# Maximum Likelihood Estimation for the Truncated Multivariate Normal Distribution ```r mle.tmvnorm(X, lower = rep(-Inf, length = ncol(X)), upper = rep(+Inf, length = ncol(X)), start = list(mu = rep(0, ncol(X)), sigma = diag(ncol(X))), fixed = list(), method = "BFGS", cholesky = FALSE, lower.bounds = -Inf, upper.bounds = +Inf, ...) ``` ## Arguments - `X`: Matrix of quantiles, each row is taken to be a quantile. - `lower`: Vector of lower truncation points, default is `rep(-Inf, length = ncol(X))`. - `upper`: Vector of upper truncation points, default is `rep( Inf, length = ncol(X))`. - `start`: Named list with elements `mu` (mean vector) and `sigma` (covariance matrix). Initial values for optimizer. - `fixed`: Named list. Parameter values to keep fixed during optimization. - `method`: Optimization method to use. See `optim` - `cholesky`: if TRUE, we use the Cholesky decomposition of `sigma` as parametrization - `lower.bounds`: lower bounds/box constraints for method "L-BFGS-B" - `upper.bounds`: upper bounds/box constraints for method "L-BFGS-B" - ``...``: Further arguments to pass to `optim` ## Details This method performs a maximum likelihood estimation of the parameters `mean` and `sigma` of a truncated multinormal distribution, when the truncation points `lower` and `upper` are known. `mle.tmvnorm()` is a wrapper for the general maximum likelihood method `mle`, so one does not have to specify the negative log-likelihood function. The log-likelihood function for a data matrix X (T x n) can be established straightforward as $$ \log L(X | \mu,\Sigma) = -T \log{\alpha(\mu,\Sigma)} + {-T/2} \log{\|\Sigma\|} -\frac{1}{2} \sum_{t=1}^{T}{(x_t-\mu)' \Sigma^{-1} (x_t-\mu)} $$ As `mle`, this method returns an object of class `mle`, for which various diagnostic methods are available, like `profile()`, `confint()` etc. See examples. In order to adapt the estimation problem to `mle`, the named parameters for mean vector elements are "mu_i" and the elements of the covariance matrix are "sigma_ij" for the lower triangular matrix elements, i.e. (j <= i). ## Returns An object of class `mle-class` ## Author(s) Stefan Wilhelm wilhelm@financial.com ## See Also `mle` and `mle-class` ## Examples ```r ## Not run: set.seed(1.2345) # the actual parameters lower <- c(-1,-1) upper <- c(1, 2) mu <- c(0, 0) sigma <- matrix(c(1, 0.7, 0.7, 2), 2, 2) # generate random samples X <- rtmvnorm(n=500, mu, sigma, lower, upper) method <- "BFGS" # estimate mean vector and covariance matrix sigma from random samples X # with default start values mle.fit1 <- mle.tmvnorm(X, lower=lower, upper=upper) # diagnostic output of the estimated parameters summary(mle.fit1) logLik(mle.fit1) vcov(mle.fit1) # profiling the log likelihood and confidence intervals mle.profile1 <- profile(mle.fit1, X, method="BFGS", trace=TRUE) confint(mle.profile1) par(mfrow=c(3,2)) plot(mle.profile1) # choosing a different start value mle.fit2 <- mle.tmvnorm(X, lower=lower, upper=upper, start=list(mu=c(0.1, 0.1), sigma=matrix(c(1, 0.4, 0.4, 1.8),2,2))) summary(mle.fit2) ## End(Not run) ```

mle.tmvnorm function