Let
be a sample (size n=1) from the exponential distribution, which has
the pdf
, where
is an unknown
parameter. Let's define a statistic as
.
Is
a sufficient
statistic for
?
Homework Answers
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Let
be a sample (size n=1) from the exponential distribution, which has
the pdf
, where...
Let be a sample (size n = 1) from the exponential distribution, which has the pdf...
Let be a sample (size n = 1) from the exponential distribution, which has the pdf where is an unknown parameter. Let's define a statistic as . Is a sufficient statistic for ? We were unable to transcribe this imagef(x: λ) = Xe We were unable to transcribe this imageT(X) = 11>2 T(X) We were unable to transcribe this image
Let X1, X2, ..., Xn be a random sample from X which has pdf depending on...
Let X1, X2, ..., Xn be a random
sample from X which has pdf
depending on a parameter
and
(i)
(ii)
where
< x <
. In both these two cases
a) write down the log-likelihood function and find a
1-dimensional sufficient statistic for
b) find the score function and the maximum likelihood estimator
of
c) find the observed information and evaluate the Fisher
information at
= 1.
f(20) We were unable to transcribe this image((z(0 – 2) - )dxəz(47)...
Let X1, X2,.......Xn be a random sample of size n from a continuous distribution symmetric about...
Let X1, X2,.......Xn be a
random sample of size n from a continuous distribution symmetric
about .
For testing H0: =
10 vs H1: <
10, consider the statistic T- =
Ri+ (1-i),
where i
=1 if Xi>10 , 0 otherwise; and
Ri+ is the rank of (Xi - 10) among
|X1 -10|, |X2-10|......|Xn
-10|.
1. Find the null mean and variance of T- .
2. Find the exact null distribution of T- for
n=5.
We were unable to transcribe this imageWe were...
Let be a random sample from , where is an unknown parameter. Show that is a...
Let be a random sample from , where is an unknown parameter. Show that is a sufficient statistics for , where is the sample variance. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image2 We were unable to transcribe this imageWe were unable to transcribe this image
Suppose n independent, identically distributed observations are drawn from an exponential () distribution, with pdf given...
Suppose n independent, identically distributed observations are
drawn from an exponential ()
distribution, with pdf given by f(x,)=,
0 < x <
.
The data are x1, x2, .. , xn
Construct a likelihood ratio hypothesis test of Ho :
vs H1:
(where
and
are known constants, with
), where the critical value is taken to be a constant c
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Let X1, X2, X3 ∼(iid) Exponential(λ). (a) Show that T(X1, X2, X3) = X1 + X2...
Let X1, X2, X3 ∼(iid) Exponential(λ).
(a) Show that T(X1, X2, X3) = X1 + X2 + X3 is a sufficient
statistic for λ.
(b) Find the MVUE for λ.
(c) Show that
is not a sufficient statistic for λ.
(d) Let
=
and find
. Give an argument for why
is not the best estimator of λ.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
Independent random samples X1, X2, . . . , Xn are from exponential distribution with pdfs...
Independent random samples X1, X2, . . . , Xn are from
exponential distribution with pdfs
, xi > 0, where λ is fixed but unknown. Let
. Here we have a relative large sample size n = 100.
(ii) Notice that the population mean here is µ = E(X1) = 1/λ ,
population variance σ^2 = Var(X1) = 1/λ^2 is unknown. Assume the
sample standard deviation s = 10, sample average
= 5, construct a 95% large-sample approximate confidence...
3. Let ,..., be independent random sample from N(), where is unknown. (i) Find a sufficient...
3. Let ,..., be
independent random sample from N(),
where is unknown.
(i) Find a sufficient statistic of .
(ii) Find the MLE of .
(iii) Find a pivotal quantity and use it to construct a
100(1–)% confidence
interval for .
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
Let denote the sample mean of an independent random sample of size 25 from the distribution...
Let denote the sample mean of an independent random sample of size 25 from the distribution whose p.d.f. is , 0 < x < 2. Find an approximation of P(1.3 < < 1.6). We were unable to transcribe this imagef (x)- We were unable to transcribe this image
Let {} be a random sample from the distribution. (a) Find a sufficient statistic for when...
Let {} be a random sample from the distribution. (a) Find a sufficient statistic for when is known (b) Find a sufficient statistic for when is known 7l beta ( α , β ) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let X1, X2, ..., Xn be a random
sample from X which has pdf
depending on a parameter
and
(i)
(ii)
where
< x <
. In both these two cases
a) write down the log-likelihood function and find a
1-dimensional sufficient statistic for
b) find the score function and the maximum likelihood estimator
of
c) find the observed information and evaluate the Fisher
information at
= 1.
f(20) We were unable to transcribe this image((z(0 – 2) - )dxəz(47)...
Let X1, X2,.......Xn be a
random sample of size n from a continuous distribution symmetric
about .
For testing H0: =
10 vs H1: <
10, consider the statistic T- =
Ri+ (1-i),
where i
=1 if Xi>10 , 0 otherwise; and
Ri+ is the rank of (Xi - 10) among
|X1 -10|, |X2-10|......|Xn
-10|.
1. Find the null mean and variance of T- .
2. Find the exact null distribution of T- for
n=5.
We were unable to transcribe this imageWe were...
Suppose n independent, identically distributed observations are
drawn from an exponential ()
distribution, with pdf given by f(x,)=,
0 < x <
.
The data are x1, x2, .. , xn
Construct a likelihood ratio hypothesis test of Ho :
vs H1:
(where
and
are known constants, with
), where the critical value is taken to be a constant c
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Let X1, X2, X3 ∼(iid) Exponential(λ).
(a) Show that T(X1, X2, X3) = X1 + X2 + X3 is a sufficient
statistic for λ.
(b) Find the MVUE for λ.
(c) Show that
is not a sufficient statistic for λ.
(d) Let
=
and find
. Give an argument for why
is not the best estimator of λ.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
Independent random samples X1, X2, . . . , Xn are from
exponential distribution with pdfs
, xi > 0, where λ is fixed but unknown. Let
. Here we have a relative large sample size n = 100.
(ii) Notice that the population mean here is µ = E(X1) = 1/λ ,
population variance σ^2 = Var(X1) = 1/λ^2 is unknown. Assume the
sample standard deviation s = 10, sample average
= 5, construct a 95% large-sample approximate confidence...
3. Let ,..., be
independent random sample from N(),
where is unknown.
(i) Find a sufficient statistic of .
(ii) Find the MLE of .
(iii) Find a pivotal quantity and use it to construct a
100(1–)% confidence
interval for .
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
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