Raoul LePage

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Raoul LePage Professor STATISTICS AND
PROBABILITY www.stt.msu.edu/lepage click on
STT351_F07
Week 8-27-07
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WEEK 8-27-07 PLAN Chapter 1 except Section
1-5. Kernel Density Estimate, pp
333-334. Homework due in class 9-5-07. No class
9-3-07 (labor day).
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Plot average heights of normal densities placed
at each data value, e.g. 10, 14. It is like
smearing each sample value, as it were a drop of
paint, according to the thickness of a normal
density. Each normal integrates to one, as does
their average the Sample Density Estimate shown
in dark.
Smoothing data , so you can see it.
normal densities at data 10, 14
Kernel Density Estimate
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The mean of a Kernel Density Estimate is equal to
the sample mean of its data.
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Making the densities narrower isolates different
parts of the data and reveals more detail.
NARROWER TENTS MORE DETAIL
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Closer view of the density by itself, with
narrow normal curves.
density
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Histograms lump data into categories (the black
boxes), not as good for continuous data.
DENSITY OR HISTOGRAM ?
density histogram
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Form of each rectangle comprising a Probability
Histogram. Example A sample of n 40 finds
three data values which are at least 30 but less
than 35 (interval 30, 35)).
height area w height 3 / 40
3/(40 5)
Histograms may radically change their shape in
response to minor changes of bin locations or
widths.

30 35 bin-width w 35 - 30
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Plot of average heights of 5 tents placed at data
12, 21, 42, 8, 9.
DENSITY FOR 12, 21, 42, 8, 9
normal density smear around datum 42
data density
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Narrower tents operate at higher resolution but
they may bring out features that are illusory.
IS DETAIL ILLUSORY ?
which do we trust ?
kinkier
smoother
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Population of N 500 compared with two samples
of n 30 each.
BEWARE OVER-FINE RESOLUTION
POP mean 32.02
population of N 500
with 2 samples of n 30
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Population of N 500 compared with two samples
of n 30 each.
BEWARE OVER-FINE RESOLUTION
sample means are close
SAM1 mean 33.03 SAM2 mean 30.60
POP mean 32.02
densities not good at fine resolution
population of N 500
with 2 samples of n 30
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The same two samples of n 30 each from the
population of 500.
WE DO BETTER AT COARSE RESOLUTION
SAM1 mean 33.03 SAM2 mean 30.60
POP mean 32.02
how about coarse resolution ?
population of N 500
with 2 samples of n 30
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The same two samples of n 30 each from the
population of 500.
WE DO BETTER AT COARSE RESOLUTION
SAM1 mean 33.03 SAM2 mean 30.60
POP mean 32.02
agreement better at coarser resolution
population of N 500
with 2 samples of n 30
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The same two samples of n 30 each from the
population of 500.
HOW ABOUT MEDIUM RESOLUTION ?
SAM1 mean 33.03 SAM2 mean 30.60
POP mean 32.02
medium resolution ?
population of N 500
with 2 samples of n 30
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The same two samples of n 30 each from the
population of 500.
HOW ABOUT MEDIUM RESOLUTION ?
SAM1 mean 33.03 SAM2 mean 30.60
POP mean 32.02
not good at medium resolution
population of N 500
with 2 samples of n 30
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A sample of only n 600 from a population of N
500 million.(medium resolution)
SAMPLING ONLY 600 FROM 500 MILLION ?
large sample of n 600 ?
POP mean 32.02
medium resolution ?
population of N 500,000
with a sample of n 600
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A sample of only n 600 from a population of N
500 million.(MEDIUM resolution)
SAMPLING ONLY 600 FROM 500 MILLION ?
sample of n 600 sample mean 32.84
mean very close
POP mean 32.02
densities are close
population of N 500,000
with a sample of n 600
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A sample of only n 600 from a population of N
500 million.(FINE resolution)
SAMPLING ONLY 600 FROM 500 MILLION ?
sample of n 600 sample mean 32.84
POP mean 32.02
FINE resolution
densities very close
population of N 500,000,000
with a sample of n 600
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TALKING POINTS

• A density is controlled by the sd, referred to as
  bandwidth, of the normal densities used to make
  it.
• 1a. You have to be content with the
  information revealed by the population density at
  your chosen bandwidth.
• 1b. Small samples zero-in fairly well on
  densities at coarse resolution, i.e. made with
  large bandwidth.
• 1c. Samples in hundreds may perform
  remarkably well, even at fine resolution, i.e.
  small bandwidth.
• 2. Histograms are notorious for being unstable
  for some data. Yet, they remain popular. Learn
  to make them by hand.
• 3. Learn to make a density for 2 to 4 data
  values by hand.

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