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

what does preconceived mean

sammie Reply

physiological Psychology

Nwosu Reply

How can I develope my cognitive domain

Amanyire Reply

why is communication effective

Dakolo Reply

Communication is effective because it allows individuals to share ideas, thoughts, and information with others.

effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.

it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.

miss

Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment

Wekolamo Reply

please i need answer

Wekolamo

because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).

Manix Reply

Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!

ARC

A child is a member of community not society elucidate ?

JESSY Reply

Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?

Simon Reply

compare and contrast skinner's perspective on personality development on freud

namakula Reply

Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning

war

explain how nature and nurture affect the development and later the productivity of an individual.

Amesalu Reply

nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.

Samuel

I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills

Zyryn Reply

good👍

Jonathan

and having a good philosophy of the world is like a sandwich and a peanut butter 👍

Jonathan

generally amnesi how long yrs memory loss

Kelu Reply

interpersonal relationships

Abdulfatai Reply

What would be the best educational aid(s) for gifted kids/savants?

Heidi Reply

treat them normal, if they want help then give them. that will make everyone happy

Saurabh

What are the treatment for autism?

Magret Reply

hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities. before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these

Jharna

continue.. sometimes due to these physical problems..the diagnosis may be misdiagnosed. treatment for autism. well it depends on the severity. since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child

Jharna

child interact with other kids under doc supervision. play therapy. speech therapy. Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game. string and beads game. the more you interact with the child the more effective

Jharna

results you'll get.. please consult a therapist to know what suits best on your child. and last as a parent. I know sometimes it's overwhelming to guide a special kid. but trust the process and be strong and patient as a parent.

Jharna

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

	43 Biology 43 Animal Reproduction Development MCQ By OpenStax Start Quiz
©flickr: Donnie	How well do you know Beanie Boos? By Angela Eckman Start Quiz
	Classical Music By Marion Cabalfin Start Quiz
	4 Psychology MCQ 2009 Final Exam By John Gabrieli Start Exam
©flickr: quinn	Neuroanatomy By George Turner Start Quiz
	Clinical Psychology MCQ By Saylor Foundation Start Quiz
	15 AP 15 Autonomic Nervous System MCQ By OpenStax Start Quiz
©flickr: Luis	Atoms By Carly Allen Start Quiz
	5 AP Key Terms 05 The Integumentary System By OpenStax Start Key Terms
	Anthropology Aesthetics Culture By Richley Crapo Start Assignment

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!