Moments of generalized order statistics appear in several areas of science and engineering. These moments are useful in studying properties of the random variables which are arranged in increasing order of importance, for example, time to failure of a computer system. The computation of these moments is sometimes very tedious and hence some algorithms are required. One algorithm is to use a recursive method of computation of these moments and is very useful as it provides the basis to compute higher moments of generalized order statistics from the corresponding lower-order moments. Generalized order statistics provides several models of ordered data as a special case. The moments of generalized order statistics also provide moments of order statistics and record values as a special case. In this research, the recurrence relations for single, product, inverse and ratio moments of generalized order statistics will be obtained for Lindley–Weibull distribution. These relations will be helpful for obtained moments of generalized order statistics from Lindley–Weibull distribution recursively. Special cases of the recurrence relations will also be obtained. Some characterizations of the distribution will also be obtained by using moments of generalized order statistics. These relations for moments and characterizations can be used in different areas of computer sciences where data is arranged in increasing order.

Several situations arise where the ordering of the data is of great importance. For example arrangement of Olympic records or magnitude of the earthquake measured on a Richter scale etc. The distributional properties of such data are studied by using specialized methods known as

Order statistics and record values are special cases of a more general class of models for ordered data, known as _{j} = _{r} with _{i} = _{j}; _{i} ≠ _{j};

The marginal distribution of _{i} = _{j};

The joint density function of two _{i} = _{j} is given by [

The marginal density function of a single _{i} ≠ _{j}; _{i} ≠ _{j}; _{1} < _{2} and _{i} = 0 and _{i} = −1.

Since the development of

The Weibull distribution, introduced by [

The density and distribution function of the Lindley–Weibull distribution are related as

The relation (

We will, now, derive expressions for recursive computation of moments of

The

These moments are not easy to compute for most of the distribution and hence the recursive computation is used for these moments. An expression for recursive computation of moments of

The above expression can be written as

Now, using (

Simplifying and re-arranging the above expression we have

The (

In the following, we will obtain an expression for recursive computation of product moments of

The above expression can be written as

Now, using (

Simplifying the above expression we have

In this section, we will give some characterizations for the Lindley–Weibull distribution in terms of simple and product moments of _{i} =

Using

_{i} =

Now using the above relation in (

Using

In this section, we have given the numerical study for moments of order statistics when the sample is available from the Lindley–Weibull distribution. We have computed mean and variance of order statistics for the Lindley–Weibull distribution using various combinations of parameters. The results are given in

In this paper, we have derived expressions for recursive computation of single, product, inverse and ratio moments of

1 | 1.1635 | |||||||||

2 | 0.7587 | 1.5683 | ||||||||

3 | 0.5886 | 1.0988 | 1.8031 | |||||||

4 | 0.4908 | 0.8822 | 1.3154 | 1.9657 | ||||||

5 | 0.4258 | 0.7507 | 1.0795 | 1.4726 | 2.0890 | |||||

6 | 0.3789 | 0.6602 | 0.9317 | 1.2274 | 1.5951 | 2.1877 | ||||

7 | 0.3431 | 0.5933 | 0.8276 | 1.0704 | 1.3452 | 1.6951 | 2.2699 | |||

8 | 0.3148 | 0.5413 | 0.7492 | 0.9582 | 1.1827 | 1.4426 | 1.7793 | 2.3399 | ||

9 | 0.2917 | 0.4996 | 0.6876 | 0.8726 | 1.0651 | 1.2768 | 1.5256 | 1.8518 | 2.4009 | |

10 | 0.2725 | 0.4651 | 0.6374 | 0.8045 | 0.9747 | 1.1556 | 1.3575 | 1.5976 | 1.9154 | 2.4549 |

1 | 0.3878 | |||||||||

2 | 0.2529 | 0.5228 | ||||||||

3 | 0.1962 | 0.3663 | 0.6010 | |||||||

4 | 0.1636 | 0.2941 | 0.4385 | 0.6552 | ||||||

5 | 0.1419 | 0.2502 | 0.3598 | 0.4909 | 0.6963 | |||||

6 | 0.1263 | 0.2201 | 0.3106 | 0.4091 | 0.5317 | 0.7292 | ||||

7 | 0.1144 | 0.1978 | 0.2759 | 0.3568 | 0.4484 | 0.5650 | 0.7566 | |||

8 | 0.1049 | 0.1804 | 0.2497 | 0.3194 | 0.3942 | 0.4809 | 0.5931 | 0.7800 | ||

9 | 0.0972 | 0.1665 | 0.2292 | 0.2909 | 0.3550 | 0.4256 | 0.5085 | 0.6173 | 0.8003 | |

10 | 0.0908 | 0.1550 | 0.2125 | 0.2682 | 0.3249 | 0.3852 | 0.4525 | 0.5325 | 0.6385 | 0.8183 |

( |
||||||||||

1 | 0.3889 | |||||||||

2 | 0.3005 | 0.4774 | ||||||||

3 | 0.2578 | 0.3858 | 0.5232 | |||||||

4 | 0.2310 | 0.3382 | 0.4335 | 0.5532 | ||||||

5 | 0.2120 | 0.3069 | 0.3851 | 0.4658 | 0.5750 | |||||

6 | 0.1976 | 0.2840 | 0.3525 | 0.4177 | 0.4898 | 0.5920 | ||||

7 | 0.1862 | 0.2663 | 0.3283 | 0.3848 | 0.4424 | 0.5088 | 0.6059 | |||

8 | 0.1767 | 0.2521 | 0.3092 | 0.3600 | 0.4095 | 0.4621 | 0.5244 | 0.6176 | ||

9 | 0.1688 | 0.2402 | 0.2937 | 0.3404 | 0.3846 | 0.4295 | 0.4783 | 0.5375 | 0.6276 | |

10 | 0.1620 | 0.2300 | 0.2806 | 0.3241 | 0.3647 | 0.4046 | 0.4461 | 0.4922 | 0.5489 | 0.6363 |

( |
||||||||||

1 | 0.1398 | |||||||||

2 | 0.1068 | 0.1728 | ||||||||

3 | 0.0911 | 0.1381 | 0.1901 | |||||||

4 | 0.0813 | 0.1203 | 0.1559 | 0.2015 | ||||||

5 | 0.0745 | 0.1088 | 0.1377 | 0.1681 | 0.2098 | |||||

6 | 0.0693 | 0.1004 | 0.1255 | 0.1499 | 0.1772 | 0.2163 | ||||

7 | 0.0652 | 0.0939 | 0.1165 | 0.1375 | 0.1591 | 0.1844 | 0.2217 | |||

8 | 0.0618 | 0.0887 | 0.1095 | 0.1282 | 0.1467 | 0.1666 | 0.1903 | 0.2261 | ||

9 | 0.0590 | 0.0844 | 0.1038 | 0.1209 | 0.1374 | 0.1542 | 0.1728 | 0.1954 | 0.2300 | |

10 | 0.0566 | 0.0808 | 0.0990 | 0.1149 | 0.1299 | 0.1448 | 0.1605 | 0.1780 | 0.1997 | 0.2333 |

1 | 1.4815 | |||||||||

2 | 0.6409 | 1.9944 | ||||||||

3 | 0.3902 | 0.9687 | 2.3418 | |||||||

4 | 0.2734 | 0.6257 | 1.2180 | 2.6107 | ||||||

5 | 0.2070 | 0.4545 | 0.8174 | 1.4233 | 2.8316 | |||||

6 | 0.1647 | 0.3528 | 0.6090 | 0.9822 | 1.5988 | 3.0196 | ||||

7 | 0.1356 | 0.2857 | 0.4812 | 0.7457 | 1.1271 | 1.7524 | 3.1836 | |||

8 | 0.1145 | 0.2385 | 0.3950 | 0.5975 | 0.8687 | 1.2568 | 1.8893 | 3.3292 | ||

9 | 0.0986 | 0.2035 | 0.3332 | 0.4958 | 0.7039 | 0.9807 | 1.3743 | 2.0127 | 3.4603 | |

10 | 0.0862 | 0.1768 | 0.2869 | 0.4218 | 0.5894 | 0.8021 | 1.0834 | 1.4817 | 2.1253 | 3.5795 |

1 | 0.1646 | |||||||||

2 | 0.0712 | 0.2216 | ||||||||

3 | 0.0434 | 0.1076 | 0.2602 | |||||||

4 | 0.0304 | 0.0695 | 0.1353 | 0.2901 | ||||||

5 | 0.0230 | 0.0505 | 0.0908 | 0.1581 | 0.3146 | |||||

6 | 0.0183 | 0.0392 | 0.0677 | 0.1091 | 0.1776 | 0.3355 | ||||

7 | 0.0151 | 0.0317 | 0.0535 | 0.0829 | 0.1252 | 0.1947 | 0.3537 | |||

8 | 0.0127 | 0.0265 | 0.0439 | 0.0664 | 0.0965 | 0.1396 | 0.2099 | 0.3699 | ||

9 | 0.0110 | 0.0226 | 0.0370 | 0.0551 | 0.0782 | 0.1090 | 0.1527 | 0.2236 | 0.3845 | |

10 | 0.0096 | 0.0196 | 0.0319 | 0.0469 | 0.0655 | 0.0891 | 0.1204 | 0.1646 | 0.2361 | 0.3977 |

( |
||||||||||

1 | 0.1121 | |||||||||

2 | 0.0676 | 0.1409 | ||||||||

3 | 0.0501 | 0.0917 | 0.1592 | |||||||

4 | 0.0404 | 0.0705 | 0.1083 | 0.1726 | ||||||

5 | 0.0341 | 0.0582 | 0.0854 | 0.1210 | 0.1831 | |||||

6 | 0.0297 | 0.0499 | 0.0716 | 0.0971 | 0.1312 | 0.1918 | ||||

7 | 0.0264 | 0.0440 | 0.0621 | 0.0823 | 0.1067 | 0.1398 | 0.1991 | |||

8 | 0.0238 | 0.0394 | 0.0552 | 0.0721 | 0.0914 | 0.1149 | 0.1471 | 0.2054 | ||

9 | 0.0218 | 0.0358 | 0.0498 | 0.0644 | 0.0806 | 0.0991 | 0.1220 | 0.1536 | 0.2110 | |

10 | 0.0201 | 0.0329 | 0.0455 | 0.0585 | 0.0724 | 0.0879 | 0.1059 | 0.1282 | 0.1593 | 0.2160 |

( |
||||||||||

1 | 0.0492 | |||||||||

2 | 0.0288 | 0.0674 | ||||||||

3 | 0.0210 | 0.0430 | 0.0788 | |||||||

4 | 0.0168 | 0.0326 | 0.0527 | 0.0869 | ||||||

5 | 0.0141 | 0.0266 | 0.0411 | 0.0601 | 0.0933 | |||||

6 | 0.0122 | 0.0227 | 0.0341 | 0.0477 | 0.0661 | 0.0985 | ||||

7 | 0.0108 | 0.0199 | 0.0294 | 0.0401 | 0.0532 | 0.0710 | 0.1029 | |||

8 | 0.0097 | 0.0177 | 0.0260 | 0.0349 | 0.0452 | 0.0578 | 0.0753 | 0.1066 | ||

9 | 0.0089 | 0.0161 | 0.0233 | 0.0310 | 0.0396 | 0.0495 | 0.0619 | 0.0790 | 0.1100 | |

10 | 0.0081 | 0.0147 | 0.0212 | 0.0280 | 0.0354 | 0.0437 | 0.0533 | 0.0654 | 0.0823 | 0.1129 |