# Prediction from Transmuted Rayleigh Distribution in the Presence of Outliers

Authors: Tahani Ahmad Aloafi
Journal: Journal of Mathematics
 (2022)

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## Abstract

The quality of the procedures used in statistical analysis depends largely on the assumed probability distribution. However, there are still many problems with data that do not follow any of the classical distributions; therefore, researchers have developed many standardized probability distributions by generalizing or transforming them. Transmuted Rayleigh distribution extends the Rayleigh distribution in the analysis of data and provides larger flexibility in modeling real data. In this article, Bayesian predictive intervals for order statistics of future observations from this distribution are obtained in the presence of outliers when the scale parameter is unknown. The slippage outlier model is utilized in addition to the two-sample prediction scheme. We shall consider two cases: (i) a single outlier in the informative sample and (ii) multiple outliers in the future sample. Numerical computations are obtained to illustrate the effect of outliers on the Bayesian predictive intervals.


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## Body

## 1. Introduction

Statistical prediction is the process by which the values for unknown observations are inferred based on current observations and other information at hand. Although the amount of literature in prediction is vast, just a little of it has been in prediction in the presence of outliers; for example, Lingappaiah [1] obtained the predictive distribution of the rth order statistics from gamma distribution when an outlier is present and Dixit [2] obtained the predictive distribution of the rth order statistics for the future sample from Weibull distribution in the presence of k outliers. Jaheen [3] attained the Bayesian prediction bounds for future observations from Burr type X distribution in the presence of a single outlier, and AL-Hussaini and Jaheen [4] obtained the Bayesian prediction bounds for some order statistics of future observations from Burr type XII distribution in the presence of a single outlier.There has been increased attention on the problem of outliers recently, but most studies have been undertaken for a single outlier. Childs [5] provided the probability density function (pdf) of the rth order statistics and joint pdf of the rth and sth order statistics in the presence of multiple-outlier models.The Rayleigh model plays a valuable role and has a wide application, such as in life-testing experiments, communication engineering, medicine, and finance. Despite this importance, there are still many problems with data that do not follow this distribution; therefore, the researchers resorted to studying the expanded and transmuted distributions of the original ones [6]. Shaw and Buckley [7] suggested the quadratic rank transmutation map approach, and many other authors have also studied this transmutation for some distributions. The main purpose of such transmuted of the existing distributions is to increase the model flexibility of the complex data structures as compared to existing probability, where the original distribution has some limitations. Among the authors who study the transmuted distributions, Merovci [8] presented the transmuted Rayleigh distribution (TRD) and studied some properties of it. Additionally, Ahmad et al. [9] introduced a new class of TRD and Dey et al. [10] explored some properties and different methods used to estimate unknown TRD parameters. Furthermore, Mahendra et al. [11] researched single- and double-acceptance sampling plans for truncated life tests based on TRD. However, nothing has appeared in the previous literature regarding prediction intervals for future observations from TRD; thus, in this article, we obtain the Bayesian predictive intervals (BPIs) for the rth future order statistic from TRD when the scale parameter is unknown. In addition, we consider the two-sample prediction scheme and the slippage outlier model. Ultimately, we are interested in two cases: the single outlier in the informative sample and the multiple outliers in the future sample.The significance of this research comes from the importance of the Bayesian inference, the effect of the presence of outliers in the samples, and the importance of transmuted distributions that are more flexible than the original distributions.


## 2. Bayesian Prediction When the Informative Sample Has a Single Outlier

LetX1,X2,…,Xn−1 be n−1 independent observations from TRD σ,λ with pdf.(1)fx;σ,λ=xσ2e−x2/2σ21−λ+2λe−x2/2σ2,x>0,σ>0,λ≤1,and cumulative density function (cdf) Fx is(2)Fx;σ,λ=1−e−x2/2σ21+λe−x2/2σ2,x>0,σ>0,λ≤1.Let Xn be an outlier observation that arises from TRD with pdf g where g=fx;σ/h,λ and where 0<h<1 (this model is a slippage outlier model).The order statistics sampleX1:n,X2:n,…,Xn:n represents the informative sample which has a single outlier. The likelihood function is the joint pdf of X1:n,X2:n,…,Xn:n which can be written as follows:(3)f1,2,…,n:n1x1,x2,…,xn=n−1!∏i=1nfxi∑i=1ngxifxi,−∞<x1<x2<…<xn<∞.(See [12].) By using (1), we have(4)f1,2,…,n:n1x1,x2,…,xn∝1σ2n∏i=1nxitiσe−∑i=1nxi2/2σ2B1σ,xi>0,σ>0,where(5)tiσ=1−λ+2λe−xi2/2σ2,B1σ=∑i=1ngxifxi=∑i=1nh2e−h2−1xi2/2σ2tiσ/htiσ,xi>0,σ>0,λ≤1.LetY1:m,Y2:m,…,Ym:m represent the future sample from TRD σ,λ, and we assume that the two samples (informative and future) are independent.Let the scale parameterσ be the only unknown parameter in TRD σ,λ, and suppose that πσ denotes a prior has the density function of σ in the following form:(6)πσ=ab2b−1Γb1σ2b+1/2e−a/2σ2,σ>0,a,b>0.This function is a square root inverted gamma distribution which “has major advantages over other distributions because of its analytical tractability, richness, and easy interpretability” [13]. Note that selecting the square root inverted gamma prior distribution for the parameter σ is equivalent to choosing gamma prior distribution for the parameter 1/σ2. The parameters a and b are chosen so as to mirror the prior knowledge about the unknown quantities. Waller et al. [14] presented two methods by which experiences, beliefs, and judgments can be used to assign values to the parameters of gamma prior distribution. One method depends on knowledge concerning the failure rate, while the other assumes information about the reliability. Both methods need to provide two percentile values which are used to solve a system of two equations and determine values for the parameters a and b of the prior distribution.The posterior density function, denoted byπ∗σ|x, is then given via multiplying (4) by πσ, as follows:(7)π∗σ|x∝1σ2n+b+1/2∏i=1n(tiσ)e−∑i=1nxi2+a/2σ2B1σ,x,a,b,σ>0,where x=x1,x2,…,xn.The density function ofYr:m,r=1,2,…,m is given by the following equation:(8)fr:my=m!r!m−r!Fyr−11−Fym−rfy,−∞<y<∞.By substituting (1) and (2) for (8) and then expanding binomially in y, we have(9)fr:my|σ=m!r−1!m−r!∑i=0m−rm−ri−1iyσ2e−y2/2σ21−λ+2λe−y2/2σ2×1−e−y2/2σ2i+r−11+λe−y2/2σ2i+r−1,y>0,σ>0,λ≤1.Once again, when expanding1−e−y2/2σ2i+r−1 and 1+λe−y2/2σ2i+r−1 binomially, we get(10)fr:my|σ∝Σ∑w=12τwλyσ2e−zwy2/2σ2,y>0,σ>0,λ≤1,where(11)Σ=∑i=0m−rm−ri∑j=0i+r−1i+r−1j∑l=0i+r−1i+r−1l−1i+jλl,τ1λ=1−λ,τ2λ=2λ,z1=j+l+1,z2=j+l+2.The predictive density function is given by the following equation:(12)f∗yr|x=∫σfr:my|σπ∗σ|xdσ=D1Σ∑w=12τwλ∫0∞y1σ2n+b+3/2e−Σi=1nxi2+a+zwy2/2σ2∏i=1ntiσB1σdσ,xi,y,σ,a,b>0,where D1 is a normalizing constant given by the following equation:(13)D1=1∑∑w=12(τwλ/zw)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσB1σdσ.The predictive survival function is obtained from (12), as follows:(14)PYr≥v|x=D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwv2/2σ2∏i=1ntiσB1σdσ.A1−ρ100% BPI for Yr=Yr:m is PLx≤Yr:m≤Ux=1−ρ, where Lx and Ux are the lower and upper Bayesian prediction bounds for Yr, respectively.We can obtain these bounds by solving the following equations numerically:(15)D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwLx2/2σ2∏i=1ntiσB1σdσ=1−ρ2,D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwUx2/2σ2∏i=1ntiσB1σdσ=ρ2.


### 2.1. Special Cases

(1)
By denotingh=1 in (15), we obtain the 1−ρ100% BPI for the future order statistic Yr:m from TRD, when there is no outlier, by solving the result equations:(16)PYr≥Lx|x=D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwLx2/2σ2∏i=1ntiσdσ=1−ρ2,PYr≥Ux|x=D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwUx2/2σ2∏i=1ntiσdσ=ρ2,where(17)D1=1∑∑w=12(τwλ/zw)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσdσ,Σ=∑i=1m−rm−ri∑j=1i+r−1i+r−1j∑l=1i+r−1i+r−1l−1i+jλl,τ1λ=1−λ,τ2λ=2λ,z1=j+l+1, and z2=j+l+2.(2)
By denotingλ=0 in (15), A 1−ρ100% BPI for the future order statistic Yr:m from the Rayleigh distribution in the presence of a single outlier in the informative sample is given by solving the following result equations:(18)PYr≥Lx|x=D1∑1z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Lx2/2σ2B1σdσ=1−ρ2,PYr≥Ux|x=D1∑1z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Ux2/2σ2B1σdσ=ρ2,where(19)Σ=∑i=1m−rm−ri∑j=1i+r−1i+r−1j−1i+j,z1=j+1,D1=1∑1/z1∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2B1σdσ,B1σ=∑i=1nh2e−h2−1xi2/2σ2.(3)
By indicatingh=1 in (18), the 1−ρ100% BPI for the future order statistic Yr:m from the Rayleigh distribution, when there is no outlier, is given by solving the following result equations:(20)PYr≥Lx|x=D1∑1z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Lx2/2σ2dσ=1−ρ2,PYr≥Ux|x=D1∑1z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Ux2/2σ2dσ=ρ2,where(21)Σ=∑i=1m−rm−ri∑j=1i+r−1i+r−1j−1i+j,z1=j+1,D1=1∑1/z1∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2dσ.(4)
By denotingr=1 in (15), the 1−ρ100% BPI for Y1:m is given by solving the result equations as follows:(22)D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwLx2/2σ2∏i=1ntiσB1σdσ=1−ρ2,D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwUx2/2σ2∏i=1ntiσB1σdσ=ρ2,where(23)D1=1∑∑w=12τλ∫0∞∫0∞y1/σ2n+b+3/2e−∑i=1nxi2+a+zwy2/2σ2∏i=1ntiσB1σdσdy,=1∑∑w=12(τwλ/zw)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσB1σdσ,Σ=∑i=0m−1m−1i∑j=0iij∑l=0iil−1i+jλl,τ1λ=1−λ,τ2λ=2λ,z1=j+l+1,z2=j+l+2.(5)
By signifyingr=m in (15), the 1−ρ100% BPI for Ym:m is given by solving the result equations as follows:(24)D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwLx2/2σ2∏i=1ntiσB1σdσ=1−ρ2,D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwUx2/2σ2∏i=1ntiσB1σdσ=ρ2,where(25)D1=1∑∑w=12τwλ∫0∞∫0∞y1/σ2n+b+3/2e−∑i=1nxi2+a+zwy2/2σ2∏i=1ntiσB1σdσdy,=1∑∑w=12(τwλ/zw)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσB1σdσ,Σ=∑j=0m−1m−1j∑l=0m−1m−1l−1jλl,τ1λ=1−λ,τ2λ=2λ,z1=j+l+1,z2=j+l+2.


### 2.2. Numerical Results

We have chosen the hyperparametersa and b as a=1 and b=1 and then generated a value of σ from (6) σ=1.2866. We have produced a sample of size 5 with λ=0.5 from TRD 0.5,1.2866.The order statistics of this sample are0.6481,1.3799,1.4946,1.51847,1.9835.We have also generated a sample of size 4 from TRD0.5,1.2866. The order statistics of this sample are 0.413329,1.27725,1.58463,3.0119. Additionally, we have used (16) to set the 95% BPI for the future observation y3:4.The results was as following: Thelower=0.5754, an upper=2.8815 and the length of BPI=2.3061. Importantly, when we generate a random value from TRD 0.5,1.2866/0.3, h=0.3, the informative sample becomes 0.6481,1.3799,1.4946,1.51847,4.59967. The box plot suggests that the last observation is an outlier. Moreover, the lower bound, the upper bound and the length of BPI are 0.84473, 4.2139, and 3.36917, respectively. Thus, we note that the length of BPI, when there is an outlier, is longer than the length of BPI when the sample does not have an outlier.


#### 2.2.1. Simulation Study

In this study, we present the lower and upper95% Bayesian prediction bounds for future observations Yr:m from TRD, where r=1 and r=m, and the lengths of BPI for different values of n and m based on order statistics. We consider the informative sample to contain a single outlier. Algorithm 1 is used.Algorithm 1:
(1)
For given values of the prior parametersa and b, where a=1 and b=1, a random value of the parameter σ is generated from the square root inverted gamma distribution which has pdf as in (6).(2)
For a chosen value ofλλ=0 and the generated value of σ obtained in Step 1, we generate a sample of size n−1, where n=5 and 10 from TRD λ,σ given by (1) and a sample of size n=1 from TRD λ,σ/h, where h=0.1,0.2,0.3,0.4,0.5. The union and order of the elements of the two samples give the order statistics sample of size n which has a single outlier and thus it is considered to be the informative sample.(3)
For a chosen value ofλ and the generated value of σ obtained in Step (1), we produce an order statistics sample of size m, where m=4,7, and 10 from TRD λ,σ. This sample is considered to be the future sample.(4)
Based on the pairs of generated order statistics samples of sizen and m, we obtain the lower and upper bounds of BPI for the future observations yr:m, where r=1 and r=m, by solving (14) numerically with ρ=0.05.(5)
We repeat the Steps 2, 3, and 4, 100 times, and then compute the average of intervals bounds and their lengths.The results are presented in Tables1 and 2.Table 1
95% BPI for Y1:m and Ym:m in the presence of a single outlier in the informative sample n=5.
mhY1YmLULengthLULength41.09411.3341.239 9.76863.2042.4350.1.27843.8543.5751.4165.6794.2630.2.16992.3522.1821.2835.1433.8600.3.13641.8871.7501.0744.3063.2320.4.11571.6011.485.90353.6232.7190.5.10731.4851.377.88093.5322.65171.0697.9887.9180.99473.4542.4590.1.21292.9462.7331.6385.4423.8040.2.13201.8271.6951.4604.852 53.3910.3.10201.4111.3091.3264.4063.080 00.4.08 8841.229 41.1401.1913.9602.7680.5.07841.086 01.0071.1523.8302.677101.0578.8195.76 1681.116 83.5862.4690.1.18212.5212.3391.6755.1333.4570.2.10511.4551.3501.6635.0953.4320.3.08971.2411.1511.5024.6033.1000.4.07621.055.97881.3574.1582.8000.5.0670.9279.86091.2993.9812.681Table 2
95% BPI for Y1:m and Ym:m in the presence of a single outlier in the informative sample n=10.
mhY1YmLULengthLULength41.08341.0971.014.87693.1872.3100.1.23393.0532.8191.3744.9963.6210.2.16022.0411.9311.1664.2383.0720.3.16022.0411.9311.0383.7732.7350.4.10051.3911.284.91553.3272.4120.5.10131.3221.221.87693.1872.31071.0710.9346.86351.0553.1762.1210.1.18301.4651.3531.5674.6493.7230.2.11222.3902.2071.3844.1072.7230.3.09131.1921.1001.3504.0052.6550.4.08171.067.98591.2033.5692.3660.5.07891.031.95211.1553.4282.273From Tables1 and 2, we observe that the length of BPI increases when there is a single outlier h≠1 and decreases as h,m, and/or n increases. We also observe that BPI for Ym:m is longer than BPI for Y1:m.


## 2.1. Special Cases

(1)
By denotingh=1 in (15), we obtain the 1−ρ100% BPI for the future order statistic Yr:m from TRD, when there is no outlier, by solving the result equations:(16)PYr≥Lx|x=D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwLx2/2σ2∏i=1ntiσdσ=1−ρ2,PYr≥Ux|x=D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwUx2/2σ2∏i=1ntiσdσ=ρ2,where(17)D1=1∑∑w=12(τwλ/zw)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσdσ,Σ=∑i=1m−rm−ri∑j=1i+r−1i+r−1j∑l=1i+r−1i+r−1l−1i+jλl,τ1λ=1−λ,τ2λ=2λ,z1=j+l+1, and z2=j+l+2.(2)
By denotingλ=0 in (15), A 1−ρ100% BPI for the future order statistic Yr:m from the Rayleigh distribution in the presence of a single outlier in the informative sample is given by solving the following result equations:(18)PYr≥Lx|x=D1∑1z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Lx2/2σ2B1σdσ=1−ρ2,PYr≥Ux|x=D1∑1z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Ux2/2σ2B1σdσ=ρ2,where(19)Σ=∑i=1m−rm−ri∑j=1i+r−1i+r−1j−1i+j,z1=j+1,D1=1∑1/z1∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2B1σdσ,B1σ=∑i=1nh2e−h2−1xi2/2σ2.(3)
By indicatingh=1 in (18), the 1−ρ100% BPI for the future order statistic Yr:m from the Rayleigh distribution, when there is no outlier, is given by solving the following result equations:(20)PYr≥Lx|x=D1∑1z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Lx2/2σ2dσ=1−ρ2,PYr≥Ux|x=D1∑1z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Ux2/2σ2dσ=ρ2,where(21)Σ=∑i=1m−rm−ri∑j=1i+r−1i+r−1j−1i+j,z1=j+1,D1=1∑1/z1∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2dσ.(4)
By denotingr=1 in (15), the 1−ρ100% BPI for Y1:m is given by solving the result equations as follows:(22)D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwLx2/2σ2∏i=1ntiσB1σdσ=1−ρ2,D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwUx2/2σ2∏i=1ntiσB1σdσ=ρ2,where(23)D1=1∑∑w=12τλ∫0∞∫0∞y1/σ2n+b+3/2e−∑i=1nxi2+a+zwy2/2σ2∏i=1ntiσB1σdσdy,=1∑∑w=12(τwλ/zw)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσB1σdσ,Σ=∑i=0m−1m−1i∑j=0iij∑l=0iil−1i+jλl,τ1λ=1−λ,τ2λ=2λ,z1=j+l+1,z2=j+l+2.(5)
By signifyingr=m in (15), the 1−ρ100% BPI for Ym:m is given by solving the result equations as follows:(24)D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwLx2/2σ2∏i=1ntiσB1σdσ=1−ρ2,D1∑∑w=12τwλzw∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zwUx2/2σ2∏i=1ntiσB1σdσ=ρ2,where(25)D1=1∑∑w=12τwλ∫0∞∫0∞y1/σ2n+b+3/2e−∑i=1nxi2+a+zwy2/2σ2∏i=1ntiσB1σdσdy,=1∑∑w=12(τwλ/zw)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσB1σdσ,Σ=∑j=0m−1m−1j∑l=0m−1m−1l−1jλl,τ1λ=1−λ,τ2λ=2λ,z1=j+l+1,z2=j+l+2.


## 2.2. Numerical Results

We have chosen the hyperparametersa and b as a=1 and b=1 and then generated a value of σ from (6) σ=1.2866. We have produced a sample of size 5 with λ=0.5 from TRD 0.5,1.2866.The order statistics of this sample are0.6481,1.3799,1.4946,1.51847,1.9835.We have also generated a sample of size 4 from TRD0.5,1.2866. The order statistics of this sample are 0.413329,1.27725,1.58463,3.0119. Additionally, we have used (16) to set the 95% BPI for the future observation y3:4.The results was as following: Thelower=0.5754, an upper=2.8815 and the length of BPI=2.3061. Importantly, when we generate a random value from TRD 0.5,1.2866/0.3, h=0.3, the informative sample becomes 0.6481,1.3799,1.4946,1.51847,4.59967. The box plot suggests that the last observation is an outlier. Moreover, the lower bound, the upper bound and the length of BPI are 0.84473, 4.2139, and 3.36917, respectively. Thus, we note that the length of BPI, when there is an outlier, is longer than the length of BPI when the sample does not have an outlier.


### 2.2.1. Simulation Study

In this study, we present the lower and upper95% Bayesian prediction bounds for future observations Yr:m from TRD, where r=1 and r=m, and the lengths of BPI for different values of n and m based on order statistics. We consider the informative sample to contain a single outlier. Algorithm 1 is used.Algorithm 1:
(1)
For given values of the prior parametersa and b, where a=1 and b=1, a random value of the parameter σ is generated from the square root inverted gamma distribution which has pdf as in (6).(2)
For a chosen value ofλλ=0 and the generated value of σ obtained in Step 1, we generate a sample of size n−1, where n=5 and 10 from TRD λ,σ given by (1) and a sample of size n=1 from TRD λ,σ/h, where h=0.1,0.2,0.3,0.4,0.5. The union and order of the elements of the two samples give the order statistics sample of size n which has a single outlier and thus it is considered to be the informative sample.(3)
For a chosen value ofλ and the generated value of σ obtained in Step (1), we produce an order statistics sample of size m, where m=4,7, and 10 from TRD λ,σ. This sample is considered to be the future sample.(4)
Based on the pairs of generated order statistics samples of sizen and m, we obtain the lower and upper bounds of BPI for the future observations yr:m, where r=1 and r=m, by solving (14) numerically with ρ=0.05.(5)
We repeat the Steps 2, 3, and 4, 100 times, and then compute the average of intervals bounds and their lengths.The results are presented in Tables1 and 2.Table 1
95% BPI for Y1:m and Ym:m in the presence of a single outlier in the informative sample n=5.
mhY1YmLULengthLULength41.09411.3341.239 9.76863.2042.4350.1.27843.8543.5751.4165.6794.2630.2.16992.3522.1821.2835.1433.8600.3.13641.8871.7501.0744.3063.2320.4.11571.6011.485.90353.6232.7190.5.10731.4851.377.88093.5322.65171.0697.9887.9180.99473.4542.4590.1.21292.9462.7331.6385.4423.8040.2.13201.8271.6951.4604.852 53.3910.3.10201.4111.3091.3264.4063.080 00.4.08 8841.229 41.1401.1913.9602.7680.5.07841.086 01.0071.1523.8302.677101.0578.8195.76 1681.116 83.5862.4690.1.18212.5212.3391.6755.1333.4570.2.10511.4551.3501.6635.0953.4320.3.08971.2411.1511.5024.6033.1000.4.07621.055.97881.3574.1582.8000.5.0670.9279.86091.2993.9812.681Table 2
95% BPI for Y1:m and Ym:m in the presence of a single outlier in the informative sample n=10.
mhY1YmLULengthLULength41.08341.0971.014.87693.1872.3100.1.23393.0532.8191.3744.9963.6210.2.16022.0411.9311.1664.2383.0720.3.16022.0411.9311.0383.7732.7350.4.10051.3911.284.91553.3272.4120.5.10131.3221.221.87693.1872.31071.0710.9346.86351.0553.1762.1210.1.18301.4651.3531.5674.6493.7230.2.11222.3902.2071.3844.1072.7230.3.09131.1921.1001.3504.0052.6550.4.08171.067.98591.2033.5692.3660.5.07891.031.95211.1553.4282.273From Tables1 and 2, we observe that the length of BPI increases when there is a single outlier h≠1 and decreases as h,m, and/or n increases. We also observe that BPI for Ym:m is longer than BPI for Y1:m.


## 2.2.1. Simulation Study

In this study, we present the lower and upper95% Bayesian prediction bounds for future observations Yr:m from TRD, where r=1 and r=m, and the lengths of BPI for different values of n and m based on order statistics. We consider the informative sample to contain a single outlier. Algorithm 1 is used.Algorithm 1:
(1)
For given values of the prior parametersa and b, where a=1 and b=1, a random value of the parameter σ is generated from the square root inverted gamma distribution which has pdf as in (6).(2)
For a chosen value ofλλ=0 and the generated value of σ obtained in Step 1, we generate a sample of size n−1, where n=5 and 10 from TRD λ,σ given by (1) and a sample of size n=1 from TRD λ,σ/h, where h=0.1,0.2,0.3,0.4,0.5. The union and order of the elements of the two samples give the order statistics sample of size n which has a single outlier and thus it is considered to be the informative sample.(3)
For a chosen value ofλ and the generated value of σ obtained in Step (1), we produce an order statistics sample of size m, where m=4,7, and 10 from TRD λ,σ. This sample is considered to be the future sample.(4)
Based on the pairs of generated order statistics samples of sizen and m, we obtain the lower and upper bounds of BPI for the future observations yr:m, where r=1 and r=m, by solving (14) numerically with ρ=0.05.(5)
We repeat the Steps 2, 3, and 4, 100 times, and then compute the average of intervals bounds and their lengths.The results are presented in Tables1 and 2.Table 1
95% BPI for Y1:m and Ym:m in the presence of a single outlier in the informative sample n=5.
mhY1YmLULengthLULength41.09411.3341.239 9.76863.2042.4350.1.27843.8543.5751.4165.6794.2630.2.16992.3522.1821.2835.1433.8600.3.13641.8871.7501.0744.3063.2320.4.11571.6011.485.90353.6232.7190.5.10731.4851.377.88093.5322.65171.0697.9887.9180.99473.4542.4590.1.21292.9462.7331.6385.4423.8040.2.13201.8271.6951.4604.852 53.3910.3.10201.4111.3091.3264.4063.080 00.4.08 8841.229 41.1401.1913.9602.7680.5.07841.086 01.0071.1523.8302.677101.0578.8195.76 1681.116 83.5862.4690.1.18212.5212.3391.6755.1333.4570.2.10511.4551.3501.6635.0953.4320.3.08971.2411.1511.5024.6033.1000.4.07621.055.97881.3574.1582.8000.5.0670.9279.86091.2993.9812.681Table 2
95% BPI for Y1:m and Ym:m in the presence of a single outlier in the informative sample n=10.
mhY1YmLULengthLULength41.08341.0971.014.87693.1872.3100.1.23393.0532.8191.3744.9963.6210.2.16022.0411.9311.1664.2383.0720.3.16022.0411.9311.0383.7732.7350.4.10051.3911.284.91553.3272.4120.5.10131.3221.221.87693.1872.31071.0710.9346.86351.0553.1762.1210.1.18301.4651.3531.5674.6493.7230.2.11222.3902.2071.3844.1072.7230.3.09131.1921.1001.3504.0052.6550.4.08171.067.98591.2033.5692.3660.5.07891.031.95211.1553.4282.273From Tables1 and 2, we observe that the length of BPI increases when there is a single outlier h≠1 and decreases as h,m, and/or n increases. We also observe that BPI for Ym:m is longer than BPI for Y1:m.


## 3. Bayesian Prediction When Outliers Are in the Future Sample

LetX1:n,X2:n,…,Xn:n be the informative sample from TRD λ,σ which has pdf as in (1). The likelihood function is given by the following:(26)f1,2,…,nx1,x2,…,xn=n!∏i=1nfxi,−∞<x1<x2<⋯<xn<∞.By substituting (1) for (26), we get(27)Lσ|x=n!1σ2n∏i=1nxi∏i=1ntiσe−∑i=1nxi2/2σ2,xi,σ>0,where(28)tiσ=1−λ+2λe−xi2/2σ2.LetY1,Y2,…,Ym−p be m−p independent observations from TRD σ,λ and Ym−p+1,…,Ymp be independent observations that arise from TRD σ/h,λ, 0<h<1 (and independent of Y1,Y2,…,Ym−p).The order statisticsY1:m,Y2:m,…,Ym:m represent the future sample which has multiple outliers.Suppose that the prior density function is as in (6).The posterior density function is then given as follows:(29)π∗σ∝1σ2n+b+1/2∏i=1ntiσe−∑i=1nxi2+a/2σ2,x,a,b>0,σ>0.The pdf ofYr:m,r=1,2,…,m in the presence of outliers is given as follows:(30)fr:mpy=∑s=max0,r−p−1minm−p−1,r−1C1FysGyr−s−11−Fym−p−s−1×1−Gyp−r+s+1fy+∑s=max0,r−pminm−p,r−1C2FysGyr−s−11−Fym−p−s×1−Gyp−r+sgy,−∞<y<∞,where(31)C1=m−p!p!s!r−s−1!m−p−s−1!p−r+s+1!,C2=m−p!p!s!r−s−1!m−p−s!p−r+s!.Please refer [5].By substituting (1) and (2) for (30) and then using binomial expansion, we get(32)fr:mpy|σ=∑s=max0,r−p−1minm−p−1,r−1C1∑a=0m−p−s−1m−p−s−1a∑b=0p−r+s+1p−r+s+1b×−1a+byσ2e−y2/2σ21−λ+2λe−y2/2σ21−e−y2/2σ21+λe−y2/2σ2s+a×1−e−h2y2/2σ21+λe−h2y2/2σ2r−s−1+b+∑s=max0,r−pminm−p,r−1C2∑a=0m−p−sm−p−sa∑b=0p−r+sp−r+sb−1a+b×h2yσ2e−h2y2/2σ21−λ+2λe−h2y2/2σ21−e−y2/2σ21+λe−y2/2σ2s+a×1−e−h2y2/2σ21+λe−h2y2/2σ2r−s−1+b,y>0,σ>0,λ≤1.Expanding binomially1−e−y2/2σ2s+a, 1+λe−y2/2σ2s+a, 1−e−h2y2/2σ2r−s−1+b, and 1+λe−h2y2/2σ2r−s−1+b, we get(33)fr:mpy=∑i=12∑i∑j=12τjλ1σ2ye−zijy2/2σ2,y>0,σ>0,λ≤1,where(34)Σ1=∑s=max0,r−p−1minm−p−1,r−1C1∑a=0m−p−s−1m−p−s−1a∑b=0p−r+s+1p−r+s+1b∑c=0s+as+ac×∑d=0s+as+ad∑e=0r−s−1+br−s−1+be∑l=0r−s−1+br−s−1+bl−1a+b+c+eλd+l,Σ2=∑s=max0,r−pminm−p,r−1C2∑a=0m−p−sm−p−sa∑b=0p−r+sp−r+sb∑c=0s+as+ac×∑d=0s+as+ad∑e=0r−s−1+br−s−1+be∑l=0r−s−1+br−s−1+bl−1a+b+c+eλd+lh2,τ1λ=1−λ,τ2λ=2λ,z11=c+d+1+h2e+l,z12=c+d+2+h2e+l,z21=c+d+h2e+l+1,and z22=c+d+h2e+l+2.The predictive density function is given as follows:(35)f∗yr|x=∫σfr:mpy|σπ∗σdσD2∑i=12∑i∑j=12τjλ∫0∞y1σ2n+b+3/2e−∑i=1nxi2+a+zijy2/2σ2∏i=1ntiσdσ,xi,y>0,σ>0,a,b>0,where(36)D2=1∑i=12∑i∑j=12(τjλ/zij)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσdσ.The predictive survival function is obtained from (35) as follows:(37)PYr≥v|x=D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zijv2/2σ2∏i=1ntiσdσ.A1−ρ100% BPI for Yr:m is given by solving the following equations numerically:(38)D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zijLx2/2σ2∏i=1ntiσdσ=1−ρ2,D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zijUx2/2σ2∏i=1ntiσdσ=ρ2.


### 3.1. Special Cases

(1)
By indicatingp=0 in (38), the 1−ρ100% BPI for the future order statistic Yr:m from TRD, when there are no outliers, can be obtained by solving the result equations:(39)D2∑1∑j=12τjλz1j∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1jLx2/2σ2∏i=1ntiσdσ=1−ρ2,D2∑1∑j=12τjλz1j∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1jUx2/2σ2∏i=1ntiσdσ=ρ2,where(40)D2=1∑1∑j=12τjλ∫0∞∫0∞y1/σ2n+b+3/2e−∑i=1nxi2+a+z1jy2/2σ2∏i=1ntiσdσdy,=1∑1∑j=12(τjλ/z1j)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσdσ,Σ1=m!m−r!r−1!∑a=0m−rm−ra∑c=0r−1+ar−1+ac×∑d=0r−1+ar−1+ad−1a+cλd,τ1λ=1−λ,τ2λ=2λ,z11=c+d+1,z12=c+d+2.(2)
By signifyingr=1 in (38), the 1−ρ100% BPI for the smallest future order statistic Y1:m from TRD, when the future sample has outliers, is given by solving the following result equations:(41)D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−Σi=1nxi2+a+zijLx2/2σ2∏i=1ntiσdσ=1−ρ2,D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−Σi=1nxi2+a+zijUx2/2σ2∏i=1ntiσdσ=ρ2,where(42)D2=1∑i=12∑i∑j=12(τjλ/zij)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσdσ,Σ1=m−p∑a=0m−p−1m−p−1a∑b=0ppb∑c=0aac×∑d=0aad∑e=0bbe∑l=0bbl−1a+b+c+eλd+l,Σ2=p∑a=0m−pm−pa∑b=0p−1p−1b∑c=0aac×∑d=0aad∑e=0bbe∑l=0bbl−1a+b+c+eλd+lh2,τ1λ=1−λ,τ2λ=2λ,z11=c+d+1+h2e+l,z12=c+d+2+h2e+l,z21=c+d+h2e+l+1,z22=c+d+h2e+l+2.(3)
By demonstratingλ=0 in (38), the 1−ρ100% BPI for the future order statistic Yr:m from the Rayleigh distribution, when the future sample has outliers, is given by solving the resulting equations as follows:(43)PYr≥Lx|x=D2∑i=12∑i1zi∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+ziLx2/2σ2dσ=1−ρ2,PYr≥Ux|x=D2∑i=12∑i1zi∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+ziUx2/2σ2dσ=ρ2,where(44)Σ1=∑s=maxd0,r−p−1minm−p−1,r−1C1∑a=0m−p−s−1m−p−s−1a∑b=0p−r+s+1p−r+s+1b×∑c=0s+as+ac∑e=0r−s−1+br−s−1+be−1a+b+c+e,Σ2=∑s=max0,r−pminm−p,r−1C2∑a=0m−p−sm−p−sa∑b=0p−r+sp−r+sb×∑c=0s+as+ac∑e=0r−s−1+br−s−1+be−1a+b+c+eh2,D2=1∑i=12∑i1/zi∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2dσ,z1=c+1+h2e,z2=c+h2e+1.(4)
By denotingp=0 in (47) or λ=0 in (39), the 1−ρ100% BPI for the future order statistic Yr:m from the Rayleigh distribution, when there are no outliers, is given by solving the result equations as follows:(45)PYr≥Lx|x=D2∑11z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Lx2/2σ2dσ=1−ρ2,PYr≥Ux|x=D2∑11z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Ux2/2σ2dσ=ρ2,where(46)Σ1=m!r−1!m−r!∑a=0m−rm−ra∑c=0r−1+ar−1+ac−1a+c,D2=1∑11/z1∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2dσ,z1=c+1.(5)
By denotingr=m in (38), the 1−ρ100% BPI for the largest future order statistic Ym:m from TRD, when the future sample has outliers, is given by solving the result equations:(47)D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zijLx2/2σ2∏i=1ntiσdσ=1−ρ2,D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zijUx2/2σ2∏i=1ntiσdσ=ρ2,where(48)D2=1∑i=12∑i∑j=12(τjλ/zij)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntidσdσ,Σ1=m−p∑c=0m−p−1m−p−1c∑d=0m−p−1m−p−1d∑e=0ppe×∑l=0ppl−1c+eλd+l,Σ2=p∑c=0m−pm−pc∑d=0m−pm−pd∑e=0p−1p−1e∑l=0p−1p−1l−1c+eλd+lh2,τ1λ=1−λ,τ2λ=2λ,z11=c+d+1+h2e+l,z12=c+d+2+h2e+l,z21=c+d+h2e+l+1,z22=c+d+h2e+l+2.


### 3.2. Numerical Results

#### 3.2.1. Simulation Study

Here, we present the95% Bayesian prediction lower and upper bounds for the future observations Yr:m from TRD, where r=1 and r=m based on order statistics, and the lengths of BPI for n=5 and m=4,7. We consider the case of outliers as in the future sample. We use the steps in Algorithm 2.Algorithm 2:
(1)
For given values of the prior parametersa and b, where a=1 and b=1, a random value of the parameter σ is generated from the square root inverted gamma distribution which has pdf as in (6).(2)
For a chosen value ofλ and the generated value of σ obtained in Step (1), we generate an order statistics sample of size n, n=5 from TRD λ,σ. This sample is considered to be the informative sample.(3)
For a chosen value ofλ and the generated value of σ obtained in Step 1, we generate a sample of size m−p, m=4,7 from TRD λ,σ and a sample of size p,p=1,2 from TRD λ,σ/h, where h=0.1,0.2,0.3,0.4,0.5.The union and arrangement of the elements of the two samples give the order statistics sample of sizem which has p outliers and it is considered to be the future sample.(4)
Based on the pairs of generated order statistics samples of sizen and m, we obtain the lower and upper bounds of BPI for the future observations Yr:m, where r=1 and r=m, by solving the equation (38) numerically with ρ=0.05.(5)
We repeat the Steps 2, 3, and 4, 100 times, to compute the average intervals bounds and their lengths.The results are presented in Table3.Table 3
95% BPI for Y1:m and Ym:m in the presence of outliers in the future sample n=5.
mphy1ymLULengthLULength401.09051.283 61.193 0.74883.131 72.372 910.1.10801.531 71.423 62.012 326.19724.185 60.2.10751.524 31.4161.407 813.612 612.204 80.3.10291.458 41.355 51.160 38.890 87.730 50.4.10281.458 11.355 21.047 56.702 55.655 00.5.10171.441 91.340 2.96635.347 94.381 620.1.12921.831 21.702 04.754 929.757 625.002 60.2.12771.811 11.683 31.407 813.612 612.204 80.3.12431.762 61.638 21.160 38.890 87.730 50.4.12051.707 81.587 31.047 56.702 55.655 00.5.11251.594 61.482 1.96635.347 94.381 6701.0686.9723.90371.017 23.532 62.515 410.1.07981.131 11.051 3.11 63226.4324.3180.2.07511.064 9.98971.510 713.13511.6240.3.07451.056 4.98191.328 08.8297.5010.4.07371.045 4.97161.198 26.4885.2900.5.07241.026 9.95441.174 15.3994.22520.1.08401.190 61.106 64.78129.4225.140.2.08151.156 01.074 42.40714.7712.650.3.08131.152 71.071 41.7439.7938.0500.4.08251.170 31.087 71.4797.4315.9520.5.08011.136 21.056 11.277 05.755 84.478 8From Table3, we note that the length of BPI increases as p increases, h and/or m decreases. The variance of TRD increases as h decreases. This implies that as h becomes small, the observations are in a large domain, so the length of BPI increases as h decreases. We also note that BPI is longer for Ym:m than for Y1:m. By comparing the length of BPI in the two cases, (i) a single outlier in the informative sample and (ii) multiple outliers in the future sample, we note that the effect of outliers appears clearly in the second case on the BPI of the future observation Ym:m although the length of BPI of Y1:m is longer in the first case than it is in the second case.


## 3.1. Special Cases

(1)
By indicatingp=0 in (38), the 1−ρ100% BPI for the future order statistic Yr:m from TRD, when there are no outliers, can be obtained by solving the result equations:(39)D2∑1∑j=12τjλz1j∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1jLx2/2σ2∏i=1ntiσdσ=1−ρ2,D2∑1∑j=12τjλz1j∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1jUx2/2σ2∏i=1ntiσdσ=ρ2,where(40)D2=1∑1∑j=12τjλ∫0∞∫0∞y1/σ2n+b+3/2e−∑i=1nxi2+a+z1jy2/2σ2∏i=1ntiσdσdy,=1∑1∑j=12(τjλ/z1j)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσdσ,Σ1=m!m−r!r−1!∑a=0m−rm−ra∑c=0r−1+ar−1+ac×∑d=0r−1+ar−1+ad−1a+cλd,τ1λ=1−λ,τ2λ=2λ,z11=c+d+1,z12=c+d+2.(2)
By signifyingr=1 in (38), the 1−ρ100% BPI for the smallest future order statistic Y1:m from TRD, when the future sample has outliers, is given by solving the following result equations:(41)D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−Σi=1nxi2+a+zijLx2/2σ2∏i=1ntiσdσ=1−ρ2,D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−Σi=1nxi2+a+zijUx2/2σ2∏i=1ntiσdσ=ρ2,where(42)D2=1∑i=12∑i∑j=12(τjλ/zij)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntiσdσ,Σ1=m−p∑a=0m−p−1m−p−1a∑b=0ppb∑c=0aac×∑d=0aad∑e=0bbe∑l=0bbl−1a+b+c+eλd+l,Σ2=p∑a=0m−pm−pa∑b=0p−1p−1b∑c=0aac×∑d=0aad∑e=0bbe∑l=0bbl−1a+b+c+eλd+lh2,τ1λ=1−λ,τ2λ=2λ,z11=c+d+1+h2e+l,z12=c+d+2+h2e+l,z21=c+d+h2e+l+1,z22=c+d+h2e+l+2.(3)
By demonstratingλ=0 in (38), the 1−ρ100% BPI for the future order statistic Yr:m from the Rayleigh distribution, when the future sample has outliers, is given by solving the resulting equations as follows:(43)PYr≥Lx|x=D2∑i=12∑i1zi∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+ziLx2/2σ2dσ=1−ρ2,PYr≥Ux|x=D2∑i=12∑i1zi∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+ziUx2/2σ2dσ=ρ2,where(44)Σ1=∑s=maxd0,r−p−1minm−p−1,r−1C1∑a=0m−p−s−1m−p−s−1a∑b=0p−r+s+1p−r+s+1b×∑c=0s+as+ac∑e=0r−s−1+br−s−1+be−1a+b+c+e,Σ2=∑s=max0,r−pminm−p,r−1C2∑a=0m−p−sm−p−sa∑b=0p−r+sp−r+sb×∑c=0s+as+ac∑e=0r−s−1+br−s−1+be−1a+b+c+eh2,D2=1∑i=12∑i1/zi∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2dσ,z1=c+1+h2e,z2=c+h2e+1.(4)
By denotingp=0 in (47) or λ=0 in (39), the 1−ρ100% BPI for the future order statistic Yr:m from the Rayleigh distribution, when there are no outliers, is given by solving the result equations as follows:(45)PYr≥Lx|x=D2∑11z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Lx2/2σ2dσ=1−ρ2,PYr≥Ux|x=D2∑11z1∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+z1Ux2/2σ2dσ=ρ2,where(46)Σ1=m!r−1!m−r!∑a=0m−rm−ra∑c=0r−1+ar−1+ac−1a+c,D2=1∑11/z1∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2dσ,z1=c+1.(5)
By denotingr=m in (38), the 1−ρ100% BPI for the largest future order statistic Ym:m from TRD, when the future sample has outliers, is given by solving the result equations:(47)D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zijLx2/2σ2∏i=1ntiσdσ=1−ρ2,D2∑i=12∑i∑j=12τjλzij∫0∞1σ2n+b+1/2e−∑i=1nxi2+a+zijUx2/2σ2∏i=1ntiσdσ=ρ2,where(48)D2=1∑i=12∑i∑j=12(τjλ/zij)∫0∞1/σ2n+b+1/2e−∑i=1nxi2+a/2σ2∏i=1ntidσdσ,Σ1=m−p∑c=0m−p−1m−p−1c∑d=0m−p−1m−p−1d∑e=0ppe×∑l=0ppl−1c+eλd+l,Σ2=p∑c=0m−pm−pc∑d=0m−pm−pd∑e=0p−1p−1e∑l=0p−1p−1l−1c+eλd+lh2,τ1λ=1−λ,τ2λ=2λ,z11=c+d+1+h2e+l,z12=c+d+2+h2e+l,z21=c+d+h2e+l+1,z22=c+d+h2e+l+2.


## 3.2. Numerical Results

### 3.2.1. Simulation Study

Here, we present the95% Bayesian prediction lower and upper bounds for the future observations Yr:m from TRD, where r=1 and r=m based on order statistics, and the lengths of BPI for n=5 and m=4,7. We consider the case of outliers as in the future sample. We use the steps in Algorithm 2.Algorithm 2:
(1)
For given values of the prior parametersa and b, where a=1 and b=1, a random value of the parameter σ is generated from the square root inverted gamma distribution which has pdf as in (6).(2)
For a chosen value ofλ and the generated value of σ obtained in Step (1), we generate an order statistics sample of size n, n=5 from TRD λ,σ. This sample is considered to be the informative sample.(3)
For a chosen value ofλ and the generated value of σ obtained in Step 1, we generate a sample of size m−p, m=4,7 from TRD λ,σ and a sample of size p,p=1,2 from TRD λ,σ/h, where h=0.1,0.2,0.3,0.4,0.5.The union and arrangement of the elements of the two samples give the order statistics sample of sizem which has p outliers and it is considered to be the future sample.(4)
Based on the pairs of generated order statistics samples of sizen and m, we obtain the lower and upper bounds of BPI for the future observations Yr:m, where r=1 and r=m, by solving the equation (38) numerically with ρ=0.05.(5)
We repeat the Steps 2, 3, and 4, 100 times, to compute the average intervals bounds and their lengths.The results are presented in Table3.Table 3
95% BPI for Y1:m and Ym:m in the presence of outliers in the future sample n=5.
mphy1ymLULengthLULength401.09051.283 61.193 0.74883.131 72.372 910.1.10801.531 71.423 62.012 326.19724.185 60.2.10751.524 31.4161.407 813.612 612.204 80.3.10291.458 41.355 51.160 38.890 87.730 50.4.10281.458 11.355 21.047 56.702 55.655 00.5.10171.441 91.340 2.96635.347 94.381 620.1.12921.831 21.702 04.754 929.757 625.002 60.2.12771.811 11.683 31.407 813.612 612.204 80.3.12431.762 61.638 21.160 38.890 87.730 50.4.12051.707 81.587 31.047 56.702 55.655 00.5.11251.594 61.482 1.96635.347 94.381 6701.0686.9723.90371.017 23.532 62.515 410.1.07981.131 11.051 3.11 63226.4324.3180.2.07511.064 9.98971.510 713.13511.6240.3.07451.056 4.98191.328 08.8297.5010.4.07371.045 4.97161.198 26.4885.2900.5.07241.026 9.95441.174 15.3994.22520.1.08401.190 61.106 64.78129.4225.140.2.08151.156 01.074 42.40714.7712.650.3.08131.152 71.071 41.7439.7938.0500.4.08251.170 31.087 71.4797.4315.9520.5.08011.136 21.056 11.277 05.755 84.478 8From Table3, we note that the length of BPI increases as p increases, h and/or m decreases. The variance of TRD increases as h decreases. This implies that as h becomes small, the observations are in a large domain, so the length of BPI increases as h decreases. We also note that BPI is longer for Ym:m than for Y1:m. By comparing the length of BPI in the two cases, (i) a single outlier in the informative sample and (ii) multiple outliers in the future sample, we note that the effect of outliers appears clearly in the second case on the BPI of the future observation Ym:m although the length of BPI of Y1:m is longer in the first case than it is in the second case.


## 3.2.1. Simulation Study

Here, we present the95% Bayesian prediction lower and upper bounds for the future observations Yr:m from TRD, where r=1 and r=m based on order statistics, and the lengths of BPI for n=5 and m=4,7. We consider the case of outliers as in the future sample. We use the steps in Algorithm 2.Algorithm 2:
(1)
For given values of the prior parametersa and b, where a=1 and b=1, a random value of the parameter σ is generated from the square root inverted gamma distribution which has pdf as in (6).(2)
For a chosen value ofλ and the generated value of σ obtained in Step (1), we generate an order statistics sample of size n, n=5 from TRD λ,σ. This sample is considered to be the informative sample.(3)
For a chosen value ofλ and the generated value of σ obtained in Step 1, we generate a sample of size m−p, m=4,7 from TRD λ,σ and a sample of size p,p=1,2 from TRD λ,σ/h, where h=0.1,0.2,0.3,0.4,0.5.The union and arrangement of the elements of the two samples give the order statistics sample of sizem which has p outliers and it is considered to be the future sample.(4)
Based on the pairs of generated order statistics samples of sizen and m, we obtain the lower and upper bounds of BPI for the future observations Yr:m, where r=1 and r=m, by solving the equation (38) numerically with ρ=0.05.(5)
We repeat the Steps 2, 3, and 4, 100 times, to compute the average intervals bounds and their lengths.The results are presented in Table3.Table 3
95% BPI for Y1:m and Ym:m in the presence of outliers in the future sample n=5.
mphy1ymLULengthLULength401.09051.283 61.193 0.74883.131 72.372 910.1.10801.531 71.423 62.012 326.19724.185 60.2.10751.524 31.4161.407 813.612 612.204 80.3.10291.458 41.355 51.160 38.890 87.730 50.4.10281.458 11.355 21.047 56.702 55.655 00.5.10171.441 91.340 2.96635.347 94.381 620.1.12921.831 21.702 04.754 929.757 625.002 60.2.12771.811 11.683 31.407 813.612 612.204 80.3.12431.762 61.638 21.160 38.890 87.730 50.4.12051.707 81.587 31.047 56.702 55.655 00.5.11251.594 61.482 1.96635.347 94.381 6701.0686.9723.90371.017 23.532 62.515 410.1.07981.131 11.051 3.11 63226.4324.3180.2.07511.064 9.98971.510 713.13511.6240.3.07451.056 4.98191.328 08.8297.5010.4.07371.045 4.97161.198 26.4885.2900.5.07241.026 9.95441.174 15.3994.22520.1.08401.190 61.106 64.78129.4225.140.2.08151.156 01.074 42.40714.7712.650.3.08131.152 71.071 41.7439.7938.0500.4.08251.170 31.087 71.4797.4315.9520.5.08011.136 21.056 11.277 05.755 84.478 8From Table3, we note that the length of BPI increases as p increases, h and/or m decreases. The variance of TRD increases as h decreases. This implies that as h becomes small, the observations are in a large domain, so the length of BPI increases as h decreases. We also note that BPI is longer for Ym:m than for Y1:m. By comparing the length of BPI in the two cases, (i) a single outlier in the informative sample and (ii) multiple outliers in the future sample, we note that the effect of outliers appears clearly in the second case on the BPI of the future observation Ym:m although the length of BPI of Y1:m is longer in the first case than it is in the second case.


## 4. Conclusion

In this paper, we presented the Bayesian prediction intervals for order statistics of future observations from TRD in the presence of outliers in two cases. These were a single outlier in the informative sample and multiple outliers in the future sample. We used the simulation study by Mathematica to calculate the length of BPI, and we found that the presence of outliers affects the length of BPI. This effect becomes clearer in the following cases: the small samples, the increasing of number of outliers in the sample, and the presence of outliers in future samples.



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