The comparative studies on reliability for Rayleigh models

The comparative studies on reliability for Rayleigh models

Ji Eun Oh ¹ · Joong Kweon Sohn ²

12 Department of Statistics, Kyungpook National University

Received 12 January 2018, revised 12 March 2018, accepted 13 March 2018

Abstract

In this paper, several methods to estimate the reliability R of a Rayleigh model as a function of given time t are proposed and studied. Many research have been done by providing some informations on parameters of a Raleigh model rather R itself. Here R is a probability to endure a certain level of stress. Thus for the Bayesian method, several priors are given to R directly. Also a bootstrap method and a maximum likelihood method are examined and compared with a Bayesian method in term of mean square errors. we find that Bayesian methods perform better than other methods.

Keywords: Bayes estimator, bootstrap estimator, maximum likelihood estimator, rayleigh distribution, reliability.

1. Introduction

Reliability is the probability that a product or system will work properly for a specified period of time under the design operating conditions without failure. So that the reliability depends on specific time as a random variable and its function from the chosen distribution can be estimated and controled for system maintenance. The study of reliability is important to develop the appropriate future plans to improve the quality and performance of the system recently. Many authors had been studied to estimate the reliability from their own statistical distribution function that they assumed.

Among all the statistical life time model, the Rayleigh distribution is the most important models and the widely used in large-scale tests in life and reliability. In this study, we assume that the life times of units follow a Rayleigh distribution, which has been widely used to describe life time reliability analyses.

The study of maximum likelihood estimation (MLE) of reliability with different distribu- tion have been discussed by several authors. The MLE is used to derive point and asymptotic confidence estimates of the unknown parameters. First Harter and Moore (1965) showed an explicit form for the MLE of based on type II censored data. Thoman et al. (1970) studied with maximum likelihood (ML) estimator and exact confidence interval for reliability and

1

Graduate student, Department of Statistics, Kyungpook National University, Daegu 41566, Korea.

2

Corresponding author: Professor, Department of Statistics, Kyungpook National University, Daegu

41566, Korea. E-mail: [email protected]

tolerance limits in the Weibull distribution. Dyer and Whisenand (1973) considered the best linear unbiased estimator of the parameter of the Rayleigh distribution. Smith et al. (1987) considered MLE and exact confidence intervals for reliability in the Weibull Distribution.

Kang and Kim (1994) considered the approximate MLE of scale parameter of weibull dis- tribution with type II censoring. Lee et al. (2014) also studied reliability estimation and ratio distribution in a general exponential distribution. Kim and Cha (2016) considered the bivariate reliability models with multiple dynamic competing risks.

Inferences for the Rayleigh distribution have been discussed by several authors. Sinha and Howlader (1983) showed the Credible and highest predictive density (HPD) intervals of the parameter and reliability of Rayleigh distribution. The approximate MLE is proofed as effi- cient as the best linear unbiased estimator of Rayleigh distribution by Balakrishnan (1989).

Bayesian estimation and prediction problems for the Rayleigh distribution based on Type II censored sample have been considered by Fernandez (2000). Wu et al. (2006) considered the Bayesian estimator and prediction intervals for future observations based on progressively type II censored samples. Kim and Han (2009) considered ML estimator, approximate ML estimator and Bayes estimation procedures for the scale parameter based on a multiply type II censored sample. Dey (2009) compared the Bayes estimates of reliability function with different loss function. Recently, Lee et all (2010) considered the reliability of right truncated Rayleigh distribution. Pak et al. and Kweon et al. (2014) considered the reliability estima- tion with Rayleigh distribution on fuzzy lifetime data and type I hybrid censored sample, respectively. Rasheed and Aaref (2016) studied reliability estimation in inverse Rayleigh distribution using precautionary loss function with MLE method and Bayesian method.

Most of studies deal with scale parameter of Rayleigh or inverse Rayleigh distribution so far according to literature reviews. We focus on estimating reliability of Rayleigh distribution and derive three estimator of reliability that not focus on the distribution parameter but on reliability parameter directly.

We use the following notation that the Rayleigh distribution with the scale parameter σ and the corresponding density function f (x; σ) for σ > 0 is given by

f (x; σ) = x

σ ² e ⁻

, x > 0, σ > 0 (1.1) and reliability function as follow.

θ = R(t) = P (X > t) = e ⁻

, x > 0, t > 0, σ > 0. (1.2) This study has two objectives. First, it aims to develop three estimators of reliability based on the MLE method, bootstrap method and Bayesian method. We also focus on reliability function to estimate not the scale parameter of Rayleigh distribution. The second objective is to compute and compare the confidence intervals of reliability using the estimators from the previous three method.

In Section 2, we calculate the maximum likelihood (ML) estimator of the reliability func-

tion θ with MLE method and derive the asymptotic distribution and the confidence interval

of θ. In order to the confidence limits for parameter, we calculate the Fisher information

matrix to obtain the asymptotic variances and covariance of the ML estimator of the pa-

rameter.

In Section 3, we generate the bootstrap samples using the ML estimator from section 2 and obtain the bootstrap estimator and confidence interval of bootstrap sample with N replications.

In Section 4, we derive the Bayes estimator of reliability with the inverse gamma prior on scale parameter and HPD confidence interval.

In Section 5, we show some results based on Monte-Carlo simulation to compare the mean squared error (MSE) of each estimator using MLE method and bootstrap method and Bayesian method in different sample size and the scale parameter σ. Also we showed some results of three kinds of confidence interval of reliability and compare the coverage probability and interval length of each estimator.

2. Maximum likelihood estimation (MLE) method

The method of maximum likelihood estimation by Harter and Moore (1965) is a commonly used procedure because it has very desirable properties.

The Rayleigh distribution applies to a non-limited positive continuous variable, the prob- ability density function and the cumulative distribution function is given by:

f (x, σ) = x

σ ² e ⁻

, x > 0, σ > 0, (2.1) F (x, σ) =1 − e ⁻

, x > 0, σ > 0, (2.2) where x is a continuous random variable defined over (0, ∞) and σ is the scale parameter.

The reliability function on specific time t which is R(t),

θ = R(t) = P (X > t) = Z t

0

F (x, σ)dx = e ⁻

, x > 0, t > 0, σ > 0.

We recalled R(t) is a function of θ, θ = e ⁻

, t > 0, σ > 0 and σ ² = − _2logθ ^t

.

To find the estimate of ML estimator of reliability function R(t), we first have to find the ML estimator of σ ² which is a scale parameter of the Rayleigh distribution.

Suppose x = (x 1 , x 2 , · · · , x n ) be an observed random sample of size n from the Rayleigh distribution, where σ is a scale parameter. The likelihood function of the observed sample is

L(σ|x) = 1 σ ²ⁿ

n

Y

i=1

x i

!

exp −

n

X

i=1

x ² _i 2σ ²

!

. (2.3)

The log-likelihood function may then be written as

l(σ|x) = log(L(σ|x)) =

n

X

i=1

logx i − 2nlogσ − P n

i=1 x ² _i

2σ ² . (2.4)

The ML estimator of σ ² can be obtained by first derivative of (2.4) with respect to σ,

σ ˆ ² = P x ² _i 2n .

The likelihood function of θ, which can be obtained by replace the σ ² = − _2logθ ^t

to (2.3) is given by

L(θ|x) =



− 2logθ t ²

 ^{n n} Y

i=1

x _i

! exp



− P x ² _i 2

 

− 2logθ t ²



(2.5)

and log-likelihood function of θ may then be written as

l(θ|x) = nlog(2logθ) − 2nlogt + X

logx i +  P x ² _i t ²



logθ. (2.6)

Calculating the first partial derivatives of (2.6) with respect to θ and equating to zero, we obtain the likelihood equation

∂l(θ|x)

∂θ = n

θlogθ + P x ² _i t ²

1 θ = 0.

Hence, the ML estimator of θ

θ ˆ M LE = exp



− t ² n P x ² _i



. (2.7)

We can get the ˆ θ M LE with invariance properties of MLE.

θ ˆ M LE = ˆ R(t) = exp



− t ² 2ˆ σ ²



= exp



− t ² n P x ² _i



. (2.8)

2.1. Asymptotic distribution and confidence interval of reliability

In this section, first we need to find the asymptotic distribution of ˆ σ ² and then we drive the asymptotic distribution of ˆ θ. Base on the asymptotic distribution of ˆ θ, we can get the asymptotic confidence interval of θ. In order to the confidence limits for parameter, we calculate the Fisher information matrix to obtain the asymptotic variances and covariance of the ML estimator of the parameter.

From the log-likelihood function in (2.6), we have

∂ ² l(θ|x)

∂θ ² = n

(1 + logθ) − P x ² _i t ²

1

θ ² . (2.9)

The Fisher information I(θ) is then obtained by taking expectation of minus (2.9). In practice, we usually estimate the I(θ) by I 0 (ˆ θ) that obtained by following

I(θ) = −E  ∂ ² l(θ|x)

∂ ² θ

 ,

I 0 (ˆ θ) = − E

 n

(1 + logθ) − P x ² _i t ²

1 θ ²



= 2nσ ² (1 + logθ) − nθ ² t ² θ ² t ² (1 + logθ) ,

V ar(ˆ θ) = 1

I 0 (ˆ θ) = θ ² t ² (1 + logθ) 2n ˆ σ ² (1 + logθ) − nθ ² t ²

.

Under some mild regularity conditions ˆ θ is approximately normally distributed with mean θ and variance d V ar(ˆ θ).

θ ∼ N (θ, d ˆ V ar(ˆ θ)).

Thus, the 100(1 − α)% approximate confidence interval for θ is

 θ − z ˆ

2

q

V ar(ˆ d θ), θ + z ˆ

2

q V ar(ˆ d θ)

 ,

V AR(ˆ [ θ) = θ ² t ² (1 + logθ) 2n ˆ σ ² (1 + logθ) − nθ ² t ² ,

= θ ² t ² (1 + logθ) P x ² _i (1 + logθ) − nθ ² t ² , where z

2

is the percentile of the standard normal distribution with right-tail probability ^α ₂ and ˆ σ ² = ^{P x} _2n

.

3. Bootstrap method

The bootstrap method introduced in Efron (1982) is a very general resampling procedure for estimating the distributions of statistics based on independent observations. The boot- strap method is shown to be successful in many situations, which is being accepted as an alternative to the asymptotic methods.

For the above example, we can easily calculate its bootstrap distribution. We can easily

imagine that the above computation becomes too complicated to compute directly if n is

large. Therefore, simple random sampling was proposed to generate bootstrap distribution.

In the bootstrap literature, a variety alternatives are suggested other than simple random sampling.

The following steps are followed to obtain bootstrap sample from Rayleigh distribution with parameter.

Step 1 : Generate random samples x 1 , x 2 , · · · , x n is a data sample drawn from Rayleigh distribution with the scale parameter σ and compute the ML estimator of σ ² , ˆ σ ² . Step 2 : Generate a bootstrap sample x ^∗ ₁ , x ^∗ ₂ , · · · , x ^∗ _n from Rayleigh distribution with σ ˆ ² . Based on bootstrap sample, compute the new estimator of scale parameter, ˆ σ ^∗ and bootstrap estimator of reliability using equation (2.8), say ˆ θ Boot .

Step 3 : Repeat Step2, NBOOT times. NBOOT is 2000 replications.

3.1. The bootstrap confidence interval of reliability

In this section, we propose to use the percentile bootstrap method based on the idea of Efron (1982). We follow the next few step to get the bootstrap interval of estimator using bootstrap method.

To generate the confidence intervals by bootstrapping, the sampling distribution of the parameter θ is simulated by sampling over and over from the current data and recomputing parameter estimates θ from each “bootstrapped” sample. The variability shown by the many θ ^∗ values gives us a hint about the variability of the one estimate θ we got from our data.

Step 1 : Generate random samples x ₁ , x ₂ , · · · , x _n is a data sample drawn from Rayleigh distribution with the scale parameter σ ² and compute the ML estimator of σ ² , ˆ σ ² . Step 2 : Using ˆ σ ² , generate a bootstrap sample x ^∗ ₁ , x ^∗ ₂ , · · · , x ^∗ _n from Rayleigh distribu- tion. Based on bootstrap sample, compute bootstrap estimate of reliability using ˆ σ ^∗ and equation (2.8), say ˆ θ Boot .

Step 3 : Repeat Step2-3 NBOOT times based on N different bootstrap samples.

Step 4 : Generate the confidence interval of which has asymptotic normal distribution with mean θ and variance d V ar(ˆ θ ^∗ ).

4. Bayesian method

Bayesian analysis is an important approach to statistics, which formally seeks use of prior information and Bayes Theorem provides the formal basis for using this information. An important pre-requisite in Bayesian estimation is the appropriate choice of prior for the parameters. However, Bayesian analysts also hard to choose which prior is better than the other. Very often, priors are chosen according to ones subjective knowledge and beliefs. In this paper, we derive the Bayes estimator with the inverse gamma prior on scale parameter and comparing with non-informative prior at simulation study.

First we develop the Bayesian estimation procedure for the estimation of reliability form

Rayleigh distribution assuming independent inverse gamma prior for the unknown model

parameters. Thus, we consider the natural conjugate prior for given by

π(σ) = β ^α

Γ(α) σ ^−(α+1) exp



− β 2σ ²



∝  1 σ

 (α+1)

exp



− β 2σ ²



, σ > 0, α > 0, β > 0, (4.1) where shape parameter α > 0 and scale parameter β > 0. This density is known as the square-root inverted gamma distribution.

By combining (2.6) and (4.1), the posterior density of σ is given by

π(σ|x) =

2  _{P x}

+β 2



Γ ^2n+α ₂
(σ ² ) ^n+α+1 exp



− P x ² _i + β 2σ ²



, (4.2)

where it is the inverse gamma density, IG(α, β) with α = ^2n+α ₂ , β = ^{P x} ₂

^+β . It is well known that the Bayes estimator of σ under squared loss function.

To estimate the reliability function θ, θ = e ⁻

, t > 0, σ > 0 and substituting σ ² = − _2logθ ^t

into (4.2). The posterior probability density function of θ is given by

π(θ|x) =

 _{P x}

+β 2



Γ ^2n+α ₂









 exp

 P x

+β

 −





 − _logθ ^t













 t ² 2



(logθ) ⁻² 1 θ ,

=

 _{P x}

+β 2



Γ ^2n+α ₂
θ

⁻¹ (−logθ)

⁻¹ , 0 < θ < 1. (4.3) The Bayes estimator of reliability function θ with the squared error loss function which is L(ˆ θ, θ) = (ˆ θ − θ) ² , ˆ θ Bayes is by Dey (2009),

θ ˆ _Bayes =E(θ|x) = Z ∞

0

θpi(θ|x)dθ

θ ˆ _Bayes =



1 + t ² P x ² _i + β

 ⁻

. (4.4)

If α = 0, β = 0, we get a non-informative prior and also if α = 1, β = 1, we get a inverse gamma prior.

4.1. The Bayesian confidence interval of relability

In order to derive the confidence interval for Bayes estimator, we use the posterior density

function of θ. We already have the posterior probability density function from previous

chapter and given by

π(θ|x) =

 _{P x}

+β 2



Γ ^2n+α ₂
θ

P x2i+β

2

−1 (−logθ)

⁻¹ , 0 < θ < 1.

From the posterior probability density function of θ, we know that as followed,

R(t) = θ = exp



− t ² 2σ ²

 ,

 P x ² _i + β t ²

 log  1

θ



∼ χ ²  2n + α 2

 .

By the logarithm of posterior probability density function and some transformation,

 _{P x}

+β t



log ¹ _θ
is chi-squared distribution with degree of freedom ^2n+α ₂
. The 100(1−α)%

HPD credible set for θ is the interval (c L , c U ) which satisfying R c

c

π(θ|x)dθ = 1 − α.

Z L 0

π(θ|x)dθ = α 2 and

Z ∞ U

π(θ|x)dθ = α 2 .

Therefore (1 − α) equal-tail credible limit c _L and c _U for the reliability θ are the solution of as followed.

Since ^{P x} _σ

∼ χ ² (2n), 100(1 − α)% equal tail interval limits c L and c U for the reliability function are given by

 P x ² _i + β t ²

 log

 1 c _U (t)



= χ ²

 2n + α 2



and  P x ² _i + β t ²

 log

 1 c _L (t)



= χ ²

 2n + α 2

 . The HPD limits H L (t) and H U (t) are calculated by Sinha and Howlader (1983),

P  P x ² _i + β t ²

 log

 1 H U



< χ ²  2n + α 2



<  P x ² _i + β t ²

 log

 1 H L



= 1 − α.

5. Simulation study

In this section, a simulation study is conducted to investigate and compare the performance of ML estimator and Bootstrap estimators and Bayes estimators (under squared error loss function) of reliability and also derived the confidence intervals for three estimators. We mainly compute MSE and average estimates of the ML estimator, Bootstrap estimator and Bayes estimator. To compare the performance of the confidence interval of reliability, we compute the coverage probability (CP) and interval lengths (IL) of each estimators.

We compared the results of estimates of reliability using MLE method, bootstrap method

and the Bayesian method with different sample sizes and values of the scale parameter : (n, σ)

= (15, 1), (30,1), (50, 1), (100, 1) and (15 2), (30, 2), (50, 2), (100, 2) using Monte-Carlo simulation.

Through this simulation study, we find the MSE and mean of each estimates so that we compare the estimates with true value of . In bootstrap sampling, we replicate bootstrap sampling process for 2000 times.

We considered two Bayes estimators with different prior that one is the inverse gamma prior and the other is the non-informative prior for the reliability function of Rayleigh distri- bution. We assume that the parameters of inverse gamma distribution as a prior distribution with α = 1, β = 1 and consider as a non-informative prior distribution with α = 0, β = 0 for Bayes estimators.

We compared the confidence interval length and coverage probability of the estimates of reliability. The confidence level taken is 1 − α = 0.95.

Table 5.1 MSE and average values (AV) θ = 0.61, σ = 1, t = 1

n θ ˆ

θ ˆ

θ ˆ

θ ˆ

(α = 1, β = 1) (α = 0, β = 0)

MSE AV MSE AV MSE AV MSE AV

15 0.0074 0.5844 0.0053 0.6677 0.0094 0.6010 0.0116 0.5743

30 0.0040 0.5992 0.0033 0.5951 0.0029 0.6062 0.0034 0.5925

50 0.0021 0.6075 0.0018 0.6109 0.0017 0.6067 0.0024 0.5992

100 0.0009 0.6053 0.0009 0.5954 0.0008 0.6052 0.0012 0.6014 θ = 0.88, σ = 2, t = 1

n θ ˆ

θ ˆ

θ ˆ

θ ˆ

(α = 1, β = 1) (α = 0, β = 0)

MSE AV MSE AV MSE AV MSE AV

15 0.00089 0.8758 0.00093 0.8336 0.00155 0.8671 0.0019 0.8672 30 0.00053 0.8785 0.00046 0.8714 0.00041 0.8757 0.00040 0.8792 50 0.00025 0.8802 0.00029 0.8858 0.00023 0.8792 0.00021 0.8793 100 0.00013 0.8819 0.00013 0.8831 0.00011 0.8810 0.00012 0.8808

* ML : Maximum likelihood estimates

* Boot : Bootstrap estimates

* Bayes : Bayes estimates with inverse gamma prior

* Bayes

: Bayes estimates with non-informative prior

Tables 5.1 shows that the MSE and average values of the three estimators. As the sample size increases, the Bayes estimator performs better than MLE overall but they all have their MSE values decreasing as sample size increase.

At small sample like n = 15, MSE values of Bayes estimates corresponds to a larger than other estimators but at sample size n = 30, 50 and 100, MSE values of Bayes estimators were smaller than others for σ = 1.

When the σ is 2, Bayes estimator has the smaller MSE than the MLE estimator as the sample size increases.

Table 5.2 summaries that the coverage probability and interval length of each estimators

with different sample size and σ. The mean of interval length decrease as sample size in-

crease and the coverage probability of bootstrap and Bayes estimators are higher than ML

estimators. The Bayes estimator is the shortest interval length.

Table 5.2 Coverage probability and interval length θ = 0.61, σ = 1, t = 1

n θ ˆ

θ ˆ

θ ˆ

θ ˆ

(α = 1, β = 1) (α = 0, β = 0)

15 0.9500 0.9400 0.9650 0.9750

0.3601 0.3231 0.4129 0.4273

30 0.9650 0.9450 0.9650 0.9650

0.2581 0.2407 0.2485 0.2511

50 0.9550 0.9555 0.9600 0.9550

0.2010 0.2012 0.1963 0.1971

100 0.9650 0.9655 0.9650 0.9750

0.1430 0.1370 0.1403 0.1402

θ = 0.88, σ = 2, t = 1

n θ ˆ

θ ˆ

θ ˆ

θ ˆ

(α = 1, β = 1) (α = 0, β = 0)

15 0.9850 0.9970 0.9800 0.9800

0.3039 0.3361 0.2042 0.1976

30 0.9550 0.9960 0.9850 0.9800

0.2122 0.2427 0.1143 0.1135

50 0.9650 0.9950 0.9650 0.9650

0.1641 0.1874 0.0875 0.0871

100 0.9650 0.9950 0.9650 0.9600

0.1162 0.1116 0.0613 0.0610

Table 5.3 Average interval estimation θ = 0.61, σ = 1, t = 1

n MLE Bootstrap Bayes

Method Method Method

(c

, c

) (c

, c

) (c

, c

) 15 (0.3514, 0.8483) (0.2918, 0.8468) (0.4546, 0.8968) 30 (0.4015, 0.8027) (0.3898, 0.8046) (0.4872, 0.8783) 50 (0.4518, 0.7566) (0.4607, 0.7616) (0.4741, 0.8614) 100 (0.4952, 0.7100) (0.4994, 0.7134) (0.4964, 0.8355) θ = 0.88, σ = 2, t = 1

n MLE Bootstrap Bayes

Method Method Method

(c

, c

) (c

, c

) (c

, c

) 15 (0.7254, 0.9899) (0.7413, 0.9772) (0.7349, 0.9397) 30 (0.7578, 0.9864) (0.7792, 0.9870) (0.7852, 0.9676) 50 (0.7880, 0.9734) (0.7970, 0.9607) (0.7990, 0.9632) 100 (0.8161, 0.9467) (0.8230, 0.9379) (0.8190, 0.9572)

Table 5.3 summaries that the average confidence interval of three estimators of reliability.

Figure 5.1 shows below that the asymptotic distribution of each estimators at different

sample size and σ.

(a) MLE estimates σ = 1 (b) MLE estimates σ = 2

(c) Bootstrap estimates σ = 1 (d) Bootstrap estimates σ = 2

(e) Bayes estimates σ = 1 (f) Bayes estimates σ = 2

Figure 5.1 Box plots and histograms of each estimates

6. Conclusion and discussion

A simulation studies were conducted to examine the performance of the different esti- mators how close to the true value of reliability and the coverage probability of confidence interval of each estimator was compared. It is shown that the asymptotic distribution of reliability depends only on time and sample size. The ML estimator close to the true value and has the small MSE when sample size is large. So the ML estimator of reliability was accurate results of simulation when the sample size is large.

Its clear that the MSE of Bayes estimator is smaller than the other two estimators except the sample size is 15. The interval length of three estimators are quite similar and the coverage probability of ML estimator is the lowest. Among three estimators, the Bayes estimators interval length is the smallest.

Based on the above discussion, we can conclude that the Bayes estimator has the smallest MSE and the shortest interval length as the sample size increases even if MSE of ML estimator performed good at small sample size. We consider that Bayes estimator is good estimator of this study.

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