KDKW.FD: Approximate maximum likelihood algorithm based on...

In johannabertl/ApproxML: Approximate maximum likelihood estimation

Description

This function approximates the maximum likelihood estimate of the multivariate parameter theta. It requires a function to simulate data and compute summary statistics at each position of the parameter space. These simulations are used to obtain estimates the likelihood in a stochastic approximation algorithm based on finite differences approximation of the gradient (Kiefer-Wolfowitz algorithm).

Usage

KDKW.FD(s.obs, theta.0, simfun, rest = matrix(c(-Inf, Inf), ncol = 2,
  nrow = length(theta.0), byrow = T), ce, gamma = 1/6, C = 0, a,
  alpha = 1, A = 0, K, nk, Hfun = bw.nrd0.mult, Hnum, kernel,
  lg = T, ...)

KDKW.FD.theta.0(theta.0, s.obs, simfun, rest = matrix(c(-Inf, Inf), ncol
  = 2, nrow = length(theta.0), byrow = T), ce, gamma = 1/6, C = 0, a,
  alpha = 1, A = 0, K, nk, Hfun = bw.nrd0.mult, Hnum, kernel,
  lg = T, ...)

KDKW.FD.s.obs(s.obs, theta.0, simfun, rest = matrix(c(-Inf, Inf), ncol =
  2, nrow = length(theta.0), byrow = T), ce, gamma = 1/6, C = 0, a,
  alpha = 1, A = 0, K, nk, Hfun = bw.nrd0.mult, Hnum, kernel,
  lg = T, ...)

Arguments

s.obs	Vector of observed summary statistics
theta.0	Vector of starting point
simfun	name of the function which is used to simulate data (further arguments to simfun are given with ...)
rest	Matrix with restrictions on the parameterspace (matrix with 2 columns that contain the lower and upper bounds for the elements of theta). Use -Inf and Inf for unrestricted parameters. The default value is an unrestricted parameter space.
ce	numeric
gamma	numeric
C	integer
a	numeric
alpha	numeric
A	integer
K	number of steps
nk	number of simulations per step
Hfun	function which computes the bandwidth matrix. Can be one of the functions in bw.r
Hnum	how often shall the bandwidth matrix be computed? Possible values: "once" or an integer indicating after how many iterations the bandwidth selection is to be renewed (e. g. 1 => bandwidth selection in each step)
kernel	kernel function
lg	logical. Use the log-likelihood instead of the likelihood?
...	further arguments which are passed on to simulate the data with simfun

Functions

• KDKW.FD.theta.0: The same function for a set of different starting values, use e. g. with apply or mclapply. It uses try catch errors.

• KDKW.FD.s.obs: The same function for a set of different summary statistics, use e. g. with apply or mclapply. It uses try catch errors.

Gain sequences

The gain sequences are defined following Spall (2013):

a_k = a/(k + A)^alpha and c_k = c/(k + C)^gamma.

Usually, C>0 is only used if the algorithm is resumed. The default values alpha=1 and gamma=1/6 are given in Spall (2013).

Author(s)

Johanna Bertl

References

• Bertl, J.; Ewing, G.; Kosiol, C. & Futschik, A. Approximate Maximum Likelihood Estimation for Population Genetic Inference. Statistical Applications in Genetics and Molecular Biology, 2017, 16, p. 291-312

• Kiefer, J. & Wolfowitz, J. Stochastic estimation of the maximum of a regression function. The Annals of Mathematical Statistics, 1952, 23, p. 462-466

• Blum, J. R. Multidimensional stochastic approximation methods. The Annals of Mathematical Statistics, 1954, 25, 737-744

• Spall, J. C. Introduction to Stochastic Search and Optimization: Estimation, Simulation and Control. Wiley, 2003

Examples

# bivariate normal distribution with sumstats x1+x2 and x1-x2:

test1 = KDKW.FD(s.obs=c(2,0), theta.0=c(50,50), simfun=SIMnormal.mult2a, rest=matrix(c(-100, 100), ncol=2, nrow=2, byrow=T), ce=1, gamma = 1/6, C=0, a = 10, alpha = 1, A=0, K=200, nk=20, Hfun = bw.nrd0.flex, Hnum="once", kernel=robust.unscaled.diagonal, lg = T, n=1)

johannabertl/ApproxML documentation built on May 22, 2019, 2:19 p.m.