Statistical Inference of Sine Inverse Rayleigh Distribution

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.

For instance, for S-G the CDF and the PDF are where gðy; nÞ considers a PDF of baseline distribution. 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 (2) in (3) as Likewise, by combining (1), (2) and (4), one obtains the corresponding PDF to (5) as where θ is a scale parameter.
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.

Fundamental Properties
We study in this part several statistical characteristics of the SIR model.

Quantile Function
If Y~SIR then, the quantile function of SIR is ! À1 s : and by taking u = 0.5 we get the median (M) as M ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi h½lnð3Þ À1 q :

Moments
Theorem 1: Assume that Y is an r.v. from SIR, thus the r th moment of SIR distribution is Proof: Assume that Y is an r. v. with pdf (6). One can determine r th moments of SIR distribution from l r ¼ By inserting the expansion cos½GðyÞ ¼ P 1 i¼0 ðÀ1Þ i ð2iÞ! ½GðyÞ 2i ; n to the previous equation then, The last equation can be rewritten as Then, The mgf of Y is obtain the incomplete moments, denoted by φ s (t), of the SIR distribution as follows, where φ s (t), defined by Using (8), φ s (t) will be as given where Àðs; tÞ ¼ R t 0 y sÀ1 e Ày dy denotes the lower incomplete gamma function. The Lorenz curve and the Bonferroni curve are generated, respectively, from following equations The Zenga curves are given by

Statistical Inference
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.

ML Estimators
To obtain the MLEs of the SIR model with a parameter θ, let Y 1 ,…, Y n be observed values of this model. The log-likelihood function denoted by ℓ, can be expressed as The ML equation of the SIR model then becomes The MLEs of θ are then obtained by equating ∂ℓ/∂θ with zero and solving simultaneously these equations.

LS and WLS Estimators
Let Y 1 , Y 2 , …, Y n be an n-sized random sample (RS) from SIR model and denote the ordered samples in the RS by Y (1) , Y (2) , …, Y (n) . The expectation and variance of this model do not depend the unknown parameter given by EðFðY ðiÞ ÞÞ ¼ i n þ 1 ; and varðFðY ðiÞ ÞÞ ¼ iðn À i þ 1Þ ðn þ 1Þ 2 ðn þ 2Þ ; In the above equations, F(Y (i) ) represents the CDF of the model while Y (i) represents the i th order statistic (OS). Thus, the LSEs can be determined by obtaining the least sum of squared errors as follows, regarding θ.

Numerical Results
We generate 3000 RS Y 1 , …, Y n of sizes n = 10, 20, 30 and 50 from SIR were generated. Three different values of the parameter θ are chosen: 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 Tabs. 1-3 and the following observations can be made.
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.

Applications to Real Data
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 −',) (B1), the Akaike information criterion (AIC) (B2), the consistent AIC (B3), the Schwarz criterion (B4) and the Hannan-Quinn information criterion (B5). The model that has lowest B1-B5 values is deemed the best one with respect to fitting the real data.
The first data set: contains the survival time (days) of 72 guinea pigs with virulent tubercle bacilli infection, originally presented in [17].
The second data set: consists the waiting time (minutes) of 100 bank customers before they were served, as presented in [18].
The third data set: contains 100 results on the breaking stress (Gba) of carbon fibers, as presented in [19].
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 Tabs. 4-6. We see that among the estimation methods adopted, ML provides the best results for the included data sets.

Conclusion
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.