Loglog: The Loglog distribution

In reliaR: Package for some probability distributions.

Description

Density, distribution function, quantile function and random generation for the Loglog distribution with shape parameter alpha and scale parameter lambda.

Usage

dloglog(x, alpha, lambda, log = FALSE)
ploglog(q, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
qloglog(p, alpha, lambda, lower.tail = TRUE, log.p = FALSE)
rloglog(n, alpha, lambda)

Arguments

x,q	vector of quantiles.
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
alpha	shape parameter.
lambda	scale parameter.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x].

Details

The loglog(Pham) distribution has density

f(x) = α ln(λ) x^{α - 1} λ^{x^α} exp{1 - λ^(x^α)}; x > 0, λ > 0, α > 0

where α and λ are the shape and scale parameters, respectively. (Pham, 2002)

Value

dloglog gives the density, ploglog gives the distribution function, qloglog gives the quantile function, and rloglog generates random deviates.

References

Pham, H.(2002). A Vtub-Shaped Hazard Rate Function with Applications to System Safety, International Journal of Reliability and Applications. ,Vol. 3, No. l, pp. 1-16.

Pham, H.(2006). System Software Reliability, Springer-Verlag.

See Also

.Random.seed about random number; sloglog for Loglog survival / hazard etc. functions;

Examples

data(sys2)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(sys2)
## alpha.est = 0.9058689 lambda.est = 1.0028228

dloglog(sys2, 0.9058689, 1.0028228, log = FALSE)
ploglog(sys2, 0.9058689, 1.0028228, lower.tail = TRUE, log.p = FALSE)
qloglog(0.25, 0.9058689, 1.0028228, lower.tail=TRUE, log.p = FALSE)
rloglog(30, 0.9058689, 1.0028228)

Example output

 [1] 0.0022032455 0.0021133440 0.0020511397 0.0019685570 0.0018755802
 [6] 0.0018590586 0.0018198975 0.0017919266 0.0017869154 0.0017798682
[11] 0.0017751687 0.0017302942 0.0017160115 0.0016838918 0.0016668051
[16] 0.0016246736 0.0016219026 0.0016162869 0.0016044964 0.0015900221
[21] 0.0015818858 0.0015422637 0.0015347865 0.0015290191 0.0015222006
[26] 0.0015132702 0.0015086772 0.0015026867 0.0014941230 0.0014841321
[31] 0.0014796375 0.0014586740 0.0014455157 0.0014421089 0.0014162513
[36] 0.0013907124 0.0013859532 0.0013719763 0.0013620725 0.0013194967
[41] 0.0013194155 0.0013169973 0.0013001643 0.0012600465 0.0012524235
[46] 0.0012430324 0.0012243413 0.0012170086 0.0012013925 0.0011250976
[51] 0.0011127267 0.0011095841 0.0010837060 0.0010704526 0.0010148497
[56] 0.0009772907 0.0009615468 0.0009252041 0.0008987561 0.0008887743
[61] 0.0008800268 0.0008694449 0.0008379232 0.0008364109 0.0007861297
[66] 0.0007823702 0.0007495651 0.0007409938 0.0007151092 0.0006411933
[71] 0.0006174557 0.0006005120 0.0005125660 0.0004892145 0.0004404733
[76] 0.0004019713 0.0003724226 0.0003273328 0.0003184076 0.0002949493
[81] 0.0002506233 0.0002392433 0.0002367308 0.0002247245 0.0002150311
[86] 0.0001796168
 [1] 0.01165072 0.01738222 0.02314510 0.03426187 0.05409331 0.05874261
 [7] 0.07148517 0.08226432 0.08435792 0.08738965 0.08946934 0.11178820
[13] 0.11986941 0.13982236 0.15143185 0.18282282 0.18501426 0.18949913
[19] 0.19909688 0.21118780 0.21811837 0.25294747 0.25967079 0.26487925
[25] 0.27105795 0.27917780 0.28336319 0.28882881 0.29665046 0.30577878
[31] 0.30988360 0.32898241 0.34090408 0.34397940 0.36713203 0.38960181
[37] 0.39373904 0.40579122 0.41423944 0.44963721 0.44970319 0.45166731
[43] 0.46519812 0.49644661 0.50222530 0.50927498 0.52308005 0.52841440
[49] 0.53962376 0.59153450 0.59952356 0.60153475 0.61782013 0.62597365
[55] 0.65886037 0.67992756 0.68849724 0.70771391 0.72122034 0.72621649
[61] 0.73055005 0.73573723 0.75083878 0.75155037 0.77455710 0.77622755
[67] 0.79052013 0.79417199 0.80499787 0.83429978 0.84322326 0.84945303
[73] 0.87998266 0.88759560 0.90284071 0.91427718 0.92269659 0.93495050
[79] 0.93729123 0.94330963 0.95414936 0.95681884 0.95740190 0.96015640
[85] 0.96234175 0.97002771
[1] 143.1209
 [1] 1.663993133 1.203556845 1.029665169 1.939245463 0.947028679 0.091929928
 [7] 0.077970872 0.002131361 0.896308782 0.937806879 3.092873985 1.461127617
[13] 1.424003948 0.486634171 1.685904863 0.359018206 1.293011426 0.237216869
[19] 0.471854346 0.126115313 0.972164674 1.550816050 0.617255194 0.478770789
[25] 0.386919980 0.354416821 0.153191645 0.067182306 0.194884849 0.908141263

reliaR documentation built on May 1, 2019, 9:51 p.m.