^{1}

^{2}

In recent years, several statistical finite mixture models have been proposed to model the lifetime data with heterogeneity. The Lindley distribution has been highlighted by many authors for these types of lifetime data analysis. This pape r introduces a new Lindley family distribution called location - based ge neralized Akash distribution (NGAD) with monotonic increasing and bathtub failure rates. The density function of NGAD is flexible to cover the left-skewed, right-skewed and symmetrical shapes with different tail-weights. Its fundament al structural properties and its ability to provide a suitable statistical model for various types of data sets are studied. The maximum likelihood (ML) method is used to estimate its unknown parameters and the performance of ML estimates are examined by a simulation study. Finally, several real-data sets with different characteristics are used to illustrate its flexibility. It is observed that NGAD provides a better fit than some other existing modified Lindley distributions.

Several standard base-line distributions are available to model the lifetime data and the failure rate function is considered as the most crucial factor for these models. Examples of such distributions are exponential, gamma, Weibull, and log-normal distributions which have different capabilities to describe the shapes of the failure rate function. While the exponential distribution has a constant failure rate, the gamma and Weibull distributions have both increasing and decreasing failure rates [

Another classical finite mixture model under the Bayesian framework is the Lindley distribution (LD) which is a finite mixture of two non-identical distributions, exponential and gamma distributions. It was introduced by Lindley (1958) [

f Λ ( λ ) = θ 2 1 + θ ( 1 + λ ) e − θ λ ; λ > 0 , θ > 0 , (1)

where θ is the shape parameter, and λ is the respective random variable. Equation (1) presents two-component mixture of exponential ( θ ) and gamma

( 2, θ ) distributions with the mixing proportion p = θ 1 + θ . The statistical properties of LD have discussed by Ghitany, Atieh, and Nadarajah (2008) and showed

that the Lindley distribution is more flexible and provides a better fit than the exponential distribution for lifetime data [

A considerable number of modifications of LD have been proposed by researchers with actual mixing components of LD based on the two-component mixture of exponential ( θ ) and gamma ( 2, θ ) with different mixing proportions. Some notables are mentioned as, Shanker and Sharma (2013) proposed a two-parameter Lindley distribution [

To increase more flexibility in this line of developments, Monsef (2016) [

f Λ ( λ ) = θ 2 θ + α ( 1 + α ( λ − β ) ) e − θ ( λ − β ) ; λ > β > 0 , θ > 0 , 1 + α λ > 0 , α + θ > 0 , (2)

where θ and α are shape parameters and β is a location parameter. Equation (2) presents two-component mixture of an exponential ( θ , β ) and gamma

( 2 , θ , β ) distributions with the mixing proportion p = θ θ + α .

On the other hand, Shanker (2015) [

f Λ ( λ ) = θ 3 θ 2 + 2 ( 1 + λ 2 ) e − θ λ ; λ > 0 ; θ > 0 , (3)

where θ is a shape parameter. Equation (3) shows two-component mixture of

exponential ( θ ) and gamma ( 3, θ ) with mixing proportion θ 2 θ 2 + 2 .

In many lifetime distributions, the location parameter may be assumed to be zero. However, the location parameter is an important parameter that provides an estimate of the failure-free period, that is the earliest time at which a failure may be observed. When we include the location parameter in a distribution, it changes the starting point of the distribution. For example, time to achieve pain relief in a patient after who has applied with a treatment method does not start from the value of zero. Then, in this application, the location parameter of the relevant distribution cannot be taken as zero.

Distribution | Abbreviation | Parameters | Failure rate | References | ||
---|---|---|---|---|---|---|

scale | shape | location | ||||

Lindley | LD | - | θ | - | increasing | Lindley (1958) [ |

Two-parameter Lindley | TwPLD | - | θ , α | - | increasing | Shanker et al. (2013) [ |

Quasi Lindley | QLD | θ | α | - | increasing | Shanker et al. (2013) [ |

Shanker | SD | - | θ | - | increasing | Shanker (2015) [ |

Akash | AD | - | θ | - | increasing | Shanker (2015) [ |

Lindley with location distribution | LwLD | - | θ , α | β | increasing | Monsef (2016) [ |

Three-parameter Lindley | ThPLD | - | θ , α , β | - | increasing | Shanker et al. (2017) [ |

& decreasing |

In this paper, we introduce and study a new three-parameter Lindley family distribution with location parameter as a modification of AD. The new distribution will be called as location based generalized Akash distribution (NGAD). The NGAD is a two-component mixture of exponential and gamma distributions which generalizes the AD. Further, a simulation study will be done to study the performance of the ML method in the parameter estimation of the NGAD.

The remaining part of this paper is organized as follows: In Section 2, we introduce the NGAD with its density and distribution functions. We present the statistical and reliability properties of NGAD in Section 3 and Section 4, respectively. The size-biased form of the NGAD is discussed in Section 5. Further, the estimation of the parameters of NGAD by using the ML method is discussed in Section 6. Finally, a simulation study is done to study the performance of the maximum likelihood estimators for NGAD, and real-world applications are used to illustrate its flexibility with the above-mentioned existing modified Lindley distributions.

In this section, we introduce the location based generalized Akash distribution with its probability density function (pdf) and cumulative distribution function (cdf). Note that the Akash distribution (3) contains only one shape parameter. To develop the new distribution first, we define a new three-parameter Lindley family distribution by including the location parameter β as a finite mixture of

exponential ( θ , β ) and gamma ( 3, θ , β ) with mixing proportion p = θ 2 θ 2 + η

under the Bayesian framework, as follows:

f Λ ( λ ; θ , β , η ) = p f 1 ( λ ; θ , β ) + ( 1 − p ) f 2 ( λ ; 3, θ , β ) ,

where, θ and η are shape parameters and

f 1 ( λ ; θ , β ) = θ e − θ ( λ − β ) , and f 2 ( λ ; 3 , θ , β ) = θ 3 ( λ − β ) 2 2 ; λ > β > 0 ; θ > 0 .

Then, the pdf of the NGAD with parameters θ , β and η is defined by:

f Λ ( λ ; θ , β , η ) = θ 3 2 ( θ 2 + η ) ( 2 + η ( λ − β ) 2 ) e − θ ( λ − β ) ; λ > β > 0 , θ > 0 , θ 2 > − η . (4)

The range of the parameters is based on the log-likelihood function. Then, NGAD has one additional shape parameter and the location parameter compared to AD's parameter. Plots in ^{rd} parameter.

from

The mode value in the unimodal case can be derived directly from the first derivative of Equation (4), and given as:

f ′ ( λ ) = θ 3 e − θ ( λ − β ) 2 ( θ 2 + η ) ( 2 η ( λ − β ) − θ ( 2 + η ( λ − β ) 2 ) ) .

Then, f ′ ( λ ) = 0 gives the mode value:

λ 0 = η ( β θ + 1 ) + η ( η − 2 θ 2 ) η θ ; η > 2 θ 2 and λ 0 > β .

The corresponding cumulative distribution function of NGAD is given by:

F Λ ( λ ) = 1 − ( 1 + η ( 2 θ λ + ( θ λ ) 2 − β θ ( 2 ( 1 + θ λ ) − β θ ) ) 2 ( θ 2 + η ) ) e − θ ( λ − β ) , (5)

where λ > β > 0 , θ > 0 , θ 2 > − η .

Note that when η = 2 , β = 0 , the NGAD reduces to the AD with parameter θ .

Here, some important statistical properties of the NGAD are derived such as r^{th} moments and related measures, moment generating and characteristic functions, and quantile function.

Some basic important characteristics of the distribution such as central tendency, dispersion, skewness, kurtosis, and index of dispersion can be studied by using the moments. The following theorem gives the r^{th} moment about the origin.

Theorem 1. The r^{th} moment about the origin of the NGAD is given by:

μ ′ r = 1 2 ( θ 2 + η ) θ r ( r e θ β Γ ( r , θ β ) ( 2 θ 2 + η ( ( r + 1 ) ( r + 2 ) − 2 β θ ( r + 1 ) + β 2 θ 2 ) ) + ( θ β ) r ( 2 θ 2 + η ( ( r + 1 ) ( r + 2 ) − r β θ ) ) ) . (6)

Proof.

μ ′ r = ∫ β ∞ λ r θ 3 2 ( θ 2 + η ) ( 2 + η ( λ − β ) 2 ) e − θ ( λ − β ) d λ = θ 3 e θ β 2 ( θ 2 + η ) ( 2 ∫ β ∞ λ r e − θ λ d λ + η ∫ β ∞ λ r ( λ − β ) 2 e − θ λ d λ ) = θ 3 e θ β 2 ( θ 2 + η ) ( 2 θ r + 1 Γ ( r + 1, θ β ) + η ( 1 θ r + 3 Γ ( r + 3, θ β ) − 2 β θ r + 2 Γ ( r + 2, θ β ) + β 2 θ r + 1 Γ ( r + 1, θ β ) ) )

= 1 2 ( θ 2 + η ) θ 2 ( r Γ ( r , θ β ) e θ β ( 2 θ 2 + η ( ( r + 1 ) ( r + 2 ) − 2 β θ ( r + 1 ) + β 2 θ 2 ) ) + ( θ β ) r ( 2 θ 2 + η ( ( r + 1 ) ( r + 2 ) − r β θ ) ) ) = 1 2 ( θ 2 + η ) θ r ( r e θ β Γ ( r , θ β ) ( 2 θ 2 + η ( ( r + 1 ) ( r + 2 ) − 2 β θ ( r + 1 ) + β 2 θ 2 ) ) + ( θ β ) r ( 2 θ 2 + η ( ( r + 1 ) ( r + 2 ) − r β θ ) ) ) .

The first four moments about the origin are derived by substituting r = 1,2,3 , and 4 in Equation (6) as:

μ ′ 1 = θ 2 ( 1 + θ β ) + η ( 3 + θ β ) ( θ 2 + η ) θ = μ ,

μ ′ 2 = θ 2 ( 2 + θ β ( 2 + θ β ) ) + η ( 12 + θ β ( 6 + θ β ) ) ( θ 2 + η ) θ 2 ,

μ ′ 3 = θ 2 ( 6 + θ β ( 6 + θ β ( 3 + θ β ) ) ) + η ( 60 + θ β ( 36 + θ β ( 9 + θ β ) ) ) ( θ 2 + η ) θ 3 , and

μ ′ 4 = θ 2 ( 24 + θ β ( 24 + θ β ( 12 + θ β ( 4 + θ β ) ) ) ) + η ( 360 + θ β ( 240 + θ β ( 72 + θ β ( 12 + θ β ) ) ) ) ( θ 2 + η ) θ 4 ,

respectively. Then, the r^{th}-order moments about the mean can be obtained by using the relationship between moments about the mean and moments about the origin; i.e.

μ r = E [ ( Λ − μ ) r ] = ∑ i = 0 r ( r i ) ( − 1 ) r − i μ ′ i μ r − i

Therefore, μ 2 = σ 2 , μ 3 , and μ 4 are obtained as:

μ 2 = − μ 2 + μ ′ 2 = θ 4 + η ( 8 θ 2 + 3 η ) ( θ 2 + η ) 2 θ 2 ,

μ 3 = 2 μ 3 − 3 μ ′ 2 μ + μ ′ 3 = 2 θ 6 + η ( 30 θ 4 + η ( 18 θ 2 + 6 η ) ) ( θ 2 + η ) 3 θ 3 , and

μ 4 = − 3 μ 4 + 6 μ ′ 2 μ 2 − 4 μ ′ 3 μ + μ ′ 4 = 9 θ 8 + η ( 192 θ 6 + η ( 306 θ 4 + η ( 216 θ 2 + 45 η ) ) ) ( θ 2 + η ) 4 θ 4 ,

respectively. Then, the the coefficient of variation (c.v), measures of skewness ( γ 1 ), kurtosis ( γ 2 ) and the Index of dispersion/Fano factor ( γ 3 ) of the NGAD can be derived as:

c . v = ( μ 2 ) 1 / 2 μ ′ 1 = θ 4 + η ( 8 θ 2 + 3 η ) θ 2 ( 1 + θ β ) + η ( 3 + θ β ) ,

γ 1 = μ 3 ( μ 2 ) 3 / 2 = 2 θ 6 + η ( 30 θ 4 + η ( 18 θ 2 + 6 η ) ) ( θ 4 + η ( 8 θ 2 + 3 η ) ) 3 2 ,

γ 2 = μ 4 μ 2 2 = 9 θ 8 + η ( 192 θ 6 + η ( 306 θ 4 + η ( 216 θ 2 + 45 η ) ) ) ( θ 4 + η ( 8 θ 2 + 3 η ) ) 2 , and

γ 3 = μ 2 μ ′ 1 = θ 4 + η ( 8 θ 2 + 3 η ) θ ( θ 2 + η ) ( θ 2 ( 1 + θ β ) + η ( 3 + θ β ) ) ,

respectively.

The moment generating function (mgf) and the characteristic function (cf) are directly associated with a probability distribution's characteristics. Further, these can be used to generate the moments of a distribution. The following theorem provides the moment generating function of the NGAD.

Theorem 2. The moment generating function say M Λ ( t ) of the NGAD is given as follows:

M Λ ( t ) = θ 3 e β t 2 ( θ 2 + η ) ( θ − t ) 3 ( 2 ( θ − t ) 2 + η ( 2 + β ( θ − t ) ( 2 + β ( θ − t ) − 2 ( 1 + β ( θ − t ) ) + β ( θ − t ) ) ) ) . (7)

Proof.

M Λ ( t ) = E ( e t λ ) = ∫ β ∞ e t λ θ 3 2 ( θ 2 + η ) ( 2 + η ( λ − β ) 2 ) e − θ ( λ − β ) d λ = θ 3 e θ β 2 ( θ 2 + η ) ( 2 ∫ β ∞ e − λ ( θ − t ) d λ + η ∫ β ∞ e − λ ( θ − t ) ( λ − β ) 2 d λ ) = θ 3 e θ β 2 ( θ 2 + η ) ( 2 e − β ( θ − t ) θ − t + η ( Γ ( 3, β ( θ − t ) ) ( θ − t ) 3 − 2 β Γ ( 2, β ( θ − t ) ) ( θ − t ) 2 + β 2 e − β ( θ − t ) θ − t ) ) = θ 3 e β t 2 ( θ 2 + η ) ( θ − t ) 3 ( 2 ( θ − t ) 2 + η ( 2 + β ( θ − t ) ( 2 + β ( θ − t ) − 2 ( 1 + β ( θ − t ) ) + β ( θ − t ) ) ) )

↓ θ → η | 0.01 | 0.10 | 0.45 | 0.95 | 1.45 | 1.95 |
---|---|---|---|---|---|---|

0.03 | 1.092 (4.825) | 1.144 (4.978) | 1.152 (4.995) | 1.153 (4.997) | 1.154 (4.998) | 1.155 (4.999) |

0.05 | 1.089 (4.727) | 1.129 (4.939) | 1.148 (4.985) | 1.151 (4.993) | 1.152 (4.995) | 1.153 (4.996) |

0.10 | 1.347 (5.333) | 1.089 (4.812) | 1.132 (4.945) | 1.143 (4.973) | 1.147 (4.982) | 1.148 (4.987) |

0.20 | 1.883 (7.842) | 1.130 (4.768) | 1.092 (4.826) | 1.116 (4.903) | 1.127 (4.933) | 1.133 (4.949) |

0.45 | 2.123 (9.790) | 1.618 (6.415) | 1.147 (4.798) | 1.083 (4.733) | 1.082 (4.772) | 1.088 (4.807) |

0.70 | 2.084 (9.621) | 1.948 (8.264) | 1.376 (5.432) | 1.171 (4.846) | 1.110 (4.741) | 1.089 (4.727) |

0.95 | 2.054 (9.415) | 2.082 (9.282) | 1.614 (6.397) | 1.330 (5.278) | 1.211 (4.937) | 1.151 (4.806) |

1.20 | 2.036 (9.285) | 2.120 (9.689) | 1.802 (7.361) | 1.501 (5.902) | 1.344 (5.326) | 1.254 (5.051) |

1.45 | 2.026 (9.205) | 2.122 (9.793) | 1.934 (8.167) | 1.655 (6.589) | 1.483 (5.831) | 1.373 (5.420) |

Similarly, the characteristic function say, ψ ( t ) of the NGAD can be derived as follows:

ψ Λ ( t ) = E ( e i t λ ) = θ 3 e β i t 2 ( θ 2 + η ) ( θ − i t ) 3 ( 2 ( θ − i t ) 2 + η ( 2 + β ( θ − i t ) ( 2 + β ( θ − i t ) − 2 ( 1 + β ( θ − i t ) ) + β ( θ − i t ) ) ) ) . (8)

The quantile function is useful for quantile estimates and random number generation. The quantile function of NGAD can be derived by solving F ( λ ) = u , 0 < u < 1 . Then, the u^{th} quantile function of NGAD is derived as:

F ( λ u ) = 1 − ( 1 + η ( 2 θ λ u + ( θ λ u ) 2 − β θ ( 2 ( 1 + θ λ u ) − β θ ) ) 2 ( θ 2 + η ) ) e − θ ( λ u − β ) = u .

⇒ ( 2 ( θ 2 + η ) + η ( 2 θ λ u + ( θ λ u ) 2 − β θ ( 2 ( 1 + θ λ u ) − β θ ) ) ) e − θ ( λ u − β ) − 2 ( θ 2 + η ) ( 1 − u ) = 0. (9)

Equation (9) is not a closed-form. However, the u^{th} quantile can be estimated and random varieties from NGAD can be generated by using the numerical methods.

By substituting u = 0.25,0.5 and 0.75 in Equation (9), the first three quartiles can be derived by solving the following equations, respectively.

( 2 ( θ 2 + η ) + η ( 2 θ λ 0.25 + ( θ λ 0.25 ) 2 − β θ ( 2 ( 1 + θ λ 0.25 ) − β θ ) ) ) e − θ ( λ 0.25 − β ) − 2 ( θ 2 + η ) 0.75 = 0 ,

( 2 ( θ 2 + η ) + η ( 2 θ λ 0.5 + ( θ λ 0.5 ) 2 − β θ ( 2 ( 1 + θ λ 0.5 ) − β θ ) ) ) e − θ ( λ 0.5 − β ) − 2 ( θ 2 + η ) 0.5 = 0 , and

( 2 ( θ 2 + η ) + η ( 2 θ λ 0.75 + ( θ λ 0.75 ) 2 − β θ ( 2 ( 1 + θ λ 0.75 ) − β θ ) ) ) e − θ ( λ 0.75 − β ) − 2 ( θ 2 + η ) 0.25 = 0.

In this study, we derive important reliability measures of NGAD such as survival function/reliability function S ( λ ) , hazard rate function/failure rate function h ( λ ) , reversed hazard rate function r ( λ ) , cumulative hazard rate function H ( λ ) , mean residual life function m ( λ ) , and inequality measures, namely Lorenz curve L ( λ ) , and Benferroni curve B ( λ ) . Further, we obtain Renyi entropy measure associated to NGAD.

The survival function and hazard rate function are crucial functions to specify a survival distribution. The survival function is the probability of surviving up to a point λ . Then, the survival function of NGAD is defined as:

S ( λ ) = p ( Λ > λ ) = 1 − F ( λ ) = ( 1 + η ( 2 θ λ + ( θ λ ) 2 − β θ ( 2 ( 1 + θ λ ) − β θ ) ) 2 ( θ 2 + η ) ) e − θ ( λ − β ) ; λ > β . (10)

It is clear that, S ( β ) = 1 and l i m λ → ∞ S ( λ ) = 0 .

The hazard rate function (hrf) is the instantaneous failure rate. It is widely used to describe the lifetime. The hrf of the NGAD is defined as:

h ( λ ) = l i m Δ λ → 0 P ( λ < Λ < λ + Δ λ | Λ > λ ) Δ λ = f ( λ ) S ( λ ) = θ 3 ( 2 + η ( λ − β ) 2 ) 2 ( θ 2 + η ) + η ( θ λ ( 2 + θ λ ) − β θ ( 2 ( 1 + θ λ ) − β θ ) ) . (11)

Note that, h ( β ) = θ 3 θ 2 + η and l i m λ → ∞ h ( λ ) = θ .

The corresponding reversed hazard rate function of NGAD is defined as:

r ( λ ) = lim Δ λ → 0 P ( λ < Λ < λ + Δ λ | Λ < λ ) Δ λ = θ 3 ( 2 + η ( λ − β ) 2 ) e − θ ( λ − β ) 2 ( θ 2 + η ) − ( 2 θ 2 + η ( 2 + θ λ ( 2 + θ λ ) − β θ ( 2 ( 1 + θ λ ) − β θ ) ) ) e − θ ( λ − β ) , (12)

and the cumulative hazard rate function of NGAD is defined as:

H ( λ ) = ∫ β λ h ( t ) d t = − ln [ S ( λ ) ] = − ln ( 1 + η ( θ λ ( 2 + θ λ ) − β θ ( 2 ( 1 + θ λ ) − β θ ) ) 2 ( θ 2 + η ) ) + θ ( λ − β ) . (13)

The residual life function is defined as the remaining lifetime of a unit of age λ > β until the time of failure. The following theorem derives the mean residual life function of NGAD.

Theorem 3. The mean residual life function of NGAD is given by:

m ( λ ) = 2 θ 2 + η ( 6 − β θ ( 4 − β θ ) + θ λ ( 4 − 2 β θ + θ λ ) ) θ ( 2 ( θ 2 + η ) + η ( θ λ ( 2 + θ λ ) − β θ ( 2 ( 1 + θ λ ) − β θ ) ) ) ; λ > β . (14)

Proof. m ( λ ) = 1 1 − F ( λ ) ∫ λ ∞ t f ( t ) d t − λ .

Now, consider the integrals separately as follows:

∫ λ ∞ t f ( t ) d t = ∫ λ ∞ t θ 3 2 ( θ 2 + η ) ( 2 + η ( t − β ) 2 ) e − θ ( t − β ) d t = θ 3 e θ β 2 ( θ 2 + η ) ( 2 ∫ λ ∞ t e − θ t d t + η ( ∫ λ ∞ t 3 e − θ t d t − 2 β ∫ λ ∞ t 2 e − θ t d t + β 2 ∫ λ ∞ t e − θ t d t ) ) = θ 3 e θ β 2 ( θ 2 + η ) ( 2 Γ ( 2 , θ λ ) θ 2 + η ( Γ ( 4 , θ λ ) θ 4 − 2 β Γ ( 3 , θ λ ) θ 3 + β 2 Γ ( 2 , θ λ ) θ 2 ) ) = e − θ ( λ − β ) 2 ( θ 2 + η ) θ ( 2 θ 2 ( 1 + θ λ ) + A )

where, A = η ( 6 − β θ ( 4 − β θ ) + θ λ ( 6 − β θ ( 4 − β θ ) + θ λ ( 3 − 2 β θ + θ λ ) ) ) .

Therefore,

m ( λ ) = 2 θ 2 ( 1 + θ λ ) + A θ ( 2 ( θ 2 + η ) + η ( θ λ ( 2 + θ λ ) − β θ ( 2 ( 1 + θ λ ) − β θ ) ) ) − λ = 2 θ 2 + η ( 6 − β θ ( 4 − β θ ) + θ λ ( 4 − 2 β θ + θ λ ) ) θ ( 2 ( θ 2 + η ) + η ( θ λ ( 2 + θ λ ) − β θ ( 2 ( 1 + θ λ ) − β θ ) ) ) .

Then, Equation (14) satisfies the following properties

m ( λ ) ≥ 0, m ( β ) = θ 2 + 3 η θ ( θ 2 + η ) , and l i m λ → ∞ m ( λ ) = 1 θ .

The Lorenz and Bonferroni curves were formulated to measure the income inequality. They are widely used in economics, reliability, demography, medicine, and insurance. The following theorem derives the Lorenz curve for NGLD.

Theorem 4. The Lorenz curve is defined for FPGLD as

L ( F ( λ ) ) = 1 − ∫ λ ∞ x f ( x ) d x μ = 1 − e − θ ( λ − β ) ( 2 θ 2 ( 1 + θ λ ) + B ) 2 ( θ 2 ( 1 + θ β ) + η ( 3 + θ β ) ) , (15)

where, B = η ( 6 + θ λ ( 6 + 3 θ λ + ( θ λ ) 2 ) − β θ ( 4 + θ λ ( 4 + 2 θ λ − β θ ) − β θ ) )

Proof. L ( F ( λ ) ) = 1 − ∫ λ ∞ x f ( x ) d x μ .

Note that

∫ λ ∞ x f ( x ) d x = ∫ λ ∞ x θ 3 2 ( θ 2 + η ) ( 2 + η ( x − β ) 2 ) e − θ ( x − β ) d x = θ 3 e θ β 2 ( θ 2 + η ) ( 2 ∫ λ ∞ x e − θ x d x + η ∫ λ ∞ x ( x − β ) 2 e − θ x d x ) = θ 3 e θ β 2 ( θ 2 + η ) ( 2 ∫ λ ∞ x e − θ x d x + η ( ∫ λ ∞ x 3 e − θ x d x − 2 β ∫ λ ∞ x 2 e − θ x d x + β 2 ∫ λ ∞ x e − θ x d x ) )

= θ 3 e θ β 2 ( θ 2 + η ) ( 2 Γ ( 2 , θ λ ) θ 2 + η ( Γ ( 4 , θ λ ) θ 4 − 2 β Γ ( 3 , θ λ ) θ 3 + β 2 Γ ( 2 , θ λ ) θ 2 ) ) = e − θ ( λ − β ) 2 ( θ 2 + η ) θ ( 2 θ 2 ( 1 + θ λ ) + η ( 6 + θ λ ( 6 + 3 θ λ + ( θ λ ) 2 ) − β θ ( 4 + θ λ ( 4 + 2 θ λ − β θ ) − β θ ) ) ) .

Therefore,

L ( F ( λ ) ) = 1 − e − θ ( λ − β ) ( 2 θ 2 ( 1 + θ λ ) + B ) 2 ( θ 2 ( 1 + θ β ) + η ( 3 + θ β ) ) ,

and the corresponding Bonferroni curve for the NGAD is defined as:

B ( F ( λ ) ) = L ( F ( λ ) ) F ( λ ) = 2 ( θ 2 + η ) ( 2 ( θ 2 ( 1 + θ β ) + η ( 3 + θ β ) ) − ( 2 θ 2 ( 1 + θ λ ) + C ) e − θ ( λ − β ) ) 2 ( θ 2 ( 1 + θ β ) + η ( 3 + θ β ) ) ( 2 ( θ 2 + η ) − ( 2 θ 2 + D ) e − θ ( λ − β ) ) , (16)

where,

C = η ( 6 + θ λ ( 6 + 3 θ λ + ( θ λ ) 2 ) − β θ ( 4 + θ λ ( 4 + 2 θ λ − β θ ) − β θ ) ) and

D = η ( 2 + θ λ ( 2 + θ λ ) − β θ ( 2 ( 1 + θ λ ) − β θ ) ) .

The Renyi entropy is a measure of variation of uncertainty measure of a distribution say H γ ( λ ) [

Theorem 5. The Renyi entropy of the NGAD is given by:

H γ ( λ ) = 1 1 − γ ln ∫ β ∞ ( f ( λ ) ) γ d λ = 1 1 − γ ln ( θ 3 γ ( θ 2 + η ) γ ∑ k = 0 γ ( γ k ) η k Γ ( 2 k + 1 ) 2 k ( γ θ ) 2 k + 1 ) ; γ ≥ 0 , γ ≠ 1. (17)

Proof.

H γ ( λ ) = 1 1 − γ ln ∫ β ∞ ( f ( λ ) ) γ d λ = 1 1 − γ ln ( θ 3 γ ( θ 2 + η ) γ ∫ β ∞ ∑ k = 0 γ ( γ k ) ( η ( λ − β ) 2 2 ) k e − γ θ ( λ − β ) d λ ) = 1 1 − γ ln ( θ 3 γ ( θ 2 + η ) γ ∑ k = 0 γ ( γ k ) ( η 2 ) k 1 ( γ θ ) 2 k + 1 ∫ β ∞ y 2 k e − y d y ) ; y = γ θ ( λ − β ) = 1 1 − γ ln ( θ 3 γ ( θ 2 + η ) γ ∑ k = 0 γ ( γ k ) η k Γ ( 2 k + 1 ) 2 k ( γ θ ) 2 k + 1 ) .

When a recording of observations with an unequal chance, the weighted distributions are used significantly. This provides more flexibility to the standard distributions incorporating sampling probabilities which are proportional to a non-negative weighted function w ( λ ) . The applications of the weighted distributions in reliability, biomedical and ecological sciences have been studied by Patil and Rao (1978) [

f Λ w ( λ ) = w ( λ ) f ( λ ) E ( w ( λ ) ) ; λ > β (18)

where E ( w ( λ ) ) = ∫ β ∞ w ( λ ) f ( λ ) d λ ; 0 < E ( w ( λ ) ) < ∞ .

When w ( λ ) = λ γ , γ > 0 , the resulting distribution is called size-biased version of NGAD with order γ , and is defined as:

f Λ s γ ( λ ) = λ γ f ( λ ) E ( λ γ ) ; λ > β , γ > 0 ,

where Λ s is the respective random variable.

The following theorem gives the density function for the sized-biased version of NGAD.

Theorem 6. The density function for r^{th} order sized-biased form of NGAD is derived as:

f Λ s γ ( λ ) = λ γ θ γ + 3 ( 2 + η ( λ − β ) 2 ) e − θ λ Γ ( γ + 1 , θ β ) ( 2 θ 2 + E ) + η e − θ β ( θ β ) γ + 1 ( γ + 2 − θ β ) ; λ > β , γ > 0 , (19)

where E = η ( ( γ + 1 ) ( γ + 2 ) − 2 θ β ( γ + 1 ) + β 2 θ 2 ) .

Proof. f Λ s γ ( λ ) = λ γ f ( λ ) E ( λ γ ) .

Note that

E ( λ γ ) = ∫ β ∞ λ γ f ( λ ) d λ = ∫ β ∞ λ γ θ 3 2 ( θ 2 + η ) ( 2 + η ( λ − β ) 2 ) e − θ ( λ − β ) d λ = θ 3 e θ β 2 ( θ 2 + η ) ( 2 ∫ β ∞ λ γ e − θ λ d λ + ∫ β ∞ λ γ ( λ − β ) 2 e − θ λ d λ ) = θ 3 e θ β 2 ( θ 2 + η ) ( Γ ( γ + 1 , θ β ) θ γ + 1 + η ( Γ ( γ + 3 , θ β ) θ γ + 3

− 2 β Γ ( γ + 2 , θ β ) θ γ + 2 + β 2 Γ ( γ + 1 , θ β ) θ γ + 1 ) ) = e θ β 2 ( θ 2 + η ) θ γ ( Γ ( γ + 1 , θ β ) ( 2 θ 2 + η ( ( γ + 1 ) ( γ + 2 ) − 2 β θ ( γ + 1 ) + ( β θ ) 2 ) ) + η ( θ β ) γ + 1 e − θ β ( γ + 2 − θ β ) ) .

Therefore,

f Λ s γ ( λ ) = λ γ θ γ + 3 ( 2 + η ( λ − β ) 2 ) e − θ λ Γ ( γ + 1 , θ β ) ( 2 θ 2 + E ) + η e − θ β ( θ β ) γ + 1 ( γ + 2 − θ β ) .

By substituting γ = 1 in Equation (19), the length-biased probability density function can be obtained as:

λ θ 4 ( 2 + η ( λ − β ) 2 ) e − θ ( λ − β ) ( 1 + θ β ) ( 2 θ 2 + η ( 6 − 4 θ β + ( β θ ) 2 ) ) + η ( θ β ) 2 ( 3 − θ β ) ; λ > β , γ > 0. (20)

In this section, the method of ML is introduced to estimate the parameters of NGAD. Further, the confidence intervals for the unknown parameters are derived.

Let λ 1 , λ 2 , ⋯ , λ n be identically and independently distributed random variables from NGAD with the likelihood function of the i^{th} sample value λ i as:

L ( θ , β , η | λ i ) = θ 3 2 ( θ 2 + η ) ( 2 + η ( λ i − β ) 2 ) e − θ ( λ i − β ) .

Hence, the log-likelihood function l = ln ( L ) is given by:

l = 3 n ln θ − ∑ i = 1 n θ ( λ i − β ) + ∑ i = 1 n ln ( 2 + η ( λ i − β ) 2 ) − n ln ( 2 ( θ 2 + η ) ) .

Then, the ML estimators of θ , β , and η , abbreviated as θ ^ , β ^ , and η ^ can be derived by equating the partial derivatives of the l with respect to each parameter to zero. Then, the systems of equations are:

∂ l ∂ θ = 3 n θ − ∑ i = 1 n ( λ i − β ) − 2 n θ θ 2 + η , ∂ l ∂ β = n θ − ∑ i = 1 n 2 η ( λ i − β ) 2 + η ( λ i − β ) 2 , and ∂ l ∂ η = ∑ i = 1 n ( λ i − β ) 2 2 + η ( λ i − β ) 2 − n θ 2 + η .

The second partial derivatives of the l are:

∂ 2 l ∂ θ 2 = − 3 n θ 2 − 2 n ( η − θ 2 ) ( θ 2 + η ) 2 , ∂ 2 l ∂ θ ∂ β = n , ∂ 2 l ∂ θ ∂ η = 2 n θ ( θ 2 + η ) 2 , ∂ 2 l ∂ β 2 = 2 η ∑ i = 1 n 2 − η ( λ i − β ) 2 ( 2 + η ( λ i − β ) 2 ) 2 , ∂ 2 l ∂ β ∂ η = − ∑ i = 1 n 4 ( λ i − β ) ( 2 + η ( λ i − β ) 2 ) 2 , and ∂ 2 l ∂ η 2 = n ( θ 2 + η ) 2 − ∑ i = 1 n ( λ i − β ) 4 ( 2 + η ( λ i − β ) 2 ) 2 .

By the asymptotic theory, the estimators are asymptotically normal 3-variate with mean ( θ , β , η ) and the observed information matrix:

I ( θ , β , η ) = ( − ∂ 2 l ∂ θ 2 − ∂ 2 l ∂ θ ∂ β − ∂ 2 l ∂ θ ∂ η − ∂ 2 l ∂ β ∂ θ − ∂ 2 l ∂ β 2 − ∂ 2 l ∂ β ∂ η − ∂ 2 l ∂ η ∂ θ − ∂ 2 l ∂ η ∂ β − ∂ 2 l ∂ η 2 )

at θ = θ ^ , β = β ^ , and η = η ^ . By the asymptotic theory, the estimates are approximately multivariate normal. Therefore, the ( 1 − a ) 100 % confidence interval for the parameters θ , β , and η are given by

θ ^ ± z a / 2 var ( θ ^ ) , β ^ ± z a / 2 var ( β ^ ) , η ^ ± z a / 2 var ( η ^ ) ,

wherein, the var ( θ ^ ) , var ( β ^ ) , and var ( η ^ ) are the variance of θ ^ , β ^ , and η ^ , respectively, and can be derived by diagonal elements of I − 1 ( θ , β , η ) and z a / 2 is the critical value at a level of significance.

In this section, we examine the performance of the parameter estimates of NGAD by ML method and asymptotic theory respect to sample size n. Here, we perform a simulation study, and further, we illustrate the flexibility of the NGAD over the LD, QLD, TwPLD, LwLD, SD, AD, and ThPLD by using several real data sets.

In this subsection, we present the simulation study results to examine the performance of the estimations by the ML method that proposed in Section 6. Equation (9) is used to carry out the simulation study by generating random samples from NGAD, where u is assumed to follow uniform distribution ( 0,1 ) . The replicates are 1000 in all the simulations with sample sizes 40,100,160, and 220. The combinations of parameter values are set at (

1) The average biases are:

2) The average MSEs are:

This subsection is considered to show the flexibility of NGAD over some other existing Lindley family distributions by fitting these to several real data sets. The Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and Kolmogorov-Smirnov Statistics (K-S Statistics) are utilized to compare the performance of LD, SD, AD, TwPLD, QLD, LwLD, and ThPLD. The ML method was used to estimate the unknown parameters. The first three real data sets that were considered for the goodness of fit of distributions are given below:

Data set 1: The following data set represents the tensile strength, measured in GPa, of 69 carbon fibres tested under tension at gauge lengths of 20 mm and reported by Bader and Priest (1982) [

1.312, 1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997, 2.006, 2.021, 2.027, 2.055, 2.063, 2.098, 2.140, 2.179, 2.224, 2.240, 2.253, 2.270, 2.272, 2.274, 2.301, 2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.490, 2.511, 2.514, 2.535, 2.554, 2.566, 2.570, 2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697, 2.726, 2.770, 2.773, 2.800, 2.809, 2.818, 2.821, 2.848, 2.880, 2.954, 3.012, 3.067, 3.084, 3.090, 3.096, 3.128, 3.233, 3.433, 3.585, 3.585.

Data set 2: The following data set reported by Lawless (1982) [

17.88, 28.92, 33, 41.52, 42.12, 45.6, 48.8, 51.84, 51.96, 54.12, 55.56, 67.8, 68.44, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, 173.4.

Data set 3: The following data set represents the tree circumferences in Marshall, Minnesota and reported by Shakil, Kibria, and Singh (2010) [

1.8, 1.8, 1.9, 2.4, 3.1, 3.4, 3.7, 3.7, 3.8, 3.9, 4.0, 4.1, 4.9, 5.1, 5.1, 5.2, 5.3, 5.5, 8.3, 13.7.

Some of the statistical measures for data set 1, 2 and 3 are given in

The fitted density plot of each distribution is shown in

Then, similarly we have fitted NGAD and LwLD to several real-data sets (Appendix) having different EK values and compare their performances.

Parameter | Average MSE (bias) | |||
---|---|---|---|---|

0.006952 | 0.006849 | 0.006763 | 0.006298 | |

(0.029578) | (0.025475) | (0.021125) | (0.019797) | |

2.274852 | 1.035007 | 0.996790 | 0.899319 | |

(0.510524) | (0.233261) | (0.201401) | (0.173923) | |

13.811090 | 12.060685 | 7.910364 | 7.140838 | |

(1.448722) | (1.343749) | (1.192976) | (1.168940) | |

0.003798 | 0.000793 | 0.000361 | 0.000356 | |

(0.017002) | (0.004542) | (0.003929) | (0.002167) | |

1.699948 | 0.258165 | 0.162436 | 0.120519 | |

(0.336223) | (0.018714) | (0.007087) | (0.003035) | |

17.132331 | 10.118250 | 4.588242 | 4.077955 | |

(2.236709) | (1.846210) | (1.514328) | (1.209076) |

Data | Sample size | Minimum value | Mean | Median | Skewness | EK |
---|---|---|---|---|---|---|

Data 1 | 69 | 1.312 | 2.451 | 2.478 | −0.028 | −0.059 |

Data 2 | 23 | 17.880 | 72.230 | 67.800 | 0.941 | 0.488 |

Data 3 | 20 | 1.800 | 4.535 | 3.950 | 2.173 | 5.353 |

Data | Distribution | Sample size | Skewness | EK | AIC | K−S statistics |
---|---|---|---|---|---|---|

Data4 | NGAD | 34 | 0.219 | −0.843 | 47.147 | 0.097 |

LwLD | 50.324 | 0.116 | ||||

Data5 | NGAD | 31 | 0.405 | −0.713 | 213.769 | 0.103 |

LwLD | 215.115 | 0.106 | ||||

Data6 | NGAD | 66 | −0.131 | 0.223 | 204.602 | 0.188 |

LwLD | 218.816 | 0.222 | ||||

Data7 | NGAD | 63 | −0.900 | 0.924 | 80.828 | 0.257 |

LwLD | 94.758 | 0.277 | ||||

Data8 | NGAD | 100 | 0.372 | 1.120 | 954.899 | 0.172 |

LwLD | 980.661 | 0.218 | ||||

Data9 | NGAD | 202 | 1.175 | 1.365 | 1915.144 | 0.091 |

LwLD | 1902.719 | 0.069 | ||||

Data10 | NGAD | 72 | 1.342 | 1.991 | 220.760 | 0.179 |

LwLD | 199.551 | 0.141 | ||||

Data11 | NGAD | 59 | 1.608 | 2.256 | 388.848 | 0.134 |

LwLD | 384.510 | 0.110 | ||||

Data12 | NGAD | 72 | 1.473 | 2.890 | 668.821 | 0.336 |

LwLD | 531.213 | 0.186 | ||||

Data13 | NGAD | 20 | 1.720 | 2.924 | 38.025 | 0.143 |

LwLD | 36.987 | 0.121 |

In this study, a new generalized Akash distribution has been introduced by incorporating the location parameter to improve the flexibility of the failure rate function. Then, its structural properties including parameter estimation have been discussed. This new distribution yields a more flexible density and failure rate shapes. Further, it has the capability to model the bathtub and monotonic increasing failure rate shapes. The simulation study indicates that the ML method performs well in the estimation of the unknown parameters for NGAD. To illustrate the theoretical findings, the real-world applications were used, and the results reveal that the NGAD is superior over AD, Lindley distribution (LD) and some other existing modified LDs that have been developed based on exponential and gamma mixtures with different mixing proportions. Based on the special characteristics, the proposed model may attract wider applications in reliability, mortality, actuarial, ecological sciences, among others.

We thank the Postgraduate Institute of Science, University of Peradeniya, Sri Lanka for providing all facilities to do this research.

The authors declare no conflicts of interest regarding the publication of this paper.

Tharshan, R. and Wijekoon, P. (2020) Location Based Generalized Akash Distribution: Properties and Applications. Open Journal of Statistics, 10, 163-187. https://doi.org/10.4236/ojs.2020.102013

Data set 4 (Mol.et al., (2012)): the data represents the level of mercury [

1.007, 1.447, 0.763, 2.010, 1.346, 1.243, 1.586, 0.821, 1.735, 1.396, 1.109, 0.993,2.007, 1.373, 2.242, 1.647, 1.350, 0.948, 1.501, 1.907, 1.952, 0.996, 1.433, 0.866, 1.049, 1.665,2.139, 0.534, 1.027, 1.678, 1.214, 0.905, 1.525, 0.763.

Data set 5 (Fuller.et al., (1994): the data represents the strength of glass of the aircraft window [

18.83, 20.80, 21.657, 23.03, 23.23, 24.05, 24.321, 25.50, 25.52, 25.80, 26.69, 26.77, 26.78, 27.05, 27.67, 29.90, 31.11, 33.20, 33.73, 33.76, 33.89, 34.76, 35.75, 35.91,36.98, 37.08, 37.09, 39.58, 44.045, 45.29, 45.381.

Data set 6 (Nichols.et al., (2006): the data are breaking stress of carbon fibers [

3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 3.56,4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 1.57, 2.67, 2.93, 3.22, 3.39, 2.81, 4.20, 3.33, 2.55, 3.31, 3.31, 2.85,1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.70, 2.03,1.89, 2.88, 2.82, 2.05, 3.65, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.35, 2.55, 2.59, 2.03, 1.61, 2.12, 3.15, 1.08, 2.56, 1.80, 2.53.

Data set 7 (Smith.et al., (1987): the data represents the strength of glass fibers [

0.55, 0.93, 1.25, 1.36, 1.49, 1.52, 1.58, 1.61, 1.64, 1.68, 1.73, 1.81, 2, 0.74, 1.04, 1.27, 1.39, 1.49, 1.53, 1.59, 1.61, 1.66, 1.68, 1.76,1.82, 2.01, 0.77, 1.11, 1.28, 1.42, 1.5, 1.54, 1.6, 1.62, 1.66, 1.69,1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.48, 1.5, 1.55, 1.61, 1.62, 1.66,1.7, 1.77, 1.84, 0.84, 1.24, 1.3, 1.48, 1.51, 1.55, 1.61, 1.63, 1.67,1.7, 1.78, 1.89.

Data set 8 (Birnbaum.et al., (1969)): the data are fatigue life of 6061T6 aluminum coupons [

70, 90, 96, 97, 99, 100, 103, 104, 104,105, 107, 108, 108, 108, 109, 109, 112, 112, 113, 114, 114, 114, 116, 119, 120,120, 120, 121, 121, 123, 124, 124, 124, 124, 124, 128, 128, 129, 129, 130, 130,130, 131, 131, 131, 131, 131, 132, 132, 132, 133, 134, 134, 134, 134, 136, 136,137, 138, 138, 138, 139, 139, 141, 141, 142, 142, 142, 142, 142, 142, 144, 144,145, 146, 148, 148, 149, 151, 151, 152, 155, 156, 157, 157, 157, 157, 158, 159,162, 163, 163, 164, 166, 166, 168, 170, 174, 201, 212.

Data set 9 (Weisberg, (2005): the data are athletes’ skin folds [

28.0, 109.1, 102.8, 104.6, 126.4, 80.3, 75.2, 87.2, 97.9, 75.1, 65.1, 171.1,76.8, 117.8, 90.2, 97.2, 99.9, 125.9, 69.9, 98, 96.8, 80.3, 74.9, 83.0, 91.0, 76.2, 52.6, 111.1,110.7, 74.7, 113.5, 99.8, 80.3, 109.5, 123.6, 91.2, 49.0, 110.2, 89.0, 98.3, 122.1, 90.4, 106.9,156.6, 101.1, 126.4, 114.0, 70.0, 77.0, 148.9, 80.1, 156.6, 115.9, 181.7, 71.6, 143.5, 200.8, 68.9,103.6, 71.3, 54.6, 88.2, 95.4, 47.5, 55.6, 62.9, 52.5, 62.6, 49.9, 57.9, 109.6, 98.5, 136.3, 103.6,102.8, 131.9, 33.8, 43.5, 46.2, 73.9, 36.8, 67, 41.1, 59.4, 48.4, 50.0, 54.6, 42.3, 46.1, 46.3, 109.0,98.1, 80.6, 68.3, 47.6, 61.9, 38.2, 43.5, 56.8, 41.6, 58.9, 44.5, 41.8, 33.7, 50.9, 40.5, 51.2, 54.4,52.3, 57.0, 65.3, 52.0, 42.7, 35.2, 49.2, 61.8, 46.5, 34.8, 60.2, 48.1, 44.5, 54.0, 44.7, 64.9, 43.8,58.3, 52.8, 43.1, 78.0, 40.8, 41.5, 50.9, 49.6, 88.9, 48.3, 61.8, 43.0, 61.1, 43.8, 54.2, 41.8, 34.1,30.5, 34.0, 46.7, 71.1, 65.9, 34.3, 34.6, 31.8, 34.5, 31.0, 32.6, 31.5, 32.6, 31.0, 33.7, 30.3, 38.0,55.7, 37.5, 112.5, 82.7, 29.7, 38.9, 44.8, 30.9, 44.0, 37.5, 37.6, 31.7, 36.6, 48, 41.9, 30.9, 52.8,43.2, 113.5, 96.9, 49.3, 42.3, 96.3, 56.5, 105.7, 100.7, 56.8, 75.9, 52.8, 47.8, 76.0, 61.2, 75.6,43.3, 49.5, 70.0, 75.7, 57.7, 67.2, 56.5, 47.6, 60.4, 34.9.

Data set 10 (Bjerkedal, (1960)): the data represents he survival times of guinea pigs [

0.1, 0.33, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 1.07, 1.08, 1.08, 1.08, 1.09, 1.12, 1.13, 1.15, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 1.36, 1.39, 1.44, 1.46, 1.53, 1.59, 1.6, 1.63, 1.63, 1.68, 1.71, 1.72, 1.76, 1.83, 1.95, 1.96, 1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54, 2.54, 2.78, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55.

Data set 11 (Mead, (2016)): the data consist the taxes revenue [

5.9, 20.4, 14.9, 16.2, 17.2, 7.8, 6.1, 9.2, 10.2, 9.6, 13.3, 8.5, 21.6, 18.5,5.1,6.7, 17, 8.6, 9.7, 39.2, 35.7, 15.7, 9.7, 10, 4.1, 36, 8.5, 8, 9.2, 26.2,21.9,16.7, 21.3, 35.4, 14.3, 8.5, 10.6, 19.1, 20.5, 7.1, 7.7, 18.1, 16.5, 11.9, 7,8.6,12.5, 10.3, 11.2, 6.1, 8.4, 11, 11.6, 11.9, 5.2, 6.8, 8.9, 7.1, 10.8.

Data set 12 (Choulakian.et al., (2001): the data are flood peaks [

1.7, 2.2, 14.4, 1.1,0.4, 20.6, 5.3,0.7, 1.9,13.0, 12.0, 9.3, 1.4, 18.7, 8.5,25.5, 11.6, 14.1, 22.1, 1.1, 2.5, 27.0, 14.4, 1.7, 37.6, 0.6,2.2,39.0, 0.3,15.0, 11.0, 7.3, 22.9, 1.7, 0.1, 1.1, 0.6, 9.0, 1.7, 7.0, 20.1, 0.4, 2.8, 14.1, 9.9, 10.4, 10.7, 30.0, 3.6, 5.6, 30.8, 13.3, 4.2, 25.5, 3.4,11.9, 21.5, 27.6, 36.4, 2.7, 64.0,1.5,2.5,27.4, 1.0,27.1, 20.2, 16.8, 5.3,9.7,27.5, 2.5.

Data set 13 (Gross.et al., (1975)): the data represent the relief times of patients [

1.1, 1.4, 1.3, 1.7, 1.9, 1.8, 1.6, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.2, 1.4, 3.0, 1.7, 2.3, 1.6, 2.0.

Data | Model | MLE | AIC | BIC | K-S Statistic | Critical value | |
---|---|---|---|---|---|---|---|

Data 1 | NGAD | 117.756 | 123.756 | 130.458 | 0.138 | ||

LwLD | 128.735 | 134.735 | 141.437 | 0.185 | |||

TwPLD | 261.735 | 265.735 | 270.203 | 0.434 | 0.164 | ||

QLD | 261.735 | 265.735 | 270.203 | 0.434 | |||

ThPLD | 263.7358 | 267.735 | 274.438 | 0.434 | |||

AD | 224.280 | 226.280 | 232.982 | 0.348 | |||

LD | 233.021 | 235.021 | 237.255 | 0.349 | |||

SD | 238.381 | 240.381 | 242.615 | 0.387 | |||

Data 2 | NGAD | 225.825 | 231.825 | 235.232 | 0.111 | ||

LwLD | 226.789 | 232.209 | 235.616 | 0.090 | |||

TwPLD | 242.873 | 246.873 | 249.144 | 0.263 | 0.283 | ||

QLD | 242.873 | 246.873 | 249.144 | 0.263 | |||

ThPLD | 242.873 | 248.873 | 252.279 | 0.263 | |||

AD | 227.058 | 229.058 | 232.465 | 0.107 | |||

LD | 231.471 | 233.471 | 234.606 | 0.149 | |||

SD | 231.060 | 233.060 | 234.196 | 0.145 | |||

Data 3 | NGAD | 83.992 | 89.992 | 92.979 | 0.171 | ||

LwLD | 82.891 | 88.891 | 91.878 | 0.166 | |||

Data 3 | TwPLD | 100.4738 | 104.4738 | 106.465 | 0.278 | 0.304 | |
---|---|---|---|---|---|---|---|

QLD | 100.4738 | 104.4738 | 106.465 | 0.278 | |||

ThPLD | 102.4738 | 106.4738 | 109.460 | 0.278 | |||

AD | 89.931 | 91.931 | 94.918 | 0.215 | |||

LD | 94.066 | 96.066 | 97.061 | 0.211 | |||

SD | 92.094 | 94.094 | 95.090 | 0.222 |