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Estimate Normal Distribution Given Set of Quantile Values

Source: R/est_norm.R
est_norm.Rd

With at least two quantile values find the mean and standard deviation of a normal distribution to match up with empirical values provided.

Usage

est_norm(q, p, weights = rep(1, length(p)), ...)

Arguments

q

quantile values.

p

probabilities corresponding to the q quantiles.

weights

relative weight of each quantile. The higher the weight the better the approximated distribution will be at fitting that quantile.

...

additional arguments passed to optim. See Details.

Value

a pedbp_est_norm object. This is a list with elements:

  • par: a named numeric vector with the mean and standard deviation for a Gaussian distribution

  • qp: a numeric matrix with two columns built from the input values of q and p

  • weights: the weights used

  • call: The call made

  • optim: result from calling optim

Details

For X ~ N(mu, sigma), Pr[X <= q] = p

Given the set of quantiles and probabilities, est_norm uses optim (with method = "L-BFGS-B", lower = c(-Inf, 0), upper = c(Inf, Inf)) to find the preferable mean and standard deviation of a normal distribution to fit the provided quantiles.

Use the weight argument to emphasize which, if any, of the provided quantiles needs to be approximated closer than others. By default all the quantiles are weighted equally.

Examples


# Example 1
q <- c(-1.92, 0.1, 1.89) * 1.8 + 3.14
p <- c(0.025, 0.50, 0.975)

x <- est_norm(q, p)
str(x)
#> List of 5
#>  $ par    : Named num [1:2] 3.31 1.72
#>   ..- attr(*, "names")= chr [1:2] "mean" "sd"
#>  $ qp     : num [1:3, 1:2] -0.316 3.32 6.542 0.025 0.5 ...
#>   ..- attr(*, "dimnames")=List of 2
#>   .. ..$ : NULL
#>   .. ..$ : chr [1:2] "q" "p"
#>  $ weights: num [1:3] 1 1 1
#>  $ call   : language est_norm(q = q, p = p)
#>  $ optim  :List of 5
#>   ..$ par        : num [1:2] 3.31 1.72
#>   ..$ value      : num 8.73e-05
#>   ..$ counts     : Named int [1:2] 7 7
#>   .. ..- attr(*, "names")= chr [1:2] "function" "gradient"
#>   ..$ convergence: int 0
#>   ..$ message    : chr "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
#>  - attr(*, "class")= chr "pedbp_est_norm"
x
#>     mean       sd 
#> 3.312550 1.721361 

plot(x)


# Example 2 -- build with quantiles that are easy to see unlikely to be from
# a Normal distribuiton
q <- c(-1.92, 0.05, 0.1, 1.89) * 1.8 + 3.14
p <- c(0.025, 0.40, 0.50, 0.975)

# with equal weights
x <- est_norm(q, p)
x
#>     mean       sd 
#> 3.469290 1.563764 
plot(x)


# weight to ignore one of the middle value and make sure to hit the other
x <- est_norm(q, p, weights = c(1, 2, 0, 1))
x
#>     mean       sd 
#> 3.621388 1.551445 
plot(x)


# equal weight the middle, more than the tails
x <- est_norm(q, p, weights = c(1, 2, 2, 1))
x
#>      mean        sd 
#> 3.3200002 0.3552437 
plot(x)