Неклассические методы вероятностного и статистического анализа моделей смеси распределений тема диссертации и автореферата по ВАК РФ 01.01.05, доктор наук Панов Владимир Александрович

• Панов Владимир Александрович
• доктор наук
• 2022

Оглавление диссертации доктор наук Панов Владимир Александрович

Contents

Introduction

Acknowledgments

1 Semiparametric estimation in mixture models

1.1 Inference in normal variance-mean Gaussian mixtures

1.1.1 Historical overview

1.1.2 Mellin transform

1.1.3 Estimation of ^

1.1.4 Estimation of G with known ^

1.1.5 Estimation of G with unknown ^

1.1.6 Numerical example

1.1.7 Real data example: diamond sizes

1.1.8 Discussion

1.2 Inference in continuous-time moving average Levy processes

1.2.1 Set-up and historical remarks

1.2.2 Setup

1.2.3 Mellin transform approach for moving average processes

1.2.4 Convergence rates

1.2.5 Example

1.2.6 Mixing properties of the Levy-based MA processes

1.2.7 Numerical example

2 Stochastic time-changed models

2.1 Estimation of the Blumenthal-Getoor indices

2.1.1 Formulation of the problem

2.1.2 Assumptions on the model

2.1.3 Several examples

2.1.4 The characteristic function of xa

2.1.5 Main idea of the estimation procedure

2.1.6 The case of known a

2.1.7 The case of unknown a

2.1.8 Numerical examples

2.2 Multivariate subordinated models

2.2.1 Subordination of stable processes: general idea

2.2.2 Subordination of stable processes for financial modelling

2.2.3 Main properties of the class of stable processes

2.2.4 Dependence structures for stable processes and related models

2.2.5 Multivariate subordination of stable processes

2.2.6 Main theorem

2.2.7 Empirical analysis

3 Construction of honest confidence sets

3.1 Density estimation

3.1.1 Statement of the problem

3.1.2 Asymptotic behaviour of the maximum of Gaussian processes

3.1.3 Projection estimates for densities

3.1.4 Relation to the extreme value theory for Gaussian processes

3.1.5 SBR-type theorem for projection density estimates

3.1.6 Sequence of accompanying laws

3.1.7 Numerical example

3.2 Confidence bands for the Levy density estimators

3.2.1 Motivation of this research

3.2.2 Projection estimates for Levy densities

3.2.3 Some assumptions on the basis

3.2.4 Relation to the theory of Gaussian processes

3.2.5 Asymptotic behaviour of the corresponding Gaussian process

3.2.6 Sequence of accompanying laws

3.2.7 Asymptotic confidence bands

3.2.8 Discussion

4 Limit laws and phase transitions in the mixture models

4.1 Preliminaries: classical REM

4.1.1 Asymptotic analysis of classical REM

4.1.2 Anderson parabolic problem

4.2 Alloy-type REM

4.2.1 Definition

4.2.2 Limit laws for the alloy-type REM

4.2.3 Free energy for the alloy-type REM

5 Analysis of the "purity" of a distribution

5.1 Statement of the problem

5.2 Discrete Dickman-Goncharov distribution

5.2.1 Definition of the Dickman-Goncharov distribution

5.2.2 Discrete Dickman-Goncharov distribution. Erdos problem

Conclusion

Notational Conventions

Abbreviations

References

Введение диссертации (часть автореферата) на тему «Неклассические методы вероятностного и статистического анализа моделей смеси распределений»

Introduction

Definition of the mixture model. A mixture of probability distributions is a distribution of a random variable £, such that

P{£ e B} = i Pa(B)dG(a), B e B(Rd), (1)

J A

where Pa is a parametric family of probability measures on (Rd, B(Rd)), the mapping a ^ Pa(B) is measurable for any B e B(Rd), A c Rk is a set of possible values of the parameter a, and G is the distribution function of this parameter. The family Pa (and the distribution of £) may depend also on some other parameters, but we omit this dependence here for simplicity. If the set A consists of a finite number of elements, that is, A = (a1,..., am}, then the mixture is called finite. In this case, (1) can be written as

m

P{e e B} = ^^(B) B e B(Rd), (2)

i=1

where ni > 0 Vi = 1..m, ^m=1 ni = 1.

Historical remarks. For finite models (2), the standard task of mathematical statistics is to estimate the elements of the support ai,i = 1..m, and the mixing distribution (n1,...,nm) based on the observations of the random variable The method of moments was applied to this problem as early as 1894 by Pearson [124] for the case when Pa is a family of one-dimensional Gaussian distributions and m = 2. Classical statistical methods for finite mixtures (method of moments, method of maximum likelihood, Bayesian approaches), as well as the issues of model identifiability, were quite well studied in the 60-70th years of the last century, see the survey by Gupta and Huang [81]. Nowadays, the most popular approach for the parameter estimation in finite mixture models is the EM algorithm, which is a method for solving an optimisation problem that arises when calculating the maximum likelihood estimate, see McLachlan and Krishnan [104].

The study of the probabilistic and statistical properties of the mixture model is a popular area of stochastic analysis. Let us mention that over the past 15 years, 6 PhD theses on the topics in this field have been defended at the Moscow State University (Gorshenin [163], Koksharov [164],

Korchagin [165], Krylov [166], Nazarov [167], Savinov [168]). The relevance of this topic is also confirmed by a large number of publications dealing with the applications of mixture models in finance, astronomy, image analysis, genomics, and in many other areas. Many recent results are collected by Fruhwirth-Schnatter et al. [68]. In fact, mixture models can be applied to the analysis of any data for which statistical clustering and classification tasks are relevant, see, e.g., McNicholas [105].

At the same time, the existing methods of statistical analysis are not applicable to some new problems that have arisen in the literature recently. Let us list the main mathematical problems considered in this thesis.

Direction of research and structure of this thesis.

Chapter 1. Semiparametric estimation in mixture models. A normal variance-mean mixture is defined as

where p e R, stands for the density of the normal distribution with mean ps and variance

s, and G is a mixing distribution on R+. As can be easily seen, a random variable X has the distribution (3) if and only if

The variance-mean mixture models play an important role in statistical modelling and have many applications. In particular, such mixtures appear as limit distributions in the asymptotic theory for dependent random variables and they are also useful for modelling data stemming from heavy-tailed and skewed distributions, see, e.g. Barndorff-Nielsen, Kent and S0rensen [15], BarndorffNielsen [13], Bingham and Kiesel [38], Bingham, Kiesel and Schmidt [39]. If G is the generalized inverse Gaussian distribution, then the normal variance-mean mixture distribution coincides with the so-called generalized hyperbolic distribution. The latter distribution has an important property that the logarithm of its density function is a smooth unimodal curve approaching linear asymptotes. This type of distributions was used to model the sizes of the particles of sand (Bagnold [9], Barndorff-Nielsen and Christensen [14]), and the diamond sizes in marine deposits

(x — ps)2

| G(ds),

(3)

2s

X = pZ + TZn, where n (0,1), Z ~ G.

(4)

in South West Africa (Barndorff-Nielsen [12]).

In Section 1.1, we study the problem of statistical inference for the mixing distribution G and the parameter p based on a sample Xi,..., Xn from the distribution with density p(-; p, G). Our methodology relies on a new idea, namely, to use the superposition of Mellin and Laplace transforms. Interestingly enough, the presented method can be also adapted for the statistical inference in more complex models, i.e., the continuous-time moving average Levy processes, which we consider in Section 1.2.

Chapter 2. Stochastic time change. In the one-dimensional case (d = 1), the concept of stochastic time change is that for some random process Lt, t > 0, the deterministic time t is replaced by a non-decreasing non-negative random process T(s), s > 0, which plays the role of random time. That is,

X(s) = LT(s), s > 0, (5)

where it is often assumed that the processes L and T are independent.

Theoretically it is known that even in the case of the Brownian motion L, the resulting class of time-changed processes is rather large and basically coincides with the class of all semimartingales (Monroe [112]). Nevertheless, the practical application of this fact for financial modelling meets two major problems: first, the change of time T can be highly intricate - for instance, if X has discontinuous trajectories (see Barndorff-Neilsen and Shiryaev [18]); second, the dependence structure between L and T can also be quite sophisticated.

The change of time can be motivated by the fact that some economic effects (e.g., nervousness of the market which is indicated by higher volatility) can be better expressed in terms of "business" time T, which may run faster than the physical one in some periods - e.g., when the number of transactions is high, see Clark [47], Ane and Geman [6], Veraart and Winkel [152]. For instance, Ane and Geman [6] show that the stock prices can be modelled by a time-changed process with L equal to the Brownian motion with drift, and T equal to the cumulative number of trades till time s. This choice of T is intuitively correct, and leads to a good understanding of the model, see Tauchen and Pitts [146] and Andersen [4].

Note that this class of models is closely related to mixtures. For instance, if the time change

process T is itself a Levy process (subordinator) and the processes T, L are independent, then the resulting process X is also a Levy process, and for any s > 0, the distribution of the process X is a mixture of probability distributions:

Suppose now that a time-changed process X is observable on the equidistant time grid

Since the model is based on two stochastic processes, L and T, it would be natural to ask whether the important characteristics of these processes can be recovered from the available observations. For instance, one of these characteristics is the Blumenthal-Getoor index, which indicates the jump activity of the process, see [134].

A significant technical difficulty lies in the type of the data - while the results for the high-frequency data (that is, for the case when the time step between two consecutive observations tends to zero) were obtained in the papers by Ait-Sahalia and Jacod (see [2], [3]), the case of low-frequency data (fixed time interval, but an infinite horizon), which is considered in this thesis, was not studied earlier, see [28]. This fact motivates our study, which we present in Section 2.1.

In Section 2.2 we turn towards a multivariate extension of the time-change process and show the performance of the model for the reproducing the dependence between stock returns.

Chapter 3. Construction of honest confidence sets. Assume that we are given a sample X1,...Xn drawn from some absolutely continuous distribution, and we wish to estimate the density p of this distribution. More precisely, for any a e (0,1), we aim to construct (1 — a)-confidence sets Cn(x) for p that are honest to a given class F of density functions in the sense

where en ^ 0 as n ^ Very often, the set Cn(x) is constructed using an estimate pn of p. In this respect, the confidence bands can be used for showing the quality of the estimates: the narrower is the confidence band, the better is the estimate.

0 < A < ... <nA with some n e N and A > 0.

(6)

Typically, the construction of confidence bands is based on the so-called SBR-type (Smirnov - Bickel - Rosenblatt) limit theorems, which yield the asymptotic behaviour of the maximal deviation of the considered estimate pn in terms of the quantity

D[pn] = sup-==-.

ueR vPvU)

The SBR-type theorems state that

sup

peF

P <! D[pn] < — + bn\ - e-

an

->■ 0,

as n —> oo

(7)

for some deterministic sequences an and bn tending to infinity as n ^ see Smirnov [141], Bickel and Rosenblatt [36], Gine, Koltchinskii and Sakhanenko [73], Gine and Nickl [74], Bull [44]. Despite the long history of studying the SBR-type theorems, facts of this type are known only for kernel density estimates and projection estimates on some types of basis (the Haar wavelets and the Battle-Lemarie wavelets). This issue is confirmed by a number of recent articles on this topic published in the the Annals of Statistics, see Gine and Nickl [74], Chernozhukov et al. [46]. In Section 3.1, we consider the construction of the honest confidence intervals based on the projection estimates, when using the basis of Legendre polynomials. The solution of this problem relies on some special asymptotic properties of non-stationary Gaussian processes, which are well studied for the special case of the nonstationarity - the so-called cyclostationarity, introduced by Konstant and Piterbarg [97] - but were not previously known in more general cases. Results of this kind are of particular interest for the analysis of mixtures of distributions, since classical methods for the construction of confidence sets in this case (for example, the method based on the kernel density estimates) lead to inadequate results.

In Section 3.2 we turn towards a similar, but more complicated statistical problem, namely, the construction of confidence bands for the densities of the Levy measure.

Chapter 4. Limit theorems and phase transitions in mixture models. The general theory of the limit distributions for the sums and for the maxima of random variables is well-described in brilliant books by Petrov [127] and Embrechts, Kluppelberg and Mikosch [60]. Nevertheless, the analysis of the limiting distribution in particular model can be rather tricky. Note, for example, that the limit laws for the random energy model (REM) introduced by

x

Derrida [56], [57] in the beginning of 80-th, were fully described only 20 years later in the papers by Bovier, Kurkova and Love [42], and Ben Arous, Bogachev and Molchanov [31]. It would be a worth mentioning that the probabilistic analysis of the REM model appears also in the context of the parabolic Anderson problem, since it leads to the study of the same random exponential sum. Asymptotic behaviour of this sum, as well as closely related concepts of intermittency and localisation, were studied in the papers by Molchanov and his co-authors [32], [69], [70], [71]. In this thesis, we aim to describe the limit laws for a new model of the REM type with a mixture distribution of energy levels. Models of this type are also motivated by the parabolic Anderson problem with the potential having a mixture distribution.

Section 4 deals with the asymptotic analysis of the alloy-type REM. It turns out that the limit behaviour of the corresponding random exponential sum drastically depends on the relation between parameters, but, as we show below, the description of the phase transitions in this model is possible.

Chapter 5. Analysis of the "purity" of a distribution. Due to the Lebesque theorem, any probability measure P can be represented as a sum of three measures

P(-) = a1Pd(0 + a2Pac(-) + asPsc(-),

where a1 + a2 + a3 = 1, a» > 0 Vi, and the measures Pd, Pac, Psc are, respectively, the measures of discrete, absolute continuous and singular distributions. For the solution of statistical estimation problems, it is important to know whether some of parameters a1,a2,a3 are equal to zero - for example, if a3 = 0 (there is no singular component), then the estimation methods are significantly simplified. The Jessen - Wintner theorem yields that the sum of a.s. convergent random sums has pure type, that is, two out of three numbers a1, a2, a3 are equal to 0 (see Proposition 5.1 below). However, for specific models, the determination of the type can be a rather challenging task. A classical example is the Erdos problem dealing with the Bernoulli convolution, which is defined as the series Z = ±Pn, where the signs are chosen randomly each with probabilities 1/2,

and p e (0,1), see Erdos [61], [62]. The term "Bernoulli convolution" is motivated by the fact that Z is a convolution of an infinite number of measures of the type + 5pn)/2. The most well-known result in this field is the fact that Z has an absolutely continuous distribution for almost all p e (1/2,1). This fact was established by Solomyak [142].

In Chapter 5, we study the question of the type of the discrete Dickman-Goncharov distribution. This question is of great interest, since the number of applications of the Dickman-Goncharov distribution is growing: new applications have appeared in mathematics (random walks on solvable groups, random graph theory, and so on) and also in biology (models of growth and evolution of unicellular populations), finance (theory of extreme phenomena in finance and insurance), physics (the model of random energy levels), and other fields.

Theoretical and practical significance

The obtained theoretical results significantly expand the methodology of the statistical estimation for the mixture models. At the same time, Levy-based models (in particular, time-changed models) are widely used to describe the dynamics of stock returns, and therefore the developed methods can be used for the solution of financial problems related to trading on the stock exchange. For example, the method of estimating of the Blumenthal - Getoor index can be used to determine the degree of reliability of a financial asset, while methods of modelling multidimensional processes that well reproduce the dependence between financial assets, are used for testing the trading algorithms on the stock exchange.

The obtained results for the Levy processes - in particular, the methods used for estimating the Blumenthal-Getoor index and the methods for modelling the multidimensional processes (see Section 2) - were included by the author in the course "Modelling of jump-type processes in economics" at the Higher School of Economics.

Approbation of the established results

The main results of this thesis were presented at the following international conferences and seminars.

1. Bernoulli-IMS World Congress in Probability and Statistics (online conference, organized by Seoul National University, July 2021), talk "The Dickman-Goncharov distribution".

2. Extreme Value Analysis Conference - EVA2019 (Zagreb, Croatia, July 2019), talk "Extremes of Gaussian non-stationary processes and improved confidence bands for densities".

3. International Workshop on Applied Probability - IWAP2018 (Budapest, Hungary, Juny 2018), talk "Multivariate subordination of stable processes".

4. German Probability and Statistics Days - GPSD2018 (Freiburg, Germany, February 2018), talk "Multivariate subordination of stable processes".

5. Conference on Ambit Fields and Related Topics (Aarhus, Denmark, August 2017), talk "Low-frequency estimation for moving-average Levy processes".

6. Probability Seminar Essen (Essen, Germany, June 2017), talk "Low-frequency estimation for moving-average Levy processes".

7. Extreme Value Analysis Conference - EVA2017 (Delft, the Netherlands, June 2017), talk "Distribution of maximal deviation for Levy density estimators".

8. World Congress in Probability and Statistics (Toronto, Canada, July 2016), talk "Low-frequency estimation of continuous-time moving average Levy processes".

9. German Probability and Statistics Days - GPSD2016 (Bochum, Germany, March 2016), talk "Statistical inference for fractional Levy processes and related models".

10. Conference on Stochastic Processes and their Applications - SPA2015 (Oxford, UK, July 2015), talk "Generalized Ornstein-Uhlenbeck process: Mellin transform of the invariant distribution and statistical inference".

11. European Meeting of Statisticians - EMS2015 (Amsterdam, the Netherlands, July 2015), talk "Statistical inference for exponential functionals of Levy processes".

12. Statistical Inference for Levy Processes (Leiden, the Netherlands, September 2014), talk "Maximal deviation distribution for projection estimates of Levy densities".

Moreover, the author has given 10 talks at the seminar "Stochastic analysis and its applications in economics" at the Faculty of mathematics of the Higher School of Economics (seminar is coordinated by Prof. V. Konakov and Prof. A. Kolesnikov) and delivered 12 talks at the workshops organised by the International laboratory of stochastic analysis and its applications of the Higher School of Economics (https://lsa.hse.ru/).

List of author's published papers on the topic of the thesis

The list consists of 12 papers, among them 6 papers are published in journals with quartiles Q1/Q2 in Web of Science ([2], [3], [4], [6], [9], [12]), and 11 papers - in journals with quartiles Q1/Q2 in Scopus. Of the published papers, three are single-authored, and two are with co-authors - graduate students at HSE.

[1] Belomestny, D. and Panov, V. Semiparametric estimation in the normal variance-mean mixture model. Statistics, 52(3):571-589, 2018.

https://www.tandfonline.com/doi/abs/10.1080/02331888.2018.1425865?journalCode=gsta20

[2] Belomestny, D., Panov, V. and Woerner, J. Low-frequency estimation of continuous-time moving average Levy processes. Bernoulli, 25(2):902-931, 2019.

https://projecteuclid.org/journals/bernoulli/volume-25/issue-2/Low-frequency-estimation -of-continuous-time-moving-average-L%C3%A9vy-processes/10.3150/17-BEJ1008.short

[3] Belomestny D., Panov V. Abelian theorems for stochastic volatility models with application to the estimation of jump activity. Stochastic Processes and their Applications. 123 (1): 15-44, 2013.

https://www.sciencedirect.com/science/article/pii/S0304414912001925

[4] Belomestny, D. and Panov, V. Estimation of the activity of jumps in time-changed Levy models. Electronic Journal of Statistics, 7:2970-3003, 2013.

https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-7/issue-none...

[5] Panov, V. Series representations for multivariate time-changed Levy models. Methodology and Computing in Applied Probability, 19(1):97-119, 2017

https://link.springer.com/article/10.1007/s11009-015-9461-8

[6] Panov, V. and Samarin, E. Multivariate asset-pricing model based on subordinated stable processes. Applied Stochastic Models in Business and Industry, 35(4):1060-1076, 2019.

https://onlinelibrary.wiley.com/doi/10.1002/asmb.2446

[7] Panov, V. Some properties of the one-dimensional subordinated stable model. Statistics and Probability Letters, 146:80-84, 2019.

https://www.sciencedirect.com/science/article/abs/pii/S0167715218303493

[8] Konakov, V., Panov, V. and Piterbarg, V. Extremes of a class of non-stationary Gaussian processes and maximal deviation of projection density estimates. Extremes, 24(3):617-651, 2021.

https://link.springer.com/article/10.1007/s10687-020-00402-2

[9] Konakov, V. and Panov, V. Sup-norm convergence rates for Levy density estimation. Extremes, 19(3):371-403, 2016.

https://link.springer.com/article/10.1007/s10687-020-00402-2

[10] Molchanov, S. and Panov, V. Limit theorems for the alloy-type random energy model. Stochastics, 91(5):754-772, 2019.

https://www.tandfonline.com/doi/full/10.1080/17442508.2018.1545841

[11] Panov, V. Limit theorems for sums of random variables with mixture distribution.

Statistics and Probability Letters, 129:379 - 386, 2017.

https://www.sciencedirect.com/science/article/abs/pii/S0167715217302213

[12] Molchanov, S. and Panov, V. The Dickman-Goncharov distribution. Russian Mathematical Surveys, 75(6):1089-1132, 2020.

https://iopscience.iop.org/article/10.1070/RM9976/meta?casa_token=2HoQHwUbDx4AAAAA: bMJWR2yeYoRJoZaZXdjCXVmbRTA9SryGJxF7ISQS-TP6vK2mojbjSaXms3rx0fT5tLV9l5ZYMCQsjyA

Заключение диссертации по теме «Теория вероятностей и математическая статистика», Панов Владимир Александрович

Conclusion

This section lists the main results of the study and discusses the novelty of the obtained results.

(1) In Chapter 1, we present the algorithm for semiparametric estimation of unknown parameters and an unknown mixing distribution for variance-mean Gaussian mixtures. It is shown that the convergence rates of the proposed estimates are determined by the properties of the Mellin transform of the density of the mixing distribution. The novelty of the presented method consists in the employing of the properties of the superposition of the Mellin and Laplace transforms. The proposed approach is a significant contribution to this topic, because, unlike the previously known estimation methods, it is not based on the solution of difficult optimisation problems [99].

The presented method was adapted for continuous-time moving average Levy processes. An algorithm was developed for the statistical estimation of the Levy measure and other parameters of the Levy process from the observations of the model, which is an integral over this process. The algorithm is new and can be applied to a wide class of models known as the ambit fields, see the survey by Podolskij [131]. The properties of exponential mixing were proved for processes from the considered class, and upper bounds for the constructed estimates were derived.

(2) Chapter 2 deals with statistical inference for the time-changed Levy processes. A method was developed for the estimation of the Blumenthal-Getoor indices of the Levy processes used in the construction of this model, based on the observations of the model itself. The novelty of the method lies in the use of the asymptotic properties of the characteristic function of the process for large values of the argument. The rates of convergence of the proposed estimates are obtained, and it is proved that these rates are optimal in the minimax sense.

We also present a new approach for the joint description of the returns of several stocks, based on the multidimensional time-changed Levy processes. For the subordinated stable processes, we developed a method for representing the processes from the considered class in the form of infinite series. It is shown that this method can be effectively used to model the returns of stock prices, and the constructed two-dimensional models reproduce well the correlations between stock returns.

(3) Chapter 3 focuses on the construction of honest confidence sets for an (unknown) distribution density based on projection estimates. It is shown that this problem is equivalent to the analysis of the asymptotic behaviour of non-stationary Gaussian processes of a certain type. From the technical side, the main difficulty is to describe the second term of the asymptotical behaviour of the maximum, and to show that the convergence rates are polynomial. The efficiency of the obtained results is illustrated by the examples of the mixture models.

We present also similar results for the density estimates of the Levy measure. The main result of this part is the presentation of the sequences of accompanying laws that approximate the distribution of the maximum deviation of the estimates for the density of the Levy measure with errors of polynomial order. It is shown that the rates of convergence given in previous papers on this topic [66] are in fact of logarithmic order.

(4) In Chapter 4, we explain the connection between the Anderson parabolic problem and the model of random energy levels. For the case, when the energy levels are modelled by the Gaussian mixtures, we study the asymptotic behaviour of the system depending on the parameters and describe the phase transitions. Our results are a contribution to the theory of random energy models, since similar facts were previously known only for the Gaussian case, see [31].

(5) Chapter 5 contains some new results on the type of the generalised Dickman-Goncharov distribution. For the case when the geometric distribution is used in the construction of this model, the relation to the Erdos problem for the Bernoulli convolution is described. It is shown that in this case, the generalised Dickmann - Goncharov distribution is absolutely continuous for almost all values of the parameter from a certain interval.

Список литературы диссертационного исследования доктор наук Панов Владимир Александрович, 2022 год

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