A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1)

1977 ◽  
Vol 9 (01) ◽  
pp. 87-104
Author(s):  
P. A. Jacobs ◽  
P. A. W. Lewis
Keyword(s):  
Linear Combination ◽  
Point Process ◽  
Moving Average ◽  
Random Variables ◽  
Stationary Sequence ◽  
Autoregressive Moving Average ◽  
Marginal Distributions ◽  
Joint Distributions ◽  
Exponential Random Variables ◽  
Exponential Sequence

A stationary sequence of random variables with exponential marginal distributions and the correlation structure of an ARMA (1, 1) process is defined. The process is formed as a random linear combination of i.i.d. exponential random variables and is very simple to generate on a computer. Moments and joint distributions for the sequence are obtained, as well as limiting properties of sums of the random variables and of the point process whose intervals have the EARMA (1, 1) structure.

A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1)

1977 ◽  
Vol 9 (1) ◽  
pp. 87-104 ◽  
Author(s):  
P. A. Jacobs ◽  
P. A. W. Lewis
Keyword(s):  
Linear Combination ◽  
Point Process ◽  
Moving Average ◽  
Random Variables ◽  
Stationary Sequence ◽  
Autoregressive Moving Average ◽  
Marginal Distributions ◽  
Joint Distributions ◽  
Exponential Random Variables ◽  
Exponential Sequence

A stationary sequence of random variables with exponential marginal distributions and the correlation structure of an ARMA (1, 1) process is defined. The process is formed as a random linear combination of i.i.d. exponential random variables and is very simple to generate on a computer. Moments and joint distributions for the sequence are obtained, as well as limiting properties of sums of the random variables and of the point process whose intervals have the EARMA (1, 1) structure.

Extending the correlation structure of exponential autoregressive–moving-average processes

1981 ◽  
Vol 18 (1) ◽  
pp. 181-189 ◽  
Author(s):  
Ed McKenzie
Keyword(s):  
Autocorrelation Function ◽  
Moving Average ◽  
Binary Sequence ◽  
Correlation Structure ◽  
Autoregressive Moving Average ◽  
Correlation Structures ◽  
Exponential Random Variables ◽  
Moving Average Processes ◽  
Negative Exponential ◽  
Exponential Sequence

Some recent constructions for the generation of dependent sequences of identically distributed negative exponential random variables with specific correlation structures are generalized. This is achieved by attributing a correlation structure to the binary sequence which controls the generation of the exponentials. The procedure causes the autocorrelation function of the exponential sequence to copy that of the binary sequence and thus be extended to include negative values and other values beyond the usual range.

Extending the correlation structure of exponential autoregressive–moving-average processes

1981 ◽  
Vol 18 (01) ◽  
pp. 181-189 ◽  
Author(s):  
Ed McKenzie
Keyword(s):  
Autocorrelation Function ◽  
Moving Average ◽  
Binary Sequence ◽  
Correlation Structure ◽  
Autoregressive Moving Average ◽  
Correlation Structures ◽  
Exponential Random Variables ◽  
Moving Average Processes ◽  
Negative Exponential ◽  
Exponential Sequence

Some recent constructions for the generation of dependent sequences of identically distributed negative exponential random variables with specific correlation structures are generalized. This is achieved by attributing a correlation structure to the binary sequence which controls the generation of the exponentials. The procedure causes the autocorrelation function of the exponential sequence to copy that of the binary sequence and thus be extended to include negative values and other values beyond the usual range.

Autoregressive moving-average processes with negative-binomial and geometric marginal distributions

1986 ◽  
Vol 18 (03) ◽  
pp. 679-705 ◽  
Author(s):  
Ed McKenzie
Keyword(s):  
Negative Binomial ◽  
Moving Average ◽  
Random Variables ◽  
Autoregressive Moving Average ◽  
Marginal Distributions ◽  
Alternative Derivation ◽  
Discrete Random Variables ◽  
Moving Average Processes ◽  
Negative Exponential ◽  
Interval Lengths

Some simple models are described which may be used for the modelling or generation of sequences of dependent discrete random variates with negative binomial and geometric univariate marginal distributions. The models are developed as analogues of well-known continuous variate models for gamma and negative exponential variates. The analogy arises naturally from a consideration of self-decomposability for discrete random variables. An alternative derivation is also given wherein both the continuous and the discrete variate processes arise simultaneously as measures on a process of overlapping intervals. The former is the process of interval lengths and the latter is a process of counts on these intervals.

Autoregressive moving-average processes with negative-binomial and geometric marginal distributions

1986 ◽  
Vol 18 (3) ◽  
pp. 679-705 ◽  
Author(s):  
Ed McKenzie
Keyword(s):  
Negative Binomial ◽  
Moving Average ◽  
Random Variables ◽  
Autoregressive Moving Average ◽  
Marginal Distributions ◽  
Alternative Derivation ◽  
Discrete Random Variables ◽  
Moving Average Processes ◽  
Negative Exponential ◽  
Interval Lengths

Some simple models are described which may be used for the modelling or generation of sequences of dependent discrete random variates with negative binomial and geometric univariate marginal distributions. The models are developed as analogues of well-known continuous variate models for gamma and negative exponential variates. The analogy arises naturally from a consideration of self-decomposability for discrete random variables. An alternative derivation is also given wherein both the continuous and the discrete variate processes arise simultaneously as measures on a process of overlapping intervals. The former is the process of interval lengths and the latter is a process of counts on these intervals.

scholarly journals The general theory of canonical correlation and its relation to functional analysis

1961 ◽  
Vol 2 (2) ◽  
pp. 229-242 ◽  
Author(s):  
E. J. Hannan
Keyword(s):  
Linear Combination ◽  
Canonical Correlation ◽  
Random Variables ◽  
Random Variable ◽  
Finite Variance ◽  
Finite Sample ◽  
Standard Description ◽  
Joint Distributions ◽  
Gaussian Case ◽  
The Relationship

The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variablesxs, and any linear combination ofqrandom variablesytinsofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, forp=q= 1, to include a description of the correlation of any function of a random variablexand any function of a random variabley(both functions having finite variance) for a class of joint distributions ofxandywhich is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space. These theorems lend themselves easily to the generalisation of the theory to situations wherepandqare not finite. In the case of Gaussian, stationary, processes this generalisation is equivalent to the classical spectral theory and corresponds to a canonical reduction of a (finite) sample of data which is basic. The theory also then extends to any number of processes. In the Gaussian case, also, the present discussion-is connected with the results of Gelfand and Yaglom [2] relating to the amount of information in one random process about another.

An exponential moving-average sequence and point process (EMA1)

1977 ◽  
Vol 14 (01) ◽  
pp. 98-113 ◽  
Author(s):  
A. J. Lawrance ◽  
P. A. W. Lewis
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Point Spectrum ◽  
Initial Conditions ◽  
Moving Average ◽  
Stationary Sequence ◽  
Exponential Distributions ◽  
Linear Combinations ◽  
Time Curve ◽  
Function Point

A construction is given for a stationary sequence of random variables {Xi } which have exponential marginal distributions and are random linear combinations of order one of an i.i.d. exponential sequence {ε i }. The joint and trivariate exponential distributions of Xi −1, Xi and Xi + 1 are studied, as well as the intensity function, point spectrum and variance time curve for the point process which has the {Xi } sequence for successive times between events. Initial conditions to make the point process count stationary are given, and extensions to higher-order moving averages and Gamma point processes are discussed.

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