# ExpoWeibull: The Exponentiated Weibull(EW) distribution In reliaR: Package for some probability distributions.

## Description

Density, distribution function, quantile function and random generation for the Exponentiated Weibull(EW) distribution with shape parameters `alpha` and `theta`.

## Usage

 ```1 2 3 4``` ```dexpo.weibull(x, alpha, theta, log = FALSE) pexpo.weibull(q, alpha, theta, lower.tail = TRUE, log.p = FALSE) qexpo.weibull(p, alpha, theta, lower.tail = TRUE, log.p = FALSE) rexpo.weibull(n, alpha, theta) ```

## 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. `theta` shape 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 Exponentiated Weibull(EW) distribution has density

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

where α and θ are the `shape` and `scale` parameters, respectively.

## Value

`dexpo.weibull` gives the density, `pexpo.weibull` gives the distribution function, `qexpo.weibull` gives the quantile function, and `rexpo.weibull` generates random deviates.

## References

Mudholkar, G.S. and Srivastava, D.K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability, 42(2), 299-302.

Murthy, D.N.P., Xie, M. and Jiang, R. (2003). Weibull Models, Wiley, New York.

Nassar, M.M., and Eissa, F. H. (2003). On the Exponentiated Weibull Distribution, Communications in Statistics - Theory and Methods, 32(7), 1317-1336.

`.Random.seed` about random number; `sexpo.weibull` for Exponentiated Weibull(EW) survival / hazard etc. functions

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10``` ```## Load data sets data(stress) ## Maximum Likelihood(ML) Estimates of alpha & theta for the data(stress) ## Estimates of alpha & theta using 'maxLik' package ## alpha.est =1.026465, theta.est = 7.824943 dexpo.weibull(stress, 1.026465, 7.824943, log = FALSE) pexpo.weibull(stress, 1.026465, 7.824943, lower.tail = TRUE, log.p = FALSE) qexpo.weibull(0.25, 1.026465, 7.824943, lower.tail=TRUE, log.p = FALSE) rexpo.weibull(30, 1.026465, 7.824943) ```

### Example output

```  [1] 0.15535555 0.32433811 0.32623451 0.36714164 0.16973307 0.25309842
[7] 0.22375718 0.29929778 0.31848810 0.25309842 0.07857711 0.38088307
[13] 0.23821739 0.23274151 0.37814691 0.22198276 0.25687482 0.40435163
[19] 0.24561019 0.04849526 0.14852624 0.37798114 0.28375795 0.27988247
[25] 0.20299742 0.28181930 0.36220878 0.33746495 0.28763957 0.23274151
[31] 0.20299742 0.31092214 0.09742125 0.21322902 0.35883653 0.21670644
[37] 0.21670644 0.30317993 0.35712705 0.17575185 0.24561019 0.38898980
[43] 0.35883653 0.35191198 0.38505383 0.31092214 0.31861535 0.40661032
[49] 0.30705567 0.40808426 0.29744629 0.15815433 0.27988247 0.27863777
[55] 0.12125393 0.32052810 0.04800260 0.15815433 0.40139284 0.35468275
[61] 0.23821739 0.34923923 0.06272063 0.02448390 0.38574668 0.35468275
[67] 0.41108127 0.22149497 0.17883686 0.38207513 0.40661032 0.20019017
[73] 0.04032162 0.37033957 0.20446402 0.18348829 0.40661032 0.37814691
[79] 0.23413986 0.08173712 0.40139284 0.00207999 0.15815433 0.37033957
[85] 0.07491453 0.35987976 0.31477639 0.05941018 0.41131239 0.39658549
[91] 0.34923923 0.16191225 0.41131239 0.35987976 0.40931946 0.40602217
[97] 0.29735516 0.30899024 0.41120976 0.16242446
[1] 0.8422438155 0.6164118467 0.6131589785 0.5332998866 0.8259973736
[6] 0.7232126432 0.7613386642 0.6569541706 0.1340486637 0.7232126432
[11] 0.9237928693 0.4996259311 0.7428627315 0.7499269617 0.2112165974
[16] 0.7635673574 0.7181129445 0.2819481154 0.7331865155 0.9537515346
[21] 0.8498398633 0.5072147292 0.6802762496 0.6859126417 0.7869320809
[26] 0.6831041345 0.5442405349 0.5932466175 0.6745622828 0.7499269617
[31] 0.7869320809 0.6386472201 0.9044996578 0.7744468176 0.5514510943
[36] 0.7701475174 0.7701475174 0.6509293851 0.5550309247 0.8190881828
[41] 0.7331865155 0.4765250492 0.5514510943 0.5656668244 0.4881363136
[46] 0.6386472201 0.6260562664 0.4047528930 0.6448270160 0.3022651669
[51] 0.1155616947 0.8391087806 0.6859126417 0.1011551591 0.0251194916
[56] 0.6228605451 0.9542340199 0.8391087806 0.2698605782 0.1745223793
[61] 0.7428627315 0.1674827518 0.0097198419 0.9769728624 0.2264978914
[66] 0.1745223793 0.3350554832 0.0660847221 0.0460665213 0.2188192456
[71] 0.4047528930 0.0555418078 0.9617235113 0.5259249483 0.0575650813
[76] 0.8101081598 0.4047528930 0.2112165974 0.0729196068 0.9205869620
[81] 0.2698605782 0.0001228907 0.8391087806 0.5259249483 0.0124689026
[86] 0.1816684182 0.6323902119 0.9429957431 0.3473925607 0.2538976948
[91] 0.1674827518 0.0392522986 0.3473925607 0.1816684182 0.3843501682
[96] 0.2900522521 0.6599374360 0.6417467844 0.3556181185 0.8343003146
[1] 1.790155
[1] 2.4583289 4.7313768 5.6982148 2.8302478 1.6173978 1.7200807 3.3184794
[8] 1.9733880 2.9305439 1.2556116 1.2311626 2.3605360 2.4232444 1.4300929
[15] 2.5499523 4.0088037 3.1565007 2.6940529 6.5123279 1.2155381 3.7420957
[22] 2.0644640 0.9040087 2.1888215 3.2125522 0.9007149 1.4249401 1.5286497
[29] 2.5812185 3.7376725
```

reliaR documentation built on May 29, 2017, 12:34 p.m.