# getBDinform: Helpers for Getting Information Matrix for MLE estimates on... In DOBAD: Analysis of Discretely Observed Linear Birth-and-Death(-and-Immigration) Markov Chains

## Description

Assume we have data that is the state at discrete time points of a linear birth-death process, which has immigration parameter constrained to be a known constant times the birth rate. After using EM Algorithm for estimating rate parameters of a linear Birth-Death process, these functions compute matrices related to the information matrix.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```getBDinform.full.SC.manual(ENplus, ENminus, L, m) getBDinform.lost.SC.manual(ENplus, ENminus, EHoldtime, ENplusSq, ENminusSq, EHoldtimeSq, ENplusNminus, ENplusHoldtime, ENminusHoldtime, L, m, beta.immig, T) getBDinform.PO.SC.manual(ENplus, ENminus, EHoldtime, ENplusSq, ENminusSq, EHoldtimeSq, ENplusNminus, ENplusHoldtime, ENminusHoldtime, L, m, beta.immig, T) ```

## Arguments

 `L` Lambda, birth rate `m` Mu, death rate `beta.immig` Immigration rate is constrained to be a multiple of the birth rate. immigrationrate = beta.immig * lambda where lambda is birth rate. `T` Amount of time process is observed for; corresponds to time window over which all the expectations are computed. `ENplus` Expectation of the N^+, the number of jumps up , conditional on the data. `ENminus` Expectation of N^-, the number of jumps down, conditional on the data. `EHoldtime` Expectation of R, the total holdtime, conditional on the data. `ENplusSq` Expectation of (N^+)^2, the square of the number of jumps up, conditional on the data. `ENminusSq` Expectation of (N^-)^2, the square of the number of jumps down, conditional on the data. `EHoldtimeSq` Expectation of (R)^2, the square of the total holdtime, conditional on the data. `ENplusNminus` Expectation of N^+ * N^-, the product of the number of jumps up and the number of jumps down, conditional on the data. `ENplusHoldtime` Expectation of N^+ * R, the product of the number of jumps up and the total holdtime, conditional on the data. `ENminusHoldtime` Expectation of N^- * R, the product of the number of jumps down and the total holdtime, conditional on the data.

## Details

Assume we have a linear-birth-death process X_t with birth parameter lambda, death parameter mu, and immigration parameter beta*lambda (for some known, real beta). We observe the process at a finite set of times over a time interval [0,T]. Can run the EM algorithm to do maximum likelihood. These functions are used to then compute pieces related to the information matrix.

See equations 3.2 and 3.3 in the Louis paper for the notation.

getBDinform.lost.SC.manual computes I_x|y.

getBDinform.full.SC.manual computes I_x.

getBDinform.PO.SC.manual computes I_y (i.e. the difference between the other two functions).

They have the "manual" suffix because the user passes in the expectations. Some of them can be computed analytically by the methods in this package, but others cannot, so those are usually done by Monte Carlo (conditional on the data) simulation.

NOTE: To make sure the answers are coherent, it is important to pass in expectations that are consistent with each other. For instance, if the expectations ENplus, ENminus, and EHoldtime are computed analytically but simulations are used to estimate the rest, then the results may be nonsense, because the values passed in were not necessarily feasible expectations all from the same measure.

## Value

Symmetric 2x2 matrix; First row/column corresponds to lambda, second corresponds to mu

Charles Doss

## Source

Louis, T A. (1982). Finding the observed information matrix when using the EM algorithm. J. Roy. Statist. Soc. Ser. B. 44 226-233.

DOBAD documentation built on May 2, 2019, 3:04 a.m.