^{1}

^{2}

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This paper introduces a new distribution based on the exponential distribution, known as Size-biased Double Weighted Exponential Distribution (SDWED). Some characteristics of the new distribution are obtained. Plots for the cumulative distribution function, pdf and hazard function, tables with values of skewness and kurtosis are provided. As a motivation, the statistical application of the results to a problem of ball bearing data has been provided. It is observed that the new distribution is skewed to the right and bears most of the properties of skewed distribution. It is found that our newly proposed distribution fits better than size-biased Rayleigh and Maxwell distributions and many other distributions. Since many researchers have studied the procedure of the weighted distributions in the estates of forest, biomedicine and biostatistics etc., we hope in numerous fields of theoretical and applied sciences, the findings of this paper will be useful for the practitioners.

Weighted distributions are suitable in the situation of unequal probability sampling, such as actuarial sciences, ecology, biomedicine biostatistics and survival data analysis. These distributions are applicable, when observations are recorded without any experiment, repetition and random process. The notion of weighted distributions has been used as a device for the collection of suitable model for observed data, during last 25 years. The idea is most applicable when sampling frame is not available and random sampling is not possible. Firstly the idea of weighted distributions was introduced by Fisher [

Let

The weighted distribution is defined as;

where w(x) is a weight function. When

In forest product research, equilibrium and length biased distributions have been used as moment distributions. Kochar and Gupta [

Oluyede [

Mir and Ahmad [

Dara [

Zahida and Munir [

The exponential distribution has a fundamental role in describing a large class of phenomena, particularly in the area of reliability theory. This distribution is commonly used to model waiting times between occurrences of rare events, lifetimes of electrical or mechanical devices. It is also used to get approximate solutions to difficult distribution problems.

The size-biased double weighted exponential distribution is given by:

where f(x) is the first weight and

Here

Thus the pdf of SDWED is

where

Distribution function of a density function is defined as:

The reverse Hazard rate function of SDWED is given by

The Mills Ratio is given by:

Using Equation (3)

Putting

and after a long simplification, the information generating function will be:

Note that the limit of the density function given in Equation (3) is as follows:

since

Taking log of Equation (3) on both sides:

Differentiating Equation (15) with respect to x, we obtain:

The mode of the SDWED is obtained by solving the nonlinear equation with respect to x:

The mode of SDWED is given in

c | λ | mode |
---|---|---|

2 | 1 | 1.543 |

2 | 2 | 0.578 |

2 | 3 | 0.336 |

2 | 4 | 0.250 |

2 | 5 | 0.200 |

c | λ | Mean | Variance | Standard Deviation |
---|---|---|---|---|

2 | 1 | 2.4 | 1.98 | 1.41 |

2 | 2 | 1.2 | 0.50 | 0.70 |

2 | 3 | 0.8 | 0.04 | 0.22 |

2 | 4 | 0.6 | 0.01 | 0.12 |

2 | 5 | 0.4 | 0.01 | 0.08 |

The r^{th} moment of SDWED is given by

for r = 1, 2, 3, 4, the first four moments about the mean are

The maximum likelihood estimation of SDWED distribution may be defined as:

Here the independent observations are

This admits the partial derivatives:

and

Equating these equations to zero, then we get:

c | |||
---|---|---|---|

2.481 | 1 | 0.0005 | 1.067 |

2.484 | 1 | 0.0049 | 1.076 |

2.480 | 2 | 0.0003 | 1.066 |

3.000 | 2 | 1.3450 | 2.778 |

3.001 | 3 | 1.0060 | 2.999 |

which can be solved simultaneously for

The asymptotic variance-covariance matrix is the inverse of

The inverse of the asymptotic covariance matrix is

The Ball Bearing Data Records

See for data set published in Lawless [

In

In this paper, Size-biased Double Weighted Exponential Distribution (SDWED) has

Distributions | |||||||||
---|---|---|---|---|---|---|---|---|---|

Size-biased Rayleigh | - | - | - | - | - | - | 46.764 | 0.708 | 0.134 |

Size-biased Maxwell | - | - | - | 40.50 | - | - | 1.693 | 0.278 | |

Weighted Weibull (size-biased) | 0.8151 | 0.604 | 4.759 | - | - | 0.1909 | 0.0332 | ||

Size-Biased Double weighted exponential distribution(SDWED) | 16.64 | 0.027 | - | 0.133 | 0.134 |

been introduced. The pdf of the SDWED has been studied as well as different reliability measures such as survival function, failure rate function or hazard function. The moments, mode, the coeff. of skewness and the coeff. of kurtosis of SDWED have been derived. For estimating the parameters of SDWED, MLE method has been used. The SDWED has been fitted to Ball Bearing data set. SDWED suggested a good fit of the data as comparing to other distributions.

Perveen, Z., Ahmed, Z. and Ahmad, M. (2016) On Size-Biased Double Weighted Exponential Distribution (SDWED). Open Journal of Statistics, 6, 917- 930. http://dx.doi.org/10.4236/ojs.2016.65076