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Rayleigh distribution (RD) has wide applications in many real life situations especially life testing, reliability analysis, medicines etc. In this paper a new weighted Rayleigh distribution named area-biased Rayleigh distribution (ARD) is introduced. Some mathematical properties of the (ARD) including cumulative distribution function, moments, skewness, kurtosis, median, mode, entropy, reliability measures as survival function and hazard function have been derived. Parameter of the ARD is estimated by method of moments (MOM), maximum likelihood (ML), and Bayesian. Properties of the estimators are developed. It is proved that the ML estimator attains the Cramer Rao lower bound. Applications of the ARD provided for some life time data sets. Kolmogorov Smirnov (K-S) test statistics is applied to check the good fit of the model.

In many real life fields such as medication, engineering and business, among others, modeling and examine lifetime data are crucial. Numerous lifetime distributions have been used to model lifetime data sets [

A random variable X is said to have the Rayleigh distribution (RD) with parameter σ if its probability density function is given by

f ( x ) = x σ 2 e − x 2 / 2 σ 2 , x > 0 ; σ > 0. (1)

while the cumulative distribution function of the Rayleigh distribution

F ( x ) = 1 − e − x 2 / 2 σ 2 (2)

where σ denote the scale parameter.

Mean = μ ′ 1 = σ π 2 (3)

μ ′ 2 = 2 σ 2 (4)

One of the generalized Rayleigh distribution is given by

f ( x ) = 2 ( 2 σ 2 ) N / 2 Γ ( N / 2 ) x N − 1 exp ( − x 2 2 σ 2 ) , x > 0 ; σ > 0 (5)

Pdf in Equation (5) is also named as chi-squared distribution with N degree of freedom and scale parameter σ.

The concept of weighted distributions was initially introduced by Fisher [

Weighted distributions had been frequently used in research related to reliability, bio-medicine, meta-analysis, econometrics, survival analysis, renewal processes, physics, ecology and branching processes can be observed in Patil and Ord [

Suppose X is a non-negative random variable with its pdf f ( x ; θ ) , θ is a parameter, then f w ( x ; θ ) distribution is weighted version of f ( x ; θ ) , and is defined as

f w ( x ; θ ) = w ( x ) f ( x ; θ ) E [ w ( x ) ] (6)

where w ( x ) is an arbitrary non-negative function. For w ( x ) = x α , α = 1 or 2 it is called size biased and area biased distributions respectively. The pdf of size-biased Rayleigh distribution is

f ( x ) = x 2 σ 3 π 2 e − x 2 / 2 σ 2 , x > 0 ; σ > 0 (7)

Using Equation (1) and Equation (6), pdf of the area biased Rayleigh distribution (ARD) is

f ( x ) = x 3 2 σ 4 e − x 2 / 2 σ 2 , x > 0 ; σ > 0. (8)

Rayleigh distribution in (1) and size-biased Rayleigh distribution in (7) are special cases of the generalized Rayleigh distribution in (6) for N = 2 and N = 3 respectively. Moreover, the newly derived area-biased Rayleigh distribution is also a special case of generalized Rayleigh distribution given in (6) for N = 4.

Cumulative distribution function (cdf) of the ARD

F ( x ) = γ ( 2 , x ′ ) , where x ′ = x 2 2 σ 2 , (9)

∫ 0 x t n − 1 e − t d t = γ ( n , x )

is lower incomplete gamma function.

Moments and Shannon EntropyThe rth moments of the ARD are

μ ′ r = r ( r + 1 ) ( 2 σ 2 ) r / 2 4 Γ ( r 2 ) , (10)

For r = 1, 2, 3, 4 in Equation (10), the first four moments of the ARD are

Mean = μ ′ 1 = 3 σ 2π 4 , (11)

μ ′ 2 = 4 σ 2 , μ ′ 3 = 15 σ 3 2 π 2 , μ ′ 4 = 24 σ 4 , (12)

First four mean moments of the ARD are

Variance = μ 2 = σ 2 ( 32 − 9 π 8 ) (13)

μ 3 = σ 3 π ( 27 π − 84 ) 8 2 , μ 4 = 3 σ 4 ( 512 + 96 π − 81 π 2 ) 64 (14)

β 1 = 4 π ( 27 π − 84 ) 2 ( 32 − 9 π ) , β 2 = 3 ( 512 + 96 π − 81 π 2 ) ( 32 − 9 π ) 2 (15)

As the expressions of β 1 & β 2 in Equation (15) are independent of σ 2 so, applying value of π , β 1 → 0.1646 & β 2 → 3.0593 . So ARD is positively skewed and leptokurtic.

Median of the ARD is

γ ( 2 , m ′ ) = 1 2 , where m ′ = m 2 2 σ 2 , (16)

∫ 0 x t n − 1 e − t d t = γ ( n , x ) is lower incomplete gamma function.

Mode of the ARD is

mode = σ 3 , (17)

The Shannon entropy of the ARD is

H ( x ) = ln ( σ ) − 1.77058 (18)

In this section parameter of ARD is estimated through method of moments (MOM) and maximum likelihood estimator (MLE).

Equating x ¯ = ∑ i = 1 n x i n and Equation (11) as x ¯ = 3 σ 2π 4 we get MOM estimator of σ

σ ˜ = 4 X ¯ 3 2π . (19)

The likelihood function of (8)

L ( σ 2 ) = ∏ i = 1 n f ( x i / σ ) = ∏ i = 1 n x i 3 2 σ 4 e − x i 2 / 2 σ 2

Applying natural logarithm as log_{e}_{ }

log ∏ i = 1 n f ( x i / σ ) = − n log 2 − n log σ 4 − ∑ i = 1 n x i 2 2 σ 2 + ∑ i = 1 n log x i 3 (20)

Taking derivative of the Equation (20), we get

d d σ 2 log ∏ i = 1 n f ( x i / σ 2 ) = − 2 n σ 2 + 1 2 σ 4 ∑ i = 1 n x i 2 , (21)

Equating (19) to zero and simplifying we get the MLE estimator of σ

σ ^ 2 = ∑ i = 1 n X i 2 4 n (22)

σ ^ = ∑ i = 1 n X i 2 4 n (23)

Theorem 3.1: If X 1 , X 2 , ⋯ , X n follows the ARD then MOM σ ˜ of σ is unbiased and have minimum variance.

Proof: Applying expectation on (19) and simplifying it we get

E ( σ ˜ ) = 4 3 2 π E (X)

E ( σ ˜ ) = σ , (24)

So σ ˜ is unbiased estimator of σ. Applying variance on (19), we get

V a r ( σ ˜ ) = V a r ( 4 X ¯ 3 2 π )

After some simplifications we get,

V a r ( σ ˜ ) = σ 2 ( 32 − 9 π ) 9 π n , (25)

As n → ∞ , V a r ( σ ˜ ) → 0 . So for large “n”, MOM σ ˜ estimator of σ have minimum variance.

Theorem 3.2: If X 1 , X 2 , ⋯ , X n follows the ARD then MLE σ ^ 2 of σ 2 is unbiased and have minimum variance.

Proof: Applying expectation on (22) and simplifying it we get

E ( σ ^ ) 2 = 1 4 n ∑ i = i n E ( X i 2 ) , where E ( X i 2 ) = μ ′ 2

E ( σ ^ 2 ) = σ 2 (26)

So σ ^ 2 is unbiased estimator of σ 2 and σ ^ is biased estimator of σ.

Now applying variance on (22), we get

V a r ( σ ^ 2 ) = V a r ( ∑ i = 1 n X i 2 4 n )

V a r ( σ ^ 2 ) = σ 4 2 n (27)

As n → ∞ , V a r ( σ ^ 2 ) → 0 . So for large “n”, MLE σ ^ 2 estimator of σ 2 have minimum variance.

Theorem 3.3: Let X = ( X 1 , X 2 , ⋯ , X n ) be a random sample from a pdf f ( x / σ 2 ) in (8), where shape parameter, under regularity conditions on f ( x / σ 2 ) for an unbiased estimator ∅ ^ ( x ) of ∅ ( σ 2 ) i.e. E ( σ ^ 2 ) = σ 2

V a r ( σ 2 ) ≥ d d σ 2 ∅ ( σ 2 ) I ( σ 2 ) (28)

where

I ( σ 2 ) = − E [ d 2 d ( σ 2 ) 2 log ∏ i = 1 n f ( x i / σ 2 ) ]

Proof: Taking second derivative of (21), we get

d 2 d ( σ 2 ) 2 log ∏ i = i n f ( x i / σ 2 ) = 2 n σ 4 − ∑ i = 1 n x i 2 σ 6 (29)

Applying expectation on (27) and simplifying it

I ( σ 2 ) = − E [ d 2 d ( σ 2 ) 2 log ∏ i = 1 n f ( x i / σ 2 ) ] = 2 n σ 4 (30)

and

∅ ( σ 2 ) = σ 2 , so d d σ 2 ∅ ( σ 2 ) = 1 (31)

Substituting (30) and (31) in (28), we get

V a r ( σ 2 ) ≥ 1 2 n / σ 4 , where V a r ( σ 2 ) = σ 4 2 n

So the unbiased estimator σ ^ 2 estimator of σ 2 attains the Cramer Lower Bound.

The posterior probability distribution function can be derived by using

f ( σ / x _ ) = f ( x _ / σ ) g ( σ ) ∫ f ( x _ / σ ) g ( σ ) (32)

Using ARD f ( x / σ ) in (8) and uniform prior g ( σ ) = 1 , 0 < σ < 1 , in (32) we get the posterior pdf of ARD as

f ( σ / x _ ) = 2 Γ ( 2 n − 1 2 ) ( 2 ∑ i = 1 n x i 2 ) 2 n − 1 2 ( 1 σ 2 ) 2 n exp ( − ∑ i = 1 n x i 2 2 σ 2 ) , σ > 0 (33)

Using (33) the Bayesian estimator of σ

σ * = E ( σ / x _ ) = ∫ σ f ( σ / x ) d σ

σ * = ∑ i = 1 n x i 2 2 Γ ( 2 n − 1 2 ) Γ ( 2 n − 1 ) (34)

The survival function of the ARD

S ( x ) = Γ ( 2 , x ′ ) (35)

where

x ′ = x 2 2 σ 2 , and Γ ( n , x ) = ∫ x ∞ t n − 1 e − t d t

is upper incomplete gamma function.

The hazard function of the ALD is

h ( x ) = x 3 exp ( − x 2 / 2 σ 2 ) 2 σ 4 Γ ( 2 , x ′ ) (36)

In this section ARD is applied on two life time data sets and compared it with Lindley distribution (LD), Exponential distribution, quasi Lindley distribution (QLD), Rayleigh distribution (RD) and size-biased Rayleigh distribution (SRD) by using Kalmogorov Smirnov (K-S) Statistic. ARD is also compared for survival function and hazard function with RD and SRD on both data 1 and only with SRD for data 2 as RD is not provided good fit for data 2.

Data set 1: This data set represents the lifetime’s data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported by Gross and Clark [

Data Set 2: This data set is the strength data of glass of the aircraft window reported by Fuller et al. [

Form

Form

1) The survival function graphs of the ARD are smoothly decreasing as compare to RD and SRD.

2) The hazard function graphs of the ARD are monotonically increasing in a smooth way as compare to RD and SRD. From

Form

1) The survival function graphs of the ARD are smoothly decreasing as compare to SRD.

2) The hazard function graphs of the ARD are monotonically increasing; it means that the instantaneous failure for the strength of aircraft window is increasing.

Model | Estimation of parameters | K-S Statistics | P-value for K-S Statistics | ||
---|---|---|---|---|---|

Data 1 | Lindley | θ ^ = 0.816118 | 0.34 | 0.294 | |

Exponential | θ ^ = 0.526316 | 0.39 | |||

Quasi Lindley | θ ^ = 1.545110 | α ^ = − 0.483393 | 0.20 | ||

Rayleigh | σ ^ 2 = 1.702 | 0.249 | |||

Size-biased Rayleigh | σ ^ 2 = 1.135 | 0.165 | |||

Area-biased Rayleigh | σ ^ 2 = 0.851 | 0.112 |

Data 2 | Lindley | θ ^ = 0.062988 | 0.33 | 0.240 | |
---|---|---|---|---|---|

Exponential | θ ^ = 0.032455 | 0.43 | |||

Quasi Lindley | θ ^ = 0.103985 | α ^ = − 0.546267 | 0.30 | ||

Rayleigh | σ ^ 2 = 500.129 | 0.287 | |||

Size-biased Rayleigh | σ ^ 2 = 333.419 | 0.209 | |||

Area-biased Rayleigh | σ ^ 2 = 250.065 | 0.157 |

In this article a new weighted single parameter Rayleigh distribution named as area-biased Rayleigh distribution (ARD) is introduced. Various properties of the ARD have been derived. It can be seen from

estimator attains the Cramer Rao lower bound. Then the model is applied into two life time data sets. Kolmogorov Smirnov (K-S) test statistic is used to see the fit good on both data sets. The value of K-S for ARD is compared with some other well-known models named Lindley, Exponential, Quasi Lindley, Size-Biased Rayleigh and it is concluded that ARD is showing better fit on such kind of data sets as comparing to these models. At the end ARD, RD and SRD are used to show the graphical trend of the survival function and hazard function on both data sets. The survival function graph of SRD is decreasing more smoothly as comparing to other models. The hazard function graph of ARD for both data sets is increasing gradually. It means that the instantaneous failures are increasing. Overall it can be seen that ARD model can be a best alternative of the other well-known models and it is showing wide applications in the field of medical.

Bashir, S. and Rasul, M. (2018) A New Weighted Rayleigh Distribution: Properties and Applications on Lifetime Time Data. Open Journal of Statistics, 8, 640-650. https://doi.org/10.4236/ojs.2018.83041