# P-O Estimator P-O(penalized method to ordinary method) estimator using QMLE or adaBayes and $L^q$ penalty. ```r poest( yuima, estimator = "qmle", start, prior=NULL, lower, upper, penalty, method = "L-BFGS-B", ... ) ``` ## Arguments - `yuima`: A 'yuima' object. - `estimator`: Charactor. 'qmle' or 'adaBayes'. - `start`: A named list. Initial suggestion for parameter values - `prior`: A list of prior distributions for the parameters specified by 'code'. See `adaBayes` for details. - `lower`: A named list for specifying lower bounds of parameters. - `upper`: A named list for specifying upper bounds of parameters. - `penalty`: Parameters for penalizing. A list of 2 sub-lists named 'drift' and 'diffusion'. Each sub-list may contain `q` , `gamma` and `r`. See example for details. - `method`: Optimization method for `qmle` or `adaBayes`. - `...`: Passed to `qmle` or `adaBayes` method. ## Details Calculate the P-O estimator for stochastic processes by using the QMLE or adaBayes function and $L^q$ penalty. ## Returns - **initial_estimator**: Result of the initial QMLE or adaBayes. - **final_estimator**: Result of the final QMLE or adaBayes. - **model**: Given yuima model. - **estimator**: Name of estimation method. - **call**: Call - **coef**: Result of estimation. - **vcov**: Vcov matrix of coefficients. - **selected_coef**: Coefficients not regarded to be 0 in the penalization step. ## Author(s) Kosuke Kito and Masahiro Kurisaki with YUIMA project Team ## References Suzuki, T. and Yoshida, N., 2020. Penalized least squares approximation methods and their applications to stochastic processes. Japanese Journal of Statistics and Data Science, 3(2), pp.513-541. Huang, J., Horowitz, J. and Ma, S., 2008. Asymptotic properties of bridge estimators in sparse high-dimensional regression models. The Annals of Statistics, 36(2). ## Examples ```r ### example1 normal case # model definition d <- 5 p <- d sol <- c(paste0("x", 1:d), "y") drift <- c(paste0("-0.2*", sol[1:d]), 0) logsigma <- paste(paste0("theta", 1:p, "*", sol[1:p]), collapse = "+") diffusion_Y <- paste0("pmin(exp(", logsigma, "), 10^5)") diffusion <- diag(0.5, d + 1) diffusion[d+1,d+1] <- diffusion_Y ymodel <- setModel(drift = drift, diffusion = diffusion, solve.variable = sol, state.variable = sol) n <- 1000 ysamp <- setSampling(Terminal = 1, n = n) yuima <- setYuima(model = ymodel, sampling = ysamp) true.par <- as.list(c(1, -1, double(p - 2))) names(true.par) <- paste0("theta", 1:p) set.seed(111) yuima <- simulate(yuima, xinit = 1, true.parameter = true.par) # other arguments definition start.par <- as.list(runif(p,-2,2)) names(start.par) <- paste0("theta", 1:p) lower.par <- as.list(rep(-2, p)) names(lower.par) <- paste0("theta", 1:p) upper.par <- as.list(rep(2, p)) names(upper.par) <- paste0("theta", 1:p) #penalty parameters drift.penalty <- list(q = 0.3, gamma = 1,r = 1.2) diffusion.penalty <- list(q = 0.3, gamma=1,r = 1.2) penalty <- list(drift = drift.penalty, diffusion = diffusion.penalty) method = "L-BFGS-B" # estimation res.po <- poest( yuima = yuima, start = start.par, lower = lower.par, upper = upper.par, penalty = penalty, rcpp = TRUE ) # show results print(res.po) summary(res.po) ### example2 not giving penalty # model definition d <- 5 p <- d sol <- c(paste0("x", 1:d), "y") drift <- c(paste0("-0.2*", sol[1:d]), 0) logsigma <- paste(paste0("theta", 1:p, "*", sol[1:p]), collapse = "+") diffusion_Y <- paste0("pmin(exp(", logsigma, "), 10^5)") diffusion <- diag(0.5, d + 1) diffusion[d+1,d+1] <- diffusion_Y ymodel <- setModel(drift = drift, diffusion = diffusion, solve.variable = sol, state.variable = sol) n <- 1000 ysamp <- setSampling(Terminal = 1, n = n) yuima <- setYuima(model = ymodel, sampling = ysamp) true.par <- as.list(c(1, -1, double(p - 2))) names(true.par) <- paste0("theta", 1:p) set.seed(111) yuima <- simulate(yuima, xinit = 1, true.parameter = true.par) # other arguments definition start.par <- as.list(runif(p,-2,2)) names(start.par) <- paste0("theta", 1:p) lower.par <- as.list(rep(-2, p)) names(lower.par) <- paste0("theta", 1:p) upper.par <- as.list(rep(2, p)) names(upper.par) <- paste0("theta", 1:p) method = "L-BFGS-B" # estimation res.po <- poest( yuima = yuima, start = start.par, lower = lower.par, upper = upper.par, rcpp = TRUE ) # when penalty or any element of penalty is not given, # the following values will be used for elements not given. # list( # drift = list(q = 1, gamma = 1, r = (3 - q + gamma) / 2), # diffusion = list(q = 1, gamma = 1, r = (3 - q + gamma) / 2), # ) # show results print(res.po) summary(res.po) ### example3 use adaBayes # model definition d <- 5 p <- d sol <- c(paste0("x", 1:d), "y") drift <- c(paste0("-0.2*", sol[1:d]), 0) logsigma <- paste(paste0("theta", 1:p, "*", sol[1:p]), collapse = "+") diffusion_Y <- paste0("pmin(exp(", logsigma, "), 10^5)") diffusion <- diag(0.5, d + 1) diffusion[d+1,d+1] <- diffusion_Y ymodel <- setModel(drift = drift, diffusion = diffusion, solve.variable = sol, state.variable = sol) n <- 1000 ysamp <- setSampling(Terminal = 1, n = n) yuima <- setYuima(model = ymodel, sampling = ysamp) true.par <- as.list(c(1, -1, double(p - 2))) names(true.par) <- paste0("theta", 1:p) set.seed(111) yuima <- simulate(yuima, xinit = 1, true.parameter = true.par) # other arguments definition estimator <- "adaBayes" start.par <- as.list(runif(p,-2,2)) names(start.par) <- paste0("theta", 1:p) prior <- as.list(rep(list(list(measure.type="code",df="dunif(z,-2,2)")), p)) names(prior) <- paste0("theta", 1:p) lower.par <- as.list(rep(-2, p)) names(lower.par) <- paste0("theta", 1:p) upper.par <- as.list(rep(2, p)) names(upper.par) <- paste0("theta", 1:p) #penalty parameters drift.penalty <- list(q = 0.3, gamma = 1,r = 1.2) diffusion.penalty <- list(q = 0.3, gamma=1,r = 1.2) penalty <- list(drift = drift.penalty, diffusion = diffusion.penalty) method <- "mcmc" mcmc <- 1000 path <- TRUE # estimation res.po <- poest( yuima = yuima, estimator = estimator, prior = prior, start = start.par, lower = lower.par, upper = upper.par, penalty = penalty, method = method, mcmc = mcmc, path = path, rcpp = TRUE) # show results print(res.po) summary(res.po) ```

poest function