A Thesis Submitted to

Council of The Administration and Economics/ Karbala
University as Partial fulfillment of the Requirements for the

Degree of Master of Science in Statistics

written By
Alaa Khalid AbdulHussein
Supervised by

Asst. Prof. DR. Sada Faydh Mohammed

This thesis aims to provide a more accurate statistical estimation of the probability density function (PDF) for survival data by employing asymmetric kernel functions. The study focuses on developing a flexible model based on the asymmetric Weibull kernel function due to its suitability for long-tailed data.

In the theoretical part, the mathematical formulation of the Weibull kernel function was developed, its main properties were analyzed, and it was theoretically compared with other commonly used kernel functions in previous studies in terms of shape, probabilistic behavior, and representational capability.

In the experimental part, two simulation studies were conducted using four sample sizes (50, 150, 200, and 400) and four different data scenarios. In the first experiment, the exponential distribution was used with varying parameter values (0.3, 0.9, 1.5, 2). In the second experiment, four distributions were employed: the exponential distribution (parameter = 1.5), the Weibull distribution (parameters = 0.5, 1.3), the Gamma distribution (parameters = 2, 2.5), and the Lognormal distribution (parameters = 0.2, 1.5).

The performance of the proposed Weibull kernel was evaluated and compared with other symmetric and asymmetric kernel functions using the Integrated Squared Error (ISE) criterion and the Silverman and Cross-Validation methods for bandwidth selection.

The results demonstrated the superiority of the Weibull kernel function in achieving the lowest ISE values, particularly for a sample size of 200. The proposed model was also applied to real survival data of catheterized patients from a public hospital, and the results showed the efficiency of the developed function through the close agreement between the estimated survival functions and the actual data. These findings confirm the effectiveness of using asymmetric kernel functions particularly the Weibull kernel—in modeling survival data. MATLAB software was used for the practical implementation of the thesis.

A Thesis Submitted to

Council of The Administration and Economics/ Karbala
University as Partial fulfillment of the Requirements for the

Degree of Master of Science in Statistics

written By
Alaa Khalid AbdulHussein
Supervised by

Asst. Prof. DR. Sada Faydh Mohammed

This thesis aims to provide a more accurate statistical estimation of the probability density function (PDF) for survival data by employing asymmetric kernel functions. The study focuses on developing a flexible model based on the asymmetric Weibull kernel function due to its suitability for long-tailed data.

In the theoretical part, the mathematical formulation of the Weibull kernel function was developed, its main properties were analyzed, and it was theoretically compared with other commonly used kernel functions in previous studies in terms of shape, probabilistic behavior, and representational capability.

In the experimental part, two simulation studies were conducted using four sample sizes (50, 150, 200, and 400) and four different data scenarios. In the first experiment, the exponential distribution was used with varying parameter values (0.3, 0.9, 1.5, 2). In the second experiment, four distributions were employed: the exponential distribution (parameter = 1.5), the Weibull distribution (parameters = 0.5, 1.3), the Gamma distribution (parameters = 2, 2.5), and the Lognormal distribution (parameters = 0.2, 1.5).

The performance of the proposed Weibull kernel was evaluated and compared with other symmetric and asymmetric kernel functions using the Integrated Squared Error (ISE) criterion and the Silverman and Cross-Validation methods for bandwidth selection.

The results demonstrated the superiority of the Weibull kernel function in achieving the lowest ISE values, particularly for a sample size of 200. The proposed model was also applied to real survival data of catheterized patients from a public hospital, and the results showed the efficiency of the developed function through the close agreement between the estimated survival functions and the actual data. These findings confirm the effectiveness of using asymmetric kernel functions particularly the Weibull kernel—in modeling survival data. MATLAB software was used for the practical implementation of the thesis.