Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Estimating α Under Interval Uncertainty Algorithm for Computing the Smallest Value of α Algorithm for Computing the Largest Value of α Estimating Parameters of Pareto Distribution Under Interval and Fuzzy Uncertainty Nitaya Buntao ********************************************************* Department of Applied Statistics King Mongkut’s University of Technology North Bangkok Thailand Email: taltanot@hotmail.com February 16, 2011 Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Estimating α Under Interval Uncertainty Algorithm for Computing the Smallest Value of α Algorithm for Computing the Largest Value of α Table of Contents Formulation of the Problem 1 Background Purpose of the Study Interval Uncertainty First Result: Estimating x 0 Under Interval Uncertainty 2 Estimating α Under Interval Uncertainty 3 Analysis of the Problem: Reducing the Problem Bounds of α via Bounds of S Algorithm for Computing the Smallest Value of α 4 Algorithm for Computing the Largest Value of α 5 Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Background Estimating α Under Interval Uncertainty Purpose of the Study Algorithm for Computing the Smallest Value of α Interval Uncertainty Algorithm for Computing the Largest Value of α Estimating Parameters of the Pareto Distribution The Pareto distribution is a power law probability distribution: the probability that X is greater than some number x is given by x α � α · ; if x > x 0 0 f X ( x ) = x α +1 0 ; if x < x 0 . Figure: Pareto probability density functions for various α with x 0 = 1 . Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Background Estimating α Under Interval Uncertainty Purpose of the Study Algorithm for Computing the Smallest Value of α Interval Uncertainty Algorithm for Computing the Largest Value of α Estimating Parameters of the Pareto Distribution Reminder: the Pareto distribution is a power law probability distribution: the probability that X is greater than some number x is given by x α � α · ; if x > x 0 0 f X ( x ) = x α +1 0 ; if x < x 0 . Estimators of parameters x 0 and α based on the observed data values x 1 , . . . , x n come from applying the Maximum Likelihood techniques: x 0 = min( x 1 , . . . , x n ) , ˆ and � n �� − 1 � x i � α = n · ˆ ln . min( x 1 , . . . , x n ) i =1 Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Background Estimating α Under Interval Uncertainty Purpose of the Study Algorithm for Computing the Smallest Value of α Interval Uncertainty Algorithm for Computing the Largest Value of α Need to take into Account Interval and Fuzzy Uncertainty In practice, we rarely know the exact values of x i . For example, in financial situations, we can take, as x i , the price of the financial instrument at the i -th moment of time – e.g., on the i -th day. However, the price does not remain stable during the day. Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Background Estimating α Under Interval Uncertainty Purpose of the Study Algorithm for Computing the Smallest Value of α Interval Uncertainty Algorithm for Computing the Largest Value of α Need to take into Account Interval and Fuzzy Uncertainty In practice, we rarely know the exact values of x i . For example, in financial situations, we can take, as x i , the price of the financial instrument at the i -th moment of time – e.g., on the i -th day. However, the price does not remain stable during the day. It is more reasonable to consider the whole range [ x i , x i ] of the daily prices instead of a single value x i . Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Background Estimating α Under Interval Uncertainty Purpose of the Study Algorithm for Computing the Smallest Value of α Interval Uncertainty Algorithm for Computing the Largest Value of α Need to take into Account Interval and Fuzzy Uncertainty In practice, we rarely know the exact values of x i . For example, in financial situations, we can take, as x i , the price of the financial instrument at the i -th moment of time – e.g., on the i -th day. However, the price does not remain stable during the day. It is more reasonable to consider the whole range [ x i , x i ] of the daily prices instead of a single value x i . We need to find the range of all resulting values of x 0 and α . Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Background Estimating α Under Interval Uncertainty Purpose of the Study Algorithm for Computing the Smallest Value of α Interval Uncertainty Algorithm for Computing the Largest Value of α Need to take into Account Interval and Fuzzy Uncertainty In practice, we rarely know the exact values of x i . For example, in financial situations, we can take, as x i , the price of the financial instrument at the i -th moment of time – e.g., on the i -th day. However, the price does not remain stable during the day. It is more reasonable to consider the whole range [ x i , x i ] of the daily prices instead of a single value x i . We need to find the range of all resulting values of x 0 and α . Estimating this range under interval uncertainty is a particular case of a general problem of interval computations. Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Background Estimating α Under Interval Uncertainty Purpose of the Study Algorithm for Computing the Smallest Value of α Interval Uncertainty Algorithm for Computing the Largest Value of α Interval Uncertainty: x i ∈ [ x i , x i ] Some of these values x i may be flukes caused by accidental errors. Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Background Estimating α Under Interval Uncertainty Purpose of the Study Algorithm for Computing the Smallest Value of α Interval Uncertainty Algorithm for Computing the Largest Value of α Interval Uncertainty: x i ∈ [ x i , x i ] Some of these values x i may be flukes caused by accidental errors. Experts can usually tell which values x i are possible. However, experts used word from a natural language. Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
Formulation of the Problem First Result: Estimating x 0 Under Interval Uncertainty Background Estimating α Under Interval Uncertainty Purpose of the Study Algorithm for Computing the Smallest Value of α Interval Uncertainty Algorithm for Computing the Largest Value of α Interval Uncertainty: x i ∈ [ x i , x i ] Some of these values x i may be flukes caused by accidental errors. Experts can usually tell which values x i are possible. However, experts used word from a natural language. To describe these natural-language statements, it is reasonable to use fuzzy logic . Nitaya Buntao ********************************************************* Department of Applied Statistics King Mong Estimating Parameters of Pareto Distribution Under Interval an
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