aarjaneiro/planetmath huggingface dataset:
Lehmer mean

Lehmer mean

Let

p

be a

real number

.

Lehmer mean

of the positive numbers

a

1

,

…

,

a

n

is defined as

L

p

⁢

(

a

1

,

…

,

a

n

)

:=

a

1

p

+

…

+

a

n

p

a

1

p

-

1

+

…

+

a

n

p

-

1

.

(1)

This

definition

fulfils both requirements set for means (

http://

planetmath

.org/Mean3

).  In the case of Lehmer mean of two positive numbers

a

and

b

we see for

a

≦

b

that

a

=

a

p

+

a

⁢

b

p

-

1

a

p

-

1

+

b

p

-

1

≦

a

p

+

b

p

a

p

-

1

+

b

p

-

1

≦

a

p

-

1

⁢

b

+

b

p

a

p

-

1

+

b

p

-

1

=

b

.

The Lehmer mean of certain numbers is the

greater

the greater is the

parametre

p

, i.e.

L

p

⁢

(

a

1

,

…

,

a

n

)

≧

L

q

⁢

(

a

1

,

…

,

a

n

)

∀

p

>

q

.

This turns out from the nonnegativeness of the

partial derivative

of

L

p

with respect to

p

; in the case

n

=

2

it writes

∂

⁡

L

p

∂

⁡

p

=

a

p

-

1

⁢

b

p

-

1

⁢

(

a

-

b

)

⁢

(

ln

⁡

a

-

ln

⁡

b

)

(

a

p

-

1

+

b

p

-

1

)

2

≧

0

.

Thus in the below list containing special cases of Lehmer mean, the is the least and the contraharmonic the greatest (cf. the

comparison of Pythagorean means

).

E.g. for two

arguments

a

and

b

, one has

•

L

0

⁢

(

a

,

b

)

=

2

⁢

a

⁢

b

a

+

b

,

harmonic mean

,

•

L

1

/

2

⁢

(

a

,

b

)

=

a

⁢

b

,

geometric mean

,

•

L

1

⁢

(

a

,

b

)

=

a

+

b

2

,

arithmetic mean

,

•

L

2

⁢

(

a

,

b

)

=

a

2

+

b

2

a

+

b

,

contraharmonic mean

.

Note.

The least (

http://planetmath.org/LeastNumber

) and the greatest of the numbers (

http://planetmath.org/GreatestNumber

)

a

1

,

…

,

a

n

may be regarded as borderline cases of the Lehmer mean, since

lim

p

→

-

∞

⁡

L

p

⁢

(

a

1

,

…

,

a

n

)

=

min

⁡

{

a

1

,

…

,

a

n

}

,

lim

p

→

+

∞

⁡

L

p

⁢

(

a

1

,

…

,

a

n

)

=

max

⁡

{

a

1

,

…

,

a

n

}

.

For proving these equations, suppose that there are exactly

k

greatest (resp. least) ones among the numbers and that those are

a

1

=

…

=

a

k

.  Then we can write

L

p

⁢

(

a

1

,

…

,

a

n

)

=

a

1

p

⁢

[

k

+

(

a

k

+

1

a

1

)

p

+

…

+

(

a

n

a

1

)

p

]

a

1

p

-

1

⁢

[

k

+

(

a

k

+

1

a

1

)

p

-

1

+

…

+

(

a

n

a

1

)

p

-

1

]

.

Letting

p

→

+

∞

(resp.

p

→

-

∞

),  this equation yields

L

p

⁢

(

a

1

,

…

,

a

n

)

⟶

a

1

.

Title

Lehmer mean

Canonical

name

LehmerMean

Date of creation

2013-03-22 19:02:06

Last modified on

2013-03-22 19:02:06

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Definition

Classification

msc 62-07

Classification

msc 11-00

Related topic

OrderOfSixMeans

Related topic

LeastAndGreatestNumber

Related topic

MinimalAndMaximalNumber

Generated on Sat Feb 10 12:02:41 2018 by

LaTeXML
Occam’s razor

Occam’s razor

The following example in

statistics

illustrates the principle of

Occam’s razor

. See

http://en.wikipedia.org/wiki/Occam's_Razor

Article on Occam’s Razor on Wikipedia for a thorough discussion, mathematical or not, of this principle, which is also known as the

principle of parsimony

.

Example in statistics:

Given the following ten pairs of hypothetical

observations

of

continuous

response variable

Y

and continuous explanatory variable

X

:

Title

Occam’s razor

Canonical

name

OccamsRazor

Date of creation

2013-03-22 14:44:14

Last modified on

2013-03-22 14:44:14

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62-07

Classification

msc 00A20

Classification

msc 62A01

Synonym

law of parsimony

\@unrecurse

Generated on Sat Feb 10 12:02:43 2018 by

LaTeXML
percentile

percentile

Given a

distribution function

F

X

of a

random variable

X

, on a

probability space

(

Ω

,

B

,

P

)

a

p

𝑡ℎ

-

percentile

of

F

X

for a given real number

p

, is a real number

r

such that

1.

P

(

X

≤

r

)

≥

p

100

,

2.

P

(

X

≥

r

)

≥

1

-

p

100

.

Remarks

.

•

The most common percentiles of a distribution function are the median (

http://planetmath.org/MedianOfADistribution

) (the

50

th

-percentile or the

second quartile

), the

lower

quartile

(the

25

th

-percentile or the first quartile), and the

upper quartile

(the

75

th

-percentile or the third quartile).

•

In practice, the quartiles are calculated as follows: calculate the median

m

first, then the median of the data points below

m

is the first quartile, and the median of the data points above

m

is the third quartile.

•

The interval between the first quartile and the third quartile is called the

interquartile range

, or

IQR

for short. Sometimes, the

difference

between the first and third quartiles is also called the IQR. Like

standard deviation

, IQR is a measure of dispersion. However, IQR is a more robust statistic.

Title

percentile

Canonical

name

Percentile

Date of creation

2013-03-22 16:17:13

Last modified on

2013-03-22 16:17:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62-07

Synonym

first quartile

Synonym

third quartile

Synonym

IQR

Defines

quartile

Defines

upper quartile

Defines

lower quartile

Defines

interquartile range

Generated on Sat Feb 10 12:02:52 2018 by

LaTeXML
estimator

estimator

Let

X

1

,

X

2

,

…

,

X

n

be samples (with

observations

X

i

=

x

i

) from a

distribution

with

probability density function

f

(

X

∣

θ

)

, where

θ

is a real-valued unknown

parameter

(

http://planetmath.org/StatisticalModel

) in

f

. Consider

θ

as a

random variable

and let

τ

⁢

(

θ

)

be its realization.

An

estimator

for

θ

is a

statistic

η

θ

=

η

θ

⁢

(

X

1

,

X

2

,

…

,

X

n

)

that is used to, loosely speaking, estimate

τ

⁢

(

θ

)

. Any value

η

θ

(

X

1

=

x

1

,

X

2

=

x

2

,

…

,

X

n

=

x

n

)

of

η

θ

is called an

estimate

of

τ

⁢

(

θ

)

.

Example

.
Let

X

1

,

X

2

,

…

,

X

n

be iid from a

normal distribution

N

⁢

(

μ

,

σ

2

)

. Here the two parameters are the mean

μ

and the

variance

σ

2

. The

sample mean

X

¯

is an estimator of

μ

, while the

sample variance

s

2

is an estimator of

σ

2

. In

addition

, sample median, sample mode, sample

trimmed mean

are all estimators of

μ

. The statistic defined by

1

n

-

1

⁢

∑

i

=

1

n

(

X

i

-

m

)

2

,

where

m

is a sample median, is another estimator of

σ

2

.

Title

estimator

Canonical

name

Estimator

Date of creation

2013-03-22 14:52:22

Last modified on

2013-03-22 14:52:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62A01

Defines

estimate

Generated on Sat Feb 10 12:03:00 2018 by

LaTeXML
F distribution

F distribution

Let

X

and

Y

be

random variables

such that

1.

X

and

Y

are

independent

2.

X

∼

χ

2

⁢

(

m

)

, the

chi-squared distribution

(

http://planetmath.org/ChiSquaredRandomVariable

) with

m

degrees of freedom

3.

Y

∼

χ

2

⁢

(

n

)

, the chi-squared distribution with

n

degrees of freedom

Define a new random variable

Z

by

Z

=

(

X

/

m

)

(

Y

/

n

)

.

Then the

distribution

of

Z

is called the

central F distribution

, or simply the

F distribution with m and n degrees of freedom

, denoted by

Z

∼

F

⁡

(

m

,

n

)

.

By transformation of the random variables

X

and

Y

, one can show that the

probability density function

of the F distribution of

Z

has the form:

f

Z

⁢

(

x

)

=

m

m

/

2

⁢

n

n

/

2

B

⁡

(

m

2

,

n

2

)

⋅

x

(

m

/

2

)

-

1

(

m

⁢

x

+

n

)

(

m

+

n

)

/

2

,

for

x

>

0

, where

B

⁡

(

α

,

β

)

is the

beta function

.

f

Z

⁢

(

x

)

=

0

for

x

≤

0

.

For a fixed

m

, say 10, below are some graphs for the probability density functions of the F distribution with

(

m

,

n

)

degrees of freedom.

The next set of graphs shows the

density functions

with

(

m

,

n

)

degrees of freedom when

n

is fixed. In this example,

n

=

10

.

If

X

∼

χ

2

⁢

(

m

,

λ

)

, the non-central chi-square distribution with m degrees of freedom and

non-centrality parameter

λ

, with

Y

and

Z

defined as above, then the distribution of

Z

is called the

non-central
F distribution with m and n degrees of freedom and non-centrality parameter

λ

.

Remarks

•

the “F” in the F distribution is given in honor of statistician R. A. Fisher.

•

If

X

∼

F

⁡

(

m

,

n

)

, then

1

/

X

∼

F

⁡

(

n

,

m

)

.

•

If

X

∼

t

⁡

(

n

)

, the t distribution with

n

degrees of freedom, then

X

2

∼

F

⁡

(

1

,

n

)

.

•

If

X

∼

F

⁡

(

m

,

n

)

, then

E

⁡

[

X

]

=

n

n

-

2

⁢

if

⁢

n

>

2

,

and

Var

⁡

[

X

]

=

2

⁢

n

2

⁢

(

m

+

n

-

2

)

m

⁢

(

n

-

2

)

2

⁢

(

n

-

4

)

⁢

if

⁢

n

>

4

.

•

Suppose

X

1

,

…

,

X

m

are random samples from a

normal distribution

with mean

μ

1

and

variance

σ

1

2

. Furthermore, suppose

Y

1

,

…

,

Y

n

are random samples from another normal distribution with mean

μ

2

and variance

σ

2

2

. Then the

statistic

defined by

V

=

σ

1

^

2

σ

2

^

2

,

where

σ

1

^

2

and

σ

1

^

2

are

sample variances

of the

X

i

′

⁢

s

and the

Y

j

′

⁢

s

, respectively, has an F distribution with m and n degrees of freedom.

V

can be used to test whether

σ

1

2

=

σ

2

2

.

V

is an example of an F test.

Title

F distribution

Canonical name

FDistribution

Date of creation

2013-03-22 14:26:56

Last modified on

2013-03-22 14:26:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62A01

Synonym

Fisher F distribution

Synonym

F-distribution

Synonym

central F-distribution

Synonym

central F distribution

Defines

non-central F distribution

Generated on Sat Feb 10 12:03:03 2018 by

LaTeXML
score function

score function

Given a

statistical model

{

f

𝐗

(

𝒙

∣

𝜽

)

:

𝜽

∈

Θ

}

with

log-likelihood function

ℓ

(

𝜽

∣

𝒙

)

, the

score function

U

is defined to be the

gradient

of

ℓ

:

U

⁢

(

𝜽

)

=

∇

⁡

ℓ

=

∂

⁡

ℓ

∂

⁡

𝜽

.

Since the score function

U

is also a function of the

random vector

𝒙

,

U

is itself a random vector.
By setting

U

to

0

, we have a system of

k

equation

(s), otherwise known as the

likelihood equation(s)

:

U

⁢

(

𝜽

)

=

(

∂

⁡

ℓ

∂

⁡

θ

1

,

…

,

∂

⁡

ℓ

∂

⁡

θ

k

)

=

(

0

,

…

,

0

)

.

If

𝜽

=

θ

is one-dimensional, then the score function is simply referred to as the

score

of

θ

.

The maximum likelihood estimate (MLE)

𝜽

^

of the parameter vector

𝜽

can usually be found by finding the solutions of the likelihood equations. The likelihood equations may also be formed by setting the gradient of the plain

likelihood function

to zero. The use of the log function often facilitates the

algebra

as many

distributions

are

exponential

in nature. For some distributions it may also be necessary to test that the solution to the likelihood equations is really a

minimum

as opposed to a

point of inflection

.

Example

.

n

independent

observations

are made from a

random variable

X

with a

Poisson distribution

with parameter

λ

. The observed values are

x

1

,

…

,

x

n

. The log-likelihood of the joint pdf is

ℓ

(

λ

∣

𝒙

)

=

∑

i

=

1

n

-

λ

+

x

i

ln

(

λ

)

-

ln

(

x

i

!

)

and so the score function is

U

⁢

(

λ

)

=

d

⁢

ℓ

d

⁢

λ

=

∑

i

=

1

n

(

-

1

+

x

i

λ

)

=

-

n

+

n

⁢

x

¯

λ

,

where

n

⁢

x

¯

=

∑

x

i

. To find the MLE of

λ

, we set

U

=

0

and solve for

λ

. So the MLE

λ

^

of

λ

=

x

¯

.

Title

score function

Canonical name

ScoreFunction

Date of creation

2013-03-22 14:28:02

Last modified on

2013-03-22 14:28:02

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62A01

Synonym

score

Synonym

score statistic

Defines

likelihood equation

Generated on Sat Feb 10 12:03:10 2018 by

LaTeXML
statistical model

statistical model

Let

𝐗

=

(

X

1

,

…

,

X

n

)

be a

random vector

with a given realization

𝐗

⁢

(

ω

)

=

(

x

1

,

…

,

x

n

)

, where

ω

is the outcome (of an

observation

or an experiment) in the

sample space

Ω

. A

statistical model

𝒫

based on

X

is a set of

probability distribution functions

of

X

:

𝒫

=

{

F

𝐗

}

.

If it is known in advance that this family of

distributions

comes from a set of

continuous distributions

, the statistical model

𝒫

can be equivalently defined as a set of

probability density functions

:

𝒫

=

{

f

𝐗

}

.

As an example, a coin is tossed

n

times and the results are observed. The

probability

of landing a head during one toss is

p

. Assume that each toss is

independent

of one another. If

𝐗

=

(

X

1

,

…

,

X

n

)

is defined to be the vector of the

n

ordered outcomes, then a statistical model based on

X

can be a family of

Bernoulli distributions

𝒫

=

{

∏

i

=

1

n

p

x

i

⁢

(

1

-

p

)

1

-

x

i

}

,

where

X

i

⁢

(

ω

)

=

x

i

and

x

i

=

1

if

ω

is the outcome that the

i

th toss lands a head and

x

i

=

0

if

ω

is the outcome that the

i

th toss lands a tail.

Next, suppose

X

is the number of tosses where a head is observed, then a statistical model based on

X

can be a family

binomial distributions

:

𝒫

=

{

(

n

x

)

⁢

p

x

⁢

(

1

-

p

)

n

-

x

}

,

where

X

⁢

(

ω

)

=

x

, where

ω

is the outcome that

x

heads (out of

n

tosses) are observed.

A statistical model is usually

parameterized

by a

function

, called a

parameterization

Θ

→

𝒫

⁢

given by

⁢

θ

↦

F

θ

⁢

so that

⁢

𝒫

=

{

F

θ

∣

θ

∈

Θ

}

,

where

Θ

is called a

parameter space

.

Θ

is usually a subset of

ℝ

n

. However, it can also be a

function space

.

In the first part of the above example, the statistical model is parameterized by

p

↦

∏

i

=

1

n

p

x

i

⁢

(

1

-

p

)

1

-

x

i

.

If the parameterization is a one-to-one function, it is called an

identifiable parameterization

and

θ

is called a

parameter

. The

p

in the above example is a parameter.

Title

statistical model

Canonical

name

StatisticalModel

Date of creation

2013-03-22 14:33:18

Last modified on

2013-03-22 14:33:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62A01

Defines

identifiable parameterization

Defines

parameter space

Generated on Sat Feb 10 12:03:14 2018 by

LaTeXML
t distribution

t distribution

Let

X

and

Y

be

random variables

such that

1.

X

and

Y

are

independent

2.

X

∼

N

⁡

(

0

,

1

)

, the

standard normal distribution

(with mean

0

and

variance

1

)

3.

Y

∼

χ

2

⁢

(

n

)

, the chi-square

distribution

with n

degrees of freedom

Define a new random variable

Z

by

Z

=

X

/

(

Y

n

)

1

2

Then the distribution of

Z

is called the

t

distribution with

n

degrees of freedom

,
denoted by

Z

∼

t

⁡

(

n

)

.

By

transformation

of the random variables

X

and

Y

, one can show that the

probability density function

of the

t

distribution of

Z

has the form:

f

Z

⁢

(

x

)

=

1

n

⁢

B

⁡

(

n

2

,

1

2

)

⁢

(

1

+

x

2

n

)

-

n

+

1

2

,

where

B

⁡

(

α

,

β

)

is the

beta function

.

Below are graphs of some

t

probability density functions for various degrees (

d

) of freedom.

d

=

1

,

d

=

2

,

d

=

3

,

d

=

500

Remarks

•

the

t

distribution is also known as the Student’s

t

distribution. The name Student came from the 19th Century research chemist William Sealy Gossett, who was employed by the brewing company Guinness to improve the yield of crops used to produce its beer. Gossett conducted agricultural experiments and used

random numbers

to help determine the sampling distribution of the data he collected. Because the brewing company wanted to keep the research results confidential for competitve reasons, Gossett had to use a pen name to publish his findings. Student was his pen name and the distribution he found turned out to be the

t

distribution.

•

t

⁡

(

1

)

=

Cauchy

⁡

(

0

,

1

)

, the

Cauchy distribution

with

parameters

0

and

1

.

•

t

⁡

(

n

)

is asymptotically (as

n

→

∞

)

N

⁡

(

0

,

1

)

, the standard normal distribution
with mean

0

and variance

1

.

•

If

X

∼

t

⁡

(

n

)

,

E

⁢

[

|

X

|

k

]

exists iff

k

<

n

. Therefore, a

t

distribution has no mean if it has
one degree of freedom. For

n

>

1

,

E

⁡

[

X

]

=

0

. For

n

>

2

,

Var

⁡

[

X

]

=

n

n

-

2

.

•

If

X

1

,

…

,

X

n

are random samples from a

normal distribution

with mean

μ

and variance

σ

2

. Let

μ

^

be the

sample mean

and

σ

^

2

the

sample variance

, then

U

=

μ

^

-

μ

σ

^

/

n

∼

t

⁡

(

n

-

1

)

Please note that the

statistic

U

does not depend on

σ

2

, and thus is used often in testing hypotheses involving comparison of the sample mean to the true mean, given a set of random samples that are normally distributed with an unknown mean and an unknown variance. This is an example of a

t

test.

Title

t distribution

Canonical

name

TDistribution

Date of creation

2013-03-22 14:26:50

Last modified on

2013-03-22 14:26:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

34

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62A01

Synonym

Student’s

t

Synonym

t-distribution

Generated on Sat Feb 10 12:03:17 2018 by

LaTeXML
factorization criterion

factorization criterion

Let

𝑿

=

(

X

1

,

…

,

X

n

)

be a

random vector

whose

coordinates

are

observations

, and whose

probability

(

density

)
function is,

f

(

𝒙

∣

θ

)

where

θ

is an
unknown parameter. Then a

statistic

T

⁢

(

𝑿

)

for

θ

is a

sufficient statistic

iff

f

can be expressed as a
product of (or

factored into

) two functions

g

,

h

,

f

=

g

⁢

h

where

g

is a function of

T

⁢

(

𝑿

)

and

θ

, and

h

is a function of

𝒙

. In symbol, we have

f

(

𝒙

∣

θ

)

=

g

(

T

(

𝑿

)

,

θ

)

h

(

𝒙

)

.

Applications

.

1.

In view of the above statement, let’s show that the

sample
mean

X

¯

of

n

independent

observations from a

normal
distribution

N

⁢

(

μ

,

σ

2

)

is a sufficient statistic for the
unknown mean

μ

. Since the

X

i

’s are independent

random
variables

, then the

probability density function

f

(

𝒙

∣

μ

)

, being the joint probability density
function of each of the

X

i

, is the product of the individual
density functions

f

(

x

∣

μ

)

:

f

(

𝒙

∣

μ

)

=

∏

i

=

1

n

f

(

x

∣

μ

)

=

∏

i

=

1

n

1

2

⁢

π

⁢

σ

2

exp

[

-

(

x

i

-

μ

)

2

2

⁢

σ

2

]

(1)

=

1

(

2

⁢

π

)

n

⁢

σ

2

⁢

n

⁢

exp

⁡

[

∑

i

=

1

n

-

(

x

i

-

μ

)

2

2

⁢

σ

2

]

(2)

=

1

(

2

⁢

π

)

n

⁢

σ

2

⁢

n

⁢

exp

⁡

[

-

1

2

⁢

σ

2

⁢

∑

i

=

1

n

x

i

2

]

⁢

exp

⁡

[

μ

σ

2

⁢

∑

i

=

1

n

x

i

-

n

⁢

μ

2

2

⁢

σ

2

]

(3)

=

h

⁢

(

𝒙

)

⁢

exp

⁡

[

n

⁢

μ

σ

2

⁢

T

⁢

(

𝒙

)

-

n

⁢

μ

2

2

⁢

σ

2

]

(4)

=

h

⁢

(

𝒙

)

⁢

g

⁢

(

T

⁢

(

𝒙

)

,

μ

)

(5)

where

g

is the last exponential expression and

h

is the rest of
the expression in

(

3

)

. By the factorization criterion,

T

⁢

(

𝑿

)

=

X

¯

is a sufficient statistic.

2.

Similarly, the above shows that the

sample variance

s

2

is
not a sufficient statistic for

σ

2

if

μ

is unknown.

3.

But, if

μ

is a known constant, then the statistic

T

⁢

(

X

1

,

…

,

X

n

)

=

1

n

-

1

⁢

∑

i

=

1

n

(

X

i

-

μ

)

2

is sufficient for

σ

2

by observing in

(

2

)

above, and
letting

h

⁢

(

𝒙

)

=

1

and

g

⁢

(

T

,

σ

2

)

be all of
expression

(

2

)

.

Title

factorization criterion

Canonical name

FactorizationCriterion

Date of creation

2013-03-22 15:02:48

Last modified on

2013-03-22 15:02:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 62B05

Synonym

factorization theorem

Synonym

Fisher-Neyman factorization theorem

Generated on Sat Feb 10 12:03:20 2018 by

LaTeXML
cluster sampling

cluster sampling

Cluster sampling

is a method for sampling from a

population

P

, in which the smallest units in

P

are first grouped into bigger units, called

clusters

, or

primary sampling units

, then a sampling procedure is performed based on these clusters. The units within each cluster are called the

secondary sampling units

. For example, when an advertisements are sent out to a sample of
potential customers from a population, it is advisable to first group these potential customers into households, before any sample is drawn, so as to avoid any household receiving more than one ad. There are in general two types of cluster sampling:

1.

single-stage cluster sampling

or

one-stage cluster sampling

: a sample is taken from the primary sampling units, in which all of the secondary sampling units are considered.

2.

second-stage cluster sampling

,

two-stage cluster sampling

, or emphsubsampling: a sample is taken from the primary sampling units; then within each primary sampling unit, a sample is taken from their secondary sampling units.

Title

cluster sampling

Canonical

name

ClusterSampling

Date of creation

2013-03-22 15:26:17

Last modified on

2013-03-22 15:26:17

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62D05

Synonym

one-stage cluster sampling

Synonym

two-stage cluster sampling

Defines

single-stage cluster sampling

Defines

second-stage cluster sampling

Defines

subsampling

Generated on Sat Feb 10 12:03:25 2018 by

LaTeXML
simple random sample

simple random sample

A sample

S

of size

n

from a

population

U

of size

N

is called
a

simple random sample

if

1.

it is a sample without replacement, and

2.

the

probability

of picking this sample is equal to the
probability of picking any other sample of size

n

from the same
population

U

.

From the first part of the definition, there are

(

N

n

)

samples of

n

items from a population of

N

items. From the
second part of the definition, the probability of any sample of size

n

in

U

is a constant. Therefore, the probability of picking a
particular simple random sample of size

n

from a population of
size

N

is

(

N

n

)

-

1

.

Remarks

Suppose

x

1

,

x

2

,

…

,

x

n

are values
representing the items sampled in a simple random sample of size

n

.

•

The

sample mean

x

¯

=

1

n

⁢

∑

i

=

1

n

x

i

is an

unbiased estimator

of the true

population mean

μ

.

•

The

sample variance

s

2

=

1

n

-

1

⁢

∑

i

=

1

n

(

x

i

-

x

¯

)

2

is an unbiased estimator of

S

2

, where

(

N

-

1

N

)

⁢

S

2

=

σ

2

is the true

variance

of
the population given by

σ

2

:=

1

N

⁢

∑

i

=

1

N

(

x

i

-

x

¯

)

2

.

•

The variance of the sample mean

x

¯

from the true
mean

μ

is

(

N

-

n

n

⁢

N

)

⁢

S

2

.

The larger the sample size,
the smaller the deviation from the true population mean. When

n

=

1

, the variance is the same as the true

population variance

.

Title

simple random sample

Canonical name

SimpleRandomSample

Date of creation

2013-03-22 15:13:01

Last modified on

2013-03-22 15:13:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62D05

Generated on Sat Feb 10 12:03:27 2018 by

LaTeXML