Mid-Truncated Burr XII distribution and its applications in order statistics

In this paper we define the Mid-Truncated Burr XII distribution and derive some of its statistical properties such as moments, moment generating function, characteristic function, likelihood function, etc. We have also obtained the recurrence relations for single and product moments of order statistics in a random sample of size n drawn from MidTruncated Burr XII distribution. Further, the characterization of Mid-Truncated distributions using conditional moments has been given.


Introduction
Many authors like Malik [10], Balakrishnan and Joshi [1], Balakrishnan et al. [2], Saran and Pushkarna [17,18,19], etc. have obtained several results for the single and product moments of order statistics from the un-truncated, left truncated, right truncated and doubly truncated distributions.Mohie El-Din and Sultan [12] have obtained recurrence relations for moments of order statistics from doubly truncated continuous distributions.They have also presented some applications of their results.Mohie El-Din et al. [11] have obtained moment generating functions of order statistics from doubly truncated exponential distribution in terms of hypergeometric function.They have also derived some recurrence relations between these moment generating functions.The truncated distributions are quite effectively used where a random variable is restricted to be observed on some range and these situations are common in various fields.For instance, in survival analysis, failures during the warranty period may not be counted.Therefore many researchers were being attracted to the problem of analysing such truncated data encountered in various disciplines, proposed the truncated versions of the usual statistical distributions, to improve a forecasting actuarial model and particularly for modelling data from insurance policies.But there exist life models which do not obey the complete or truncated distribution, for example, in Microbiology (cf.Tortora et al. [21]), when a bacterial strain is inoculated into a liquid growth medium, the population is counted at intervals, it is possible to plot a bacterial growth curve.There are three basic phases of growth: the log, stationary, and death phases.During log phase, the bacterial cells are most metabolically active and are preferred for industrial purposes.During stationary phase we find that the number of cells will still remain constant due to the number of bacterial death balances with the number of new cells.Consequently during this phase no investigation is required.After the stationary phase the most bacterial cells will die because of the exclusion of nutrients and lead to accumulation of waste products.In such live models mid-truncated distributions are quite effectively used where the random variable is restricted

Mid-truncated distribution
We define the Mid-truncated distribution as follows: Let Y be a continuous random variable with baseline probability density function (pdf) g(y), cumulative distribution function (cdf) G(y), define X as a corresponding mid-truncated variable, of the random variable Y with pdf f (x).We define which is called the Mid-truncated density function and, Q 1 and P 1 are the points of mid truncation of the baseline distribution under consideration.Also we assume that and Then equation (2.1) can be written as (2.4)

k-th moment
The k-th moment of a mid-truncated random variable X (denoted by µ (k) * ) is given by: Integrating by parts, we get ] , ( where µ (k) is the k-th moment of the un-truncated distribution.

Moment generating function and characteristic function
The moment generating function of a mid-truncated random variable X (denoted by M * (t)) is given by: Integrating by parts, we get where M (t) is the moment generating function of the un-truncated random variable.
Similarly, the characteristic function of the mid-truncated random variable is given by: where Φ(t) is the characteristic function of the un-truncated random variable.

Distribution function
The distribution function of a mid-truncated random variable X is given by: (2.7)

Generating data
If the distribution function of the mid-truncated random variable exists we can use the inverse transform method to generate the data.To generate a random variable X with distribution function F (x) as given in (2.7), draw U 1 , U 2 from U (0, 1) and then solve each of the following equations with respect to x.

Likelihood function
For n observations x 1 , x 2 , . . ., x n from a mid-truncated distribution, the likelihood function is given by: , where r is the number of (2.9) MID-TRUNCATED BURR XII DISTRIBUTION AND ITS APPLICATIONS IN ORDER STATISTICS

Mid-truncated Burr XII distribution
The probability density function (pdf) of Burr distribution (type XII) is given by and the cumulative distribution function (cdf) is given by Then the probability density function (pdf) of mid-truncated Burr distribution (type XII) is given by and the cumulative distribution function (cdf) is given by (3.4) Using (3.3) and (3.4), we get the relation between pdf and cdf as (3.5) Using (2.2) and (2.3), we get Using (3.6) and (3.7), equations (3.3) and (3.4) can be rewritten as and (3.9) The mid-truncated Burr XII density function for β = 2, λ = 4, ν = 0.5, p = q = 0.5 truncated at Q 1 = 1.5 and P 1 = 3.25 is provided in Figure 1.The mid-truncated Burr XII density function for β = 2.6, λ = 0.5, ν = 2, p = 0.8 and q = 0.2 truncated at Q 1 = 1.9 and P 1 = 5.9 is provided in Figure 2.  Let X 1 , X 2 , . . ., X n be a random sample of size n from the mid-truncated Burr XII distribution defined in (3.3) and let X 1:n ≤ X 2:n ≤ . . .≤ X n:n be the corresponding order statistics.Thus the probability density function (pdf) of X r:n (1 ≤ r ≤ n) is given by: where c r:n = n! (r−1)!(n−r)! .The joint density function of order statistics X r:n and where David and Nagaraja [4]).
The single moments of order statistics X r:n (1 ≤ r ≤ n) are given by Similarly, the product moments of X r:n and X s:n (1 ≤ r < s ≤ n) are given by x j y k f r,s:n (x, y)dydx. (3.13)

k-th moment
The k-th moment of the mid-truncated Burr XII distribution can be obtained using (2.5) and is given by: Substituting the values of G(Q 1 ), G(P 1 ) and G(x) from equation (3.2) and solving, we get is the incomplete beta function and can be obtained using tables given by Pearson et al. [16].
To investigate the effect of the shape parameters β, ν and scale parameter λ on the mid-truncated Burr XII density function we have computed mean, variance, skewness and kurtosis for different values of the parameters, which are presented in following Tables 1 to 3.

Moment generating function and characteristic function
The moment generating function of the mid-truncated Burr XII distribution is given by: Integrating by parts, we get

Substituting the value of
2), and solving we get is the incomplete beta function and can be obtained using tables given by Pearson et al. [16].
Similarly the characteristic function of the mid-truncated Burr XII distribution can be evaluated.

Distribution function
The cumulative distribution function of the mid-truncated Burr XII distribution (as in equation (3.9)) is given by (3.14)

Generating data
To generate random variables from mid-truncated Burr XII distribution, we use equations (3.14) and (2.8), and we get Solving the above two equations for x, we get

Likelihood function
For n observations x 1 , x 2 , . . ., x n from mid-truncated Burr XII distribution the likelihood function is obtained from equation (2.9) and is given by: 4. Recurrence relations for single moments of order statistics from mid-truncated Burr XII distribution In this section we will derive recurrence relations for single moments of order statistics from mid-truncated Burr XII distribution as defined in Section 3.

Proof
Relations in (4.1) and (4.2) may be proved by following exactly the same steps as those in proving Theorem 2 which is presented below.
and, for k > β, β ∈ Z + , ν > 1, λ, β > 0, we have where r:n is the same as is the incomplete beta function of first kind and can be obtained by using tables by Pearson et al. [16].

Proof
To prove the above theorem, first we prove the following two lemmas.

Lemma 2
If for 1 ≤ r ≤ n and k = 1, 2, . .., then, it may be written as and where is the incomplete beta function of first kind, which can be found out using tables given by Pearson et al. [16].

Proof
Relation in (4.11) may be proved by following exactly the same steps as those in proving (4.12) which is presented below.
Expanding (1 − F (x)) n−r binomially in powers of F (x) and then substituting the same in (4.10), we get for 2 ≤ r < n, On putting j = l + r, we get Substituting the value of F (x) from (3.4), we get binomially, we get , by taking where is the incomplete beta function and can be obtained using tables given by Pearson et al. [16].
Substituting the value of J from (4.14) in (4.13), it leads to the result (4.12) for the case 2 ≤ r < n.The result (4.12) for the case r = n can similarly be established.

Proof of the main Theorem 2
Using (3.5) in (4.5), we get which on simplification and using (3.12) gives where the values of I (k) r:n , for different values of r, are given in (4.11) and (4.12).This completes the proof of (4.4).Further, on putting k = β in (4.4), it reduces to (4.3).

Recursive Algorithm
Utilizing the knowledge of recurrence relations obtained above one can evaluate the moments of order statistics from mid-truncated Burr XII distribution for β ∈ Z + , k ≥ β, ν > 1 and λ > 0. Algorithm is as following: n−1 (which have already been calculated) and µ Similarly for even values of m, one can evaluate µ

Recurrence relations for product moments of order statistics from mid-truncated Burr XII distribution
Theorem 3 For 1 ≤ r < s ≤ n, x < y and j, k > 0, we have where

Proof
To prove the above theorem, first we evaluate the following term, From equation (3.5) we know for Now consider, where α = r + i, and which is evaluated in Lemma 2.

Remark 2
The results deduced for the mid-truncated Weibull distribution, so obtained, are in agreement with the results of Okasha et al. [14].
Tables 1 to 3. From these tables we observe that by keeping any two parameters fixed, the increase in the value of the third parameter results in the decrease of the values of the measures calculated.
Also, due to the flexibility of the considered distribution, i.e., mid-truncated Burr XII distribution, which contains, as special sub models, mid-truncated Weibull distribution, mid-truncated exponential distribution, midtruncated log-logistic distribution, among others, in accommodating different forms of risk functions, this midtruncated Burr XII distribution is appropriate for a variety of problems modelling lifetime data.

Figure 1 .
Figure 1.Probability density function for mid-truncated Burr XII distribution.

Figure 2 .
Figure 2. Probability density function for mid-truncated Burr XII distribution.
180 MID-TRUNCATED BURR XII DISTRIBUTION AND ITS APPLICATIONS IN ORDER STATISTICS

. 13 )
MID-TRUNCATED BURR XII DISTRIBUTION AND ITS APPLICATIONS IN ORDER STATISTICSNow consider
, by utilizing the above calculated values.In this way one can calculate µ , . . . in a recursive manner.Case 2: when k ̸ = mβ, m ∈ Z + From equation (4.2) we observe that µ (k) 1:n can be evaluated by the knowledge of µ (k−β) 1:n and after the recursive use of (4.2) (α − 1) times (where α is the greatest integer satisfying the condition (k − αβ) > 0), we conclude that µ (k) 1:n is a function of µ (k−αβ) 1:n .Thus in this case, by the knowledge of µ (k−αβ) 1:n one can evaluate values of µ (k) 1:n for all n and k in a simple recursive manner.From equation (4.3) we observe that to evaluate µ (k) r:n we should have the knowledge of only one value with sample size n i.e. µ (k−αβ) r:n(where α is the greatest integer satisfying the condition (k − αβ) > 0) and some moments of lower orders.Using the above information one can calculate µ , . . . in a recursive manner.

Table 3 .
β = 1 and λ = 1.5 as given in Lemma 2 and putting the values of Q 1 and P 1 in (4.15), we get