Moments - Parametric Quantities of Interest

A set of functions that calculate the theoretical moments (expectation, variance, skewness, excess kurtosis) and other important parametric functions (median, mode, entropy, Fisher information) of a distribution.

Usage

moments(x)

mean(x, ...)

median(x, na.rm = FALSE, ...)

mode(x)

var(x, y = NULL, na.rm = FALSE, use)

sd(x, na.rm = FALSE)

skew(x, ...)

kurt(x, ...)

entro(x, ...)

finf(x, ...)

Arguments

x: a Distribution object.
...: extra arguments.
y, use, na.rm: arguments in mean and var standard methods from the stats package not used here.

Value

Numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

Details

Given a distribution, these functions calculate the theoretical moments and other parametric quantities of interest. Some distribution-function combinations are not available; for example, the sd() function is available only for univariate distributions.

The moments() function automatically finds the available methods for a given distribution and results all of the results in a list.

Technical Note: The argument of the moment functions does not follow the naming convention of the package, i.e. the Distribution object is names x rather than distr. This is due to the fact that most of the generics are already defined in the stats package (mean, median, mode, var, sd), therefore the first argument was already named x and could not change.

Functions

median(): Median
mode(): Mode
var(): Variance
sd(): Standard Deviation
skew(): Skewness
kurt(): Kurtosis
entro(): Entropy
finf(): Fisher Information (numeric or matrix)

Examples

# -----------------------------------------------------
# Beta Distribution Example
# -----------------------------------------------------

# Create the distribution
a <- 3
b <- 5
D <- Beta(a, b)

# ------------------
# dpqr Functions
# ------------------

d(D, c(0.3, 0.8, 0.5)) # density function
#> [1] 2.268945 0.107520 1.640625
p(D, c(0.3, 0.8, 0.5)) # distribution function
#> [1] 0.3529305 0.9953280 0.7734375
qn(D, c(0.4, 0.8)) # inverse distribution function
#> [1] 0.3205858 0.5167578
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself
#>   [1] 2.1866737 1.8607350 0.4562757 0.9761423 1.7081981 2.2178898 2.2024303
#>   [8] 1.3233835 0.7386163 0.8441121 1.1062408 2.0541726 2.2961593 1.1132598
#>  [15] 1.8216994 1.9852110 0.9665665 2.1362498 0.8801992 1.2630775 1.8903052
#>  [22] 1.4761686 1.6241568 2.0172665 1.8199744 0.5491640 2.1152278 0.6348995
#>  [29] 2.3044960 1.7998164 1.6363645 1.4808840 2.2620830 2.2039299 2.1933650
#>  [36] 2.2940351 1.9724819 2.2828936 1.5994340 0.8200878 2.2275840 2.2623429
#>  [43] 1.1391907 2.3043038 1.7004710 0.7990487 1.6704638 1.4793582 1.9839090
#>  [50] 2.3005360 2.2111064 2.2846296 2.2517941 1.9750353 1.9328287 1.8069511
#>  [57] 2.0779686 1.6015518 1.2822302 2.1692320 1.9953819 0.7185783 1.6557242
#>  [64] 2.1560955 2.3043668 0.1997865 1.8153208 1.3722137 1.5286853 2.2872361
#>  [71] 0.5299206 1.2593337 1.2536151 1.4549079 0.2337613 1.2417615 2.2050286
#>  [78] 2.1147822 1.7532197 1.6525623 0.9795870 2.2934107 0.5808408 2.2015315
#>  [85] 1.4792728 2.2918547 1.6580528 1.8619862 2.1378689 2.0677331 2.2525593
#>  [92] 1.2954350 1.8349490 2.3013482 1.4205110 2.2828708 2.1617535 2.2931346
#>  [99] 0.7713010 2.0074225

# ------------------
# Moments
# ------------------

mean(D) # Expectation
#> [1] 0.375
var(D) # Variance
#> [1] 0.02604167
sd(D) # Standard Deviation
#> [1] 0.1613743
skew(D) # Skewness
#> [1] 0.3098387
kurt(D) # Excess Kurtosis
#> [1] 0.04
entro(D) # Entropy
#> [1] -0.4301508
finf(D) # Fisher Information Matrix
#>            shape1      shape2
#> shape1  0.2617971 -0.13313701
#> shape2 -0.1331370  0.08818594

# List of all available moments
mom <- moments(D)
mom$mean # expectation
#> [1] 0.375

# ------------------
# Point Estimation
# ------------------

ll(D, x)
#> [1] 43.31318
llbeta(x, a, b)
#> [1] 43.31318

ebeta(x, type = "mle")
#> $shape1
#> [1] 2.89074
#> 
#> $shape2
#> [1] 5.044725
#> 
ebeta(x, type = "me")
#> $shape1
#> [1] 2.811521
#> 
#> $shape2
#> [1] 4.904165
#> 
ebeta(x, type = "same")
#> $shape1
#> [1] 2.862696
#> 
#> $shape2
#> [1] 4.99343
#> 

mle(D, x)
#> $shape1
#> [1] 2.89074
#> 
#> $shape2
#> [1] 5.044725
#> 
me(D, x)
#> $shape1
#> [1] 2.811521
#> 
#> $shape2
#> [1] 4.904165
#> 
same(D, x)
#> $shape1
#> [1] 2.862696
#> 
#> $shape2
#> [1] 4.99343
#> 
e(D, x, type = "mle")
#> $shape1
#> [1] 2.89074
#> 
#> $shape2
#> [1] 5.044725
#> 

mle("beta", x) # the distr argument can be a character
#> $shape1
#> [1] 2.89074
#> 
#> $shape2
#> [1] 5.044725
#> 

# ------------------
# Estimator Variance
# ------------------

vbeta(a, b, type = "mle")
#>          shape1   shape2
#> shape1 16.44844 24.83272
#> shape2 24.83272 48.83039
vbeta(a, b, type = "me")
#>          shape1   shape2
#> shape1 17.64848 26.56970
#> shape2 26.56970 51.39394
vbeta(a, b, type = "same")
#>          shape1   shape2
#> shape1 16.57719 24.96198
#> shape2 24.96198 49.01071

avar_mle(D)
#>          shape1   shape2
#> shape1 16.44844 24.83272
#> shape2 24.83272 48.83039
avar_me(D)
#>          shape1   shape2
#> shape1 17.64848 26.56970
#> shape2 26.56970 51.39394
avar_same(D)
#>          shape1   shape2
#> shape1 16.57719 24.96198
#> shape2 24.96198 49.01071

v(D, type = "mle")
#>          shape1   shape2
#> shape1 16.44844 24.83272
#> shape2 24.83272 48.83039