We study in this manuscript a new one-parameter model called sine inverse Rayleigh (SIR) model that is a new extension of the classical inverse Rayleigh model. The sine inverse Rayleigh model is aiming to provide more fitting for real data sets of purposes. The proposed extension is more flexible than the original inverse Rayleigh (IR) model and it hasmany applications in physics and medicine. The sine inverse Rayleigh distribution can havea uni-model and right skewed probability density function (PDF). The hazard rate function (HRF) of sine inverse Rayleigh distribution can be increasing and J-shaped. Several of thenew model’s fundamental characteristics, namely quantile function, moments, incompletemoments, Lorenz and Bonferroni Curves are studied. Four classical estimation methods forthe population parameters, namely least squares (LS), weighted least squares (WLS), maximum likelihood (ML), and percentile (PC) methods are discussed, and the performanceof the four estimators (namely LS, WLS, ML and PC estimators) are also compared bynumerical implementations. Finally, three sets of real data are utilized to compare the behavior of the four employed methods for finding an optimal estimation of the new distribution.
Reference [
and
Much effort has been invested in the literature on estimating the IR model; see, for instance, [
In last years, several extensions for the IR model were established by means of various generalization methods such as beta IR, transmuted IR (TIR), modified IR, transmuted modified IR, Kumaraswamy exponentiated IR, weighted IR and odd Fréchet IR and half logistic IR models as mentioned in [
In the last years, many different statisticians are attracted by generated families of distributions as: sine generated (S-G) by [
For instance, for S-G the CDF and the PDF are
where
Now, we put forward a novel lifetime model with one parameter named sine inverse Rayleigh (SIR) distribution, whose CDF with parameter θ is obtained by employing
where
When the random variable Y has an SIR model, one can define X’s hazard rate function (HRF), reversed HRF, cumulative HRF, and survival function (SF) as
The remaining parts of this manuscript are presented as follows. Section 2 introduces structural characteristics of SIR distribution including; quantile function, moments, incomplete moments, and Lorenz and Bonferroni curves. Section 3 discusses some estimators for SIR distribution parameters on the basis of four different methods of estimations of least squares (LS), weighted least squares (WLS), maximum likelihood (ML), and percentile (PC). Simulation schemes are performed in Section 4. Three sets of data for real-life applicationare utilized for comparing the behavior of the four methods of estimating the new distribution in Section 5. Several concluding remarks close the work.
We study in this part several statistical characteristics of the SIR model.
If Y~SIR then, the quantile function of SIR is
By inserting the expansion
Let
The mgf of Y is
Using
The Lorenz curve and the Bonferroni curve are generated, respectively, from following equations
The Zenga curves are given by
and
The population parameters involved in the SIR model can be estimated by using four different methods of estimation namely; ML, LS, WLS and PC methods.
To obtain the MLEs of the SIR model with a parameter θ, let
The ML equation of the SIR model then becomes
The MLEs of
Let
In the above equations,
The WLSEs of θ can then be derived by calculating the minimum of the sum
Let
regarding θ.
We generate 3000 RS
The parameter θ’s ML, LS, WLS and PC estimates are calculated. Subsequently, the MSEs of the estimate of the unknown parameter are determined. Numerical results are mentioned in The MSEs of ML estimates of θ are the lowest among all determined MSEs in almost every case. The MSEs of all the estimates decrease with increasing sample sizes.
MLEs | LSEs | WLSEs | PCEs | |||||
---|---|---|---|---|---|---|---|---|
Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | |
10 | 0.527 | 0.018 | 0.521 | 0.023 | 0.519 | 0.022 | 0.458 | 0.020 |
20 | 0.513 | 0.008 | 0.509 | 0.009 | 0.511 | 0.009 | 0.473 | 0.010 |
30 | 0.509 | 0.005 | 0.506 | 0.006 | 0.503 | 0.006 | 0.475 | 0.007 |
50 | 0.506 | 0.003 | 0.505 | 0.004 | 0.504 | 0.0032 | 0.477 | 0.004 |
MLEs | LSEs | WLSEs | PCEs | |||||
---|---|---|---|---|---|---|---|---|
Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | |
10 | 0.841 | 0.047 | 0.836 | 0.061 | 0.834 | 0.057 | 0.742 | 0.052 |
20 | 0.821 | 0.021 | 0.821 | 0.027 | 0.810 | 0.0213 | 0.745 | 0.026 |
30 | 0.814 | 0.013 | 0.811 | 0.015 | 0.808 | 0.014 | 0.751 | 0.017 |
50 | 0.807 | 0.008 | 0.805 | 0.009 | 0.807 | 0.0084 | 0.763 | 0.010 |
MLEs | LSEs | WLSEs | PCEs | |||||
---|---|---|---|---|---|---|---|---|
Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | |
10 | 1.585 | 0.169 | 1.555 | 0.201 | 1.553 | 0.182 | 1.386 | 0.177 |
20 | 1.543 | 0.077 | 1.533 | 0.080 | 1.521 | 0.078 | 1.404 | 0.083 |
30 | 1.525 | 0.048 | 1.522 | 0.061 | 1.519 | 0.052 | 1.413 | 0.064 |
50 | 1.516 | 0.027 | 1.512 | 0.034 | 1.510 | 0.030 | 1.434 | 0.037 |
Further we employ three real data sets for assessing the goodness-of-fit of the SIR model with the purpose of comparing the performance of the considered estimation methods.
We consider criteria including maximum likelihood (denoted by −
The parameters of SIR are estimated by the four estimation methods, namely the ML, LS, WLS and PC estimation methods. The efficiency of the estimation methods is the same in the three data sets as shown in
Method | B1 | B2 | B3 | B4 | B5 | ||
---|---|---|---|---|---|---|---|
MLE | |||||||
LSE | 2.377 | 388.36 | 778.72 | 780.435 | 782.533 | 778.894 | |
WLSE | 2.3774 | 388.394 | 778.789 | 780.503 | 782.601 | 778.963 | |
PCE | 0.051 | 517.874 | 1038 | 1039 | 1042 | 1038 |
Method | B1 | B2 | B3 | B4 | B5 | ||
---|---|---|---|---|---|---|---|
MLE | |||||||
LSE | 66.133 | 693.104 | 1388 | 1390 | 1392 | 1388 | |
WLSE | 66.097 | 692.908 | 1388 | 1390 | 1392 | 1388 | |
PCE | 4.117 | 684.116 | 1370 | 1051 | 1053 | 1370 |
Method | B1 | B2 | B3 | B4 | B5 | ||
---|---|---|---|---|---|---|---|
MLE | |||||||
LSE | 6.799 | 258.26 | 518.52 | 520.52 | 522.628 | 518.643 | |
WLSE | 6.4747 | 255.018 | 512.036 | 512.036 | 513.09 | 512.077 | |
PCE | 1.368 | 348.028 | 698.056 | 698.056 | 699.11 | 698.097 |
We study a new model called SIR. Some fundamental characteristics of the model are investigated. We estimate the model parameters according to the ML, LS, WLS and PC methods. Numerical experiments are carried out in order to comparatively explore the performance of the four different estimation methods. Numerical results show that ML performs better than LS, WLS and PC in almost all considered situations. Applications on three sets of real data indicate that the the ML method is superior in terms of the fits to the LS, WLS and PC methods.
The author is grateful to the academic editor and all four reviewers for their valuable comments that have contributed to significantly improve the quality on the article. I would like to thank TopEdit (