Dear All, I need to fit a custom probability density (based on the symmetric beta distribution B(shape, shape), where the two parameters shape1 and shape2 are identical) to my data. The trouble is that I experience some problems also when dealing with the plain vanilla symmetric beta distribution. Please consider the code at the end of the email. In the code, dbeta1 is the density of the beta distribution for shape1=shape2=shape. In the code, dbeta2 is the same quantity written explicitly, without the normalization factor (which should not matter at all if we talk about maximizing a quantity). I then generate some random numbers according to Beta(0.2, 0.2) and I try to estimate the shape parameter using 1) fitdistr from MASS 2) mle from stats4 Results: generally speaking I have non-sense estimates of the shape parameter when I use dbeta2 instead of dbeta1 and I do not understand why. On top of that, mle crashes with dbeta2 and often I have numerical problems depending on how I seed the x sequence of random numbers. I must be misunderstanding something, so any suggestion is appreciated. Cheers Lorenzo ######################################################################### library(MASS) library(stats4) dbeta1 <- function(x, shape, ...) dbeta(x, shape, shape, ...) dbeta2 <- function(x, shape){ res <- x^(shape-1)*(1-x)^(shape-1) return(res) } LL1 <- function(shape){ R <- dbeta1(x, shape) res <- -sum(log(R)) return(res) } LL2 <- function(shape){ R <- dbeta2(x, shape) res <- -sum(log(R)) return(res) } set.seed(124) x <-rbeta(1000, 0.2, 0.2) fit_dbeta1 <- fitdistr( x , dbeta1, start=list(shape=0.5) , method="Brent", lower=c(0), upper=c(1)) print("estimate of shape from fit_dbeta1 is") print(fit_dbeta1$estimate) fit_dbeta2 <- fitdistr( x , dbeta2, start=list(shape=0.5) , method="Brent", lower=c(0), upper=c(1)) print("estimate of shape from fit_dbeta2 is") print(fit_dbeta2$estimate) fit_LL1 <- mle(LL1, start=list(shape=0.5)) print("estimate of from fit_LL1") print(summary(fit_LL1)) ## this does not work fit_LL2 <- mle(LL2, start=list(shape=0.5)) [[alternative HTML version deleted]]

Stop right there and rethink! The normalization factor depends on the parameter that you are maximizing over. -pd> On 21 Dec 2017, at 11:29 , Lorenzo Isella <lorenzo.isella at gmail.com> wrote: > > In the code, dbeta1 is the density of the beta distribution for > shape1=shape2=shape. > In the code, dbeta2 is the same quantity written explicitly, without the > normalization factor (which should not matter at all if we talk about > maximizing a quantity). >-- Peter Dalgaard, Professor, Center for Statistics, Copenhagen Business School Solbjerg Plads 3, 2000 Frederiksberg, Denmark Phone: (+45)38153501 Office: A 4.23 Email: pd.mes at cbs.dk Priv: PDalgd at gmail.com