Normal Suite: One Sample, Two Sided, Unknown Variance
Description
Generates the suite of functions related to the one sample normal experiment with a twosided alternative hypothesis of interest when the variance is unknown.
Usage
1  norm1UV.2sided(theta0, prob, mu, scale, shape, rate)

Arguments
theta0 
Scalar. The critical value of the mean under the null hypothesis: theta==theta0. 
prob 
Scalar. The prior probability of the null hypothesis. Must be a value between 0 and 1. 
mu 
Scalar. The mean of the normal prior density on theta under the alternative
hypothesis. See documentation for 
scale 
Scalar. Used to determine the standard deviation for the normal prior
density on theta under the alternative hypothesis. The standard deviation
is equal to 
shape 
Scalar. The shape parameter for the gamma prior on the inverse of the
unknown standard deviation 
rate 
Scalar. The rate parameter for the gamma prior on the inverse of the
unknown standard deviation 
Details
norm1UV.2sided
is used to generate a suite of functions for a
onesample normal experiment with a twosided alternative hypothesis when the
variance is unknown. That is, when
X ~ Normal(theta,sigma2)
H0: theta == theta0 vs. H1: theta != theta0
using the following prior on theta and sigma2
pi(thetasigma2) = u*I(theta==theta0) + (1u)*I(theta!=theta0)Normal(mu,(scale*sigma)^2),
pi(sigma2) = InverseGamma(shape,rate),
where Normal(mu,tau2) is Normal density with mean mu
and variance
tau2
and u is the prior probability of the null hypothesis
(prob
).
The functions that are generated are useful in examining the prior and
posterior densities of the parameters theta
and sigma2
, as well
as constructing the Bayes Factor and determining the sample size via an
average error based approach.
The arguments of norm1UV.2sided
are passed to each of the additional
functions upon their creation as default values. That is, if mu
is
set to 1 in the call to norm1UV.2sided
, each of the functions returned
will have the defaualt value of 1 for mu
. If an argument is not
specified in the call to norm1UV.2sided
, then it remains a required
parameter in all functions created.
Value
norm1UV.2sided
returns a list of 5 functions:
logm 
Returns a list of three vectors: the log marginal density under
the null hypothesis ( logm(xbar, s2, n, theta0, prob, mu, scale, shape, rate)

logbf 
Returns a vector. The value of the log Bayes Factor given the observed data provided and the prior parameters specified. The function has the following usage: logbf(xbar, s2, n, theta0, prob, mu, scale, shape, rate) For details on the arguments, see 
prior 
Returns a vector. The value of the prior density. The function has the following usage: prior(theta, sigma2, theta0, prob, mu, scale, shape, rate)

post 
Returns a vector. The value of the posterior density. The function has the following usage: post(theta, sigma2, xbar, s2, n, theta0, prob, mu, scale, shape, rate)

ssd.norm1UV.2sided 
Sample size calculations for this particular setup. The function has the following usage: ssd.norm1UV.2sided(alpha, w, theta0, prob, mu, scale, shape, rate, m = 2500, minn = 3, maxn = 1000, all = FALSE) See 
See Also
binom1.1sided
,binom1.2sided
,
binom2.1sided
,binom2.2sided
,
norm1KV.1sided
,norm1KV.2sided
,
norm2KV.2sided
,ssd
,BAEssd
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  ############################################################
# Generate the suite of functions for a onesample normal
# with a twosided test. Consider the hypothesis
# H0: theta==0 vs. H1: theta!=0
#
# with a normal prior for theta with prior mean 2 and
# scale of 1/3 for the standard deviation. The prior proability
# of the null hypothesis is set to 0.5. The prior density
# on sigma2 is taken to be InverseGamma with parameters
# 11 and 30 for the shape and rate.
# generate suite
f8 < norm1UV.2sided(theta0=0,prob=0.5,mu=2,scale=(1/3),shape=11,rate=30)
# attach suite
attach(f8)
# calculate the Bayes Factor for the following observed data
# n = 30, xbar = 1, s2 = 2
logbf(xbar=1,s2=2,n=30)
# perform sample size calculation with TE bound of 0.25 and weight 0.5
ssd.norm1UV.2sided(alpha=0.25,w=0.5)
# detain suite
detach(f8)
