Estimation theory for growth and immigration rates in a multiplicative process

1972 ◽  
Vol 9 (2) ◽  
pp. 235-256 ◽  
Author(s):  
C. C. Heyde ◽  
E. Seneta
Keyword(s):  
Basic Assumption ◽  
Probability Generating Function ◽  
Autoregressive Process ◽  
Estimation Theory ◽  
Strong Law ◽  
First Order ◽  
Watson Process ◽  
Branching Process With Immigration ◽  
Historical Antecedents ◽  
Strongly Consistent

This paper deals with the simple Galton-Watson process with immigration, {Xn} with offspring probability generating function (p.g.f.) F(s) and immigration p.g.f. B(s), under the basic assumption that the process is subcritical (0 < m ≡ F'(1–) < 1), and that 0 < λ ≡ B'(1–) < ∞, 0 < B(0) < 1, together with various other moment assumptions as needed. Estimation theory for the rates m and λ on the basis of a single terminated realization of the process {Xn} is developed, in that (strongly) consistent estimators for both m and λ are obtained, together with associated central limit theorems in relation to m and μ ≡ λ(1–m)–1 Following this, historical antecedents are analysed, and some examples of application of the estimation theory are discussed, with particular reference to the continuous-time branching process with immigration. The paper also contains a strong law for martingales; and discusses relation of the above theory to that of a first order autoregressive process.

Estimation theory for growth and immigration rates in a multiplicative process

1972 ◽  
Vol 9 (02) ◽  
pp. 235-256 ◽  
Author(s):  
C. C. Heyde ◽  
E. Seneta
Keyword(s):  
Basic Assumption ◽  
Probability Generating Function ◽  
Autoregressive Process ◽  
Estimation Theory ◽  
Strong Law ◽  
First Order ◽  
Watson Process ◽  
Branching Process With Immigration ◽  
Historical Antecedents ◽  
Strongly Consistent

This paper deals with the simple Galton-Watson process with immigration, {Xn } with offspring probability generating function (p.g.f.) F(s) and immigration p.g.f. B(s), under the basic assumption that the process is subcritical (0 &lt; m ≡ F'(1–) &lt; 1), and that 0 &lt; λ ≡ B'(1–) &lt; ∞, 0 &lt; B(0) &lt; 1, together with various other moment assumptions as needed. Estimation theory for the rates m and λ on the basis of a single terminated realization of the process {Xn } is developed, in that (strongly) consistent estimators for both m and λ are obtained, together with associated central limit theorems in relation to m and μ ≡ λ(1–m)–1 Following this, historical antecedents are analysed, and some examples of application of the estimation theory are discussed, with particular reference to the continuous-time branching process with immigration. The paper also contains a strong law for martingales; and discusses relation of the above theory to that of a first order autoregressive process.

scholarly journals Discrete Pseudo Lindley Distribution: Properties, Estimation and Application on INAR(1) Process

2021 ◽  
Vol 26 (4) ◽  
pp. 76
Author(s):  
Muhammed Rasheed Irshad ◽  
Christophe Chesneau ◽  
Veena D’cruz ◽  
Radhakumari Maya
Keyword(s):  
Generating Function ◽  
Simulation Study ◽  
Probability Generating Function ◽  
Autoregressive Process ◽  
Discrete Version ◽  
Unknown Parameters ◽  
Lindley Distribution ◽  
First Order ◽  
Mathematical Properties

In this paper, we introduce a discrete version of the Pseudo Lindley (PsL) distribution, namely, the discrete Pseudo Lindley (DPsL) distribution, and systematically study its mathematical properties. Explicit forms gathered for the properties such as the probability generating function, moments, skewness, kurtosis and stress–strength reliability made the distribution favourable. Two different methods are considered for the estimation of unknown parameters and, hence, compared with a broad simulation study. The practicality of the proposed distribution is illustrated in the first-order integer-valued autoregressive process. Its empirical importance is proved through three real datasets.

scholarly journals Robust Trend Estimation for AR(1) Disturbances

2016 ◽  
Vol 34 (2) ◽  
Author(s):  
Roland Fried ◽  
Ursula Gather
Keyword(s):  
Robust Estimation ◽  
Linear Trend ◽  
Autoregressive Process ◽  
Trend Estimation ◽  
First Order ◽  
Repeated Median

We discuss the robust estimation of a linear trend if the noise follows an autoregressive process of first order. We find the ordinary repeated median to perform well except for negative correlations. In this case it can be improved by a Prais-Winsten transformation using a robust autocorrelation estimator.

A new mixed first-order integer-valued autoregressive process with Poisson innovations

2020 ◽  
Author(s):  
Daniel L. R. Orozco ◽  
Lucas O. F. Sales ◽  
Luz M. Z. Fernández ◽  
André L. S. Pinho
Keyword(s):  
Autoregressive Process ◽  
First Order

Effect of Preview on Digital Pursuit Control Performance

1978 ◽  
Vol 20 (3) ◽  
pp. 371-377 ◽  
Author(s):  
Tarald O. Kvålseth
Keyword(s):  
Human Performance ◽  
Autoregressive Process ◽  
Control Task ◽  
Error Performance ◽  
First Order ◽  
Pursuit Control ◽  
Significant Performance ◽  
Rms Error ◽  
Different Characteristics ◽  
Reference Input

The effect of preview on human performance during a digital pursuit control task was analyzed for different preview spans and different characteristics of the reference input. The data from eight subjects revealed that the RMS error performance improved substantially from the case of no preview to that of one preview point, while the use of additional preview points did not result in any further significant performance improvement. The benefit of preview was most clearly established when the reference input was generated by a purely random process as opposed to a first-order autoregressive process (with the parameter α = 0.95). The RMS error increased when the variance of the reference input increased. The error appeared to be normally distributed with a tendency towards a negative bias.

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