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Inference. Mary M. Whiteside, Ph.D. Nonparametric Statistics. Two Sides of Inference. Parametric Interval estimation, xbar Hypothesis testing, m 0 Nonparametric Interval estimates, EDF Hypothesis testing, P(X<Y) > P(X>Y). Meaning of Nonparametric. Not about parameters

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inference

Inference

Mary M. Whiteside, Ph.D.

Nonparametric Statistics

two sides of inference
Two Sides of Inference
  • Parametric
    • Interval estimation, xbar
    • Hypothesis testing, m0
  • Nonparametric
    • Interval estimates, EDF
    • Hypothesis testing, P(X<Y) > P(X>Y)
meaning of nonparametric
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 - 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
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
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
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
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 statistic1
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 deviationsX/n
x in the type 1 sample
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
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
Estimation
  • Definitions
    • EDF
    • pth sample quantile
    • sample mean, variance, and standard deviation
    • unbiased estimators (S2 and s2)
intervals for parameter estimation
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
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 estimator1
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
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
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