circ}-1-\mathrm{i}w){\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)\overline{{\mathcal{M}}[G](u^{\circ}+\mathrm{i}w)}}\\\
\hskip 128.0374pt\times
x^{2(-u^{\circ}+1)-\mathrm{i}(v-w)}\left(1-\frac{|v|}{T}\right)\left(1-\frac{|w|}{T}\right)\,dv\,dw,$

where the subtrahend is a real positive number. Therefore,

${\mathbb{E}}\left[\Lambda_{1}(X_{1},x)\overline{\Lambda_{1}(X_{1},x)}\right]\\\
\leq
x^{2(-u^{\circ}+1)}\int\limits_{-T}^{T}\int\limits_{-T}^{T}x^{-\mathrm{i}(v-w)}\frac{\sqrt{{\mathcal{M}}[F_{\text{mix}}](2u^{\circ}-1+\mathrm{i}(v-w))}}{(u^{\circ}-1+\mathrm{i}v){\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)}\left(1-\frac{|v|}{T}\right)\\\
\times\frac{\sqrt{{\mathcal{M}}[F_{\text{mix}}](2u^{\circ}-1+\mathrm{i}(v-w))}}{\overline{(u^{\circ}-1+\mathrm{i}w){\mathcal{M}}[G](u^{\circ}+\mathrm{i}w)}}\left(1-\frac{|w|}{T}\right)\,dv\,dw$

Finally, by the Cauchy-Schwarz inequality,

${\mathbb{E}}\left[\Lambda_{1}(X_{1},x)\overline{\Lambda_{1}(X_{1},x)}\right]\leq
x^{2(-u^{\circ}+1)}\int\limits_{-T}^{T}\int\limits_{-T}^{T}\frac{|{\mathcal{M}}[F_{\text{mix}}](2u^{\circ}-1+\mathrm{i}(v-w))|}{((u^{\circ}-1)^{2}+v^{2})|{\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)|^{2}}\\\
\hskip 256.0748pt\times\left(1-\frac{|v|}{T}\right)^{2}\,dv\,dw\\\ \hskip
102.43008pt\leq
x^{2(-u^{\circ}+1)}\int\limits_{-T}^{T}\Bigl{[}\frac{(1-|v|/T)^{2}}{((u^{\circ}-1)^{2}+v^{2})|{\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)|^{2}}\Bigr{.}\\\
\Bigl{.}\hskip
156.49014pt\times\int\limits_{-\infty}^{\infty}|{\mathcal{M}}[F_{\text{mix}}](2u^{\circ}-1+\mathrm{i}(v-w))|\,dw\Bigr{]}\,dv\\\
\hskip 112.38829pt\leq
x^{2(-u^{\circ}+1)}\int\limits_{-T}^{T}\frac{1}{((u^{\circ}-1)^{2}+v^{2})|{\mathcal{M}}[G](u^{\circ}+\mathrm{i}v)|^{2}}\,dv\\\
\times\int\limits_{-\infty}^{\infty}|{\mathcal{M}}[F_{\text{mix}}](2u^{\circ}-1+\mathrm{i}w)|\,dw.$

Combining the last inequality with (26), we arrive at the desired result.

## References

  * Asgharian, M., Carone, M. and Fakoor, V., [2012] Asgharian, M., Carone, M. and Fakoor, V. (2012).  Large-sample study of the kernel density estimators under multiplicative censoring.  The Annals of Statistics, 40(1):159–187.
  * Belomestny, D. and Goldenshluger, A., [2020] Belomestny, D. and Goldenshluger, A. (2020).  Nonparametric density estimation from observations with multiplicative measurement errors.  Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 56(1):36–67.
  * Belomestny, D. and Goldenshluger, A., [2021] Belomestny, D. and Goldenshluger, A. (2021).  Density deconvolution under general assumptions on the distribution of measurement errors.  The Annals of Statistics, 49(2):615–649.
  * Belomestny, D. and Panov, V., [2015] Belomestny, D. and Panov, V. (2015).  Statistical inference for generalized Ornstein-Uhlenbeck processes.  Electronic Journal of Statistics, 9(2):1974–2006.
  * Belomestny, D. and Schoenmakers, J., [2015] Belomestny, D. and Schoenmakers, J. (2015).  Statistical Skorohod embedding problem: Optimality and asymptotic normality.  Statistics & Probability Letters, 104:169–180.
  * Belomestny, D., Comte, F., and Genon – Catalot, V., [2016] Belomestny, D., Comte, F., and Genon – Catalot, V. (2016).  Nonparametric Laguerre estimation in the multiplicative censoring model.  Electronic Journal of Statistics, 10(2):3114–3152.
  * Brunel, E., Comte, F. and Genon-Catalot, V., [2016] Brunel, E., Comte, F. and Genon-Catalot, V. (2016).  Nonparametric density and survival function estimation in the multiplicative censoring model.  Test, 25(3):570–590.
  * Comte, F. and Dion, C., [2016] Comte, F. and Dion, C. (2016).  Nonparametric estimation in a multiplicative censoring model with symmetric noise.  Journal of Nonparametric Statistics, 28(4):768–801.
  * Delaigle, A., [2021] Delaigle, A. (2021).  Deconvolution kernel density estimation.  In Handbook of Measurement Error Models, pages 185–220. Chapman and Hall/CRC.
  * Fournier, N. and Printems, J., [2010] Fournier, N. and Printems, J. (2010).  Absolute continuity for some one-dimensional processes.  Bernoulli, 16(2):343–360.
  * Kawata, T., [1972] Kawata, T. (1972).  Fourier analysis in probability theory.  Academic Press.
  * Lin, Z. and Bai, Z., [2011] Lin, Z. and Bai, Z. (2011).  Probability inequalities.  Springer Science & Business Media.
  * Lukacs, E., [1970] Lukacs, E. (1970).  Characteristic functions.  Griffin London, Second edition.
  * Meister, A., [2009] Meister, A. (2009).  Deconvolution problems in nonparametric statistics.  Lecture Notes in Statistics, Springer, Berlin, Heidelberg, pages 5–138.
  * Miguel, S., Comte, F. and Johannes, J., [2021] Miguel, S., Comte, F. and Johannes, J. (2021).  Linear functional estimation under multiplicative measurement errors.  arXiv:2111.14920.
  * Van Es, B., Klaassen, C.A.J., Oudshoorn, K., [2000] Van Es, B., Klaassen, C.A.J., Oudshoorn, K. (2000).  Survival analysis under cross-sectional sampling: length bias and multiplicative censoring.  Journal of Statistical Planning and Inference, 91(2):295–312.
  * Van Es, B., Spreij, P., Van Zanten, H., [2003] Van Es, B., Spreij, P., Van Zanten, H. (2003).  Nonparametric volatility density estimation.  Bernoulli, 9(3):451–465.
  * Vardi, Y., [1989] Vardi, Y. (1989).  Multiplicative censoring, renewal processes, deconvolution and decreasing density: nonparametric estimation.  Biometrika, 76(4):751–761.
  * Zhang, C.-H., [1990] Zhang, C.-H. (1990).  Fourier methods for estimating mixing densities and distributions.  The Annals of Statistics, pages 806–831.