Basic Concepts of Non-Parametric Methods ( Statistics )

This gives the basic description of Non-Parametric Methods . This is one of the important topic in Statistics and also for Mathematics and for Researchers-Scientists .

1. Non-Parametric Methods Presented by Hasnat Israq Islamic University, Bangladesh 1

2. Contents Classification of Hypothesis Test Basic Concept of Non-Parametric Test Assumptions Difference between Parametric & Non-parametric Test Why do we use Non-parametric Test Goodness of Fit Test Types Chi-square Goodness of Fit Test Kolmogorov-Smirnov Test Empirical Distribution Function Properties of Empirical Distribution Function Tests Based on Run Run & Length of the Run Different Types of Test Based on Runs Sign Test Rank Order Statistics Linear Rank Statistics Difference between Rank Order Statistics & Linear Rank Statistics

3. Classification of Hypothesis Test Parametric Test t-Test, f-Test, z-test etc Kolmogorov-Smirnov test, Rank Sum Test Non-Parametric Test

4. Basic Concept of Non-Parametric Test Non-parametric Test: Don’t make any assumption about the form of the frequency function of the parent population. Population parameters are unknown. Assumptions: The assumptions for Non-parametric tests are given below- Sample observations are independent. Lower order moments exist. Population is symmetrical . .

5. SL. NO. Parametric Test Non-parametric Test Information about population is completely known. No information about the population is available. 2. Basic assumptions are made from the parent population is normal. No assumptions are made regarding the population. 3. Null hypothesis is made on parameters of the population distribution. The null hypothesis is free from parameters. 4. They can be used when the data are Interval and ratio. They can be used when the data are nominal or ordinal. Parametric Test Vs Non-parametric Test

6. Why do we use Non-parametric test ? Readily comprehensible, very simple and easy to apply. Used to test hypothesis that don’t involve population parameters. Results are needed in a hurry and calculations must be done in hand. Researchers with minimum preparation in mathematics and statistics usually find the concepts and methods of Non-parametric procedures easy to understand.

7. Goodness of Fit Test Types of Goodness of Fit Test Chi-square Goodness of Fit Test Kolmogorov-Smirnov Test Goodness of Fit Test: Goodness of fit test is a testing procedure of nonparametric test. To check the compatibility of a set of observed sample values with a normal distribution or any other distribution. These tests are designed for a null hypothesis about the form of the cumulative distribution function or probability function of the parent population from which the sample is drawn.

8. Chi-square Goodness of Fit Test Hypothesis: A single random sample size n is drawn from a population with unknown cumulative distribution function . Now we wish to test the null hypothesis for all , where is completely specified, against alternative for all x.     H ₒ:     H ₁ :   Decision Rule: The test statistics is less than null hypothesis then we accept it otherwise reject. Chi-square Goodness of Fit Test: Designed for the null hypothesis concerning the discrete distribution Compares the observed frequencies with the frequencies expected under the null hypothesis.

9. Kolmogorov-Smirnov Test Kolmogorov-Smirnov Test: The Kolmogorov-Smirnov one sample statistic is based on the differences between the hypothesized cumulative distribution function and the empirical distribution function of the sample for all x. The test statistic is   Hypothesis: Assume we have the random sample x ₁, x₂, x₃,…..,xₙ we want to test the hypothesis for all x where is completely specified continuous distribution, against alternative for all x.     H ₒ:   H ₁ :  

10. Empirical Distribution Function Empirical Distribution Function: The cumulative relative distribution function of a random sample is called the empirical distribution function, may be considered an estimate of the population cdf for the given observed values. Properties of Empirical Distribution Function: S ₙ(x) is sometimes called the statistical image of the population. It is a random variable . It is a consistent estimator of F ₓ(x).  

11. Tests Based on Run Continued… Run: A run is defined to be a succession of one or more identical symbols. The number of elements in a run is referred to as the length of the run. The maximum number of elements is known as the longest run. Example: Suppose we observe the arrangement of five males and five females in the line to be M FF MMM F M FF Here the number of run is 6, the longest run is MMM and length of the longest run is 3.

12. Different types of Tests Based on Runs: Tests of randomness of an ordered sequence can be tested by the theory of runs. Types of test based on runs are Test based on the total number of runs. Test based on the length of the longest run. Test based on runs up and down. Test based on ranks. to be Continued… Where we use run test? Run analysis useful in time series analysis and quality control studies.

13. Sign test: Based on the sign (+ or -) of observed difference. Used to test the probability of a (+) sign equal to the probability of sign (-). Simplest nonparametric test. Sign Test Assumptions: Observations are independent. Observation come from symmetrical distribution.

14. Rank Order Statistics Rank Order Statistics: The rank order statistics for a random sample are any set of constants which indicate the order of the observations. If Xᵢ( i =1,2,3,……..N) be a random sample then the rank order statistics for these random samples are r(xᵢ) A functional form of the rank order statistics, r(xᵢ)= =1+ where,       Continued…

15. to be Continued… Rank order statistics follows discrete uniform distribution and it’s distribution is defined as, P[r(xᵢ)= j ]= ; j =1,2,3,…N   It is distribution free. It is usually useful in Non-parametric inference.

16. Linear Rank Statistics Linear Rank Statistics: Many commonly used two sample rank tests can be classified as linear combinations of certain indicator variables for the combined order samples. Such functions are often called linear rank statistics. A linear function of this indicator variable is called a linear rank statistics and thus the linear rank statistics can be written as, T N (z)=  

17. SL. NO. Rank Order Statistics Linear Rank Statistics Rank order statistics is used in single sample problem. It is used for two samples problem of combined order sample. 2. Rank order statistics can’t be expressed in terms of indicator variable. It can be expressed in terms of linear combination of an indicator variable for combined sample. 3. This test provides the information of the single population. This test provides the information about difference between two populations. Rank Order Statistics Vs Linear Rank Statistics

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