# Calculate Quasi-Likelihood and Maximum Likelihood Estimator for Linear State Space Model A function for the quasi-maximum likelihood estimation of linear state space models, extending the `qmle` function. ```r qmle.linear_state_space_model( yuima, start, lower, upper, method = "L-BFGS-B", fixed = list(), envir = globalenv(), filter_mean_init, explicit = FALSE, drop_terms = 0, ... ) ``` ## Arguments - `yuima`: A yuima object. The class of `yuima@model` must be `yuima.linear_state_space_model`. - `start`: Initial values for the parameters to be passed to the optimizer. - `lower`: A named list specifying the lower bounds of the parameters. - `upper`: A named list specifying the upper bounds of the parameters. - `method`: The optimization method to be used. See `optim`. - `fixed`: A named list of parameters to be held fixed during optimization. - `envir`: An environment in which the model coefficients are evaluated. - `filter_mean_init`: Initial values of unobserved variables for the filter calculation. - `explicit`: A logical value. If `TRUE`, use the explicit formula to solve the filitering equation. - `drop_terms`: A numeric value. The specified number of initial terms in the filtering result will be excluded from the quasi-likelihood function calculation. - `...`: Additional arguments to be passed to the `optim` method. See Examples. ## Returns A `yuima.linear_state_space_qmle-class` object, extending the `yuima.qmle-class` class. - **model**: A `yuima.linear_state_space_model` object. - **drop_terms**: A numeric value representing the number of terms ignored during optimization. - **explicit**: A logical value provided as an argument. - **mean_init**: A numeric value provided as the argument `filter_mean_init`. - **...**: Additional slots inherited from the `yuima.qmle` class. ## References Kurisaki, M. (2023). Parameter estimation for ergodic linear SDEs from partial and discrete observations. Stat Inference Stoch Process, 26, 279-330. Bini, D., Iannazzo, B., & Meini, B. (2012). Numerical Solution of Algebraic Riccati Equations. Society for Industrial and Applied Mathematics. ## Author(s) YUIMA TEAM ## Examples ```r ### Set model drift <- c("a*X", "X") diffusion <- matrix(c("b", "0", "0", "sigma"), nrow = 2) # Use `model.class="linearStateSpaceModel"` implicitly. ymodel <- setModel( drift = drift, diffusion = diffusion, solve.variable = c("X", "Y"), state.variable = c("X", "Y"), observed.variable = "Y" ) ### Set data T <- 100 N <- 50000 n <- N h <- T / N true.par <- list( a = -1.5, b = 0.3, sigma = 0.053 ) tmp.yuima <- simulate( ymodel, true.parameter = true.par, sampling = setSampling(n = N, Terminal = T) ) ydata <- tmp.yuima@data rm(tmp.yuima) ### Set yuima variable_data_mapping <- list( #"X" = NA, "Y" = 2 ) yuima <- setYuima(model = ymodel, data = ydata, variable_data_mapping = variable_data_mapping) # Estimate upper.par <- list( a = 1, b = 5, sigma = 1 ) lower.par <- list( a = -10, b = 0.01, sigma = 0.001 ) start.par <- list( a = 0.5, b = 4, sigma = 0.9 ) filter_mean_init <- 0 res <- qmle.linear_state_space_model( yuima, start = start.par, upper = upper.par, lower = lower.par, filter_mean_init = filter_mean_init ) ```

qmle_linear_state_space_model function