One-Parameter Lindley Distribution

Distribution function for the one-parameter Lindley distribution with parameter theta.

Usage

dlindley(x, theta = 1, log = FALSE)

plindley(q, theta = 1, lower.tail = TRUE, log.p = FALSE)

qlindley(p, theta = 1, lower.tail = TRUE, log.p = FALSE)

rlindley(n, theta = 1)

Arguments

x

a single value or vector of positive values.

theta

distribution parameter value. Default is 1.

log, log.p

logical; If TRUE, probabilities p are given as log(p). If FALSE, probabilities p are given directly. Default is FALSE.

q

a single value or vector of quantiles.

lower.tail

logical; If TRUE, (default), \(P(X \leq x)\) are returned, otherwise \(P(X > x)\) is returned. Default is TRUE.

p

a single value or vector of probabilities.

n

number of random values to generate.

Value

dlindley gives the density, plindley gives the distribution function, qlindley gives the quantile function, and rlindley generates random deviates.

The length of the result is determined by n for rlindley, and is the maximum of the lengths of the numerical arguments for the other functions.

Details

Probability density function (PDF) $$f(x\mid \theta )=\frac{\theta ^{2}}{(1+\theta )}(1+x)e^{-\theta x}$$

Cumulative distribution function (CDF) $$F(x\mid \theta ) = 1 - \left(1+ \frac{\theta x}{1+\theta }\right)e^{-\theta x}$$

Quantile function (Inverse CDF) $$ Q(p\mid\theta) = -1 - \frac{1}{\theta} - \frac{1}{\theta} W_{-1}\!\left((1+\theta)(p-1)e^{-(1+\theta)}\right) $$

where \(W_{-1}()\) is the negative branch of the Lambert W function.

The moment generating function (MGF) is: $$M_X(t)=\frac{\theta^2(\theta-t+1)}{(\theta+1)(\theta-t)^2}$$

The distribution mean and variance are: $$\mu=\frac{\theta+2}{\theta(1+\theta)}$$ $$\sigma^2=\frac{\mu}{\theta+2}\left(\frac{6}{\theta}-4\right)-\mu^2$$

Examples

x <- seq(0, 5, by = 0.1)
p <- seq(0.1, 0.9, by = 0.1)
q <- c(0.2, 3, 0.2)
dlindley(x, theta = 1.5)
#>  [1] 0.000000000 0.852100897 0.800083678 0.746024937 0.691502661 0.637694846
#>  [7] 0.585460310 0.535404756 0.487934623 0.443300846 0.401634288 0.362974327
#> [13] 0.327291799 0.294507328 0.264505885 0.237148255 0.212280011 0.189738448
#> [19] 0.169357892 0.150973677 0.134425085 0.119557434 0.106223522 0.094284540
#> [25] 0.083610591 0.074080899 0.065583793 0.058016508 0.051284873 0.045302912
#> [31] 0.039992388 0.035282311 0.031108444 0.027412793 0.024143116 0.021252450
#> [37] 0.018698645 0.016443944 0.014454571 0.012700355 0.011154385 0.009792681
#> [43] 0.008593906 0.007539091 0.006611389 0.005795854 0.005079239 0.004449808
#> [49] 0.003897178 0.003412165 0.002986656
dlindley(x, theta=0.5, log=TRUE)
#>  [1]      -Inf -1.746449 -1.709438 -1.679395 -1.655287 -1.636294 -1.621756
#>  [8] -1.611131 -1.603973 -1.599906 -1.598612 -1.599822 -1.603302 -1.608850
#> [15] -1.616291 -1.625469 -1.636248 -1.648508 -1.662140 -1.677049 -1.693147
#> [22] -1.710357 -1.728609 -1.747837 -1.767984 -1.788997 -1.810826 -1.833427
#> [29] -1.856758 -1.880783 -1.905465 -1.930772 -1.956675 -1.983144 -2.010155
#> [36] -2.037682 -2.065703 -2.094197 -2.123144 -2.152524 -2.182322 -2.212519
#> [43] -2.243101 -2.274053 -2.305361 -2.337011 -2.368993 -2.401293 -2.433902
#> [50] -2.466807 -2.500000
plindley(q, theta = 1.5)
#> [1] 0.1702836 0.9688948 0.1702836
plindley(q, theta = 0.5, lower.tail = FALSE)
#> [1] 0.9651599 0.4462603 0.9651599
qlindley(p, theta = 1.5)
#> [1] 0.1145570 0.2376153 0.3721440 0.5222804 0.6942468 0.8982645 1.1532321
#> [8] 1.5010908 2.0740189
qlindley(p, theta = 0.5)
#> [1] 0.5440196 1.0430722 1.5435368 2.0718301 2.6536848 3.3240888 4.1429816
#> [8] 5.2395404 7.0163914

set.seed(123154)
rlindley(5, theta = 1.5)
#> Warning: 'n' must be a non-negative integer
#> [1] 1.40543787 0.97518942 0.42922042 0.06554152 0.24698987
rlindley(5, theta = 0.5)
#> Warning: 'n' must be a non-negative integer
#> [1] 2.5089347 2.5340820 1.4101438 0.1856275 3.6330299