New modification of three parameter Sujatha distribution and its SPC applications

Introduction

The concept of weighted probability distributions plays an important role in various research areas related to Reliability, Biomedicine, Ecology and Branching process for the improvement of proper statistical models. The concept of weighted probability models attracted a lot of researchers to contemplate and carry out research on this topic. Fisher (1934) [4] introduced the concept of weighted models in connection with his studies on how the methods of ascertainment can influence the form of distribution of recorded observations. Later, it has been modified by Rao (1965) [10] in a unified manner where the observations fall into nonexperimental, non-replicated and non-random categories. The weighted distribution reduces to length biased as the weight function considers only the length of units of interest. More generally, when the sampling mechanism selects units with probability proportional to measure of the unit size, the resulting distribution is called size biased. Size biased distributions are special case of more general form known as weighted distributions. The concept of length biased sampling was introduced by Cox (1969) [3] and Zelen (1974) [16] . Patil and Rao (1978) [9] studied weighted distributions and size biased sampling with applications to wild life population and human families. Van Deusen (1986) [15] arrived at size biased distribution theory independently and applied it in fitting assumed distributions to data arising from horizontal point sampling. Various authors have studied different probability models and illustrated their applications in different fields. Jayakumar and Elangovan (2019) [6] studies area biased Ailamujia distribution with applications in bladder cancer data. Hassan, Akhter and Para (2019) [5] discussed on weighted quasi exponential distribution with properties and applications. Sharma et al. (2017) [13] presented the length and area biased Maxwell distribution. Mudasir and Ahmad (2018) [8] study the characterization and estimation of length biased Nakagami distribution. Bashir and Rasul (2016) [1] introduced the Poisson area biased Lindley distribution and its applications on biological data. Rao and Pandey (2021) [11] studied the parameter estimation of length biased weighted Frechet distribution via Bayesian approach. The three parameter Sujatha distribution is a newly introduced continuous lifetime model proposed by John and Aniefiok (2020) [7] and the proposed distribution is a modification of two parameter Sujatha distribution. Its several statistical properties such as shape of the pdf, hazard rate function, moment generating function, order statistics and raw moments have been derived and discussed. The parameters are also estimated by using the method of maximum likelihood estimation. Tesfay and Shanker (2018) [14] introduced the two parameter Sujatha distribution studied its various statistical properties and estimate its parameters through method of moments and method of maximum likelihood estimation.

Area Biased Three Parameter Sujatha (ABTPS) Distribution

The probability density function of three parameter Sujatha (TPS) distribution is given by

and the cumulative distribution function of three parameter Sujatha distribution is given by

If X be the non-negative random variable with probability density function ) (x f and ) (x w be its non-negative weight function, then the probability density function of weighted random variable w X is given by

For various weighted models, we have different choices of weight function w(x). When w(x) = x c , the resulting distribution is termed as weighted distribution. In this paper, we have to find the area biased version of three parameter Sujatha distribution. So consequently, we will take c = 2 in weights x c in order to obtain the area biased three parameter Sujatha distribution and its pdf is given by

By substituting equations ( 1) and (4) in equation ( 3), we will obtain the probability density function of area biased three parameter Sujatha distribution

and the cumulative distribution function of area biased three parameter Sujatha distribution can be obtained as

Survival Analysis

In this section, we will discuss the survival function, failure rate and reverse hazard rate functions of the area biased three parameter Sujatha distribution.

Survival function

The survival function of area biased three parameter Sujatha distribution can be obtained as

Hazard function

The hazard function is also known as failure rate or force of mortality and is given by

Reverse hazard function

The reverse hazard function is given by

Statistical Properties

In this section, we will discuss various structural properties of area biased three parameter Sujatha distribution which include moments, harmonic mean and moment generating function.

Moments

Let the random variable X following area biased three parameter Sujatha distribution with parameters θ, λ and α, then the r th order moment of proposed model can be obtained as

After the simplification of equation ( 8), we obtain

Putting r = 1, 2, 3 and 4 in equation ( 9), we will obtain the first four moments of area biased three parameter Sujatha distribution as

Harmonic mean

The harmonic mean for the proposed area biased three parameter Sujatha distribution can be obtained as

Moment Generating Function and Characteristic Function

Let X be the random variable following area biased three parameter Sujatha distribution with parameters θ, λ and α, then the MGF of X can be obtained as

Similarly, the characteristic function of area biased three parameter Sujatha distribution can be obtained as

Order Statistics

Order statistics play an important role in many practical as well as theoretical areas. Let X (1) , X (2) ,…,X (n) be the order statistics of a random sample X 1 , X 2 ,…, X n drawn from a continuous population with probability density function f x (x) and cumulative distribution function F

x (x), then the probability density function of r th order statistics X (r) is given by

By using the equations ( 5) and (7) in equation ( 10), we will obtain the probability density function of r th order statistics of area biased three parameter Sujatha distribution as

Therefore, the probability density function of higher order statistic X (n) of area biased three parameter Sujatha distribution can be obtained as

and the probability density function of first order statistic X

(1) of area biased three parameter Sujatha distribution can be obtained as

Bonferroni and Lorenz Curves

The bonferroni and Lorenz curves are also termed as classical or income distribution curves are applied in different fields like reliability, medicine, insurance and demography. The bonferroni and Lorenz curves are given by

Maximum Likelihood Estimation and Fisher's Information Matrix

In this section, we will discuss the method of maximum likelihood estimation to estimate the parameters of area biased three parameter Sujatha distribution and also studies its Fisher's information matrix. Let X 1 , X 2 ,…,X n be a random sample of size n from the area biased three parameter Sujatha distribution, then the likelihood function is given by

The log likelihood function is given by (11) By differentiating the log likelihood equation (11) with respect to parameters θ, λ and α. we must satisfy the normal equations as

The above likelihood equations are too complicated to solve it algebraically. Therefore, we use R and wolfram mathematics for estimating the required parameters of the proposed distribution.

In order to obtain the confidence interval, we use the asymptotic normality results. We have that if

. We can state the result as follows:

I I

Application

In this section, we have used a real life data set in area biased three parameter Sujatha distribution to show that the area biased three parameter Sujatha distribution provides a better fit over three parameter Sujatha, two parameter Sujatha and Sujatha distributions. The real life data set is given below as. The following data set collected by Balakrishnan is related with behavioral science which represents the General rating of Affective symptoms for preschoolers (GRASP) measures behavioral and emotional problems of children, which can be classified with depressive condition or not according to the scale. The study conducted by authors in a city located at the south part of chile has allowed collecting real data corresponding to the scores of GRASP scale of children with frequencies which are given in parenthesis in table 1 as. [2] 19 (6) 20( 15) 21( 14) 22( 9) 23( 12) 24( 10) 25 (6) 26( 9) 27( 8) 28( 5) 29( 6) 30( 4

Where k is the number of parameters in the statistical model, n is the sample size and -2logL is the maximized value of loglikelihood function under the considered model. From table 2 given above, it can be easily seen that the area biased three parameter Sujatha distribution have the lesser AIC, BIC, AICC and -2logL values as compared to three parameter Sujatha, two parameter Sujatha and Sujatha distributions. Hence, it can be concluded that the area biased three parameter Sujatha distribution leads to a better fit over three parameter Sujatha, two parameter Sujatha and sujatha distributions.

Control Limits for Area Biased Three Parameter Sujatha (ABTPS) Distribution

Numerical Example

The illustration of control limits is considered for illustrating the applications of the method proposed in the above. The Control limits using Area biased three parameter Sujatha distribution are obtained by using simulated data set for parameters 𝛼, 𝜆 𝑎𝑛𝑑 θ. From table 1-4, that for the stable label of the parameter 𝛼. the LCL becomes negative, it may be rounded to 0. The Area Biased Three Parameter Sujatha Distribution control chart are shown in Fig. 1 for 𝛼 = 0.02, λ = 0.07 and θ = 0.04. It is also pointed out that, the explanations of the process control are essential or else the process will be in control. That is, depending on the manufacturing products, the manufacturing Engineers would set the parameter values based on the type of data they are dealing with as shown in table 1-4, the greater parameter values, the greater the control limits.

Conclusion

The present study deals with a new distribution termed as area biased three parameter Sujatha distribution. The proposed new distribution is generated by using the area biased technique to the baseline distribution. Its different statistical properties which include moments, harmonic mean, survival function, hazard rate function, reverse hazard rate function, order statistics, Bonferroni and Lorenz curves have been discussed. Its parameters have also been estimated by employing the method of maximum likelihood estimation. Finally, a new distribution has been examined and analysed by using a real-life data set and it is found from the result that the proposed area biased three parameter Sujatha distribution leads to a better fit than the three parameter Sujatha, two parameter Sujatha and Sujatha distributions.

References

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