Distribution.dfNameType"Finite Discrete", "Discrete",
"Continuous", and "Mixed".Support.Minmin.Support.Maxmax.Estimation.Method(s)"mle" (maximum likelihood), "mme"
(method of moments), "mmue" (method of moments based on the
unbiased estimate of variance), "mvue" (minimum variance unbiased),
"qmle" (quasi-mle), etc., or some combination of these. In
cases where an estimator is more than one kind, a slash (/) is
used to denote all methods covered by the single estimator. For example,
for the Binomial distribution, the sample proportion is the maximum
likelihood, method of moments, and minimum variance unbiased estimator,
so this method is denoted as "mle/mme/mvue". See the help files
for the specific function listed under
Estimating Distribution Parameters for an
explanation of each of these estimation methods.Quantile.Estimation.Method(s)Estimation.Method(s). See the help files for the specific
function listed under
Estimating Distribution Quantiles for an
explanation of each of these estimation methods.Prediction.Interval.Method(s)Singly.Censored.Estimation.Method(s)Multiply.Censored.Estimation.Method(s)Number.parametersParameter.1Parameter.1, Parameter.2, ..., Parameter.5 are
character vectors containing the names of the distribution parameters
(see the column labeled Parameters in the table below). If a
distribution has $n$ parameters and $n < 5$, then the columns
labeled Parameter.n+1, ..., Parameter.5 are empty. For
example, the Normal distribution has only two parameters
associated with it (mean and sd), so the fields in
Parameter.3, Parameter.4, and Parameter.5 are
empty.Parameter.2Parameter.1Parameter.3Parameter.1Parameter.4Parameter.1Parameter.5Parameter.1Parameter.1.MinParameter.1.Min,
Parameter.2.Min, ...,
Parameter.5.Min are character
vectors containing the minimum values that can be assumed by the
distribution parameters (see the column labeled Parameter Range(s)
in the table below). The reason these are character vectors instead of numeric vectors is
because some parameters have a lower bound of 0 but must be
strictly bigger than 0 (e.g., the parameter sd for the
Normal distribution), in which case the lower bound is
.Machine$double.eps, which may vary from machine to machine.
Also, some parameters have a lower bound that depends on the value of
another parameter. For example, the parameter max for a
Uniform distribution is bounded below by the value of the
parameter min. If a distribution has $n$ parameters and $n < 5$, then the
columns labeled Parameter.n+1.Min, ..., Parameter.5.Min
have the missing value code (NA). For example, the Normal
distribution has only two parameters associated with it (mean
and sd) so the fields in
Parameter.3.Min, Parameter.4.Min, and Parameter.5.Min
have NAs in them.Parameter.2.MinParameter.1.MinParameter.3.MinParameter.1.MinParameter.4.MinParameter.1.MinParameter.5.MinParameter.1.MinParameter.1.MaxParameter.1.Max,
Parameter.2.Max, ...,
Parameter.5.Max are character
vectors containing the maximum values that can be assumed by the
distribution parameters (see the column labeled Parameter Range(s)
in the table below). The reason these are character vectors instead of numeric vectors is
because some parameters have an upper bound that depends on the value
of another parameter. For example, the parameter min for a
Uniform distribution is bounded above by the value of the
parameter max. If a distribution has $n$ parameters and $n < 5$, then the
columns labeled Parameter.n+1.Max, ..., Parameter.5.Max
have the missing value code (NA). For example, the Normal
distribution has only two parameters associated with it (mean
and sd) so the fields in
Parameter.3.Max, Parameter.4.Max, and Parameter.5.Max
have NAs in them.Parameter.2.MaxParameter.1.MaxParameter.3.MaxParameter.1.MaxParameter.4.MaxParameter.1.MaxParameter.5.MaxParameter.1.Maxdabb, pabb, qabb, and
rabb, where abb is the abbreviated name of the
distribution (see table below). These functions are described in the
help file with the name of the distribution (see the first column of the
table below). For example, the help file for Beta describes the
behavior of dbeta, pbeta, qbeta,
and rbeta. For most distributions, there is also an associated function for
estimating the distribution parameters, and the form of the names of
these functions is eabb, where abb is the
abbreviated name of the distribution (see table below). All of these
functions are listed in the help file
Estimating Distribution Parameters. For example,
the function ebeta estimates the shape parameters of a
Beta distribution based on a random sample of observations from
this distribution.
For some distributions, there are functions to estimate distribution
parameters based on Type I censored data. The form of the names of
these functions is eabbSinglyCensored for
singly censored data and eabbMultiplyCensored for
multiply censored data. All of these functions are listed under the heading
Estimating Distribution Parameters in the help file
Censored Data.
Table 1a. Available Distributions: Name, Abbreviation, Type, and Range
| Name | Abbreviation | Type |
| Range | Beta | beta |
| Continuous | $[0, 1]$ | |
| Binomial | binom |
Finite |
| $[0, size]$ | ||
| Discrete | (integer) | |
| Cauchy | cauchy |
Continuous |
| $(-\infty, \infty)$ | ||
| Chi | ||
chi |
Continuous | $[0, \infty)$ |
| Chi-square | chisq |
|
| Continuous | $[0, \infty)$ | |
| Exponential | exp |
Continuous |
| $[0, \infty)$ | ||
| Extreme | ||
evd |
Continuous | $(-\infty, \infty)$ |
| Value | ||
| F | ||
f |
Continuous | $[0, \infty)$ |
| Gamma | gamma |
|
| Continuous | $[0, \infty)$ | |
| Gamma | gammaAlt |
Continuous |
| $[0, \infty)$ | (Alternative) | |
| Generalized | gevd |
Continuous |
| $(-\infty, \infty)$ | Extreme | |
| for $shape = 0$ | Value | |
| $(-\infty, location + \frac{scale}{shape}]$ | ||
| for $shape > 0$ | ||
| $[location + \frac{scale}{shape}, \infty)$ | ||
| for $shape < 0$ | ||
| Geometric | geom |
Discrete |
| $[0, \infty)$ | ||
| (integer) | ||
| Hypergeometric | hyper |
Finite |
| $[0, min(k,m)]$ | ||
| Discrete | (integer) | |
| Logistic | logis |
Continuous |
| $(-\infty, \infty)$ | ||
| Lognormal | ||
lnorm |
Continuous | $[0, \infty)$ |
| Lognormal | lnormAlt |
|
| Continuous | $[0, \infty)$ | (Alternative) |
| Lognormal | lnormMix |
|
| Continuous | $[0, \infty)$ | Mixture |
| Lognormal | lnormMixAlt |
|
| Continuous | $[0, \infty)$ | Mixture |
| (Alternative) | ||
| Three- | ||
lnorm3 |
Continuous | $[threshold, \infty)$ |
| Parameter | ||
| Lognormal | ||
| Truncated | lnormTrunc |
Continuous |
| $[min, max]$ | Lognormal | |
| Truncated | lnormTruncAlt |
Continuous |
| $[min, max]$ | Lognormal | |
| (Alternative) | ||
| Negative | nbinom |
|
| Discrete | $[0, \infty)$ | Binomial |
| (integer) | ||
| Normal | norm |
|
| Continuous | $(-\infty, \infty)$ | |
| Normal | normMix |
Continuous |
| $(-\infty, \infty)$ | Mixture | |
| Truncated | normTrunc |
Continuous |
| $[min, max]$ | Normal | |
| Pareto | pareto |
Continuous |
| $[location, \infty)$ | ||
| Poisson | ||
pois |
Discrete | $[0, \infty)$ |
| (integer) | ||
| Student's t | ||
t |
Continuous | $(-\infty, \infty)$ |
| Triangular | tri |
|
| Continuous | $[min, max]$ | |
| Uniform | unif |
Continuous |
| $[min, max]$ | ||
| Weibull | ||
weibull |
Continuous | $[0, \infty)$ |
| Wilcoxon | wilcox |
|
| Finite | $[0, m n]$ | Rank Sum |
| Discrete | (integer) | |
| Zero-Modified | zmlnorm |
|
| Mixed | $[0, \infty)$ | Lognormal |
| (Delta) | ||
| Zero-Modified | ||
zmlnormAlt |
Mixed | $[0, \infty)$ |
| Lognormal | ||
| (Delta) | ||
| (Alternative) | ||
| Zero-Modified | zmnorm |
|
| Mixed | $(-\infty, \infty)$ | Normal |
Table 1b. Available Distributions: Name, Parameters, Parameter Default Values, Parameter Ranges, Estimation Method(s)
| Default | Parameter | ||
| Estimation | Name | Parameter(s) | Value(s) |
| Range(s) | Method(s) | Beta | shape1 |
| $(0, \infty)$ | mle, mme, mmue | ||
shape2 |
$(0, \infty)$ | ||
ncp |
0 |
$(0, \infty)$ | |
| Binomial | size |
||
| $[0, \infty)$ | mle/mme/mvue | ||
prob |
$[0, 1]$ | ||
| Cauchy | location |
0 |
|
| $(-\infty, \infty)$ | scale |
||
1 |
$(0, \infty)$ | ||
| Chi | df |
$(0, \infty)$ | |
| Chi-square | df |
||
| $(0, \infty)$ | |||
ncp |
0 |
$(-\infty, \infty)$ | |
| Exponential | rate |
1 |
|
| $(0, \infty)$ | mle/mme | ||
| Extreme | |||
location |
0 |
$ (-\infty, \infty)$ | mle, mme, mmue, pwme |
| Value | scale |
1 |
$(0, \infty)$ |
| F | df1 |
||
| $(0, \infty)$ | |||
df2 |
$(0, \infty)$ | ||
ncp |
0 |
$(0, \infty)$ | |
| Gamma | shape |
||
| $(0, \infty)$ | mle, bcmle, mme, mmue | ||
scale |
1 |
$(0, \infty)$ | |
| Gamma | mean |
||
| $(0, \infty)$ | mle, bcmle, mme, mmue | (Alternative) | cv |
1 |
$(0, \infty)$ | ||
| Generalized | location |
0 |
$(-\infty, \infty)$ |
| mle, pwme, tsoe | Extreme | scale |
1 |
| $(0, \infty)$ | Value | shape |
|
0 |
$(-\infty, \infty)$ | ||
| Geometric | prob |
$(0, 1)$ | |
| mle/mme, mvue | |||
| Hypergeometric | m |
||
| $[0, \infty)$ | mle, mvue | ||
n |
$[0, \infty)$ | ||
k |
$[1, m+n]$ | ||
| Logistic | location |
||
0 |
$(-\infty, \infty)$ | mle, mme, mmue | |
scale |
1 |
$(0, \infty)$ | |
| Lognormal | meanlog |
0 |
|
| $(-\infty, \infty)$ | mle/mme, mvue | sdlog |
|
1 |
$(0, \infty)$ | ||
| Lognormal | mean |
exp(1/2) |
$(0, \infty)$ |
| mle, mme, mmue, | (Alternative) | cv |
sqrt(exp(1)-1) |
| $(0, \infty)$ | mvue, qmle | ||
| Lognormal | |||
meanlog1 |
0 |
$(-\infty, \infty)$ | |
| Mixture | sdlog1 |
1 |
$(0, \infty)$ |
meanlog2 |
0 |
||
| $(-\infty, \infty)$ | sdlog2 |
||
1 |
$(0, \infty)$ | ||
p.mix |
0.5 |
$[0, 1]$ | |
| Lognormal | mean1 |
exp(1/2) |
|
| $(0, \infty)$ | Mixture | cv1 |
|
sqrt(exp(1)-1) |
$(0, \infty)$ | (Alternative) | |
mean2 |
exp(1/2) |
$(0, \infty)$ | |
cv2 |
sqrt(exp(1)-1) |
$(0, \infty)$ | |
p.mix |
0.5 |
||
| $[0, 1]$ | |||
| Three- | |||
meanlog |
0 |
$(-\infty, \infty)$ | lmle, mme, |
| Parameter | sdlog |
1 |
$(0, \infty)$ |
| mmue, mmme, | Lognormal | threshold |
0 |
| $(-\infty, \infty)$ | royston.skew, | ||
| zero.skew | |||
| Truncated | meanlog |
0 |
$(-\infty, \infty)$ |
| Lognormal | sdlog |
1 |
|
| $(0, \infty)$ | min |
||
0 |
$[0, max)$ | ||
max |
Inf |
$(min, \infty)$ | |
| Truncated | mean |
exp(1/2) |
|
| $(0, \infty)$ | Lognormal | cv |
|
sqrt(exp(1)-1) |
$(0, \infty)$ | (Alternative) | |
min |
0 |
$[0, max)$ | |
max |
Inf |
$(min, \infty)$ | |
| Negative | size |
||
| $[1, \infty)$ | mle/mme, mvue | Binomial | |
prob |
$(0, 1]$ | ||
mu |
$(0, \infty)$ | ||
| Normal | mean |
||
0 |
$(-\infty, \infty)$ | mle/mme, mvue | |
sd |
1 |
$(0, \infty)$ | |
| Normal | mean1 |
0 |
|
| $(-\infty, \infty)$ | Mixture | sd1 |
|
1 |
$(0, \infty)$ | ||
mean2 |
0 |
$(-\infty, \infty)$ | |
sd2 |
1 |
$(0, \infty)$ | |
p.mix |
0.5 |
||
| $[0, 1]$ | |||
| Truncated | |||
mean |
0 |
$(-\infty, \infty)$ | |
| Normal | sd |
1 |
$(0, \infty)$ |
min |
-Inf |
||
| $(-\infty, max)$ | max |
||
Inf |
$(min, \infty)$ | ||
| Pareto | location |
$(0, \infty)$ | |
| lse, mle | shape |
1 |
|
| $(0, \infty)$ | |||
| Poisson | |||
lambda |
$(0, \infty)$ | mle/mme/mvue | |
| Student's t | df |
||
| $(0, \infty)$ | ncp |
||
0 |
$(-\infty, \infty)$ | ||
| Triangular | min |
0 |
$(-\infty, max)$ |
max |
1 |
||
| $(min, \infty)$ | mode |
||
0.5 |
$(min, max)$ | ||
| Uniform | min |
0 |
$(-\infty, max)$ |
| mle, mme, mmue | max |
1 |
|
| $(min, \infty)$ | |||
| Weibull | |||
shape |
$(0, \infty)$ | mle, mme, mmue | |
scale |
1 |
$(0, \infty)$ | |
| Wilcoxon | m |
||
| $[1, \infty)$ | Rank Sum | ||
n |
$[1, \infty)$ | ||
| Zero-Modified | meanlog |
0 |
|
| $(-\infty, \infty)$ | mvue | Lognormal | sdlog |
1 |
$(0, \infty)$ | (Delta) | |
p.zero |
0.5 |
$[0, 1]$ | |
| Zero-Modified | mean |
exp(1/2) |
|
| $(0, \infty)$ | mvue | Lognormal | cv |
sqrt(exp(1)-1) |
$(0, \infty)$ | (Delta) | |
p.zero |
0.5 |
$[0, 1]$ | |
| (Alternative) | |||
| Zero-Modified | mean |
0 |
|
| $(-\infty, \infty)$ | mvue | Normal | sd |
1 |
$(0, \infty)$ | ||
p.zero |
0.5 |
$[0, 1]$ |