Using the wrapped distribution to estimatethe survival functionWith practicalapplication

A Thesis Submitted toCouncil of The Administration and Economics/ KarbalaUniiersity as Partial fulfillment of the Requirements for theDegree of Master of Science in StatisticsPresented

by The student Alaa Sami HadiSuperiised

By Ass. Prof. Dr. Sherouk Abdul Redha Al-Sabbah

Learn about cyclic statistics (circular statistics), a branch specialized in analyzing data that represent angular directions. Angular statistics is a special case that deals with data falling on a circle in many applications.

In our daily lives, we often encounter data measured in cyclic angular units, such as temperatures, bird migration directions, and wind directions in meteorology. Such data are sometimes challenging to analyze biologically. This data can include the full range, i.e., (0,2π)(0, 2\pi)(0,2π), and is called resonant data.

When a mathematical model is required to study such data, this study introduces a new circular data distribution called Mira’s participation (encapsulated-Mira). This distribution links linear random variables and circular variables within the period (0,2π)(0, 2\pi)(0,2π).

The study explores multiple mechanical and structural functions of the proposed distribution, including:

• Probabilistic and cumulative effect functions
• Survival function
• Risk
• Mean
• Skewness coefficient
• Kurtosis (flatness)

The characteristics of the distribution are analyzed using three estimation methods:

1. Maximum Likelihood (ML)
2. Least Squares (LS)
3. Cramer von Mises (CVM)

To compare these methods and identify the best estimator for the proposed distribution, the Monte Carlo simulation method was employed. Experiments were conducted with different sample sizes (25, 50, 75, 100) using the Mean Squared Error (MSE) to evaluate the estimators. Results showed that the Cramer von Mises method (CVM) provided the best estimation of parameters and the survival function for the Mira-Cabbage distribution.

Finally, the proposed distribution was applied to real data consisting of 100 observations representing Femur-Tibia Angle (FTA) features, commonly associated with bowed stems disease in the province of Karbala. Using the CVM method, the experimental results confirmed that this method was superior among.

Using the wrapped distribution to estimatethe survival functionWith practicalapplication

A Thesis Submitted toCouncil of The Administration and Economics/ KarbalaUniiersity as Partial fulfillment of the Requirements for theDegree of Master of Science in StatisticsPresented

by The student Alaa Sami HadiSuperiised

By Ass. Prof. Dr. Sherouk Abdul Redha Al-Sabbah

Learn about cyclic statistics (circular statistics), a branch specialized in analyzing data that represent angular directions. Angular statistics is a special case that deals with data falling on a circle in many applications.

In our daily lives, we often encounter data measured in cyclic angular units, such as temperatures, bird migration directions, and wind directions in meteorology. Such data are sometimes challenging to analyze biologically. This data can include the full range, i.e., (0,2π)(0, 2\pi)(0,2π), and is called resonant data.

When a mathematical model is required to study such data, this study introduces a new circular data distribution called Mira’s participation (encapsulated-Mira). This distribution links linear random variables and circular variables within the period (0,2π)(0, 2\pi)(0,2π).

The study explores multiple mechanical and structural functions of the proposed distribution, including:

• Probabilistic and cumulative effect functions
• Survival function
• Risk
• Mean
• Skewness coefficient
• Kurtosis (flatness)

The characteristics of the distribution are analyzed using three estimation methods:

1. Maximum Likelihood (ML)
2. Least Squares (LS)
3. Cramer von Mises (CVM)

To compare these methods and identify the best estimator for the proposed distribution, the Monte Carlo simulation method was employed. Experiments were conducted with different sample sizes (25, 50, 75, 100) using the Mean Squared Error (MSE) to evaluate the estimators. Results showed that the Cramer von Mises method (CVM) provided the best estimation of parameters and the survival function for the Mira-Cabbage distribution.

Finally, the proposed distribution was applied to real data consisting of 100 observations representing Femur-Tibia Angle (FTA) features, commonly associated with bowed stems disease in the province of Karbala. Using the CVM method, the experimental results confirmed that this method was superior among