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Vladimir Pozdnyakov

Publications and source records attributed to Vladimir Pozdnyakov.

16 recordsLinked to original sources
arXiv2026

A Time-Varying Branching Process Approach to Model Self-Renewing Cells

Stem cells, through their ability to produce daughter stem cells and differentiate into specialized cells, are essential in the growth, maintenance, and repair of biological tissues. Understanding the dynamics of cell populations in the proliferation process not only uncovers proliferative properties of stem cells, but also offers insight into tissue development under both normal conditions and pathological disruption. In this paper, we develop a continuous time branching process model with time-dependent offspring distribution to characterize stem cell proliferation process. We derive analytical expressions for mean, variance, and autocovariance of the stem cell counts, and develop likelihood-based inference procedures to estimate model parameters. Particularly, we construct a forward algorithm likelihood to handle situations when some cell types cannot be directly observed. Simulation results demonstrate that our estimation method recovers the time-dependent division probabilities with good accuracy.

stat.AP↗
arXiv2025

On a Variation of Gambler's Ruin Problem

Assume that letters (from a finite alphabet) in a text form a Markov chain. We track two distinct words, $U$ and $D$. A gambler gains 1 point for each occurrence of $U$ (including overlapping occurrences) and loses 1 point for each occurrence of $D$ (also including overlapping occurrences). We determine the probability of gaining $A$ points before losing $B$ points, where $A$ and $B$ are integers. Additionally, we find the expected waiting time until one of the two events -- gaining $A$ points or losing $B$ points -- occurs.

math.PR↗
arXiv2025

Martingale Approach to Gambler's Ruin Problem for Correlated Random Walks

The gambler's ruin problem for correlated random walks (CRW), both with and without delays, is addressed using the Optional Stopping Theorem for martingales. We derive closed-form expressions for the ruin probabilities and the expected game duration for CRW with increments $\{1,-1\}$ and for symmetric CRW with increments $\{1,0,-1\}$ (CRW with delays). Additionally, a martingale technique is developed for general CRW with delays. The gambler's ruin probability for a game involving bets on two arbitrary patterns is also examined.

math.PR↗
arXiv2025

High Dimensional Space Oddity

In his 1996 paper, Talagrand highlighted that the Law of Large Numbers (LLN) for independent random variables can be viewed as a geometric property of multidimensional product spaces. This phenomenon is known as the concentration of measure. To illustrate this profound connection between geometry and probability theory, we consider a seemingly intractable geometric problem in multidimensional Euclidean space and solve it using standard probabilistic tools such as the LLN and the Central Limit Theorem (CLT).

math.PR↗
arXiv2024

Recurrent Events Modeling Based on a Reflected Brownian Motion with Application to Hypoglycemia

Patients with type 2 diabetes need to closely monitor blood sugar levels as their routine diabetes self-management. Although many treatment agents aim to tightly control blood sugar, hypoglycemia often stands as an adverse event. In practice, patients can observe hypoglycemic events more easily than hyperglycemic events due to the perception of neurogenic symptoms. We propose to model each patient's observed hypoglycemic event as a lower-boundary crossing event for a reflected Brownian motion with an upper reflection barrier. The lower-boundary is set by clinical standards. To capture patient heterogeneity and within-patient dependence, covariates and a patient level frailty are incorporated into the volatility and the upper reflection barrier. This framework provides quantification for the underlying glucose level variability, patients heterogeneity, and risk factors' impact on glucose. We make inferences based on a Bayesian framework using Markov chain Monte Carlo. Two model comparison criteria, the Deviance Information Criterion and the Logarithm of the Pseudo-Marginal Likelihood, are used for model selection. The methodology is validated in simulation studies. In analyzing a dataset from the diabetic patients in the DURABLE trial, our model provides adequate fit, generates data similar to the observed data, and offers insights that could be missed by other models.

stat.ME↗
arXiv2024

On the Convergence of Credit Risk in Current Consumer Automobile Loans

Loan seasoning and inefficient consumer interest rate refinance behavior are well-known for mortgages. Consumer automobile loans, which are collateralized loans on a rapidly depreciating asset, have attracted less attention, however. We derive a novel large-sample statistical hypothesis test suitable for loans sampled from asset-backed securities to populate a transition matrix between risk bands. We find all current risk bands eventually converge to a super-prime credit, despite remaining underwater. Economically, our results imply borrowers forwent \$1,153-\$2,327 in potential credit-based savings through delayed prepayment. We present an expected present value analysis to derive lender risk-adjusted profitability. Our results appear robust to COVID-19.

q-fin.ST↗
arXiv2023

Pricing Time-to-Event Contingent Cash Flows: A Discrete-Time Survival Analysis Approach

Prudent management of insurance investment portfolios requires competent asset pricing of fixed-income assets with time-to-event contingent cash flows, such as consumer asset-backed securities (ABS). Current market pricing techniques for these assets either rely on a non-random time-to-event model or may not utilize detailed asset-level data that is now available with most public transactions. We first establish a framework capable of yielding estimates of the time-to-event random variable from securitization data, which is discrete and often subject to left-truncation and right-censoring. We then show that the vector of discrete-time hazard rate estimators is asymptotically multivariate normal with independent components, which has not yet been done in the statistical literature in the case of both left-truncation and right-censoring. The time-to-event distribution estimates are then fed into our cash flow model, which is capable of calculating a formulaic price of a pool of time-to-event contingent cash flows vis-á-vis calculating an expected present value with respect to the estimated time-to-event distribution. In an application to a subset of 29,845 36-month leases from the Mercedes-Benz Auto Lease Trust 2017-A (MBALT 2017-A) bond, our pricing model yields estimates closer to the actual realized future cash flows than the non-random time-to-event model, especially as the fitting window increases. Finally, in certain settings, the asymptotic properties of the hazard rate estimators allow investors to assess the potential uncertainty of the price point estimates, which we illustrate for a subset of 493 24-month leases from MBALT 2017-A.

q-fin.RM↗
arXiv2022

Estimating a distribution function for discrete data subject to random truncation with an application to structured finance

Proper econometric analysis should be informed by data structure. Many forms of financial data are recorded in discrete-time and relate to products of a finite term. If the data comes from a financial trust, it will often be further subject to random left-truncation. While the literature for estimating a distribution function from left-truncated data is extensive, a thorough literature search reveals that the case of discrete data over a finite number of possible values has received little attention. A precise discrete framework and suitable sampling procedure for the Woodroofe-type estimator for discrete data over a finite number of possible values is therefore established. Subsequently, the resulting vector of hazard rate estimators is proved to be asymptotically normal with independent components. Asymptotic normality of the survival function estimator is then established. Sister results for the left-truncating random variable are also proved. Taken together, the resulting joint vector of hazard rate estimates for the lifetime and left-truncation random variables is proved to be the maximum likelihood estimate of the parameters of the conditional joint lifetime and left-truncation distribution given the lifetime has not been left-truncated. A hypothesis test for the shape of the distribution function based on our asymptotic results is derived. Such a test is useful to formally assess the plausibility of the stationarity assumption in length-biased sampling. The finite sample performance of the estimators is investigated in a simulation study. Applicability of the theoretical results in an econometric setting is demonstrated with a subset of data from the Mercedes-Benz 2017-A securitized bond.

math.ST↗
arXiv2020

Moving-Resting Process with Measurement Error in Animal Movement Modeling

Statistical modeling of animal movement is of critical importance. The continuous trajectory of an animal's movements is only observed at discrete, often irregularly spaced time points. Most existing models cannot handle the unequal sampling interval naturally and/or do not allow inactivity periods such as resting or sleeping. The recently proposed moving-resting (MR) model is a Brownian motion governed by a telegraph process, which allows periods of inactivity in one state of the telegraph process. The MR model shows promise in modeling the movements of predators with long inactive periods such as many felids, but the lack of accommodation of measurement errors seriously prohibits its application in practice. Here we incorporate measurement errors in the MR model and derive basic properties of the model. Inferences are based on a composite likelihood using the Markov property of the chain composed by every other observed increments. The performance of the method is validated in finite sample simulation studies. Application to the movement data of a mountain lion in Wyoming illustrates the utility of the method.

stat.ME↗
arXiv2019

On Occupation Time for On-Off Processes with Multiple Off-States

The need to model a Markov renewal on-off process with multiple off-states arise in many applications such as economics, physics, and engineering. Characterization of the occupation time of one specific off-state marginally or two off-states jointly is crucial to understanding such processes. We derive the exact marginal and joint distributions of the off-state occupation times. The theoretical results are confirmed numerically in a simulation study. A special case when all holding times have Levy distribution is considered for the possibility of simplification of the formulas.

math.PR↗
arXiv2018

On estimation for Brownian motion governed by telegraph process with multiple off states

Brownian motion whose infinitesimal variance changes according to a three-state continuous time Markov Chain is studied. This Markov Chain can be viewed as a telegraph process with one on state and two off states. We first derive the distribution of occupation time of the on state. Then the result is used to develop a likelihood estimation procedure when the stochastic process at hand is observed at discrete, possibly irregularly spaced time points. The likelihood function is evaluated with the forward algorithm in the general framework of hidden Markov models. The analytic results are confirmed with simulation studies. The estimation procedure is applied to analyze the position data from a mountain lion.

stat.ME↗
arXiv2013

Can Tests for Jumps be Viewed as Tests for Clusters?

We investigate the utility in employing asymptotic results related to a clustering criterion to the problem of testing for the presence of jumps in financial models. We consider the Jump Diffusion model for option pricing and demonstrate how the testing problem can be reduced to the problem of testing for the presence of clusters in the increments data. The overarching premise behind the proposed approach is in the isolation of the increments with considerably larger mean pertaining to the jumps from the ones which arise from the diffusion component. Empirical verification is provided via simulations and the test is applied to financial datasets.

math.ST↗
arXiv2013

Asymptotics of a Clustering Criterion for Smooth Distributions

We develop a clustering framework for observations from a population with a smooth probability distribution function and derive its asymptotic properties. A clustering criterion based on a linear combination of order statistics is proposed. The asymptotic behavior of the point at which the observations are split into two clusters is examined. The results obtained can then be utilized to construct an interval estimate of the point which splits the data and develop tests for bimodality and presence of clusters.

math.ST↗
arXiv2013

Asymptotics of the Empirical Cross-over Function

We consider a combination of heavily trimmed sums and sample quantiles which arises when examining properties of clustering criteria and prove limit theorems. The object of interest, which we call the Empirical Cross-over Function, is an L-statistic whose weights do not comply with the requisite regularity conditions for usage of ex- isting limit results. The law of large numbers, CLT and a functional CLT are proven.

math.ST↗
arXiv2012

A Systematic Martingale Construction with Applications to Permutation Inequalities

We illustrate a process that constructs martingales from raw material that arises naturally from the theory of sampling without replacement.The usefulness of the new martingales is illustrated by the development of maximal inequalities for permuted sequences of real numbers. Some of these inequalities are new and some are variations of classical inequalities like those introduced by A. Garsia in the study of rearrangement of orthogonal series.

math.PR↗