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Inference
Mary M. Whiteside, Ph.D.
Nonparametric Statistics
Parametric
Two Sides of
Inference
Interval estimation, xbar
Hypothesis testing, m0
Nonparametric
Interval estimates, EDF
Hypothesis testing, P(X<Y) > P(X>Y)
Meaning of Nonparametric
Not about parameters
Methods for non-normal distributions
Methods for ordinal data
Data Scales
Nominal, categorical, qualitative
Ordinal
Interval
Ratio - natural zero
Random Sample - Type 1
Random sample from a finite population
Simple
Stratified
Cluster
Inferences are about the finite population
Audit comprised of a sample from a
population of invoices
Public opinion polls
QC samples of delivered goods
Random Sample - Type 2
Observations of (iid) random variables
Inferences are about the probability
distributions of the random variables
Weekly average miles per gallon for your
new Lexus
Chi square tests of independence in medical
treatment offered men and women
Effect of female literacy on infant mortality
worldwide
Transition from data sets
to distributions
All random variables, by definition, have
probability functions (pmf or pdf) and
cumulative probability distributions
Random variables defined on a random
sample (Type 1 or 2) are called statistics
with probability distributions that are
called sampling distributions
Sampling Distributions
Statistics support both sides of inference
Estimators - random variables used to
create interval estimates
Test statistics - random variables used to
test hypotheses
Consider Xbar - a
parametric statistic
 Type I sample - subset of invoices where X = sales tax
paid on an invoice randomly selected from a finite
population
Xbar is the average sales tax of n randomly selected
invoices
Xbar is an estimator of m, the average sales tax paid
for the population of invoices (with standard deviation
s)
Xbar is a test statistic for testing hypotheses
H0: m = m0
Xbar is a random variable with sampling distribution
asymptotically normal as n increases with mean m and
standard deviation sn
Consider Xbar - a
parametric statistic
 Type 2 sample - the complete set of miles per gallon
observations made by you since buying your Lexus
where X = mpg for your Lexus in a given week
Xbar is the average mpg for n observations of X
Xbar is an estimator of the expected value (mX) of the
RV X
Xbar is a test statistic for testing hypotheses
H0: m = m0
Xbar is a random variable with sampling distribution
asymptotically normal as n increases with mean mX
and standard deviation sX/ n
X in the Type 1 sample
If X from a Type 1 sample is regarded as
a random variable, then it has the discrete
uniform distribution
Prob [X = x] = 1/N for all x in the
population (where the N values of x are
assumed to be unique)
Order statistics of rank k - a
nonparametric statistic
the kth order statistic is the kth smallest
observation
the first order statistic is the smallest
observation in a sample
the nth order statistic is the largest
Large body of literature on sampling
distributions of order statistics
Estimation
Definitions
EDF
pth sample quantile
sample mean, variance, and standard
deviation
unbiased estimators (S2 and s2)
Intervals for parameter
estimation
(point estimate - r*standard error of the
estimator, point estimate +q*standard
error of the point estimate) where r is the
a/2 quantile and q is the (1-a/2) quantile
from the sampling distribution of the
estimator
r equals -q in symmetric distributions with
mean 0 (z = +/- 1.96 or t = +/-2.02581)
r does not equal -q in skewed distributions
such as Chi squared and F
Sampling distribution of
the estimator
Parametric procedures - Assumed normal
or normal based from the Central Limit
Theorem and sample size
Xbar is approximately normal if n is large
Xbar is t if X is normal and s is unknown
Xbar’s distribution is unknown if X’s
distribution is unknown and n is small
Sampling distribution of
the estimator
Nonparametric distribution-free
procedures I.e. the sampling distribution
of the statistic (estimator or test statistic)
is “free” from the distribution of X
rank order statistics
bootstrapped distributions - a/2 and 1-a/2
quantiles
Parametric vs nonparametric
sampling distributions
Exact distributions with approximate
models
Exact distributions with exact models (but
usually small samples)
or
Asymptotic distributions with exact
models