Skip to main content

Statistical Inference I ( Theory of estimation : Efficiency)

🔖Statistical Inference I ( Theory of estimation : Efficiency) 

In this article we see the  terms: 

I. Efficiency.

II. Mean Square Error.

III. Consistency.


📚Efficiency: 

We know that  two unbiased estimator of parameter gives rise to infinitely many unbiased estimators of parameter. there if one of parameter have two estimators then the problem is to choose one of the best estimator among the class of unbiased estimators. in that case we need to some other criteria to to find out best estimator. therefore, that situation  we check the variability of that estimator, the measure of variability of estimator T around it mean is Var(T). hence If T is an Unbiased estimator of parameter then it's variance gives good precision. the variance is smaller then it give's greater precision.


📑 i. Efficient estimator: An estimator T is said to be an Efficient Estimator of 𝚹, if T is unbiased estimator of  𝛉. and it's variance is less than any other estimator. i.e. Var(T) < Var(T*).

where T* is any other estimator of  𝛉.

🔖ii. Relative Efficiency: If T1 and T2 are two unbiased estimators of parameter  𝛉 and E(T₁²) < ∞ and  E(T₂²) < ∞ then relative efficiency of T1  with respect to  T2 is denoted by Efficiency( T1 ,T2 ).

and T1  )/ V(T2 )

Remark: i. if relative efficiency =1 it means that  T1 and T2 are equal efficient estimators.

ii. If Relative Efficiency ( T1 and T2 ) < 1, it means that V( T1 ) <  V( T2 ). 

i.e. T1 is more efficient than T2 .

iii. If Relative Efficiency ( T1  and T2 ) > 1, it means that V( T1 ) >  V( T2 ). 

i.e. T2 is more efficient than  T1 

Example: 1. if X1 and X2 are two independent observation from normal distribution with mean 𝚹 and variance  𝛔² , then show thatT1 = ( X1 + X2)/2 , T2 = ( X1 + 2X2 )/3 are unbiased estimators of 𝚹 and also find the relative Efficiency (  T1 ,T2 ).

Solution: if  X1  and X2 are two independent observation from normal distribution with mean 𝚹 and variance  𝛔² 

E( X1 ) = E( X2 ) = 𝚹.

we have T1 = (X1 + X2)/2 , T2 = (X1 + 2X2 )/3

E(T1 ) = E[(X1 + X2 )/2] , E(T2 )= E [(X1 + 2X2 )/3].  we have E(X1 ) = E(X2) = 𝚹.

E(T1 ) = E[(𝚹+𝚹)/2] , E(T2 )= E [(𝚹 + 2𝚹 )/3

E(T1 ) = 2𝚹/2 , E(T2 )=  3𝚹 /3

E(T1 ) = 𝚹 , E(T2 ) =  𝚹 

Thus  T1 &T2 are unbiased estimators of parameter 𝚹.

now we obtain Variation of  T1 &T2 

V(T1 ) =V[ (X1 + X2 )/2] , V(T2)= (X1 + 2X2 )/3

V(T1 ) =[V(X1 )+V( X2 )] /4 , V(T2 )=[ V(X1 )+ 4V(X2 )] /3

V(T1 ) =[V(X1 )+V( X2 )] /4 , V(T2)=[ V(X1 )+ 4V(X2 )] /9

V(T1 ) =[𝛔² +𝛔² ] /4 , V(T2)=[ 𝛔² + 4𝛔² ] /9

V(T1 ) =2𝛔² /4 , V(T2)=5𝛔² /9

V(T1 ) =𝛔² /2 , V(T2)=5𝛔² /9

relative Efficiency  T1 relative to T2 is 

E (  T1 ,T2) =  V(  T1 )/V(T2)

                    = (𝛔² /2 ) /(5𝛔² /9)

                    = 9/10

                    = 0.9 < 1

therefore  E (  T1 ,T2) < 1 , means V( T1 ) =𝛔² /2  < V(T2)=5𝛔² /9 

Hence T1 is more efficient than T2 

 

Example 2. if X1, X2, ........Xn. random sample form normal distribution with mean 𝚹 and variance  𝛔² ,    T1 = sample mean i.e. x̄, and T2= sample median. find the  Efficiency of  T1 relative to T2 .   given that variance of median is  = V( T2)= (𝝅 𝛔² )/2n.    (for large n)

solution:  X1, X2, ........Xn. random sample form normal distribution with mean 𝚹 and variance  𝛔² then we have V(x̄) = V( T1 ) = 𝛔² /n. and  V( T2 )= (𝝅 𝛔² )/2n.

therefore the  Efficiency of T1 relative to T2  = E (  T1 ,T2 ) =  V( T1 )/V(T2 )

E ( T1 ,T2 ) =  V( T1 )/V(T2) 

                       =[ 𝛔² /n] / [ (𝝅 𝛔² )/2n]

                        = [𝛔² /n] x[2n/((𝝅 𝛔² )]

                        = (2/𝝅 )

                        =  0.6369 < 1 

                  therefore  E (   T1 ,T2 ) < 1 , means V( T1  ) =𝛔² /n <  V( T2  )= 𝝅𝛔² /2n

Hence  T1  is more efficient than  T2  

i.e. sample mean is more efficient than sample median.


III. Mean Square Error (M. S. E.)

let   T1  &  T2   be the  two estimators of parameter 𝚹, where  T1   is a unbiased estimator of 𝚹 &   T2  is a biased estimator of 𝚹. suppose the possible value of    T1  is spread   around  𝚹 and the value of  T2  is near to the parameter.  therefore in this case  T2   may be preferred than that of  T1  .

that case we need to study the variability of estimator around the parameter. for this we have to find out the variance of the estimator, which measure the variability of estimator around it's mean. if  T1  is an unbiased estimator of it's variance gives a good measure of precision. but if   T2   is biased estimator of parameter then variability of  T around 𝚹. as measure of it's precision and it is called as Mean Square Error (M. S. E.).

 Mean Square Error : an estimator T is a said to be a good estimator of 𝚹 if it's mean square error is minimum.

 i.e. E(T-𝚹)² ≤ E(T*-𝚹)²

where 𝚹. = T and T* is any other estimator .

this criteria known as mean square error.


Result: show that M. S. E. of T = MSE(T) = Var(T) + b² (T,𝚹) where b(.) is biased estimator.

Proof: 

 by definition of M. S. E. (T) = E(T-𝚹.)²

                                                = E[T-E(T)+E(T)-𝚹]²

   = E{T-E(T)}²+E{E(T)-𝚹}² + E{[T-E(T)]x[E(T)-𝚹]}

                                          = E{T-E(T)}²+E{E(T)-𝚹}² 

                                                 = Var(T) + b² (T,𝚹)

{Var(T)  = E{T-E(T)}²,   b² (T,𝚹) =E{E(T)-𝚹}² }

therefore  M. S. E. of T = MSE(T) = Var(T) + b² (T,𝚹).


Definition: Mean Square Error (M. S. E.) :

The M.S.E.  of estimator T of parameter 𝚹  is defined as 

    M. S. E. of T = MSE(T) = Var(T) + b² (T,𝚹)     if  T is biased estimator of  𝚹.

    M. S. E. of T = MSE(T) = Var(T)            

          if an T unbiased estimator of 𝚹.

this is the definition of M.S.E.


Example: 1. if   T1 & T2 be the two unbiased estimators of parameter 𝚹 with variance 𝞂1² and 𝞂2² respectively and having correlation ρ then find the best liner unbiased combination of T1 & T2  and also find the expression of variance of such combination.

Solution: we have two estimators  T1 & T2 are unbiased estimators of parameter 𝚹 with variance 𝞂1² and  𝞂2².

 E(T1 ) = 𝚹 , E(T2 ) =  𝚹 

V(T1 ) =  𝞂1² , V(T2)=  𝞂2².

and correlation is ρ

consider a linear combination of estimators   T1 & T2 is T = ɑ T1 + (1-ɑ) T2

now taking expectation E(T)
E(T)  = E[ɑ T1 + (1-ɑ) T2]
           =ɑ E[T1 ]+ (1-ɑ)E[ T2]
            = ɑ 𝚹 + (1-ɑ) 𝚹 
                =  ɑ 𝚹 + 1𝚹-ɑ𝚹 
            = 𝚹
E(T) = 𝚹
therefore T is an unbiased estimator of 𝚹
now we find variance of T
Var(T) =  V[ɑ T1 + (1-ɑ) T2]
   =ɑ²V( T1 ) + (1-ɑ)² V(T2) +2ɑ(1-ɑ)COV( T1, T2 )
  V(T)= ɑ² 𝞂1²+ (1-ɑ)²  𝞂2² +2ρ 𝞂1𝞂2 ɑ(1-ɑ)
           we find the value of ɑ and 1-ɑ the variance of (T)is minimise.
for that we find [dV(T) / da] = 0 and  [d²V(T) /da²] > 0.
now,  [dV(T) / da] = 0  i.e. differentiate w.r.t. a

 dV(T) / da  = {d/da} [  ɑ² 𝞂1²+ (1-ɑ)²  𝞂2² +2ɑ(1-ɑ)ρ 𝞂1 𝞂2]
=> 2ɑ 𝞂1² -2 (1-ɑ)  𝞂2² +2(1-2ɑ)ρ 𝞂1 𝞂2  = 0
              ɑ 𝞂1² - (1-ɑ)  𝞂2² + (1-2ɑ)ρ 𝞂1 𝞂2 = 0 
      ɑ 𝞂1² -  𝞂2² +ɑ 𝞂2² + ρ 𝞂1 𝞂2 -2ɑ ρ 𝞂1 𝞂2  = 0
          ɑ( 𝞂1²+𝞂2²-2  ρ 𝞂1 𝞂2 ) -  𝞂2² + ρ 𝞂1 𝞂2 = 0
                         ɑ( 𝞂1²+𝞂2²-2  ρ 𝞂1 𝞂2 ) =  𝞂2² - ρ 𝞂1 𝞂2 
    ɑ =  ( 𝞂2² - ρ 𝞂1 𝞂2 ) / ( 𝞂1²+𝞂2²-2ρ 𝞂1 𝞂2 )       
therefore    
 1-  ɑ= ( 𝞂1² - ρ 𝞂1 𝞂2 )/ ( 𝞂1²+𝞂2²-2 ρ 𝞂1 𝞂2 ) 
and 
d²V(T) /da² = 
{d/da} [ 2ɑ 𝞂1² -2 (1-ɑ)  𝞂2² +2(1-2ɑ)ρ 𝞂1 𝞂2 ]
                    = 2𝞂1²+2𝞂2²-4  ρ 𝞂1 𝞂2
                      =    2( 𝞂1²+𝞂2²-2 ρ 𝞂1 𝞂2 )
                    =2[V( T1 ) +V(T2) -2COV( T1, T2 )]
                    = 2V( T1 - T2 ) > 0
hence variance of (T) is minimum at the value of ɑ =  ( 𝞂2² - ρ 𝞂1 𝞂2 ) / ( 𝞂1²+𝞂2²-2ρ 𝞂1 𝞂2 )       
  1-  ɑ=( 𝞂1² - ρ 𝞂1 𝞂2 )/( 𝞂1²+𝞂2²-2 ρ 𝞂1 𝞂2 )  
for variance of T we put the values of  ɑ =  ( 𝞂2² - ρ 𝞂1 𝞂2 ) / ( 𝞂1²+𝞂2²-2ρ 𝞂1 𝞂2 )       
     1-  ɑ =  ( 𝞂1² - ρ 𝞂1 𝞂2 ) / ( 𝞂1²+𝞂2²-2 ρ 𝞂1 𝞂2 ).
now find the variance of T 
Var(T) =  V[ɑ T1 + (1-ɑ) T2]
   =ɑ²V( T1 ) + (1-ɑ)² V(T2) +2ɑ(1-ɑ)COV( T1, T2 )
 V(T)   = ɑ² 𝞂1²+ (1-ɑ)²  𝞂2² +2ɑ(1-ɑ) ρ 𝞂1 𝞂2.





Note that: Smaller M.S.E. i.e. variance is small means greater precision. hence while comparing two estimators T1 & T2 of  𝞗  we choose an estimator with smaller M.S.E. this used to modifies the formula of efficiency

Efficiency of T1 & T2 = M.S.E(T1 ) / M.S.E(T2 )

If efficiency (T1 , T2 ) < 1
then T1  is more efficient estimator than  T2 
i.e.  M.S.E(T1 )  <  M.S.E(T2 )

Locally Minimum Variance Unbiased Estimator (MVUE).
Definition: Let 𝞗0 ∈ 𝚯 and U(𝞗0) be the class of all unbiased estimates of θ such that     E(T²) <∞. AndT0∈ U(𝞗0) then T0 is called locally minimum variance unbiased estimator(LMVUE).
OR simply minimum variance unbiased estimator( SMVUE). IF E(T0-θ0)² ≤E(T -θ0)² holds for all T.

Uniformly Minimum Variance Unbiased Estimator (UMVUE)
Definition: Let U be the set of all unbiased estimates of T     of 𝞗 ∈ 𝚯, such that  E(T²) <∞. and estimate T0 ∈ U is called uniformly minimum variance unbiased estimator of 𝞗 if 
E(T0-θ)² ≤E(T -θ)² for all 𝞗 ∈ 𝚯 and every T Ɛ U.

Now we see the Definition of Minimum Variance Unbiased Estimator (MVUE)
Definition: if a statistics T=T(X1, X2, .....Xn) based on a Random sample of size n such that i) T is unbiased estimator for Ɣ(θ) for all 𝞗 ∈ 𝚯  ii) it has smallest variance among the class of all unbiased estimators of  Ɣ(θ), then T  is called  Minimum Variance Unbiased Estimator (MVUE)
OR
T is MVUE of  Ɣ(θ) if 
i)E(T) = Ɣ(θ)  for all 𝞗 ∈ 𝚯
ii) V(T) ≤ V(T*)
where T* is any other unbiased estimator of   Ɣ(θ).


Result: If UMVUE exist then it is unique.
Proof : Suppose that  T1 & T2 are two uniformly minimum variance unbiased estimator of parameter θ then 
we have     E(T1) =θ,   E(T2) =θ   for all θ ∈ 𝚯
and V(T1) =  V(T2)    for all θ ∈ 𝚯         ............1
now we define new estimator T =(T1 + T2 ) /2  which is unbiased estimator 
since E(T ) = E[(T1 + T2 ) /2]
                    =  (θ+θ)/2
                     =2θ /2
                    =θ
And variance of T 
Var(T) = Var[(T1 + T2 ) /2]
            = {V(T1 ) +V( T2 ) + 2 Cov( T1 , T2)}/4
            = {2V(T1 ) + 2 ⍴ √(V(T1 ) xV( T2 ))} / 4
            = { V(T1 ) +  ⍴ V(T1 ) } / 2
            = { V(T1 )  (1+ ⍴)} / 2          ...............2
where   ⍴  is the Karl Pearson coefficient of correlation between T1 & T2 .
Since T1 is minimum variance unbiased estimator of  θ.
then 
 V(T1 ) ≤ V( T ) 
i.e.   V(T1 ) ≤ { V(T1 )  (1+ ⍴)} / 2   
      V(T1 ) ≤ { V(T1 )  (1+ ⍴)} / 2   
 1≤  (1+ ⍴) / 2   
 2 ≤  (1+ ⍴)   
 1≤  ⍴ 
we have |⍴ | ≤  1 therefore we must have ⍴  = 1.
i.e.   T1 & T2 have linear relationship of the form 
 T1  = ɑ + 𝜷T2.........3
Where  ɑ and  𝜷 are constant independent on parameterθ
E( T1 ) = E(ɑ + 𝜷T2)
E( T1 ) = E(ɑ )+ E(𝜷T2)
θ   = ɑ + 𝜷θ. ........4
now variance 

 V(T1  )=V( ɑ + 𝜷T2)
   V(T1  )=V( 𝜷T2)  
  V(T1  )=𝜷²V(T2)
therefore   𝜷² = 1
𝜷=+-1
𝜷 =1 ................5
because ⍴  = 1. T1 & T2 are positive 
from equation 4 and 5 we get
θ   = ɑ + θ
 ɑ = 0
putting 𝜷 =1,ɑ = 0 in equation 3
 T1  = 0 + 1xT2
     T1  = T2 
Thus if UMVUE exist, it is unique.

III. Consistency:

 Consistency

 In statistical inference, estimators are used to approximate unknown population parameters based on sample data. For an estimator to be considered “good,” it should possess certain desirable properties such as unbiasedness, efficiency, and consistency.

While unbiasedness ensures that an estimator gives the correct value on average, and efficiency ensures that it has the smallest possible variance among unbiased estimators, these properties alone are not sufficient.
It is also essential that the estimator becomes more accurate as the sample size increases. This is where the property of consistency becomes important.

Consistency refers to the long-run reliability of an estimator. An estimator is said to be consistent if it converges to the true value of the population parameter as the sample size increases indefinitely.

Definition: Let X1, X2, …..Xn be a r. s. of size n from the distribution f(x), let θ be a unknown parameter then the sequence of estimator T=t(x1,x2,….xn) is said to be consistent estimator for θ for every ε>0 Tn converges to θ I probability for each θ. Or following holds

 P{ | Tn – θ|≤ε} –› 1 as n –›∞

Or

P{ | Tn – θ|>ε} –› 0 as n –›∞

Note:  Therefore the consistent estimator is close to θ with probability  one for large sample size. i.e.  n –i›∞

Here  Tn is said to be converges in probability to θ it is denoted as Tn –› θ

Ex. 1. If X1, X2, ……..Xn is a r. s. from the distribution having mean μ and finite variance σ2 then show that sample mean is consistent estimator of μ.

Answer: here given that E(X) = μ and var (X) = σ2

We known that sample mean = x̄ = ∑ (xi) / n

E(x̄) = μ and var (x̄) = σ2 /n

Now by chebychev’s inequality

P{ | x̄ – μ|≤ε} ≥ 1- [var (x̄) / ε2 ]

P{ | x̄ – μ|≤ε} ≥ 1- [σ2 /nε2 ]

By taking limit as n –›∞

lim(n –›∞ ) P{ | x̄ – μ|≤ε} ≥ lim(n –›∞ ) (1- [σ2 /nε2 ])

lim(n –›∞ )P{ | x̄ – μ|≤ε} ≥ 1- lim(n –›∞ ) [σ2 /nε2 ]

lim(n –›∞ ) P{ | x̄ – μ|≤ε} ≥1- 0

lim(n –›∞ ) P{ | x̄ – μ|≤ε} ≥ 1

we known probability of any event is not greater than 1

hence lim(n –›∞ ) P{ | x̄ – μ|≤ε}= 1

x̄ –› μ

therefore the sample mean is consistent estimator for μ

 

 Sufficient condition for consistency:

If T= t(X1, X2, …….Xn) is a sequence of estimator such that

i.     E(Tn) =θ as n –›∞

ii.   V(Tn ) =0 as n –›∞ 

then Tn is Consistent estimator for θ.

 

 

 

 

 

 


                


 
 






Labels:

Comments

Post a Comment

Popular posts from this blog

B. Com. -I Statistics Practical No. 1 Classification, tabulation and frequency distribution –I: Qualitative data.

  Shree GaneshA B. Com. Part – I: Semester – I OE–I    Semester – I (BASIC STATISTICS PRACTICAL-I) Practical: 60 Hrs. Marks: 50 (Credits: 02) Course Outcomes: After completion of this practical course, the student will be able to: i) apply sampling techniques in real life. ii) perform classification and tabulation of primary data. iii) represent the data by means of simple diagrams and graphs. iv) summarize data by computing measures of central tendency.   LIST OF PRACTICALS: 1. Classification, tabulation and frequency distribution –I: Qualitative data. 2. Classification, tabulation and frequency distribution –II : Quantitative data. 3. Diagrammatic representation of data by using Pie Diagram and Bar Diagrams. 4. Graphical representation of data by using Histogram, Frequency Polygon, Frequency Curve and     Locating Modal Value. 5. Graphical representation of data by using Ogive Curves and Locating Quartile Values....
Post a Comment
Read more

Index Number

 Index Number      Introduction  We seen in measures of central tendency the data can be reduced to a single figure by calculating an average and two series can be compared by their averages. But the data are homogeneous then the average is meaningful. (Data is homogeneous means data in same type). If the two series of the price of commodity for two years. It is clear that we cannot compare the cost of living for two years by using simple average of the price of the commodities. For that type of problem we need type of average is called Index number. Index number firstly defined or developed to study the effect of price change on the cost of living. But now days the theory of index number is extended to the field of wholesale price, industrial production, agricultural production etc. Index number is like barometers to measure the change in change in economics activities.   An index may be defined as a " specialized  average designed to measure the...
2 comments
Read more

Non- Parametric Test: Run Test

Non- Parametric Test  A Non-Parametric tests is a one of the part of Statistical tests that non-parametric test does not assume any particular distribution for analyzing the variable. unlike the parametric test are based on the assumption like normality or other specific distribution  of the variable. Non-parametric test is based on the rank, order, signs, or other non-numerical data. we know both test parametric and non-parametric, but when use particular test? answer is that if the assumption of parametric test are violated such as data is not normally distributed or sample size is small. then we use Non-parametric test they can used to analyse categorical data  or ordinal data and data are obtained form in field like psychology, sociology and biology. For the analysis use the  some non-parametric test that are Wilcoxon signed-ranked test, mann-whiteny U test, sign test, Run test, Kruskal-wallis test. but the non-parametric test have lower statistical power than ...
4 comments
Read more

Statistical Inference

 Statistical Inference: The Power of Statistical Inference in Data Analysis Statistical Inference: Drawing Meaningful Conclusions from Data In a data analysis, The statistical inference is a  a powerful tool for drawing meaningful conclusions from a sample of data  and making inferences about a larger population. It enables us to make confident predictions, understand relationships, and uncover valuable insights that can inform decision-making and shape various fields of study. At its core, statistical inference involves using statistical methods to analyse sample data and extend the findings to the broader population. This approach is necessary because it is often impractical or impossible to collect data from every individual or element of interest. Instead, we carefully select a representative sample and employ statistical techniques to infer information about the larger population. The first aspect of statistical inference is estimation. Estimation allows us to estima...
1 comment
Read more

B. Com. I OE - II Practical No. 3 Scatter Plot.

  Practical No. 3   Scatter Plot.  Correlation In previous blog we discussed about the measure central tendency and Dispersion to use to study the variable. the correlation is a statistical concept that allows us to measure and understand the relationship between two or more variables. it provided a valuable information about that variables. e.g. price and demand of commodity, income and expenditure of family, height and weight of group of persons. their we use the relation   between this two variables. in above examples we see the one variable increases other variable is also changes in same or opposite direction.  definition: Correlation is statistical tool which study the relationship between two or more variables. for analysis of correlation various method and techniques are used.  example: i. Demand and supply of product                               ...
Post a Comment
Read more

Median test

 Non- Parametric test Median test Median test is also a Non-Parametric test and it is alternative to Parametric T test. The median test is used when we are interested to check the two independent sample have same median or not. It is useful when data is discrete or continuous and if data is in small size.  Assumptions:  I) the variable under study is ordinal scale II) the variable is random and Independent. The stepwise procedure for computation of median test for two independent sample : Step I :- firstly we define the hypothesis Null Hypothesis is the two independent sample have same median.  Against Alternative Hypothesis is the two independent sample have different median.  Step II :- In this step we combine two sample data. And calculating the median of combined data. Step III :- after that for testing hypothesis we constructing the (2x2) contingency table. For that table we divide the sample into two parts as number of observation above and below to the ...
Post a Comment
Read more

Non-Parametric Test 1.Sign test

Types of Non-Parametric Test : the non-parametric test is Broadly divided into three categories as: 1. One-sample Test  2.Two-sample test 3. K-sample test In this blog we discuss the one-sample tests,  there are four types of one-sample non-parametric test as follows:  1. Run Test. 2.Sign Test 3. Wilcoxon Sing-Rank test 4. Kolmogorov-smirnov test. first we discuss the sign test. 1.Sign Test The sign test id one of the very simplest non-parametric test. the test is based on signs (i.e. plus and minus signs) hence is called sign test. the non-parametric test are alternative to the parametric one, therefore the sign test is alternative to parametric T-test, is used to test the median of population instead of mean of population. (i.e. Parametric T-test is used to testing mean of population and sign test is used to testing the median of the population they are alternative to each other when the data follows assumption of T-test we use parametric t-test, but i some situation th...
4 comments
Read more

Statistical Inference Practical: Point Estimation by Method of Moment

 
Post a Comment
Read more

Statistical Inference I ( Theory of Estimation) : Unbiased it's properties and examples

 📚Statistical Inference I Notes The theory of  estimation invented by Prof. R. A. Fisher in a series of fundamental papers in around 1930. Statistical inference is a process of drawing conclusions about a population based on the information gathered from a sample. It involves using statistical techniques to analyse data, estimate parameters, test hypotheses, and quantify uncertainty. In essence, it allows us to make inferences about a larger group (i.e. population) based on the characteristics observed in a smaller subset (i.e. sample) of that group. Notation of parameter: Let x be a random variable having distribution function F or f is a population distribution. the constant of  distribution function of F is known as Parameter. In general the parameter is denoted as any Greek Letters as θ.   now we see the some basic terms :  i. Population : in a statistics, The group of individual under study is called Population. the population is may be a group of obj...
Post a Comment
Read more

B. Com. I OE - II Practical No. 4 Karl Pearson's correlation coefficient

  Practical no. 4 Karl Pearson's correlation coefficient  Correlation In previous blog we discussed about the measure central tendency and Dispersion to use to study the variable. the correlation is a statistical concept that allows us to measure and understand the relationship between two or more variables. it provided a valuable information about that variables. e.g. price and demand of commodity, income and expenditure of family, height and weight of group of persons. their we use the relation   between this two variables. in above examples we see the one variable increases other variable is also changes in same or opposite direction.  definition: Correlation is statistical tool which study the relationship between two or more variables. for analysis of correlation various method and techniques are used.  example: i. Demand and supply of product                           ...
Post a Comment
Read more