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.
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.
Measurements of the flux densities of the supernova remnant (SNR) G74.9+1.2 (CTB 87) at frequencies of 4840 and 8450 MHz were carried out with the RT-32 radio telescope of the Svetloye observatory of ...the Institute of Applied Astronomy, Russian Academy of Sciences (IAA RAS), in 2018–2019. The data contain signs of the presence of a source of a variable component in the radio emission on a time scale of a month or more. The flux densities of G74.9+1.2 over the time interval 1959.7–2010 are determined from published data, which allow the intensity of G74.9+1.2 to be compared with standard sources. All the data are presented in a single system based on the exact scale of “artificial moon” (AM) fluxes. A refined spectrum of SNR G74.9+1.2 was obtained. The totality of available data is approximated by two power-law sections with different spectral indices:
at frequencies
and
at
. The projections of two power-law sections intersect at a frequency
MHz. The break in the radio spectrum of the source, considering its age (more than 4000 years), could form as a result of synchrotron losses. The increase in the steepness of the spectrum close to 0.5 above the frequency
is an argument in favor of such an assumption. The totality of data obtained during measurements on the RT-32 and on the basis of published works allows us to state that the variable component in the G74.9+1.2 radio emission on all time scales is much less pronounced compared to younger PWNs. As a possible mechanism for the observed variability, a reconnection of the magnetic field lines in the pulsar magnetosphere is proposed.
Consider a Bienayme-Galton-Watson process with generation-dependent immigration, whose mean and variance vary regularly with non negative exponents α and β, respectively. We study the estimation ...problem of the offspring mean based on an observation of population sizes. We show that if β <2α, 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.
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.
The spectrum of G11.2-0.3 has been refined by bringing the published intensity measurements to the “artificial moon” flux scale, and the dynamics of its changes on different time scales from 0.4 to ...more than ~50 years has been investigated. An increase in the fluxes of radio emission of G11.2-0.3 for ≥30 years at 3 cm
cm with a frequency dependence was found: the average rate of changes decreases proportionally to
, and at frequencies
GHz, the increase gave way to a decrease. Measurements with the RT-32 radio telescope of the Svetloe observatory (IAA RAS) in 2013–2019 showed a decrease in fluxes of G11.2-0.3 against the background of rapid nonstationary changes with an average rate of (
) %/year at a wavelength
cm and (
%/year at
cm. The stages of growth and decline of fluxes are separated by an epoch
. The spectrum of G11.2-0.3 is the spectra sum of the shell and the plerion, with each of its parameters determined by the method developed for the 1972.5 epoch. The values of the spectral indices α1 of the shell and α2 of PWN are obtained:
and
. The dynamics of radio emission from the remnant reflects the scenario of interaction between the shock wave and CSM. Possible reasons for evolutionary and non-stationary changes are discussed.
We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, ...we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is where α(n) denotes the mean number of immigrating individuals in the nth generation.
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.
In applications of branching processes, usually it is hard to obtain samples of a large size. Therefore, a bootstrap procedure allowing inference based on a small sample size is very useful. ...Unfortunately, in the critical branching process with stationary immigration the standard parametric bootstrap is invalid. In this paper, we consider a process with non-stationary immigration, whose mean and variance vary regularly with nonnegative exponents
α
and
β
, respectively. We prove that
1
+
2
α
is the threshold for the validity of the bootstrap in this model. If
β
<
1
+
2
α
, the standard bootstrap is valid and if
β
>
1
+
2
α
it is invalid. In the case
β
=
1
+
2
α
, the validity of the bootstrap depends on the slowly varying parts of the immigration mean and variance. These results allow us to develop statistical inferences about the parameters of the process in its early stages.
It is known that conditional least squares estimator (CLSE) of the offspring mean for the process with a stationary immigration is not asymptotically normal. In the paper, we demonstrate that for the ...process with non-stationary immigration it may have a normal limit distribution. Considering a discrete time branching process
Z
(
n
) with time-dependent immigration, whose mean and variance vary regularly with nonnegative exponents
α
and
β
, respectively, we show that 1+2
α
is the threshold for asymptotic normality of the estimator. It will be proved that if
β
<1+2
α
, the estimator is asymptotically normal with two different normalizing factors, and if
β
>1+2
α
its limiting distribution is not normal, but can be expressed in terms of certain functionals of the time-changed Wiener process. When
β
=1+2
α
, the limiting distribution depends on the behavior of the slowly varying parts of the mean and variance. We derive all possible limit distributions of the weighted CLSE based on observations {
Z
(
r
+1),
Z
(
r
+2),…,
Z
(
n
)} as
n
→∞ and
r
=
n
ε
, 0≤
ε
<1. Conditions guaranteeing the strong consistency of the proposed estimator will be derived.
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