This article aims to introduce a generalization of the inverse Rayleigh distribution known as exponentiated inverse Rayleigh distribution (EIRD) which extends a more flexible distribution for modeling life data. Some statistical properties of the EIRD are investigated, such as mode, quantiles, moments, reliability, and hazard function. We describe different methods of parametric estimations of EIRD discussed by using maximum likelihood estimators, percentile based estimators, least squares estimators, and weighted least squares estimators and compare those estimates using extensive numerical simulations. The performances of the proposed methods of estimation are compared by Monte Carlo simulations for both small and large samples. To illustrate these methods in a practical application, a data analysis of real-world coating weights of iron sheets is obtained from the ALAF industry, Tanzania, during January-March, 2018. ALAF industry uses aluminum-zinc galvanization technology in the coating process. This application identifies the EIRD as a better model than other well-known distributions in modeling lifetime data.
In this research life time distribution known as exponentiated inverse Rayleigh distribution (EIRD) was developed and it can be used in reliability estimation and statistical quality control techniques. The Rayleigh distribution is originated from two parameter Weibull distribution and it is appropriate model for life-testing studies. It can be shown by transformation of random variable that if the random variable (r. v)
Suppose X is a random variable following inverse Rayleigh distribution with scale parameter
Nadarajah and Kotz [
The cumulative density function (CDF) of the EIRD is given by
Hence for the exponentiated inverse Rayleigh distribution with the scale parameter
Reliability and hazard functions of EIRD are as follows:
Probability density function of EIRD with different shape parameters.
Cumulative density function of EIRD with different shape parameters.
Curve of reliability function of EIRD with different shape parameters.
Curve of hazard function of EIRD with different values of shape parameter.
In this section, we furnish some significant statistical and mathematical properties of the EIRD such as moments, moment generating function, ordered statistics, mode, quantiles, median, skewness, and kurtosis.
The
The moment generating function is given by
Moments of order statistics have great role in quality control testing and reliability to predict time to fail of a certain item by considering few early failures. Suppose
Mode can be obtained by using the following approach:
Quantiles are very needful for estimation purposes, basically quantile estimators, and also it is used in simulation. The
Suppose U
There are different methods which are used to find skewness and kurtosis in a certain distribution. The most common method is the one which uses moments of the distribution, but in EIR distribution we have first moment only. Due to this reason the appropriate approach of finding kurtosis and skewness is by using quantiles.
When
In this section, different methods which can be used in estimation of parameters
Maximum likelihood estimation (MLE) method is mostly used in many writings. This is because estimates MLE satisfy many properties of good estimator. Some of the properties are consistency, asymptotic efficiency, and invariance property. Let
Information matrix (I) of EIRD is given by
This method was suggested by Swain et al. [
This method was originally proposed by Kao [
In this part, a Monte Carlo simulation study is conducted to evaluate the performance of different estimation method for estimating unknown parameters of EIRD. The performance of the different estimators is evaluated in terms of mean square error (MSE). The simulation is conducted by using R- software, 10000 random samples of EIRD was generated with values of n = (20, 40, 50, 100) while choosing
Average bias values of estimates and their MSE (in brackets) when n=20.
Parameters | MLE’s | LSE’S | WLSE’s | PCE’s |
---|---|---|---|---|
| 0.0720(0.0474) | -0.0050(0.0637) | 0.0065(0.0552) | 0.3384(0.5745) |
| 0.0667(0.0359) | 0.0226(0.0491) | 0.0262(0.0397) | 0.0968(0.0578) |
| 0.0452(0.0225) | -0.0108(0.0295) | -0.0027(0.0258) | -0.0068(0.0347) |
| 0.2957(0.6249) | 0.1433(1.7474) | 0.1402(1.1410) | 0.0733(0.4290) |
| 0.0419(0.0197) | -0.0109(0.0259) | -0.0032(0.0227) | -0.01400(0.0247) |
| 0.4445(1.3893) | 0.2432(5.4847) | 0.2289(3.3600) | 0.0736(0.7986) |
| 0.0400(0.0181) | -0.0109(0.0238) | -0.0034(0.0208) | -0.0172(0.0202) |
| 0.6120(2.6197) | 0.3703(14.2642) | 0.3383(8.3954) | 0.0770(1.3550) |
| 0.1439(0.1896) | -0.0100(0.2549) | 0.0130(0.2208) | 0.6767(2.2979) |
| 0.0667(0.0359) | 0.0226(0.0491) | 0.0262(0.0397) | 0.0968(0.0578) |
| 0.0904(0.0901) | -0.0216(0.1181) | -0.0053(0.1032) | -0.0136(0.1386) |
| 0.2957(0.6250) | 0.1433(1.7474) | 0.1402(1.1410) | 0.0733(0.4290) |
| 0.0839(0.0790) | -0.0218(0.1037) | -0.0064(0.0906) | -0.0280(0.0989) |
| 0.4445(1.3893) | 0.2432(5.4847) | 0.2289(3.3600) | 0.0736(0.7986) |
| 0.0800(0.0723) | -0.0217(0.0952) | -0.0053(0.1032) | -0.0344(0.0807) |
| 0.6120(2.6197) | 0.3703(14.2642) | -0.8598(1.8606) | 0.0770(1.3550) |
Average bias values of estimates and their MSE (in brackets) when n=40.
Parameters | MLE’s | LSE’S | WLSE’s | PCE’s |
---|---|---|---|---|
| 0.0331(0.0203) | -0.0080(0.0271) | 0.0027(0.0231) | 0.3920(0.5673) |
| 0.0304(0.0126) | 0.0067(0.0154) | 0.0122(0.0136) | 0.0792(0.0328) |
| 0.0208(0.0103) | -0.0087(0.0133) | -0.0011(0.0114) | -0.0273(0.0267) |
| 0.1300(0.1877) | 0.0375(0.2478) | 0.0553(0.2118) | -0.0195(0.2064) |
| 0.0193(0.0090) | -0.0085(0.0118) | -0.0013(0.0101) | -0.0291(0.0181) |
| 0.1930(0.3942) | 0.0612(0.5325) | 0.0856(0.4511) | -0.0505(0.3764) |
| 0.0184(0.0083) | -0.0083(0.0108) | -0.0014(0.0092) | -0.0292(0.0143) |
| 0.2630(0.7065) | 0.0898(0.9731) | 0.12067(0.8183) | -0.0819(0.6130) |
| 0.0661(0.0813) | -0.0159(0.1086) | 0.0055(0.0922) | 0.7841(2.2692) |
| 0.0304(0.0126) | 0.0067(0.0155) | 0.0122(0.0136) | 0.0792(0.0328) |
| 0.0415(0.0410) | 0.0174(0.0533) | -0.0021(0.0456) | -0.0546(0.1066) |
| 0.1299(0.1877) | 0.0375(0.2478) | 0.0553(0.2118) | -0.0195(0.2064) |
| 0.0386(0.0361) | -0.0170(0.0470) | -0.0023(0.0402) | -0.0581(0.0723) |
| 0.1930(0.3942) | 0.0612(0.5325) | 0.0856(0.4511) | -0.0505(0.3764) |
| 0.0368(0.0331) | -0.0167(0.0432) | -0.0021(0.0456) | -0.0585(0.0573) |
| 0.2631(0.7065) | 0.0898(0.9731) | -0.9447(1.1011) | -0.0819(0.6130) |
Average bias values of estimates and their MSE (in brackets) when n=50.
Parameters | MLE’s | LSE’S | WLSE’s | PCE’s |
---|---|---|---|---|
| 0.0264(0.0154) | -0.0058(0.0215) | 0.0036(0.0181) | 0.4028(0.5785) |
| 0.0234(0.0091) | 0.0046(0.0112) | 0.0094(0.0097) | 0.0744(0.0281) |
| 0.0166(0.0078) | -0.0066(0.0105) | 0.0001(0.0089) | -0.0327(0.0251) |
| 0.0992(0.1302) | 0.0261(0.1717) | 0.0422(0.1420) | -0.0405(0.1726) |
| 0.0155(0.0069) | -0.0065(0.0093) | -0.0002(0.0079) | -0.0330(0.0169) |
| 0.1470(0.2695) | 0.0431(0.3623) | 0.0650(0.2962) | -0.0776(0.3123) |
| 0.0147(0.0063) | -0.0064(0.0085) | -0.0003(0.0072) | -0.0311(0.0130) |
| 0.1999 (0.4770) | 0.0636(0.6509) | 0.0915(0.5277) | -0.1094(0.5007) |
| 0.0527(0.0615) | -0.0116(0.0860) | 0.0072(0.0722) | 0.8057(2.3139) |
| 0.0234(0.0091) | 0.0046(0.0112) | 0.0094(0.0097) | 0.0744(0.0281) |
| 0.0333(0.0312) | -0.0133(0.0421) | 0.0002(0.0358) | -0.0654(0.1003) |
| 0.0992(0.1302) | 0.0261(0.1718) | 0.04226(0.0142) | -0.0405(0.1726) |
| 0.0309(0.0275) | -0.0130(0.0371) | -0.0003(0.03150) | -0.0660(0.0674) |
| 0.1469(0.2695) | 0.0431(0.3623) | 0.0650(0.2962) | -0.0776(0.3123) |
| 0.0295(0.0252) | -0.0128(0.0341) | 0.0002(0.0358) | -0.0621(0.0518) |
| 0.1999(0.4770) | 0.0636(0.6509) | -0.9579(1.0577) | -0.1094(0.5007) |
Average bias values of estimates and their MSE (in brackets) when n=100.
Parameters | MLE’s | LSE’S | WLSE’s | PCE’s |
---|---|---|---|---|
| 0.0123(0.0071) | -0.0037(0.0102) | 0.0026(0.0083) | 0.4887(0.6267) |
| 0.0110(0.0040) | 0.0016(0.0051) | 0.0051(0.0044) | 0.0732(0.0201) |
| 0.0078(0.0037) | -0.0038(0.0051) | 0.0007(0.0042) | -0.0465(0.0217) |
| 0.0464(0.0546) | 0.0096(0.0732) | 0.0218(0.0608) | -0.0759(0.1151) |
| 0.0073(0.0033) | -0.0037(0.0045) | 0.0005(0.0037) | -0.0398(0.0133) |
| 0.0686(0.1113) | 0.0162(0.1506) | 0.0332(0.1244) | -0.1157(0.2009) |
| 0.0070(0.0030) | -0.0036(0.0041) | 0.0004(0.0034) | -0.0343(0.0095) |
| 0.0931(0.1943) | 0.0242(0.2650) | 0.0462(0.2181) | -0.1487(0.3103) |
| 0.2463(0.02854) | -0.0073(0.0407) | 0.0052(0.0333) | 0.9773(2.5066) |
| 0.0110(0.0040) | 0.0016(0.0051) | 0.0051(0.0044) | 0.0732(0.0201) |
| 0.0156(0.0148) | -0.0076(0.0202) | 0.0013(0.0168) | -0.0930(0.0868) |
| 0.0464(0.0546) | 0.0096(0.0732) | 0.0218(0.0608) | -0.0759(0.1151) |
| 0.0146(0.0131) | -0.0074(0.0178) | 0.0010(0.0148) | -0.0799(0.0533) |
| 0.0686(0.1113) | 0.0162(0.1506) | 0.0332(0.1244) | -0.1157(0.2009) |
| 0.0139(0.0120) | -0.0072(0.0164) | 0.0013(0.0168) | -0.0687(0.0379) |
| 0.0931(0.1943) | 0.0242(0.2650) | -0.9783(1.0173) | -0.1487(0.3103) |
We have also obtained bias and MSE estimates of reliability function and hazard rate functions using MLE. In Table
Bias and MSE of reliability and hazard functions obtained from different values of
| Parameter | RF | HF | |||
---|---|---|---|---|---|---|
| | Bias | MSE | Bias | MSE | |
20 | 1.0 | 0.5 | -0.0084 | 0.0062 | 0.0301 | 0.0077 |
1.0 | 1.5 | 0.0017 | 0.0003 | 0.1378 | 0.1375 | |
1.0 | 2.5 | 0.0009 | 0.0000 | 0.2873 | 0.5794 | |
2.0 | 0.5 | -0.0028 | 0.0079 | 0.0212 | 0.0050 | |
2.0 | 1.5 | -0.0044 | 0.0028 | 0.1103 | 0.0928 | |
2.0 | 2.5 | 0.0012 | 0.0005 | 0.2356 | 0.3965 | |
40 | 1.0 | 0.5 | -0.0045 | 0.0031 | 0.0138 | 0.0028 |
1.0 | 1.5 | 0.0010 | 0.0001 | 0.0608 | 0.0419 | |
1.0 | 2.5 | 0.0005 | 0.0000 | 0.1238 | 0.1585 | |
2.0 | 0.5 | -0.0019 | 0.0038 | 0.0099 | 0.0019 | |
2.0 | 1.5 | -0.0024 | 0.0015 | 0.0491 | 0.0295 | |
2.0 | 2.5 | 0.0007 | 0.0003 | 0.1025 | 0.1134 | |
50 | 1.0 | 0.5 | -0.0033 | 0.0025 | 0.0106 | 0.0020 |
1.0 | 1.5 | 0.0008 | 0.0001 | 0.0464 | 0.0292 | |
1.0 | 2.5 | 0.0004 | 0.0000 | 0.0941 | 0.1074 | |
2.0 | 0.5 | -0.0013 | 0.0030 | 0.0077 | 0.0014 | |
2.0 | 1.5 | -0.0018 | 0.0012 | 0.0376 | 0.0208 | |
2.0 | 2.5 | 0.0006 | 0.0002 | 0.0780 | 0.0777 | |
100 | 1.0 | 0.5 | -0.0016 | 0.0012 | 0.0050 | 0.0009 |
1.0 | 1.5 | 0.0004 | 0.0001 | 0.0217 | 0.0123 | |
1.0 | 2.5 | 0.0002 | 0.0000 | 0.0439 | 0.0439 | |
2.0 | 0.5 | -0.0007 | 0.0015 | 0.0036 | 0.0006 | |
2.0 | 1.5 | -0.0009 | 0.0006 | 0.0176 | 0.0089 | |
2.0 | 2.5 | 0.0003 | 0.0001 | 0.0365 | 0.0322 |
In this section we investigate two data sets of coating weights (gm/m∧2) from ALAF industry, Tanzania. ALAF industry is the part of Safal group which is the major producer of the most accepted and trusted steel roofing brand. Safal group is operated in the 11 countries which are found in the Eastern and Southern part of the African continent. Safal group brought a major advanced coating technology in Africa with four coating mills located at Kenya, Uganda, Tanzania, and South Africa. ALAF industry is the one of the coating mills of Safal group, and then it deals with improving the quality steel roofing. There are several processes which are done in order to improve the quality of steel roofing; one of those processes is coating process. ALAF industry uses aluminum-zinc galvanization technology in the coating process. We analyze two data sets to illustrate the model validity on EIRD. For the first data set of 72 observations on coating weight by chemical method on top center side (TCS) and for the second data set of 72 observations on coating weight by chemical method on bottom center side (BCS), the data sets are presented in Table
Coating weight by chemical method on Tcs and Bcs.
Tcs |
|
36.8 47.2 35.6 36.7 55.8 58.7 42.3 37.8 55.4 45.2 31.8 48.3 45.3 48.5 52.8 45.4 49.8 48.2 54.5 50.1 48.4 44.2 41.2 47.2 39.1 40.7 40.3 41.2 |
30.4 42.8 38.9 34.0 33.2 56.8 52.6 40.5 40.6 45.8 58.9 28.7 37.3 36.8 40.2 58.2 59.2 42.8 46.3 61.2 58.4 38.5 34.2 41.3 42.6 43.1 42.3 54.2 |
44.9 42.8 47.1 38.9 42.8 29.4 32.7 40.1 33.2 31.6 36.2 33.6 32.9 34.5 33.7 39.9 |
|
Bcs |
|
45.5 37.5 44.3 43.6 47.1 52.9 53.6 42.9 40.6 34.1 42.6 38.9 35.2 40.8 41.8 49.3 38.2 48.2 44.0 30.4 62.3 39.5 39.6 32.8 48.1 56.0 47.9 39.6 |
44.0 30.9 36.6 40.2 50.3 34.3 54.6 52.7 44.2 38.9 31.5 39.6 43.9 41.8 42.8 33.8 40.2 41.8 39.6 24.8 28.9 54.1 44.1 52.7 51.5 54.2 53.1 43.9 |
40.8 55.9 57.2 58.9 40.8 44.7 52.4 43.8 44.2 40.7 44.0 46.3 41.9 43.6 44.9 53.6 |
To verify that this EIRD model is suitable for the data sets which have been used, we have used negative log-likelihood value, Akaike information criteria (AIC), Bayesian information criteria (BIC), K-S distance, and p value,
where
The values of goodness-of-fit measures, the values of negative likelihood obtained by MLE method, and their standard errors (in brackets) of the models fitted are given in the Tables
Statistics of the goodness-of-fit, MLE, and standard errors (SE) for Bcs.
Model | | AIC | BIC | K-S | P-value | Parameter | Estimates (SE) |
---|---|---|---|---|---|---|---|
EIRD | 497.14 | 501.14 | 505.69 | 0.089 | 0.6 | | 18.2307 (5.4750) |
| 78.5609 (4.0494) | ||||||
IRD | 596.16 | 598.16 | 600.43 | 0.39 | 1.00E-09 | | 41.8013 (2.4632) |
RD | 595.14 | 597.12 | 599.41 | 0.36 | 3.00E-08 | | 31.4872 (1.8553) |
IWD | 520.72 | 524.72 | 529.27 | 0.16 | 0.04 | | 4.8144 (3.3126e-02) |
| 4.8266e+07 (8.3886e+03) | ||||||
GIERD | 495.17 | 499.17 | 503.73 | 0.12 | 0.3 | | 505.6008 (279.3040) |
| 289.2005 (26.8417) |
Statistics of the goodness of fit, MLE, and standard errors (SE) for Tcs.
Model | | AIC | BIC | K-S | P-value | Parameter | Estimates (SE) |
---|---|---|---|---|---|---|---|
EIRD | 503.66 | 507.66 | 512.22 | 0.061 | 1 | | 13.1767 (3.5726) |
| 73.0053 (3.7615) | ||||||
IRD | 593.60 | 595.60 | 597.88 | 0.37 | 9.00E-09 | | 40.8745 (2.4086) |
RD | 593.79 | 595.79 | 598.06 | 0.36 | 1.00E-08 | | 31.0157 (1.8276) |
IWD | 512.73 | 516.73 | 521.28 | 0.18 | 0.02 | | 4.0927 (3.3893e-02) |
| 2.3069e+06 (8.3925e+03) | ||||||
GIERD | 505.78 | 509.78 | 514.33 | 0.089 | 0.6 | | 245.9686 (118.6173) |
| 252.0199 (23.1886) |
The strength of fit of EIRD is compared with the other four distributions, namely: IRD, RD, IWD, and GIED, and the results in Tables
Histograms and theoretical densities and Q-Q plot for BCS.
Histograms and theoretical densities and Q-Q plot for TCS.
The parameters of EIRD are estimated by using four different estimation methods, known as, maximum likelihood, least square, weighted least square, and percentile estimation method. The efficiency of the estimation methods differs from one data set to another as shown in Tables
The goodness of fit statistics for parameter estimated under various methods for Tcs.
Method | | | | K-S | p-value |
---|---|---|---|---|---|
MLE | 13.1805 | 73.0100 | 236.9226 | 0.0612 | 0.9502 |
LSE | 10.3468 | 69.7534 | 238.6676 | 0.0544 | 0.9835 |
WLSE | 11.0723 | 70.6468 | 238.0821 | 0.0552 | 0.9805 |
PCE | 11.8341 | 71.6815 | 237.7409 | 0.0611 | 0.9510 |
The goodness of fit statistics for parameter estimated under various methods for Bcs.
Method | | | | K-S | p-value |
---|---|---|---|---|---|
MLE | 18.2130 | 78.5517 | 237.9956 | 0.0887 | 0.6230 |
LSE | 18.5736 | 79.1091 | 238.6264 | 0.0965 | 0.5139 |
WLSE | 17.7225 | 78.6110 | 238.8775 | 0.0986 | 0.4853 |
PCE | 17.1623 | 77.9676 | 238.4305 | 0.0913 | 0.5852 |
EIRD which is derived from this study performs well; from the diagram of the PDF it can be shown that the distribution is positively skewed and the CDF shows the increasing pattern as other distributions. Also by using reliability function it can be seen that the distribution can be used in lifetime studies since reliability graph tends to decrease as the time increases. The hazard function shows the upside down bath-tub curve shape. The unique characteristic of the distribution has only one moment and kurtosis and skewness are found in terms of quantile.
Four methods of estimation was used in parameter estimation; the methods are maximum likelihood, least square, weighted least square, and percentile estimation. From the simulation study it is observed that the method of maximum likelihood is the best compared to other methods since it has minimum value of MSE. Also findings revile that all methods are consistent since the values of bias and MSE decrease as sample size increases. The data set of coating weights for March, 2018 from ALAF industry is used to study the performance of the proposed distribution. It is shown that EIRD is better performed more than the existing distributions namely: IRD, RD, IWD, and GIED.
R codes for estimating parameters of exponentiated inverse Rayleigh distribution # Probability density function deird<-function(x,sigma,alpha) p<-(2 return(p) # cumulative distribution function peird<-function(x,sigma,alpha) d<-1-(1-exp(-(sigma/x)∧2))∧alpha return(d) # quantile function qeird<-function(p,sigma,alpha) q<-sigma/(-log(1-(1-p)∧(1/alpha)))∧(1/2) return(q)
# random number generation reird<-function(n,sigma,alpha) x<-qeird(p=runif(n),sigma,alpha) return(x) # Pdf Curves x <- seq(0, 3, by=.001) plot(x, deird(x, 1,0.5), type=''l'', ylim=c(0,3.5), ylab=''Density'', main='' '',lwd=3) lines(x, deird(x, 1,1.5), col=2,lwd=3) lines(x, deird(x, 1.0, 2.5), col=3,lwd=3) lines(x, deird(x, 1.0, 5.0), col=4,lwd=3) #legend(3,1.7, legend = c(''exp legend(par('usr') c(c(as.expression(substitute(EIRD(sigma==1, alpha==0.5))), as.expression(substitute(EIRD(sigma= =1,alpha==1.5))), as.expression(substitute(EIRD(sigma= =1,alpha==2.5))), as.expression(substitute(EIRD(sigma= =1,alpha==5.0))))), lty =c(2,2,2,2), lwd =c(3,3,3,3), pch=c(20,20,20,20), col=c(1,2,3,4))
# CDF curves x <- seq(0, 3, by=.001) plot(x, peird(x, 1,0.5), type=''l'', ylim=c(0,1.1), ylab=''Cumulaive density'', main='' '',lwd=2) lines(x, peird(x, 1,1.5), col=2,lwd=3) lines(x, peird(x, 1.0, 2.5), col=3,lwd=3) lines(x, peird(x, 1.0, 5.0), col=4,lwd=3) #legend(3,1.7, legend = c(''exp legend(1.5,0.4, c(c(as.expression(substitute(EIRD (sigma==1,alpha==0.5))), as.expression(substitute(EIRD(sigma ==1,alpha==1.5))), as.expression(substitute(EIRD(sigma ==1,alpha==2.5))), as.expression(substitute(EIRD(sigma ==1, alpha==5.0))))), lty =c(2,2,2,2), lwd =c(3,3,3,3), pch=c(20,20,20,20), col=c(1,2,3,4))
# Reliability curves x <- seq(0, 3, by=.001) plot(x, rteird(x, 1,0.5), type=''l'', ylim=c(0,1.0), ylab=''R(t)'',xlab=''t'', main='' '',lwd=3) lines(x, rteird(x, 1,1.5), col=2,lwd=3) lines(x, rteird(x, 1.0, 2.5), col=3,lwd=3) lines(x, rteird(x, 1.0, 5.0), col=4,lwd=3) #legend(3,1.7, legend = c(''exp legend(par('usr') c(c(as.expression(substitute(EIRD(sigma ==1, alpha==0.5))), as.expression(substitute(EIRD(sigma ==1, alpha==1.5))), as.expression(substitute(EIRD(sigma ==1, alpha==2.5))), as.expression(substitute(EIRD(sigma ==1, alpha==5.0))))), lty =c(2,2,2,2), lwd =c(3,3,3,3), pch=c(20,20,20,20), col=c(1,2,3,4)) # Hazard function curves x <- seq(0, 3, by=.001) plot(x, heird(x, 1,0.5), type=''l'', ylim=c(0,6), ylab=''h(t)'',xlab=''t'', main='' '',lwd=3) lines(x, heird(x, 1,1.5), col=2,lwd=3) lines(x, heird(x, 1.0, 2.5), col=3,lwd=3) lines(x, heird(x, 1.0, 5.0), col=4,lwd=3) #legend(3,1.7, legend = c(''exp legend(par('usr') c(c(as.expression(substitute(EIRD(sigma ==1, alpha==0.5))), as.expression(substitute(EIRD(sigma ==1, alpha==1.5))), as.expression(substitute(EIRD(sigma ==1, alpha==2.5))), as.expression(substitute(EIRD(sigma ==1, alpha==5.0))))), lty =c(2,2,2,2), lwd =c(3,3,3,3), pch=c(20,20,20,20), col=c(1,2,3,4))
### To find estimation EIRD_MLE<-function(B,n,psigma,palpha) require(maxLik) old = options(digits=20) set.seed(2018) #Setting seed o<-0 ite<-0 mps <- matrix(nrow=B,ncol=2)
rIR<-function(n,sigma,alpha) U<-runif(n,0,1) t<-sigma/sqrt(-log(1-((1-U)∧(1/alpha)))) return(t)
LL_eird1<-function(theta,x) n<-length(x) sigma<-theta alpha<-theta -n (theta mles<-optim(c(55,8.52), LL_eird1,x=Tds, hessian=TRUE,control= list(maxit=10000)) ################## Monte Carlo Simulation ####################### while(o<B) amps<-NULL x<-rIR(n, psigma, palpha) mles<-try(optim(c(psigma, palpha), LL_eird1,x=x, hessian=TRUE,control= list(maxit=10000)) if(is.double(mles o<-o+1 mps[o,] <- mles # cat(o,'' '',ite+1, ''∖n'') ite<-ite+1 old = options(digits=4) mediaemv<-c(mean(mps round(mediaemv,4) sbias<-mean(mps abias<-mean(mps smse<-sbias∧2+(sd(mps amse<-abias∧2+(sd(mps print(c(n,psigma, palpha,sbias,abias,smse,amse))
Exponentiated Inverse Rayleigh Distribution
Random variable
Probability Density Function
Cumulative Density Function
Maximum Likelihood Estimator
Least Square Estimator
Percentile Estimator
Weighted Least Square Estimator
Mean Square Error.
The used data sets are given in the article.
The authors declare that they have no conflicts of interest.
Methodology and computations are done by GS Rao and writing and data collection done by M. Sauda. Both authors read and approved the final manuscript.