# Functions to compute a and simulate from a triangular probability mass function In Balabdaoui et al (2012) the triangular density, defined as [REMOVE_ME]$$ p_x^{a, b, c, d, e} = c(x - a) / (b - a) [REMOVE_ME_2]$$ for $x \in \{a, \ldots, c\}$ and [REMOVE_ME]$$ p_x^{a, b, c, d, e} = (e - c)(x - b) / (d - b) + c [REMOVE_ME_2]$$ for $x \in \{c, \ldots, d\}$, was used to illustrate the limit process of the log-concave MLE. In order to provide the code to generate the limit process figure in Balabdaoui et al (2012, see the example in `logConDiscrMLE` for the code to generate that figure) the functions `dTriangular` and `rTriangular` are included in this package. Note that `rTriangular` uses the rejection sampling algorithm in Devroye (1987) which was specifically developed to generate random numbers from a log-concave PMF. ## Description In Balabdaoui et al (2012) the triangular density, defined as $$ p_x^{a, b, c, d, e} = c(x - a) / (b - a) $$ for $x \in \{a, \ldots, c\}$ and $$ p_x^{a, b, c, d, e} = (e - c)(x - b) / (d - b) + c $$ for $x \in \{c, \ldots, d\}$, was used to illustrate the limit process of the log-concave MLE. In order to provide the code to generate the limit process figure in Balabdaoui et al (2012, see the example in `logConDiscrMLE` for the code to generate that figure) the functions `dTriangular` and `rTriangular` are included in this package. Note that `rTriangular` uses the rejection sampling algorithm in Devroye (1987) which was specifically developed to generate random numbers from a log-concave PMF. ```r dTriangular(a, b, c, d, e) rTriangular(n, a, b, c, d, e) ``` ## Arguments - `a`: Left endpoint of triangular pmf. - `b`: Mode of triangular pmf. - `c`: Height at mode. - `d`: Left endpoint. - `e`: Height at left endpoint. - `n`: Number of random numbers to generate. ## Returns `dTriangular` returns a vector containing the value of the PMF at all values in $x \in \{a, \ldots, d\}$. `rTriangular` returns a list containing the elements: - **rand**: Vector with generated random numbers of length $n$. - **x**: Vector $(a, \ldots, d)$. - **dens**: Value of the pmf at $x$. ## References Balabdaoui, F., Jankowski, H., Rufibach, K., and Pavlides, M. (2013). Maximum likelihood estimation and confidence bands for a discrete log-concave distribution. **J. R. Stat. Soc. Ser. B Stat. Methodol.**, 75 (4), 769--790. Devroye, L. (1987). A simple generator for discrete log-concave distributions. **Computing**, 39 , 87-91. ## Author(s) Kaspar Rufibach (maintainer) kaspar.rufibach@gmail.com [http://www.kasparrufibach.ch](http://www.kasparrufibach.ch) Fadoua Balabdaoui fadoua@ceremade.dauphine.fr [http://www.ceremade.dauphine.fr/~fadoua](http://www.ceremade.dauphine.fr/~fadoua) Hanna Jankowski hkj@mathstat.yorku.ca [http://www.math.yorku.ca/~hkj](http://www.math.yorku.ca/~hkj) Kathrin Weyermann ## Note This function is used to generate the plot of the limit process in the help file for the function `logConDiscrMLE`. ## Examples ```r ## ------------------------------------------------------------- ## compute values of triangular density and simulate from it ## ------------------------------------------------------------- a <- 1 b <- 7 c <- 8 d <- 11 e <- 2 n <- 10 ^ 2 ## support x <- seq(a, d, by = 1) ## true density dens <- dTriangular(a, b, c, d, e) logdens <- log(dens) rand <- rTriangular(n, a, b, c, d, e)$rand ## does the same as rTriangular() rand2 <- sample(x = a:d, size = n, prob = dens, replace = TRUE) ```

Triangular function