Chapter 3 Flashcards

(14 cards)

1
Q

define the likelihood function L(theta) for single observation

A

For a given observation x, we call theta->L(theta;x)= p(x;theta) for x discrete and f(x;theta) for x continuous

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2
Q

define the maximum likelihood estimate of theta

A

the mle is the value of theta that maximises the likelihood function L(theta;x)

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3
Q

for a single observation how do we find theta hat mle

A

for x=x1 we have L(theta;x1)=p(x1;theta) or =f(x1;theta). we then maximise L(theta;x1) by differentiating and setting equal to 0. Remember theta E (0,1). Then show it is a maximum

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4
Q

for multiple independent observations how do we find theta hat mle

A
set L(theta)=L(theta;x1,x2,...,xn)= Px1,x,...,xn(x1,x2,...,xn)= Px1(x1;theta)Px2(x2;theta)...Pxn(xn;theta)
then differentiate ands et equal to 0 and solve for theta
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5
Q

define likelihood function for multiple observations

A
Assume the data x1,..,xn are the observed values of random variables X1,...Xn whose joint distribution depends on one or more
unknown parameters θ. The likelihood function L(θ) ≡ L(θ; x1, x2, . . . , xn) is
the joint probability mass function (discrete case) or joint probability density
function (continuous case) regarded as a function of the unknown parameter θ
for these fixed numerical values of x1, x2, . . . , xn.
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6
Q

define mle for multiple observations

A
For observed values {x1, . . . , xn}, the maximum likelihood estimator
(mle) θ hat mle(x1, . . . xn) is the value of θ which maximises the likelihood
function L(θ; x1, . . . , xn).
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7
Q

define log-likelihood function

A
For observed values {x1, . . . , xn} and associated likelihood
function L(θ) ≡ L(θ; x1, x2, . . . , xn), the log-likelihood function is defined as
l(θ) := log L(θ), where log is the natural logarithm (and we take log 0 = −∞)
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8
Q

likelihood function for simple random sample

A

If X1, X2, . . . , Xn, is a random sample of size n from a distribution
with probability mass function p (x; θ) (or probability density function
f(x; θ)) then the Xi are i.i.d. and their joint distribution factorises into the
product of marginals. Thus for a random sample
L(θ) ≡ L(θ; x1, x2, . . . , xn) =
p(x1; θ) p(x2; θ)· · · p(xn; θ) (discrete scenario)
f(x1; θ) f(x2; θ)· · · f(xn; θ) (continuous scenario

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9
Q

for observations taken froma simple random sample what does the log-likelihood function=

A

(sum
i=1-n log p(xi; θ). (discrete scenario)

sum
i=1-n log f(xi; θ). (continuous scenario)

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10
Q

what is the likelihood equation

A

∂/∂θ l(θ) = sum i=1-n ∂/∂θ log f(xi; θ) = 0

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11
Q

procedure of calculating theta hat mle in the random sample case

A
  1. Calculate ∂/∂θlogf(x;θ)
  2. compute the sum ∂/∂θlogf(x;θ)
  3. set sum =0 and θhat mle is the value satisfying the likelihood equation
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12
Q

the invariable property of mle

A

if the quantity of interest is a function t(θ) of θ the mle of t(θ) is the plug-in estimate t(θ)hat=t(θhat)

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13
Q

procedure of calculating mle for multiple parameters alpha and beta

A

for two parameters α and β, the αhat mle and βhat mle are the simultaneous solutions to the two likelihood equations
0 = sum i=1-n ∂/∂αlog f(xi; α, β)
and
0 = sum i=1-n ∂/∂β log f(xi; α, β).

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14
Q

if the density is not regular how to work out mle

A

the likelihood can be maximise at one endpoint of the interval so find Lθ and differentiate or see if it is a decreasing/increasing function.

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