Fitting a normal distribution in R

Question

I'm using the following code to fit the normal distribution. The link for the dataset for "b" (too large to post directly) is :

link for b

setwd("xxxxxx")
library(fitdistrplus)

require(MASS)
tazur <-read.csv("b", header= TRUE, sep=",")
claims<-tazur$b
a<-log(claims)
plot(hist(a))

After plotting the histogram, it seems a normal distribution should fit well.

f1n <- fitdistr(claims,"normal")
summary(f1n)

#Length Class  Mode   
#estimate 2      -none- numeric
#sd       2      -none- numeric
#vcov     4      -none- numeric
#n        1      -none- numeric
#loglik   1      -none- numeric

plot(f1n)

Error in xy.coords(x, y, xlabel, ylabel, log) :

'x' is a list, but does not have components 'x' and 'y'

I get the above error when I try to plot the fitted distribution, and even the summary statistics are off for f1n.

Would appreciate any help.

Answer 1

Looks like you are making confusion between MASS::fitdistr and fitdistrplus::fitdist .

• MASS::fitdistr returns object of class "fitdistr", and there is no plot method for this. So you need to extract estimated parameters and plot the estimated density curve yourself.
• I don't know why you load package fitdistrplus , because your function call clearly shows you are using MASS . Anyway, fitdistrplus has function fitdist which returns object of class "fitdist". There is plot method for this class, but it won't work for "fitdistr" returned by MASS .

I will show you how to work with both packages.

## reproducible example
set.seed(0); x <- rnorm(500)

---

Using MASS::fitdistr

No plot method is available, so do it ourselves.

library(MASS)
fit <- fitdistr(x, "normal")
class(fit)
# [1] "fitdistr"

para <- fit$estimate
#         mean            sd 
#-0.0002000485  0.9886248515 

hist(x, prob = TRUE)
curve(dnorm(x, para[1], para[2]), col = 2, add = TRUE)

---

Using fitdistrplus::fitdist

library(fitdistrplus)
FIT <- fitdist(x, "norm")    ## note: it is "norm" not "normal"
class(FIT)
# [1] "fitdist"

plot(FIT)    ## use method `plot.fitdist`

Answer 2

Review of previous answer

In the previous answer I did not mention the difference between two methods. In general, if we opt for maximum likelihood inference I would recommend using MASS::fitdistr , because for many basic distributions it performs exact inference instead of numerical optimization. Doc of ?fitdistr made this rather clear:

For the Normal, log-Normal, geometric, exponential and Poisson
distributions the closed-form MLEs (and exact standard errors) are
used, and ‘start’ should not be supplied.

For all other distributions, direct optimization of the
log-likelihood is performed using ‘optim’.  The estimated standard
errors are taken from the observed information matrix, calculated
by a numerical approximation.  For one-dimensional problems the
Nelder-Mead method is used and for multi-dimensional problems the
BFGS method, unless arguments named ‘lower’ or ‘upper’ are
supplied (when ‘L-BFGS-B’ is used) or ‘method’ is supplied
explicitly.

On the other hand, fitdistrplus::fitdist always performs inference in a numerical way, even if exact inference exists. Sure, the advantage of fitdist is that more inference principle is available:

Fit of univariate distributions to non-censored data by maximum
likelihood (mle), moment matching (mme), quantile matching (qme)
or maximizing goodness-of-fit estimation (mge).

---

Purpose of this answer

This answer is going to explore exact inference for normal distribution. It will have a theoretical flavour, but there is no proof of likelihood principle; only results are given. Based on these results, we write our own R function for exact inference, which can be compared with MASS::fitdistr . On the other hand, to compare with fitdistrplus::fitdist , we use optim to numerically minimize negative log-likelihood function.

This is a great opportunity to learn statistics and relatively advanced use of optim . For convenience, I will estimate the scale parameter: variance, rather than standard error.

---

Exact inference of normal distribution

---

Writing inference function ourselves

The following code is well commented. There is a switch exact . If set FALSE , numerical solution is chosen.

## fitting a normal distribution
fitnormal <- function (x, exact = TRUE) {
  if (exact) {
    ################################################
    ## Exact inference based on likelihood theory ##
    ################################################
    ## minimum negative log-likelihood (maximum log-likelihood) estimator of `mu` and `phi = sigma ^ 2`
    n <- length(x)
    mu <- sum(x) / n
    phi <- crossprod(x - mu)[1L] / n  # (a bised estimator, though)
    ## inverse of Fisher information matrix evaluated at MLE
    invI <- matrix(c(phi, 0, 0, phi * phi), 2L,
                   dimnames = list(c("mu", "sigma2"), c("mu", "sigma2")))
    ## log-likelihood at MLE
    loglik <- -(n / 2) * (log(2 * pi * phi) + 1)
    ## return
    return(list(theta = c(mu = mu, sigma2 = phi), vcov = invI, loglik = loglik, n = n))
    }
  else {
    ##################################################################
    ## Numerical optimization by minimizing negative log-likelihood ##
    ##################################################################
    ## negative log-likelihood function
    ## define `theta = c(mu, phi)` in order to use `optim`
    nllik <- function (theta, x) {
      (length(x) / 2) * log(2 * pi * theta[2]) + crossprod(x - theta[1])[1] / (2 * theta[2])
      }
    ## gradient function (remember to flip the sign when using partial derivative result of log-likelihood)
    ## define `theta = c(mu, phi)` in order to use `optim`
    gradient <- function (theta, x) {
      pl2pmu <- -sum(x - theta[1]) / theta[2]
      pl2pphi <- -crossprod(x - theta[1])[1] / (2 * theta[2] ^ 2) + length(x) / (2 * theta[2])
      c(pl2pmu, pl2pphi)
      }
    ## ask `optim` to return Hessian matrix by `hessian = TRUE`
    ## use "..." part to pass `x` as additional / further argument to "fn" and "gn"
    ## note, we want `phi` as positive so box constraint is used, with "L-BFGS-B" method chosen
    init <- c(sample(x, 1), sample(abs(x) + 0.1, 1))  ## arbitrary valid starting values
    z <- optim(par = init, fn = nllik, gr = gradient, x = x, lower = c(-Inf, 0), method = "L-BFGS-B", hessian = TRUE)
    ## post processing ##
    theta <- z$par
    loglik <- -z$value  ## flip the sign to get log-likelihood
    n <- length(x)
    ## Fisher information matrix (don't flip the sign as this is the Hessian for negative log-likelihood)
    I <- z$hessian / n  ## remember to take average to get mean
    invI <- solve(I, diag(2L))  ## numerical inverse
    dimnames(invI) <- list(c("mu", "sigma2"), c("mu", "sigma2"))
    ## return
    return(list(theta = theta, vcov = invI, loglik = loglik, n = n))
    }
  }

We still use the previous data for testing:

set.seed(0); x <- rnorm(500)

## exact inference
fit <- fitnormal(x)

#$theta
#           mu        sigma2 
#-0.0002000485  0.9773790969 
#
#$vcov
#              mu    sigma2
#mu     0.9773791 0.0000000
#sigma2 0.0000000 0.9552699
#
#$loglik
#[1] -703.7491
#
#$n
#[1] 500

hist(x, prob = TRUE)
curve(dnorm(x, fit$theta[1], sqrt(fit$theta[2])), add = TRUE, col = 2)

Numerical method is also rather accurate, except that the variance covariance does not have exact 0 off the diagonal:

fitnormal(x, FALSE)

#$theta
#[1] -0.0002235315  0.9773732277
#
#$vcov
#                 mu       sigma2
#mu     9.773826e-01 5.359978e-06
#sigma2 5.359978e-06 1.910561e+00
#
#$loglik
#[1] -703.7491
#
#$n
#[1] 500