Statistical terminology (Page 4/4)

Econometrics for honors students Page 4 / 4

Log normal distribution.

The continuous random variable x has log normal distribution if y has a normal distribution and

x = e^{y} .

Thus, if

y \sim N (μ, σ^{2}),

then the pdf of a log normal distribution is

f (x) = {\begin{cases} \frac{1}{x σ \sqrt{2 π}} e^{- \frac{{(\ln (x) - μ)}^{2}}{2 σ^{2}}},  for x > 0 \\ 0 otherwise \end{cases}} .

The mean and variance of x are

μ_{x} = e^{μ + \frac{σ^{2}}{2}}

and

σ_{x}^{2} = (e^{σ^{2}} - 1) e^{2 μ + σ^{2}} .

Because the distribution is skewed downward for variances over 1, the log normal distribution is sometimes used to describe income distributions (where there are relatively few very wealthy people and incomes generally are positive. Figure 4 shows the graphs of the pdf and cumulative functions for the log normal distributions for two values of σ .

Graph of the log-normal distribution for two values of σ. — The two panels illustrate the log-normal distribution for two values of σ ..

Gamma distribution.

A positive random variable x has a gamma distribution if its pdf is

f (x) = \frac{1}{Γ (α) β^{α}} x^{α - 1} e^{- \frac{x}{β}}

for

x > 0

and 0 elsewhere.

Γ (α)

is known as the gamma function and is defined to be

Γ (α) = \int_{0}^{\infty} y^{α - 1} e^{- y} d y = (α - 1)! .

The gamma function is often used to model waiting times like waiting for death. Its mean and variance are given by

μ = α β

and

σ^{2} = α β^{2} .

Chi-square distribution.

A chi-square distribution (

χ^{2} (k)

) is the sum of k independent standard normal random variables and is a special case of the gamma distribution (with

α = \frac{k}{2}

and

β = 2

). The pdf of a chi-square distribution with k degrees of freedom is

f (x) = \frac{1}{2^{\frac{k}{2}} Γ (\frac{k}{2})} x^{\frac{k}{2} - 1} e^{- \frac{x}{2}}

where x >0. Its mean and variance are

μ = k

and

σ^{2} = 2 k .

y = \sum_{i = 1}^{k} x_{i}^{2}

where the x _i 's are independently drawn from the standard normal distribution (N(1, 0)), then

y_{i} \sim χ^{2} (k) .

Student's t-distribution.

Consider two random variables, x and v . Assume that

x \sim N (0, 1)

and

v \sim χ^{2} (r)

and are stochastically independent. Then the random variable

t = \frac{w}{\sqrt{\frac{v}{r}}}

has the t-distribution with r degrees of freedom . The pdf and cumulative function of t are

f (t) = \frac{Γ (\frac{r + 1}{2})}{\sqrt{r π} Γ (\frac{r}{2})} {(1 + \frac{t^{2}}{r})}^{- (\frac{r + 1}{2})}

and

F (t) = \frac{1}{2} + t Γ (\frac{t}{2}) .

The mean and variance of the distribution are 0 for

r > 1

and

\frac{r}{r - 2}

for

t > 2,

respectively. The mean of the t-distribution is undefined for

t \leq 1.

The variance of the distribution is

\infty

for

1 < r \leq 2

and undefined for

r \leq 1.

The t-distribution plays a prominent role in hypothesis testing that is well-known to all undergraduate economics majors.

F distribution.

Consider two stochastically independent chi-square random variable such that

u \sim χ^{2} (r_{1})

and

v \sim χ^{2} (r_{2})

and

u, v > 0.

The new random variable

f = \frac{\frac{u}{r_{1}}}{\frac{v}{r_{2}}}

has a F-distribution with

r_{1}

and

r_{2}

degrees of freedom. The pdf for the F-distribution is

g (f) = \frac{Γ (\frac{r_{1} + r_{2}}{2}) (\frac{r_{1}}{r_{2}})}{Γ (\frac{r_{1}}{2}) Γ (\frac{r_{2}}{2})} \frac{f^{\frac{r_{1}}{2} - 1}}{{(1 + \frac{r_{1} f}{r_{2}})}^{\frac{r_{1} + r_{2}}{2}}} .

The F-distribution is used in testing if population variances are equal and in performing likelihood ratio tests.

Multinomial distribution.

Consider the n random variables

x_{1}, x_{2}, \dots, x_{n}

where each variable has a normal distribution—that is,

x_{i} \sim N (μ_{i}, σ_{i}^{2})

and the covariance between of the variables is

σ_{i j} = E [(x_{i} - μ_{i}) (x_{j} - μ_{j} .)]

We can arrange the variances and covariances into a n -by- n matrix where

Σ = [\begin{matrix} σ_{1}^{2} & σ_{12} & \dots & σ_{1 n} \\ σ_{21} & σ_{2}^{2} & \dots & σ_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ σ_{n 1} & σ_{n 2} & \dots & σ_{n}^{2} \end{matrix}]

that is known as the variance-covariance matrix. Define the vector

(x - μ) = (\begin{matrix} x_{1} - μ_{1} \\ ⋮ \\ x_{n} - μ_{n} \end{matrix})

and

{(x - μ)}^{'}

as its transpose. Then,

{(x - μ)}^{'} Σ (x - μ) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} (x_{i} - μ_{i}) (x_{j} - μ_{j}) σ_{i j},

where

σ_{i i} = σ_{i}^{2} .

| Σ |

is the determinant of the variance-covariance matrix, then the pdf for the joint distribution of these random variables is

f (x_{1}, x_{2}, \dots, x_{n}) = \frac{1}{{(2 π)}^{n / 2} {| Σ |}^{\frac{1}{2}}} e^{- \frac{1}{2} {(x - μ)}^{'} Σ (x - μ)} .

If the random variables are stochastically independent the covariances are equal to 0 and the pdf becomes

f (x_{1}, x_{2}, \dots, x_{n}) = \frac{1}{{(2 π)}^{n / 2} {(\prod_{i = 1}^{n} σ_{1}^{2})}^{\frac{1}{2}}} e^{- \frac{1}{2} \sum_{i = 1}^{n} \frac{{(x_{i} - μ_{i})}^{2}}{σ_{i}^{2}}} .

If the n random variables are all drawn from the same normal distribution with a mean of μ and a variance of

σ^{2},

then the pdf simplifies to

f (x_{1}, x_{2}, \dots, x_{n}) = \frac{1}{{(2 π σ^{2})}^{n / 2}} e^{- \frac{1}{2 σ^{2}} \sum_{i = 1}^{n} {(x_{i} - μ)}^{2}} .

Characteristics of an estimator of a population parameter θ

Finite estimators

Bias.

The bias of an estimator is defined to be

B (\hat{θ}) = E (\hat{θ}) - θ .

An estimator is unbiased if and only if

B (\hat{θ}) = 0.

Mean square error.

The mean square error (MSE) of an estimator is defined to be

M S E (\hat{θ}) = E [{(\hat{θ} - θ)}^{2}] .

It is relatively easy to show that

M S E (\hat{θ}) = V (\hat{θ}) + {(B (\hat{θ}))}^{2} .

Often a biased estimator with a smaller MSE may be preferred to an unbiased estimator with a relatively larger MSE.

Efficiency.

An estimator

\hat{θ}

is relatively more efficient than

\tilde{θ}

if and only if

V (\hat{θ}) < V (\tilde{θ}) .

Generally, we would prefer to use the most efficient estimator available (if it is unbiased).

Asymtoptic estimators

Plim.

x_{n}

converges to a constant, c , if

\lim_{n \to \infty} \Pr (| x_{n} - c | > ε) = 0

for any positive

ε .

We can write this relationship as

p \lim x_{n} = c .

Greene Greene, William H. (1990). Econometric Analysis (New York: Macmillan Publishing Company): 103. offers this example of plim: Suppose $x_{n}$ equals 0 with probability $1 - (\frac{1}{n})$ and n with probability $(\frac{1}{n}) .$ As n increases, the second point becomes more remote from the first point. However, at the same time the probability of observing the second point becomes more and more unlikely. This effect is shown in Figure 5 where as n increases the probability distribution concentrates more and more on 1.

This is an illustration of the concept of plim. — The probability x = 1 is the area of the gray box centered on 1 for n = 5; the gray area plus the blue area for n = 10; and the sum of the gray, blue, and red areas for n = 20; the probability x = n is the area of the box centered on n.

Consistency.

The estimator

\hat{θ}

is a consistent estimator of θ if and only if

p \lim \hat{θ} = θ .

Asymmtotically unbiased.

An estimator

\hat{θ}

is an asymtotically unbiased estimator of θ if

\lim_{n \to \infty} E [\hat{θ}] = θ .

Questions & Answers

how does Neisseria cause meningitis

Nyibol Reply

what is microbiologist

Muhammad Reply

what is errata

Muhammad

is the branch of biology that deals with the study of microorganisms.

Ntefuni Reply

What is microbiology

Mercy Reply

studies of microbes

Louisiaste

when we takee the specimen which lumbar,spin,

Ziyad Reply

How bacteria create energy to survive?

Muhamad Reply

Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments

_Adnan

But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?

Muhamad

they make spores

Louisiaste

what is sporadic nd endemic, epidemic

Aminu Reply

the significance of food webs for disease transmission

Abreham

food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.

Mark

explain assimilatory nitrate reduction

Esinniobiwa Reply

Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).

Elkana

This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products

Elkana

Examples of thermophilic organisms

Shu Reply

Give Examples of thermophilic organisms

Shu

advantages of normal Flora to the host

Micheal Reply

Prevent foreign microbes to the host

Abubakar

they provide healthier benefits to their hosts

ayesha

They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!

Mark

what is cell

faisal Reply

cell is the smallest unit of life

Fauziya

cell is the smallest unit of life

Akanni

Innocent

cell is the structural and functional unit of life

Hasan

is the fundamental units of Life

Musa

what are emergency diseases

Micheal Reply

There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️

_Adnan

define infection ,prevention and control

Innocent

I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life

Lubega

Heyy Lubega hussein where are u from?

_Adnan

en français

Adama

which site have a normal flora

ESTHER Reply

Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract

Safaa

skin

Asiina

skin,Oral,Nasal,GIt

Sadik

How can Commensal can Bacteria change into pathogen?

Sadik

How can Commensal Bacteria change into pathogen?

Sadik

all

Tesfaye

by fussion

Asiina

what are the advantages of normal Flora to the host

Micheal

what are the ways of control and prevention of nosocomial infection in the hospital

Micheal

what is inflammation

Shelly Reply

part of a tissue or an organ being wounded or bruised.

Wilfred

what term is used to name and classify microorganisms?

Micheal Reply

Binomial nomenclature

adeolu

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Econometrics for honors students' conversation and receive update notifications?

Ask

	SCJP Online Exam 310-065 By Prateek Ashtikar Start Quiz
©flickr: Tim	Hazard Quiz By Royalle Moore Start Quiz
	Negotiations Conflict Management BUS403 By Charles Jumper Start Quiz
	General Chemistry I MCQ By Joanna Smithback Start Quiz
©flickr: Bertram	Chemistry Ch 1 2 Test 1 By Madison Christian Start Quiz
	Fluid Mechanics MCQ By Stephanie Redfern Start Quiz
	8 Physiotherapy MMT Review By Rhodes Start Flashcards
	2 Lec:2 Randomized Controlled Trials By Janet Forrester Start Quiz
	17 AP 17 Endocrine System MCQ By OpenStax Start Quiz
©flickr: Luis	Chemistry Final Review By Madison Christian Start Exam

Statistical terminology (Page 4/4)

Log normal distribution.

The log-normal distribution.

Gamma distribution.

Chi-square distribution.

Student's t-distribution.

F distribution.

Multinomial distribution.

Characteristics of an estimator of a population parameter θ

Finite estimators

Bias.

Mean square error.

Efficiency.

Asymtoptic estimators

Plim.

Example of plim.

Consistency.

Asymmtotically unbiased.

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!