The length-biased weighted frechet distribution: Properties and estimation

Introduction

The Frechet distribution has many applications ranging from life testing through rainfall, queues, wind speeds, earthquakes, sea currents, horse racing, floods and track race records. Historically it was first presented by Maurice Frechet (1927) [10] . Many authors have studied the Frechet distribution; for examples, Mubarak (2012) [18] obtained the maximum likelihood and least squares estimates for Frechet distribution based on progressive type-II censoring.

Abbas & Yincai (2012) [1] compared the maximum likelihood, probability weighted moments and Bayes estimates for the scale parameter of Frechet distribution in complete samples. Nasir & Aslam (2015) [20] estimated the shape parameter of Frechet distribution via Bayesian scheme based on different prior distribution and various loss functions. Reyad et al. (2016) [25] obtained the QE-Bayes and E-Bayes estimates of scale parameter corresponding to the Frechet distribution. Recently, some extensions of the Fréchet distribution are introduced. Nadarajah & Kotz (2003) [19] presented the exponentiated Fréchet distribution. Mahmoud & Mandouh (2013) [15] propsed the transmuted Fréchet distribution. Mead et al. (2016) [16] suggested the beta exponential Fréchet distribution. Afify et al. (2016) [2] introdued the Weibull Fréchet distribution. The probability density function (pdf) of the Frechet distribution is ( ; , ) , 0 , , 0

Where  and  are the scale and shape parameters respectively. The theory of weighted distributions gives a unified approach to dealing with model specification and data interpretation problems. Weighted distributions always occur in researches related to survival analysis, analysis of intervention data, ecology and biomedicine (refer to Stene (1981) [27] , Gupta & Nadarajah (1985) [11] and Patil & Taillie (1989)) [22] . The idea of weighted distributions was first emerged by Fisher (1934) [9] . Nermerous studies on weighted distributions were published, for example; Rao (1965) [23] explained that weighted distributions can be used when the recorded observations cannot be considered as a random sample from original distributions, which means that sometimes it is not possible to work with truly random sample from the underlying population. Gupta & Tripathi (1996) [12] introduced the weighted bivariate logarithmic series distribution.

Shaban & Boudrissa (2007) [26] presented the length biased Weibull distribution and estimated its parameters. Das & Roy (2011) [8] pointed out the length biased weighted generalized Rayleigh distribution and studied its properties. Rashwan (2013) [24] introduced the generalized gamma length biased distribution and investigated its properties. Ahmed et al. (2016) [4] suggested the length biased weighted Lomax distributionwith its properties and estimated its paramaters. Suppose that X is a non-negative random variable and its probability density function (pdf) is () fx, then the pdf of the weighted random variable w X is called the weighted distribution which is given by

where () wx be a non-negative weight function and

We can note that different choices of the weight function ()

gave different weighted distributions. If ()

, then the resulting distribution is called length-biased distribution with pdf given by () ( ) , 0 ()

The main objective of this paper is to propose a new distribution that is called length-biased weighted Frechet (LBWF) distribution and to study some of its various statistical properties. In addition, the parameters of the new distribution are estimated by using the moments and maximum likelihood estimation methods. This paper is organized as follows: In Section 2, the length-biased weighted Frechet distribution is defined. The limit of the LBWF distribution is studied in Section 3. The survival analysis is discussed in Section 4. In Section 5, some statistical properties of the the new model are investigated. In Section 6, the moment and maximum likelihood estimates for the model parameters are obtained. Some concluding remarks have been given in the last Section.

Length-Biased Weighted Frechet Distribution

In this section, we derive the pdf and the cumulative distribution function (cdf) of the LBWF. The mean of the Frechet distribution given in Eq. ( 1) is

Then, the pdf of LBWF can be obtained by using Eq. (1) and Eq. (4) in Eq. (3) to be 1 ( ) ( ; , ) ,

The cdf of the LBWF is given by

where the upper incomplete gamma function is

The Limit of the Length-Biased Weighted Frechet Distribution

The limit of the length-biased weighted Frechet distribution when 0 x  is 0 and when x  is 0. We can show this by taking the limit of Eq. ( 5) as follows:

(1 1 ) Similarly, as x , we can observe that by replacing the limit 0 x  with x , we get (1 1 )

Survival Analysis

In this section, we obtain the reliability function, the hazard function and the inverse hazard function for the LBWF distribution.In general, the reliability function, () Rx of a random variable X can be given from

Using Eq. ( 6) in the above equation, we get

Also, the hazard function () hx , of random variable X is defined as follows:

Subsituting from Eq. ( 5) and Eq. ( 7) into the above equation yields

In addition, the reverse hazard function () r hx, can be obtained as



Inserting Eq. ( 5) and Eq. ( 6) into the last equation yields

Statistical Properties

In this section, we present a mathematical treatment of the proposed distribution such as; mean, variance, coefficient of variation, harmonic mean, raw and incomplete moments, skewness, kurtosis, mode, moment and probability generating functions, reliability, hazard and reverse hazard functions, Lorenz, Bonferroni and Zenga curves, Renyi of entropy, order statistics and record statistics.

Raw Moments

Suppose X is a random variable distributed according to LBWF distribution, then the raw moments is given by

Consequently, we can establish a recurrence relation of raw moments for the LBWF distribution to be

Therefore, the variance and the standard deviation are given by

Incomplete Moments

Suppose

X is a random variable obeying the LBWF distribution, then the th r incomplete moments denoted as ()

mz can be obtained as follows:

in the previous integration, then we get

Harmonic Mean

The harmonic mean of the LBWF distribution is derived as follows:

Coefficient of Variation

The coefficient of variation of the LBWF distribution is given by

Skewness and Kurtosis

The coefficients of skewness and kurtosis of the LBWF distribution are given respectively as follows:

Moment and Probability Generating Functions

The moment generating function of LBWF distribution, say ()

x Mt can be obtained as follows:

Likewise, the probability generating function denoted as [] ()

x Mt of the LBWF distribution can be derived as below

Mode

The mode of the LBWF distribution can be obtained by taking the logarithm of Eq. ( 5) as

Differentiating the above equation with respect to x and equating this derivate to zero, we get 1 lo g ( ; , ) 0

Solving the previous nonlinear equation for x , then the mode is given by

Lorenz, Bonferroni and Zenga Curves

The Lorenz and Bonferroni curves have been used in different fields such as demography, insurance, reliability, medicine and economics [for more details see Kleiber & Kotz (2003)] [13] . The Lorenz curve, () F Lx is defined by Oluyede & Rajasooriya (2013) [21] as follows:

in the above equation, then we get

is given by Oluyede & Rajasooriya (2013) [21] as below:



Using Eq. ( 6) and Eq. ( 22) in the above, then we have

The Zenga curve () Ax is presented by Oluyede & Rajasooriya (2013) [21] as follows:

. Then, for the new model we have

Where the lower incomplete gamma function is

Rényi Entropy

The entropy of a random variable X is a measure of randomness contained in a probability distribution. The Rényi entropy is defined as

 and 0   . Using Eq. ( 5) yields

Therefore, the Rényi entropy is given from

Order Statistics

Order statistics play an important role in probability and statistics. Let

Then, the pdf the th k order LBWF random variable: kn X can be obtained by using Eqs.( 5) and ( 6) in Eq. ( 26) to be

Therefore, the pdf of the 1 th order LBWF random variable 1:n X is given by

Also, the pdf of the th n order LBWF random variable: nn X is given by

Moreover, the joint distribution of two order statistics ::

Then, for the LBWF distribution we have

will be called an upper (lower) record value if its value exceeds (is lower than) that of all previous observations. The pdf of () Ur X , the th r upper record is given as [see Ahsanullah (1995) and Arnold et al. (1998)] [5,6]  

Then, the pdf of the th r upper record LBWF random variable () Ur X can be obtained by using Eqs. ( 5) and. (6) in Eq. ( 32) to be

Furthermore, the joint distribution of the first n upper record values  

is given by [see Ahsanullah (1995)] [5] 1 () 1, 2 , ...,

Then, for the LBWF distribution we get

Furthermore, the pdf of () Lr X , the th r lower record is given as [see Ahsanullah (1995) and Arnold et al. (1998)] [5,6]  

Consequently, for the LBWF distribution we obtain

Moreover, the joint distribution of the first n lower record values  

Then, for the proposed model we get

Methods of Estimation

In this section we describe two estimation methods to obtain the estimates for the model parameters associated to the LBWF distribution.

Moments Estimation

The moment estimators of  and  denoted as ˆMME



and ˆMME  respectively can be obtained by equating population moments with the sample moments as follows:

Using Eq. (42) in Eq. (41), we obtain

Solving Eq. (43) for  by numerical methods, then we get ˆMME  and then substituting in Eq. ( 42) to obtain ˆMME  .

Maximum Likelihood Estimation

x be a random sample from the LBWF, then the corresponding likelihood function is given by

Taking the logarithm of Eq. (44), we have

Now; differentiate Eq. ( 45) with respect to  and  , then equate this derivative to zero, we obtain the following normal equations:

lo g ( ; ) lo g 1 1 lo g lo g lo g 0

1 1 lo g ( ; , ) ln ( ) ln ( )

is the polygamma function.

Conclusion

This paper introduced a new distribution named the length-biased weighted Frechet distribution. Different properties of this distribution were studied including the mean, variance, coefficient of variation, harmonic mean, raw and incomplete moments, skewness, kurtosis, mode, moment and probability generating functions, reliability, hazard and reverse hazard functions, Lorenz, Bonferroni and Zenga curves, Renyi of entropy, order statistics and record statistics. The parameters of the new distribution are estimated by using moment and maximum likelihood approaches and the observed Fisher information matrix is obtained.

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