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. Presented by Hasnat Israq
Islamic University, Bangladesh
1

2. Content
s 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

4.  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
1. 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.

6. – 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:
– 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.  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 𝑥, where is completely specified, against alternative
for all x.
Fₓ Hₒ: Fₓ x = Fₒ(x)
Fₒ H₁: Fₓ x ≠ Fₒ(x)
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:
The Kolmogorov-Smirnov one sample statistic is based on the differences
between the hypothesized cumulative distribution function Fₒ(x)and the
empirical distribution function of the sample for all x. The test statistic
is
Dₙ = sup|Sₙ x − Fₓ(x) |
 Hypothesis:
Assume we have the random sample x₁, x₂, x₃,…..,xₙ we want to test the
hypothesis
for all x where Fₒ(x) is completely specified continuous
distribution, against alternative for all x.
Sₙ(x)
Hₒ: Fₓ x = Fₒ(x)
H₁: Fₓ x ≠ Fₒ(x)

10.  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:
– Sₙ(x) is sometimes called the statistical image of the population.
– It is a random variable.
– It is a consistent estimator of Fₓ(x).
Sₙ x =
0
k
n
1
if
if
if
x < X₍₁₎
X₍ₙ₎ < x < X₍ₙ₊₁₎
x ≥ X₍ₙ₎
; k = 1,2, … , n − 1

11. 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.
 Assumptions:
– Observations are independent.
– Observation come from symmetrical distribution.

14.  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, S u =
0
1
𝑖𝑓
𝑖𝑓
𝑢 < 0
𝑢 ≥ 0𝑗=1
𝑁
𝑆 xᵢ − xⱼ
𝑗≠𝑖
𝑁
𝑆 xᵢ − xⱼ
Continued…

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

16.  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,
TN(z)=
j=1
N
aᵢzᵢ

17. SL. NO. Rank Order Statistics Linear Rank Statistics
1. 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.