Asymptotic inference for non-supercritical partially observed branching processes

© 2017 Elsevier B.V. To estimate the offspring mean of a branching process one needs observed population sizes up to some generation. However, in applications very often not all individuals existing in the population are observed. Therefore the question about possibility of estimating the population mean based on partial observations is of interest. In existing literature this problem has been studied assuming that the process never becomes extinct, which is possible only in supercritical case. In the paper we consider it in subcritical and critical processes with a large number of initial ancestors. We prove that the Harris type ratio estimator remains consistent, if we have observations of a binomially distributed subsets of the population. To obtain the asymptotic normality of the estimator we modify the estimator using a “skipping” method. The proofs use the law of large numbers and the central limit theorem for random sums in the case when the number of terms and the terms in the sum are not independent

Conditional least squares estimators for the offspring mean in a subcritical branching process with immigration

Consider a Bienayme-Galton-Watson process with generation-dependent immigration, whose mean and variance vary regularly with non negative exponents and , respectively. We study the estimation problem of the offspring mean based on an observation of population sizes. We show that if \u3c2, the conditional least squares estimator (CLSE) is strongly consistent. Conditions which are sufficient for the CLSE to be asymptotically normal will also be derived. The rate of convergence is faster than n 1/2, which is not the case in the process with stationary immigration. © 2012 Copyright Taylor and Francis Group, LLC

Random sums and partially observed branching processes

© Nova Science Publishers, Inc. In the paper we consider a random sum of a double array of independent random variables. We provide limit theorems for the joint distribution of the random sum and the number of summands in various assumptions on the asymptotic behavior of the number of terms. Further, we apply these limit theorems in study of the following modification of a discrete-time branching process. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals change their offspring distributions. Using our limit theorems for the random sum we derive asymptotic distributions for the vector of inspected and partially observed branching processes in cases when the inspected process is subcritical, critical and supercritical

Estimation of the offspring mean in a supercritical branching process with non-stationary immigration

In this paper, we consider the conditional least squares estimator (CLSE) of the offspring mean of a branching process with non-stationary immigration based on the observation of population sizes. In the supercritical case, assuming that the immigration variables follow known distributions, conditions guaranteeing the strong consistency of the proposed estimator will be derived. The asymptotic normality of the estimator will also be proved. The proofs are based on direct probabilistic arguments, unlike the previous papers, where functional limit theorems for the process were used. © 2011 Elsevier B.V

Estimation of the mean in partially observed branching processes with general immigration

© 2018, Springer Science+Business Media B.V., part of Springer Nature. In the paper we investigate asymptotic properties of the branching process with non-stationary immigration which are sufficient for a natural estimator of the offspring mean based on partial observations to be strongly consistent and asymptotically normal. The estimator uses only a binomially distributed subset of the population of each generation. This approach allows us to obtain results without conditions on the criticality of the process which makes possible to develop a unified estimation procedure without knowledge of the range of the offspring mean. These results are to be contrasted with the existing literature related to i.i.d. immigration case where the asymptotic normality depends on the criticality of the process and are new for the fully observed processes as well. Examples of applications in the process with immigration with regularly varying mean and variance and subcritical processes with i.i.d. immigration are also considered

Bootstrap for Critical Branching Process with Non-Stationary Immigration

2000 Mathematics Subject Classification: Primary 60J80, Secondary 62F12, 60G99.In the critical branching process with a stationary immigration the standard parametric bootstrap for an estimator of the offspring mean is invalid. We consider the process with non-stationary immigration, whose mean and variance α(n) and β(n) are finite for each n ≥ 1 and are regularly varying sequences with nonnegative exponents α and β, respectively. It turns out that if α(n) → ∞ and β(n) = o(nα2(n)) as n → ∞, then the standard parametric bootstrap procedure leads to a valid approximation for the distribution of the conditional least squares estimator. We state a theorem which justifies the validity of the bootstrap. By Monte-Carlo and bootstrap simulations for the process we confirm the theoretical findings. The simulation study highlights the validity and utility of the bootstrap in this model as it mimics the Monte-Carlo pivots even when generation size is small

Estimation of the offspring mean in a branching process with non stationary immigration

© 2016, © Taylor & Francis Group, LLC. In the paper, we consider a natural estimator of the offspring mean of a branching process with non stationary immigration based on observation of population sizes and number of immigrating individuals to each generation. We demonstrate that using a central limit theorem for multiple sums of dependent random variables it is possible to derive asymptotic distributions for the estimator without prior knowledge about the behavior (criticality) of the reproduction process. Before the three cases of criticality have been considered separately. Assuming that the immigration mean and variance vary regularly, conditions guaranteeing the strong consistency of the proposed estimator is also derived

Shrinkage Estimation Using Ranked Set Samples

The purpose of this article is two-fold. First, we consider the ranked set sampling (RSS) estimation and testing hypothesis for the parameter of interest (population mean). Then, we suggest some alternative estimation strategies for the mean parameter based on shrinkage and pretest principles. Generally speaking, the shrinkage and pretest methods use the non-sample information (NSI) regarding that parameter of interest. In practice, NSI is readily available in the form of a realistic conjecture based on the experimenter\u27s knowledge and experience with the problem under consideration. It is advantageous to use NSI in the estimation process to construct improved estimation for the parameter of interest. In this contribution, the large sample properties of the suggested estimators will be assessed, both analytically and numerically. More importantly, a Monte Carlo simulation is conducted to investigate the relative performance of the estimators for moderate and large samples. For illustrative purposes, the proposed methodology is applied to a published data set. © 2011 King Fahd University of Petroleum and Minerals

Avifauna of lake systems in Syr Darya river delta (Cartma lake)

© 2016, International Journal of Pharmacy and Technology. All rights reserved.The results of avifauna study in Cartma lake, as the part of the restored ornithocomplex of Aral Sea region. Cartma lake makes the part of the left bank lake system of Syrdarya river delta, which was the bay of the Aral Sea until 1967. Due to the catastrophic fall of the water level in Syr Darya River and the subsequent separation of Aral Sea into small pieces, Cartma lake ceased to perform the function of direct communication between the Aral Sea and the river Syrdarya. The positive process which takes place during the past decade and associated with the water-bearing role of Syrdarya for the maintaining of coastal ecosystems contributes to the restoration of the Aral Sea region unique objects. In 2012 the lake systems of Syrdarya delta are included in the list of wetlands of global significance lands protected by the international Ramsar Convention. The studies were conducted to examine the avifauna of Cartma lake. The task was to determine the status of bird species existence in the area of coastal strip recovery. The classification of aquatic and semi-aquatic habitats of the lake Cartma is proposed. The lake is characterized by a relatively small set of birds and low population density of waterbirds. Wetland birds are the dominants according to habitat density. Three species listed in the Red Book of the Republic of Kazakhstan were noted. The delta of Syr Darya and Northern Aral Sea is used as a stopping point for migrating birds, taking into account the geographical location of a region combining the wintering periods in South Asia and Africa with Siberia. The details are new ones for an object under study