On the Mixture of Normal and Half-Normal Distributions

In this research, we studied the mixture of normal and half-normal distributions and introduced some properties for this mixture. In particular, we derived the mean, median, andmode of the mixture of normal and half-normal distributions.We also focused on exploring the Bayesian estimation of parameters of the mixture of normal and half-normal distributions by using dierent methods and then, using type-I censored sample units, presented a simulation study on the mixture of normal and halfnormal distributions.


Introduction
In this research work, we will use the mixture model with normal and half-normal distributions. e nite mixture is one of most popular models and tools in statistics to model data with subgroups [1], where the nite mixture model has been used in di erent areas of application, such as clustering [2] and classi cation [3] as well as other areas. For more details on mixture distributions, see Everitt and Hand [4] and Peel and McLachlan [1]. Finite mixture can be used as a tool to mixing di erent distributions to modeling data, for example, see Zhai et al. [5] and Ni et al. [6]. Knowing that using the mixture model with statistical distribution creates a new distribution with its properties, we have proposed in this paper to use a mixture model with two di erent component mixtures where the rst component mixture follows the normal distribution, while the second mixture component follows the half-normal distribution. ere has been considerable research into using the mixture model with normal distribution as well as into using it with halfnormal distribution. For the mixture of normal distribution, Day [7] estimated the mixture components for normal distribution, while [8] showed some results for the estimation of the maximum likelihood for the normal mixture.
Stephens and Phil [9] presented Bayesian methods to study the mixture of normal distribution. For the mixture of two half-normal components, Sindhu et al. [10] showed the Bayesian inference using type-I censoring. In this work, we will show the properties of the mixture of normal and halfnormal distributions including the mean, mode, and the median, as well as the maximum likelihood for the mixture distributions. We will also show some results on the Bayesian estimation for the mixture of normal and halfnormal distributions using type-I censoring. e structure of our paper is as follows. In Section 2, we show some of the important properties of the mixture of normal and halfnormal distributions. In Section 3, we estimate the maximum likelihood of the mixture. In Section 4, we present a simulation study on the mixture, and in Section 5, we state our conclusions.

e Expected Value and Variance and Moments.
In this section, we show some properties of the mixture of normal and half-normal distributions, such as the mean, variance, skewness, kurtosis, and the moments. e moments of the mixture of normal and half-normal can be given by solving the next equation where r is indicated to be the rate of the moments: where the first component of the mixture follows the normal distribution and the second component of the mixture follows the half-normal distribution, and for short notation, we rewrite (9) as follows: where 2 Mathematical Problems in Engineering Lemma 1. e moments about zero for any odd numbers of the mixture of normal and half-normal distributions can be given by Proof. Here we will explore the moments about zero for the mixture of normal and half-normal distributions (10) by focusing on I 1 and I 2 . When r is odd number, let r � 2n + 1, and suppose y � x/σ⇒x � σy⇒dx � σdy; then, we get that Now, by using (13) and (14) in (10), we can say that the odd moments of the mixture of normal and half-normal distributions can be given by e moments about the zero of any even number of the mixture of normal and half-normal distributions can be given by Proof. Here we will explore the moments about zero for the mixture of normal and half-normal distributions (10) by focusing on I 1 and I 2 . When r is even number, let r � 2n; then, we get that and Mathematical Problems in Engineering 3 By using (17) and (18) in (10), then the even moments of the mixture of normal and half-normal distributions can be given by Now we derived the first four moments of the mixture of normal and half-normal distributions as follows: where the mean of the mixture of normal and half-normal distributions can be obtained by μ 1 ′ � m 2 σ 2 ��� 2/π √ and the variance, skewness, and kurtosis can be given by solving the following equation, respectively: □

Maximum Likelihood Function
In this section, ordinary type-I censoring is performed using a fixed life-test termination time; more details about censoring can be found in Bernardo and Smith [11]. In the proposed mixture model, it is assumed that n units are employed in the life test under a fixed termination time T. When the test is proceeding, it can be noticed that out of h units, v units fail by the termination time T; however, the residual units (h − v) remain working. In several real-life cases, the failure objects can be considered as parts of the first and second subpopulations, according to the sampling scheme suggested by Mendenhall and Hader [12]. Hence, out of v units, v 1 is considered to be a member of subpopulation I and v 2 is considered to be a member of subpopulation II. It can be observed that v 1 and units remain working without giving information about the population. Assuming that the time of failure of the i th unit from the subpopulation is represented by the likelihood function is represented as in the following [13]: Mathematical Problems in Engineering If we assume then we obtain

Bayesian Estimation by Using the Prior Function.
In this section, prior distribution, Bayesian estimation, and loss function will be briefly discussed. One of the important settings in Bayesian analysis is the prior selection for the unknown parameters. Also, the experimental data are regarded as the other important component that forms the relationship between loss function and the prior distribution. e Bayesian estimation (BE) for the distribution parameters under loss functions such as the squared error loss function is studied. e parameters are assumed to have gamma priors (see (32) for computing the estimates of the parameters under SE loss function). e mixture of normal and half-normal distributions involves mixing the proportion parameter and two scale parameters. In this mixture model, the joint prior distribution of the model parameters σ 1 , σ 2 is assumed to be independent, whereas b represents the hyperparameter, and m 1 is uniformly distributed with a range of 0 to one, and the joint prior distribution function can be expressed as follows: Likelihood function (25) and prior distribution (26) are combined and result in the following joint posterior density function of the parameters σ 1 , σ 2 and m 1 : Mathematical Problems in Engineering e marginal distribution of the parameter σ 1 can be obtained by integrating (28) with respect to σ 2 and m 1 , as follows: In a similar way, the marginal distribution of σ 2 and m 1 can be obtained, as follows: 6 Mathematical Problems in Engineering

Markov Chain Monte Carlo Algorithm (MCMC).
As we can see, equations (27)-(31) cannot be valuated due to their high complexity. So, we turned to using the Metropolis-Hastings algorithm which is one of the MCMC algorithms. For more details about MCMC, see Gilks et al. [14]. We also made use of this algorithm to find the credible interval. e BE of the function of parameters U � U(Θ), Θ � (σ 1 , σ 2 , m 1 ) under the SE loss function (LF) is given by Actually, the integration in (32) cannot be solved, so we must use the MCMC algorithm to evaluate the integration in (32) for the three parameters (see Algorithm 1). en, the BE of U(σ 1 , σ 2 , m 1 ) using MCMC under SE is where M is the burn-in period.

Simulation Study
Here we performed a simulation study to illustrate the behavior of the proposed estimators using the methods for the mixture components of normal and half-normal distributions that developed in the previous sections. In this part of the paper, we made a simulation study to estimate the parameters of our mixture distributions using the classical and Bayesian estimation methods under a type-I censoring scheme. We used different censoring times and sample sizes: we used T � 13, 15 { }. Samples of different sample sizes h � 100, 80, 60, 40 { } were generated from the mixture of normal distribution and half-normal distribution. In our study, the generation of random variables was based on computer simulation, and the Mathematica software was used. Probabilistic mixing was used to simulate the mixture data. In order to generate the mixture model, the uniform distribution u(0, 1) was used to generate a random number. If u < m 1 , we assume that the observation was taken from F 1 . Otherwise, if u > m 1 , we assume that the observation was taken from F 2 . e hyperparameter values were selected in such a manner that the prior mean became the approximate expected value of the corresponding parameters. All tables and results are tabulated in the following tables.
is simulation was performed using 1000 iterations for MLE and 10, 000 iterations for the Bayesian analysis. To investigate the effect of changing the sample size, the time was set to be T � 13, while the sample sizes were varying (h � 100, 80, 60, 40 e second attempt was to increase the time T � 15. Here we found that in the most cases, the MSE decreased as the sample size gets larger, as shown in Tables 5-8. e concluding remarks shown below were based on the simulation study:

Conclusion
In this research work, we explored the mixture of normal and half-normal distributions and described their properties. We also used different methods to estimate the parameters of the mixture of normal and half-normal distributions and presented a simulation study. We used a variety of different estimation methods and concluded that the Bayesian estimates provided smaller MSEs than the classical method. e smallest CI was the most credible CI, according to the length of the CI. From the results obtained using a simulated study, we also found some reasonable findings, for example, the MSE and CI length of all parameters decrease as the sample size increases. Moreover, the bias behaves in the same manner. Another important finding was that increasing censoring times affects the MSE and CI length by decreasing them. We intend in future works to introduce a novel mixture distribution that can act as a superior model for fitting different kinds of data. We could also extend our work to apply some acceleration models in this distribution and thereby find solutions for some engineering industrial data.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.