6.5. Multiparameter Case: Testing 395(a)If the constantbis defined by the equationP(X≤b)=0.90, find the mle of
b.(b)Ifcis given constant, find the mle ofP(X≤c).6.4.7. The data filenormal50.rdacontains a random sample of sizen=50for
the situation described in Exercise 6.4.6. Download this data in R and obtain a
histogram of the observations.
(a)In Part (b) of Exercise 6.4.6, letc=58andletξ=P(X≤c). Basedonthe
data, compute the estimated value of the mle forξ. Compare this estimate
with the sample proportion, ˆp, of the data less than or equal to 58.(b)The R functionbootstrapcis64.Rcomputes a bootstrap confidence interval
for the mle. Use this function to compute a 95% confidence interval forξ.
Compare your interval with that of expression (4.2.7) based on ˆp.6.4.8.Consider Part (a) of Exercise 6.4.6.
(a)Using the data of Exercise 6.4.7, compute the mle ofb. Also obtain the
estimate based on 90th percentile of the data.(b)Edit the R functionbootstrapcis64.Rto compute a bootstrap confidence
interval forb. Then run your R function on the data of Exercise 6.4.7 to
compute a 95% confidence interval forb.6.4.9.Consider two Bernoulli distributions with unknown parametersp 1 andp 2 .If
Y andZequal the numbers of successes in two independent random samples, each
of sizen, from the respective distributions, determine the mles ofp 1 andp 2 if we
know that 0≤p 1 ≤p 2 ≤1.
6.4.10.Show that ifXifollows the model (6.4.14), then its pdf isb−^1 f((x−a)/b).
6.4.11.Verify the partial derivatives and the entries of the information matrix for
the location and scale family as given in Example 6.4.4.6.4.12.Suppose the pdf ofXis of a location and scale family as defined in Example
6.4.4. Show that iff(z)=f(−z), then the entryI 12 of the information matrix is 0.
Then argue that in this case the mles ofaandbare asymptotically independent.
6.4.13.SupposeX 1 ,X 2 ,...,Xnare iidN(μ, σ^2 ). Show thatXifollows a location
and scale family as given in Example 6.4.4. Obtain the entries of the information
matrix as given in this example and show that they agree with the information
matrix determined in Example 6.4.3.
6.5 MultiparameterCase:Testing......................
In the multiparameter case, hypotheses of interest often specifyθto be in a sub-
region of the space. For example, supposeXhas aN(μ, σ^2 ) distribution. The full
space is Ω ={(μ, σ^2 ):σ^2 > 0 ,−∞<μ<∞}. This is a two-dimensional space.