The Uniform Distribution.

Density, distribution, quantile, random number generation and parameter estimation functions for the uniform distribution on the interval $[a,b]$. Parameter estimation can be based on an unweighted i.i.d. sample only and can be performed analytically or numerically.

dUniform(x, a = 0, b = 1, params = list(a, b), ...)

pUniform(q, a = 0, b = 1, params = list(a, b), ...)

qUniform(p, a = 0, b = 1, params = list(a, b), ...)

rUniform(n, a = 0, b = 1, params = list(a, b), ...)

eUniform(X, w, method = c("analytic.MLE", "moments", "numerical.MLE"), ...)

lUniform(X, w, a = 0, b = 1, params = list(a, b), logL = TRUE, ...)

Arguments

x, q: A vector of quantiles.
a, b: Boundary parameters.
params: A list that includes all named parameters.
...: Additional parameters.
p: A vector of probabilities.
n: Number of observations.
X: Sample observations.
w: An optional vector of sample weights.
method: Parameter estimation method.
logL: logical;if TRUE, lUniform gives the log-likelihood, otherwise the likelihood is given.

Value

dUniform gives the density, pUniform the distribution function, qUniform the quantile function, rUniform generates random deviates, and eUniform estimates the parameters. lUniform provides the log-likelihood function.

Details

If a or b are not specified they assume the default values of 0 and 1, respectively.

The dUniform(), pUniform(), qUniform(),and rUniform() functions serve as wrappers of the standard dunif, punif, qunif, and runif functions in the stats package. They allow for the parameters to be declared not only as individual numerical values, but also as a list so parameter estimation can be carried out.

The uniform distribution has probability density function $$p_x(x) = 1/(b-a)$$ for $a \le x \le b$. The analytic maximum likelihood parameter estimates are as given by Engineering Statistics Handbook. The method of moments parameter estimation option is also avaliable and the estimates are as given by Forbes et.al (2011), p.179.

The log-likelihood function for the uniform distribution is given by $$l(a,b|x) = -n log(b-a)$$

Note

The analytical maximum likelihood estimation of the parameters $a$ and $b$ is calculated using the range and mid-range of the sample. Therefore, only unweighted samples are catered for in the eUniform distribution when the method analytic.MLE is selected.

References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, chapter 26, Wiley, New York.

Engineering Statistics Handbook

Forbes, C. Evans, M. Hastings, N. & Peacock, B. (2011) Statistical Distributions, 4th Ed, chapter 40, Wiley, New Jersey.

Author

Haizhen Wu and A. Jonathan R. Godfrey.
Updates and bugfixes by Sarah Pirikahu.

Examples

# Parameter estimation for a distribution with known shape parameters
X <- rUniform(n=500, a=0, b=1)
est.par <- eUniform(X, method="analytic.MLE"); est.par
#> 
#> Parameters for the Uniform distribution. 
#> (found using the  analytic.MLE method.)
#> 
#>  Parameter     Type    Estimate
#>          a boundary 0.001445471
#>          b boundary 0.997848941
#> 
#> 
plot(est.par)


# Histogram and fitted density
den.x <- seq(min(X),max(X),length=100)
den.y <- dUniform(den.x,a=est.par$a,b=est.par$b)
hist(X, breaks=10, probability=TRUE, ylim = c(0,1.2*max(den.y)))
lines(den.x, den.y, col="blue")  # Original data
lines(density(X), lty=2)         # Fitted curve


# Extracting boundary parameters
est.par[attributes(est.par)$par.type=="boundary"]
#> $a
#> [1] 0.001445471
#> 
#> $b
#> [1] 0.9978489
#> 

# log-likelihood
lUniform(X,param = est.par)
#> [1] 1.801506

# Example of parameter estimation for a distribution with
# unknown parameters currently been sought after.