It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a discrete time branching process with immigration is not asymptotically normal. If the ...offspring variance tends to zero, it is normal with normalization factor
n
2
/
3
. We study a situation of its asymptotic normality in the case of non-degenerate offspring distribution for the process with time-dependent immigration, whose mean and variance vary regularly with non-negative exponents
α
and
β
, respectively. We prove that if
β
<
1
+
2
α
, the CLSE is asymptotically normal with two different normalization factors and if
β
>
1
+
2
α
, its limit distribution is not normal but can be expressed in terms of the distribution of certain functionals of the time-changed Wiener process. When
β
=
1
+
2
α
the limit distribution depends on the behavior of the slowly varying parts of the mean and variance.
We consider a sequence of discrete time branching processes with generation-dependent immigration, where the offspring mean tends to its critical value 1. Using a martingale approach, we prove ...functional limit theorems for suitable normalized fluctuations of the process around its mean when the mean number of immigrating individuals tends to infinity. The limiting processes are deterministically time-changed Wiener processes with three different non-linear time change functions, depending on the behavior of the mean and the variance of the number of immigrants. For the normalized sequence of processes we obtain a deterministic approximation. Consequences related to the maxima and the total progeny of the process will be discussed.
We consider a sequence of discrete time nearly critical branching processes with time-dependent immigration. Using a martingale approach, we prove that when the immigration mean tends to infinity ...depending on the time of immigration, the suitable normalized sequence can be approximated in Skorokhod metric by a deterministic process. Consequences related to the maxima and the total progeny of the process will be discussed.
It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a branching process with stationary immigration is not asymptotically normal. If the ...offspring variance tends to zero, it is normal with norming factor n
3/2
. We study the process with a non degenerate offspring distribution and time-dependent immigration, whose mean and variance vary regularly with non negative exponents, α and β, respectively. We propose new weighted CLSE using more flexible weights and prove that if β < 1 + 2α, it is asymptotically normal with two different norming factors and if β > 1 + 2α, its limiting distribution is not normal but can be expressed in terms of certain functionals of the time-changed Wiener process. When β = 1 + 2α, the limiting distribution depends on the behavior of the slowly varying parts of the mean and variance. Conditions guaranteeing the strong consistency of the proposed estimator will be derived.
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A mobile VLBI station prototype based on the mechanical part of the TESLA satellite communications system antenna has been created at the Svetloe observatory of the IAA. The main solutions are ...presented for transforming the satellite communications system antenna into the antenna of the VLBI radio telescope, which permanently change its spatial orientation in accordance with the principles of VLBI observations. The processes of installation and preparation for VLBI observations are described and the results of measurement of the main characteristics are reported.
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.
We consider a model of age-dependent branching stochastic process that takes into account the incubation period of the life of individuals. We demonstrate that such processes may be treated as a ...two-type branching process with a periodic mean matrix. In the case when the Malthusian parameter does not exist study of the process requires additional restrictions on the life and incubation time distributions which define so called subexponential family (Athreya, K. 1972. Branching Processes, Springer, New York). We obtain certain new properties of subexponential distributions, in particular, describe a subclass, which is closed with respect to convolution. Using these results we derive asymptotic behavior of the first and second moments and of the probability of nonextinction. We also prove a limit theorem for the process conditioned on nonextinction.
In this paper, we investigate the asymptotic behavior of a triangular array of branching processes with non-stationary immigration. In the nearly critical case, we prove weak convergence of properly ...normalized and scaled branching processes with immigration to a deterministic function when the immigration process is generated by dependent random variables.
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 non-negative exponents α and β, respectively. We prove that if α(n)→∞ and β (n)=o(nα
2
(n)) as n→∞, then the standard parametric bootstrap procedure leads to a valid approximation for the distribution of the conditional least-squares estimator in the sense of convergence in probability. Monte Carlo and bootstrap simulations for the process confirm the theoretical findings in the paper and highlight the validity and utility of the bootstrap as it mimics the Monte Carlo pivots even when generation size is small.
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’s 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.
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