A new Size Biased Distribution with Applications in Engineering and Medical Science

Lot of work has been done in this field. Gove (2003) reviewed some of the more recent results on size-biased distributions pertaining to parameter estimation in forestry. Warren (1975) was the first to apply the size biased distributions in connection with sampling wood cells. More recently; these distributions were used to recover the distribution of canopy heights from airborne laser scanner measurements. Das and Roy (2011) discussed the length-biased Weighted Generalized Rayleigh distribution with its properties. Patil and Ord (1976) introduced the concept of size-biased sampling and weighted distributions by identifying some of the situations where the underlying models retain their form. Ayesha, (2017) discussed the Size Biased Lindley Distribution as a new life time distribution and discussed its various statistical properties. Shankar (2017) discussed a Size-Biased Poisson-Shanker Distribution and its applications to handle various count data sets. Recently, Shanker & Shukla (2018) discussed a generalized size-biased Poisson-Lindley distribution and Its Applications to model size distribution of freely-forming small group.


INTRODUCTION
studied a new probability distribution named as Ailamujia distribution for several engineering applications and discussed its various characteristics. Its probability density function and cumulative density function is given respectively as follows: (1.2) Lot of work has been done in this field. Gove (2003) reviewed some of the more recent results on size-biased distributions pertaining to parameter estimation in forestry. Warren (1975) was the first to apply the size biased distributions in connection with sampling wood cells. More recently; these distributions were used to recover the distribution of canopy heights from airborne laser scanner measurements. Das

SIZE BIASED AILAMUJIA DISTRIBUTION
If X is a non negative random variable with probability density function (pdf)   x f , then the probability density function of the size biased random variable sb X is given by: The probability density of size biased Ailamujia distribution is given as: x E The corresponding cdf of size biased Ailamujia distribution is obtained as:   Fig. 1 illustrates that the density function of size biased Ailamujia distribution is positively skewed. Fig .2 gives the graphical overview of cdf plot of SBAD. As the value of λ increases, the initial rise of cdf curve also increases.

QUANTILE AND RANDOM NUMBER GENERATION FROM SBAD
Inverse cdf method is one of the methods used for the generation of random numbers from a particular distribution. In this method the random numbers from a particular distribution are generated by solving the equation obtained on equating the cdf of a distribution to a number u. The number u is itself being generated from uniform distribution i.e., U(0,1). Thus following the same procedure for the generation of random numbers from the SBAD, we will proceed as On solving the equation (3.1) for x, we will obtain the required random number from the SBAD. Equation (3.1) will be solved for x using R-software.

RELIABILITY ANALYSIS
In this sub section, we have obtained the reliability, hazard rate, reverse hazard rate and Mills ratio of the proposed size biased Ailamujia model.

Survival function of SBAD
The reliability function is defined as the probability that a system survives beyond a specified time. It is also referred to as survival or survivor function of the distribution. It can be computed as complement of the cumulative distribution function of the model. The reliability function or the survival function of size biased Ailamujia distribution is calculated as: The graphical representation of the reliability function for the size biased Ailamujia distribution is shown in fig. 3 © 2018, IJSRMSS All Rights Reserved 69

Hazard Function
The hazard function is also known as hazard rate, instantaneous failure rate or force of mortality is given as: ; ; .

Reverse Hazard Rate and Mills Ratio
The reverse hazard rate and the Mills ratio of the size biased Ailamujia distribution are respectively given as:

STATISTICAL PROPERTIES
In this section, the different structural properties of the proposed size biased Ailamujia model have been evaluated. These include moments, mode, harmonic mean, moment generating function and characteristic function 5.1 Moments Suppose X is a random variable following size biased Ailamujia distribution with parameter , and then the rth moment for a given probability distribution is given by If we put r=1 in eq. (5.1), we get the mean of size-biased Ailamujia distribution which is given by: If we put r=2 in eq. (5.1), we have Thus, the variance of size biased Ailamujia distribution is given as: Using the values of eq. ii) Kurtosis It may be defined as the degree of peakedness of the density curve. The formula for obtaining the kurtosis of a distribution in terms of moments is given as: After using eq. (5.8) and eq. (5.4) in eq. (5.11), we have

5.3Mode of size-biased Ailamujia Distribution
In order to calculate the mode of the size biased Ailamujia model we take the logarithm of the probability density function of the size biased model: Differentiating the above equation and equating it zero, we obtain the mode as follows:

5.4Harmonic mean
The harmonic mean for the proposed model is computed as:

5.5Moment generating function and characteristic function
This section deals with the derivation of moment generating function and characteristic function of the size biased Ailamujia distribution. By the definition of moment generating function, we have: Similarly, the characteristic function of size biased Ailamujia distribution is computed as:

SHANNON'S ENTROPY OF SIZE-BIASED AILAMUJIA DISTRIBUTION
The Shannon entropy of a random variable X is a measure of the uncertainty and is given by ) (x f is the probability function of the random variable X. The Shannon's entropy of the size biased Ailamujia model as follows: On solving (6.2) we have, Using the values of (6.3) and (5.1) in (6.1)

ORDER STATISTICS
be the ordered statistics of the random sample n X X X X ,.... , , 3  Using the equations (2.1) and (2.2), the probability density function of rth order statistics of size biased Ailamujia distribution is given by: Then, the pdf of first order   1 X size biased Ailamujia distribution is given by:

ESTIMATION OF PARAMETERS
In this section, we estimate the parameters of the size biased Ailamujia distribution using different methods.

Method of Moments
In order to obtain sample moments, we replace population moments with sample moments:

Method of Maximum Likelihood Estimation
This is one of the most useful method for estimating the different parameters of the distribution. Let The maximum likelihood estimator and Moments method coincide for estimation of parameter  of the size biased Ailamujia distribution.

Simulation Study of ML Estimators
In this section, we study the performance of ML estimators for different sample sizes (n=25,50, 75, 100,150, 300). We have employed the inverse CDF technique for data simulation for size biased Ailamujia distribution using R software. The process was repeated 500 times for calculation of bias, variance and MSE. For different values of parameters of size biased Ailamujia distribution, decreasing trend is being observed in average bias, variance and MSE as we increase the sample size. Hence, the performance of ML estimators is quite well, consistent in case of size biased Ailamujia distribution.

APPLICATIONS OF SIZE-BIASED AILAMUJIA DISTRIBUTIONS
In this section, we present the goodness of fit of Size biased Ailamujia distribution using maximum likelihood estimate of the parameter on two data sets and the fit has been compared with Ailamujia distribution. For testing the goodness of fit of Size biased Ailamujia distribution over Ailamujia distribution, following two data sets have been considered.  where k is the number of parameters in the statistical model, n is the sample size and -logL is the maximized value of the loglikelihood function under the considered model. From Table 3, it has been observed that size biased Ailamujia distribution have the lesser AIC, AICC, -logL and BIC values as compared to Ailamujia Distribution. Hence we can conclude that the size biased Ailamujia distribution leads to a better fit than the Ailamujia distribution. Kolmogorov Smirnov p-value suggests that size biased Ailamujia distribution fits statistically better than Ailamujia distribution to a data set given in table 2.  Model comparison criterion like AIC, AICC, BIC, -logL suggest that size biased Ailamujia distribution leads to a better fit than the Ailamujia distribution (See table 4). Also it is clear that Kolmogorov Smirnov p-value is statistically significant in case of size biased Ailamujia distribution where as Kolmogorov Smirnov p-value is statistically not significant in case of Ailamujia distribution. Hence, we conclude that data given in table 4 follows size biased Ailamujia distribution and is appropriate for analyzing this data.

CONCLUSION
This manuscript deals with the size-biased Ailamujia distribution and studies its different statistical properties. In this paper, moments, mode, harmonic mean, survival function and hazard rate, method of moments and maximum likelihood estimates of the parameters have been obtained. The newly proposed model has been applied to the different real life data sets and the results obtained prove that the proposed size biased Ailamujia model is better fit than its sub models.