Inference

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Inference

• Mary M. Whiteside, Ph.D.
• Nonparametric Statistics

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Two Sides of Inference

• Parametric
• Interval estimation, xbar
• Hypothesis testing, m0
• Nonparametric
• Interval estimates, EDF
• Hypothesis testing, P(XltY) gt P(XgtY)

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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

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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

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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

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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

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Sampling Distributions

• Statistics support both sides of inference
• Estimators - random variables used to create
  interval estimates
• Test statistics - random variables used to test
  hypotheses

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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

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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

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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)

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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

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Estimation

• Definitions
• EDF
• pth sample quantile
• sample mean, variance, and standard deviation
• unbiased estimators (S2 and s2)

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Intervals for parameter estimation

• (point estimate - rstandard error of the
  estimator, point estimate qstandard 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

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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
• Xbars distribution is unknown if Xs
  distribution is unknown and n is small

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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

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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