_{1}

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The purpose of this paper is to introduce a size biased Lindley distribution which is a special case of weighted distributions. Weighted distributions have practical significance where some types of biased occur in a density function, i.e. probability is proportional to the size of the variate, that’s why the proposed version of size biased Lindley is designed for such situations more reasonably and more precisely. Principle properties of the density function are also discussed in this paper such as moments, measure of skewness, kurtosis, moment generating function, characteristics generating function, coefficient of variation, survival function and hazard function which are derived for understanding the structure of the proposed distribution more briefly.

Weighted distributions are required when the recorded observation from an event cannot randomly sample from actual distribution. This happens when the original observation damaged as well as an event occur in non-observability manner. Due to these inappropriate situations, resulting values are reduced, and units or events do not have same chances of occurrences as if they follow the exact distribution.

Let the original observation x has pdf f ( x ) then in case of any biased in sampling appropriate weighted function, say w ( x ) which is a function of random variable will be introduced to model the situation.

Then new density function f w ( x ) will be given by Equation (1), where f w represent a weighted distribution where w is considered as weighted function

f w ( x ) = w ( x ) f ( x ) / w (1)

where w ( x ) is considered as normalizing factor which is utilized to create total probability or area under the curve, equal to 1. If w ( x ) is constant term, then f w ( x ) = f ( x ) .

The Lindley distribution introduced with two parameters by Shanker et al. (2013) [

The moment distributions have random variable x with its weighted function f(x) and normalizing factor is E(x) to make total area is to be 1.

Mathematically,

g ( x ) = x f ( x ) E ( x )

Some structural properties discussed by using simple algebraic methods whereas some results of primary and size biased density function are compared based on random samples for each density function. For data simulation and calculation of results based on these samples r programming language is used. Both functions are compared based on these results of simulation, for different values of parameter θ .

1) One parameter Lindley distribution

A one parameter Lindley distribution with parameter θ is defined by its probability density function given as.

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

Plot of probability function of Lindley distribution (see

2) Raw moments

The r t h moments about origin of one parameter Lindley distribution is given by Equation (3)

μ ′ r = r ! ( θ + r + 1 ) θ r ( 1 + θ ) , r = 1 , 2 , 3 , ⋯ (3)

Taking r = 1 , 2 , 3 and 4 in this equation the first four moments about origin is obtained as

μ ′ 1 = θ + 2 θ ( 1 + θ )

μ ′ 2 = 2 θ 2 ( θ + 3 1 + θ )

μ ′ 3 = 6 θ 2 ( θ + 4 1 + θ )

μ ′ 4 = 24 ( θ + 5 ) θ 4 ( θ + 5 )

3) Moments about mean of one parameter Lindley distribution

Then central moments are obtained as,

μ 1 = θ + 2 θ ( 1 + θ )

μ 2 = ( θ 2 + 4 θ + 2 ) θ 2 ( θ + 1 ) 2

μ 3 = 2 ( θ 3 + 6 θ 2 + 6 θ + 2 ) θ 3 ( θ + 1 ) 3

Lindley distribution | μ_{1} | μ_{2} | μ_{3} | μ_{4} | Std. Dev |
---|---|---|---|---|---|

θ = 0.1 | 19.09091 | 199.1736 | 3998.497 | 239006.2 | 14.11289 |

θ = 0.5 | 3.333333 | 7.555556 | 31.40741 | 362.0741 | 2.748737 |

θ = 0.9 | 1.695906 | 2.192127 | 5.195381 | 32.24576 | 1.480583 |

θ = 1.3 | 1.103679 | 0.994396 | 1.656285 | 6.953597 | 0.9971941 |

θ = 1.7 | 0.8061002 | 0.5548673 | 0.712556 | 2.247497 | 0.7448942 |

θ = 2.1 | 0.6298003 | 0.3494565 | 0.3647844 | 0.9184176 | 0.5911485 |

θ = 2.5 | 0.5142857 | 0.2383673 | 0.2093528 | 0.4376736 | 0.4882287 |

θ = 2.9 | 0.4332449 | 0.1720659 | 0.1302924 | 0.2325485 | 0.4148083 |

θ = 3.3 | 0.3735025 | 0.1295714 | 0.08615088 | 0.1340034 | 0.3599603 |

μ 4 = 3 ( 3 θ 4 + 24 θ 3 + 44 θ 2 + 32 θ + 8 ) θ 4 ( θ + 1 ) 4

4) Cumulative distribution function of Lindley distribution

Cdf of the Lindley distribution is given by Equation (4)

F ( x ) = ∫ 0 x f ( x ) d ( x )

∫ 0 x θ 2 ( 1 + x ) e − θ x 1 + θ d ( x )

This gives,

F ( x ) = 1 − e − θ x [ 1 + θ x 1 + θ ] (4)

Plot of cumulative distribution function of Lindley distribution (see

5) Moment generating function of Lindley distribution

M x ( t ) = θ 2 ( 1 − t + θ ) ( 1 + θ ) ( t + θ ) 2

6) Characteristic generating function of Lindley distribution

M x ( i t ) = θ 2 1 + θ ( θ − i t + 1 ) ( i t − θ ) 2

7) Skewness, Kurtosis and Coefficient of variation of Lindley distribution (see

Skewness = 2 ( θ 3 + 6 θ 2 + 6 θ + 2 ) ( θ 2 + 4 θ + 2 ) 3 / 2

Kurtosis = 3 ( 3 θ 4 + 24 θ 3 + 44 θ 2 + 32 θ + 8 ) ( θ 2 + 4 θ + 2 ) 2

C .V = θ 2 + 4 θ + 2 θ + 2

8) Size biased Lindley distribution

The probability density function of size biased Lindley distribution is given as

f x ( x , θ ) = g ( x ) = x f ( x , θ ) E ( x )

⇒ g ( x ; θ ) = θ 3 x ( 1 + x ) e − θ x 2 + θ , x > 0 (5)

Plot of probability function of size biased Lindley distribution (see

9) Raw moments of size biased Lindley distribution

μ ′ r = θ 2 θ r + 2 [ ( r + 1 ) ! ( θ + r + 2 ) ] (6)

Taking r = 1 , 2 , 3 and 4 in this equation the first four moments about origin is obtained as

μ ′ 1 = 2 ( θ + 3 ) θ ( θ + 2 )

Lindley distribution | Skewness of LD | Kurtosis of LD | CV of LD |
---|---|---|---|

0.1 | 1.42249 | 6.024845 | 0.7392464 |

0.5 | 1.512281 | 6.342561 | 0.8246211 |

0.9 | 1.600732 | 6.710286 | 0.8730337 |

1.3 | 1.670306 | 7.032192 | 0.9035183 |

1.7 | 1.723992 | 7.299966 | 0.9240714 |

2.1 | 1.765821 | 7.520627 | 0.9386284 |

2.5 | 1.798906 | 7.702946 | 0.9493337 |

2.9 | 1.825478 | 7.854595 | 0.9574452 |

3.3 | 1.847123 | 7.981736 | 0.9637429 |

μ ′ 2 = 6 ( θ + 4 ) θ 2 ( θ + 2 )

μ ′ 3 = 24 ( θ + 5 ) θ 3 ( θ + 2 )

μ ′ 4 = 120 ( θ + 6 ) θ 4 ( θ + 2 )

10) Moments about mean of size biased Lindley distribution (see

Central moments of size biased Lindley distribution are obtained as:

μ 1 = 2 ( θ + 3 ) θ ( θ + 2 )

μ 2 = 2 ( θ 2 + 6 θ + 6 ) θ 2 ( θ + 2 ) 2

SBLD | μ_{1} | μ_{2} | μ_{3} | μ_{4} | Std. Dev |
---|---|---|---|---|---|

θ = 0.1 | 29.52381 | 299.7732 | 5999.784 | 449591.7 | 17.31396 |

θ = 0.5 | 5.6 | 11.84 | 47.872 | 708.4032 | 3.44093 |

θ = 0.9 | 2.988506 | 3.584798 | 8.148448 | 65.90233 | 1.297429 |

θ = 1.3 | 2.004662 | 1.683321 | 2.675344 | 14.75241 | 1.297429 |

θ = 1.7 | 1.494436 | 0.9650163 | 1.181765 | 4.916901 | 0.9823524 |

θ = 2.1 | 1.184669 | 0.6207837 | 0.6188595 | 1.025826 | 0.7878983 |

θ = 2.5 | 0.9777778 | 0.4306173 | 0.3620521 | 1.002462 | 0.6562144 |

θ = 2.9 | 0.8304011 | 0.3150689 | 0.2290128 | 0.5418929 | 0.56131 |

θ = 3.3 | 0.7204117 | 0.2398822 | 0.1535249 | 0.3168071 | 0.4897777 |

μ 3 = 4 ( θ 3 + 9 θ 2 + 180 θ + 12 ) θ 3 ( θ + 2 ) 3

μ 4 = 24 ( θ 4 + 12 θ 3 + 42 θ 2 + 60 θ + 30 ) θ 4 ( θ + 2 ) 4

11) Cumulative distribution function of size biased Lindley distribution

Cdf of size biased Lindley distribution is given by,

G ( x ) = ∫ 0 x g ( x ) d ( x )

G ( x ) = ∫ 0 x θ 3 x ( 1 + x ) e − θ x 2 + θ d ( x )

G ( x ) = θ 3 2 + θ ∫ 0 x x ( 1 + x ) e − θ x d ( x )

This gives

G ( x ) = 2 − 2 e − θ x − ( 2 + x θ ) + θ ( 1 − e − θ x ( 1 + x θ ) ) θ + 2 (7)

Please see

12) Moment generating function of size biased Lindley distribution

M .G .F = ( − 2 + t − θ ) θ 3 ( t − θ 3 ) ( θ + 2 )

13) Characteristics generating function of size biased Lindley distribution

C .F = ( − 2 + i t − θ ) θ 3 ( i t − θ 3 ) ( θ + 2 )

14) Skewness, Kurtosis and Coefficient of variation of size biased Lindley distribution (see

Skewness = β 1 = 4 ( θ 3 + 9 θ 2 + 18 θ + 12 ) { 2 ( θ 2 + 6 θ + 6 ) } 3 2

θ | Skewness SBLD | Kurtosis of SBLD | CV of SBLD |
---|---|---|---|

0.1 | 1.155969 | 5.003023 | 0.5864406 |

0.5 | 1.175044 | 5.053324 | 0.6144518 |

0.9 | 1.200544 | 5.128277 | 0.6335461 |

1.3 | 1.224981 | 5.206302 | 0.6472056 |

1.7 | 1.246606 | 5.279858 | 0.6573401 |

2.1 | 1.265265 | 5.34658 | 0.6650788 |

2.5 | 1.28125 | 5.406111 | 0.6711283 |

2.9 | 1.294946 | 5.458867 | 0.6759504 |

3.3 | 1.306718 | 5.505525 | 0.6798582 |

Kurtosis = β 2 = 24 ( θ 4 + 12 θ 3 + 42 θ 2 + 60 θ + 30 ) { 2 ( θ 2 + 6 θ + 6 ) } 2

C .V = δ μ ′ 1 = 2 ( θ 2 + 6 θ + 6 ) 2 ( θ + 3 )

15) Survival function of size biased Lindley distribution

S ( t ) = e − θ t θ + 2 [ θ + 2 − t θ 2 + 2 t θ + t 2 θ 2 ]

16) Hazard function of size biased Lindley distribution

H ( t ) = θ 3 t ( 1 + t ) θ + 2 − t θ 2 + 2 θ t + t 2 θ 2

LD | Mean | Variance | Standard deviation | Median | Skewness | kurtosis |
---|---|---|---|---|---|---|

θ = 0.1 | 17.280 | 128.6998 | 11.34459 | 14.960 | 0.7519824 | 2.899294 |

θ = 0.5 | 3.221 | 7.572646 | 2.751844 | 2.485 | 2.485 | 6.332139 |

θ = 0.9 | 1.610 | 2.119268 | 1.455771 | 1.245 | 1.671199 | 7.427447 |

= 1 | 1.403 | 1.582653 | 1.258035 | 1.042 | 1.62406 | 6.789065 |

θ = 1.3 | 1.054 | 0.979174 | 0.9895322 | 0.765 | 1.776654 | 7.588966 |

θ = 1.7 | 0.7706 | 0.5534715 | 0.7439567 | 0.5500 | 1.761394 | 4.358188 |

θ = 2.1 | 0.5875 | 0.3416315 | 0.5844925 | 0.4200 | 1.8265 | 7.129288 |

θ = 2.5 | 0.4656 | 0.2213501 | 0.4704785 | 0.3050 | 2.072667 | 9.192622 |

θ = 2.9 | 0.4014 | 0.1666295 | 0.4082028 | 0.2650 | 2.119914 | 9.657423 |

θ = 3.3 | 0.3386 | 0.1222347 | 0.3496208 | 0.2150 | 2.038068 | 8.910684 |

SBLD | Mean | Variance | Standard deviation | Median | Skewness | Kurtosis |
---|---|---|---|---|---|---|

θ = 0.1 | 24.570 | 132.92 | 11.5291 | 23.280 | 0.2377253 | 2.158117 |

θ = 0.5 | 5.615 | 12.35464 | 3.514917 | 4.925 | 1.158568 | 5.12711 |

θ = 0.9 | 2.950 | 3.502266 | 1.871434 | 2.550 | 1.278998 | 5.439581 |

θ = 1 | 2.647 | 2.879649 | 1.696953 | 2.310 | 1.265753 | 5.273339 |

θ = 1.3 | 1.974 | 1.617517 | 1.271816 | 1.750 | 1.220679 | 5.268042 |

θ = 1.7 | 1.482 | 0.9617643 | 0.9806958 | 1.330 | 1.052395 | 4.459788 |

θ = 2.1 | 1.160 | 0.6304327 | 0.7939979 | 1.000 | 1.178295 | 5.009183 |

θ = 2.5 | 0.9508 | 0.4314958 | 0.6568834 | 0.7950 | 1.346804 | 6.400068 |

θ = 2.9 | 0.7987 | 0.3249239 | 0.570021 | 0.6800 | 1.240408 | 4.868574 |

θ = 3.3 | 0.6867 | 0.239743 | 0.4896356 | 0.5700 | 1.435663 | 6.086738 |

By comparing the results in above tables, it is noted that mean, median, and standard deviation all these measures are greater in magnitude for size biased distribution as compared to actual distribution for respective values of parameter.

Ayesha, A. (2017) Size Biased Lindley Distribution and Its Properties a Special Case of Weighted Distribution. Applied Mathematics, 8, 808-819. https://doi.org/10.4236/am.2017.86063