• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

Renáta Géczi-Papp, Roland Szilágyi

Estimation

2nd – 3rd

seminar

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

Estimation

Classical Bayesian

Least Squares Maximum Robostness

Method Likelihood

1. Point estimators

2. Interval estimators

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

Basic Terms I

• Point Estimate: the value of estimator; a single number that is

used to estimate an unknown parameter

• Confidence Level: specific percentage π

• Confidence Interval (CI): an interval estimate is a range of

values used to estimate a population parameter

orΘ → ± ΔΘ

π P ul

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

Basic Terms IIΘ → ± ΔΘ

• Maximum Error: Δ =

• Standard Error: standard deviation of the estimators

• zπ : when test statistics are approximately normally distributed for

large samples; n 100

• tπ : Student's t-distribution is a probability distribution that arises

in the problem of estimating the mean of a normally distributed

population when the sample size is small;

n < 100

error standard or t z π

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

Basic Terms III

• Degrees of freedom (df)– The number of values in the final calculation of a statistic that

are free to vary.

– The number of independent pieces of information that go intothe estimate of a parameter.

– In general, the degrees of freedom of an estimate is equal tothe number of independent scores that go into the estimate (n)minus the number of parameters estimated as intermediatesteps in the estimation of the parameter itself.

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

Notations

Population Sample

elements X1, X2, … , XN, … x1, x2, … , xn

mean μ

standard deviation σ s

proportion P p

x

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

A.) μ – mean or expected value of the population

1.) normal population, σ known

2.) normal population, σ unknown

σ z x μ x π

s t x μ xπ

N

n-1

nσx

N

n-1

nsx

s

%10N

n If

Zπ in excel: =NORM.S.INV(probability) 𝐏𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲:(𝛑 + 𝟏)

𝟐

tπ in excel: =T.INV(probability;deg_freedom) 𝐏𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲:(𝛑 + 𝟏)

𝟐

Degree of freedom: n-1

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

B) P – the proportion of the population

= population proportion is equal to the number of elementsin the population belonging to the category of interest,divided by the total number of elements in the population

In case of large sample, when n ≥ 100!

s t p p π

tπ in excel: =T.INV(probability;deg_freedom)

𝐏𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲:(𝛑 + 𝟏)

𝟐Degree of freedom: n-1

𝐬𝐩 =(𝐩 ∗ 𝐪)

𝐧 − 𝟏𝐪 = 𝟏 − 𝐩

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

C) σ – the standard deviation of the population

Only in the case when the population distribution is normal!

π

χ

s 1-n σ

χ

s 1-n P

22α

22

22α-1

2

𝛂 = 𝟏 − 𝛑

𝝌𝟐in excel: =CHISQ.INV(probability;deg_freedom)

Degree of freedom: n-1 𝐏𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲: 𝟏 −𝜶

𝟐𝒐𝒓

𝜶

𝟐

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

Descriptive statistics in excel

Turn on the add-in:

File/Options/Add-ins/Manage excel Add-ins– Go/Analysis Toolpack

Use it:

Data/Data analysis/Descriptivestatistics

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

Determination of sample size

Same confidence level:

New confidence level:

𝒏𝒏𝒆𝒘 = 𝒏𝒐𝒍𝒅 ∗𝟏

𝒑𝒓𝒐𝒑.𝒐𝒇 𝒓𝒆𝒎𝒂𝒊𝒏𝒊𝒏𝒈𝒎𝒂𝒙 𝒆𝒓𝒓𝒐𝒓

𝟐

𝒏𝒏𝒆𝒘 = 𝒏𝒐𝒍𝒅 ∗𝒎𝒂𝒙.𝒆𝒓𝒓𝒐𝒓𝒐𝒍𝒅𝒎𝒂𝒙.𝒆𝒓𝒓𝒐𝒓𝒏𝒆𝒘

𝟐

• Faculty of Economics• Gazdaságelméleti és Módszertani Intézet

Thanks for your attentionstgpren@uni-miskolc.hu