A Cox process <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mtext>Cox</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math> directed by a stationary random measure <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ξ</mml:mi></mml:math> has second
moment <mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>var</mml:mtext><mml:mtext> </mml:mtext><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mtext>Cox</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mtext>E</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ξ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mtext>var</mml:mtext><mml:mtext> </mml:mtext><mml:mi>ξ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math>, where by
stationarity <mml:math id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>E</mml:mtext><mml:mo>(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>]</mml:mo><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mo>(</mml:mo><mml:mtext>const</mml:mtext><mml:mo>.</mml:mo><mml:mo>)</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo></mml:math><mml:math id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>E</mml:mtext><mml:mo>(</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mtext>Cox</mml:mtext></mml:msub><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>]</mml:mo><mml:mo>)</mml:mo></mml:math>, so long-range dependence (LRD) properties of
<mml:math id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mtext>Cox</mml:mtext></mml:mrow></mml:msub></mml:math> coincide with LRD properties of the random measure <mml:math id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ξ</mml:mi></mml:math>.
When <mml:math id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ξ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi>A</mml:mi></mml:msub><mml:mrow><mml:msub><mml:mi>ν</mml:mi><mml:mrow><mml:mi>J</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mstyle><mml:mtext>d</mml:mtext><mml:mi>u</mml:mi></mml:math> is determined by a density that depends
on rate parameters <mml:math id="E9" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ν</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>𝕏</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> and the current state <mml:math id="E10" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>⋅</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:math>
of an <mml:math id="E11" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝕏</mml:mi></mml:math>-valued stationary irreducible Markov renewal process (MRP) for
some countable state space <mml:math id="E12" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝕏</mml:mi></mml:math> (so <mml:math id="E13" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is a stationary semi-Markov
process on <mml:math id="E14" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝕏</mml:mi></mml:math>), the random measure is LRD if and only if each (and then
by irreducibility, every) generic return time <mml:math id="E15" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">X</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> of the
process for entries to state <mml:math id="E16" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>j</mml:mi></mml:math> has infinite second moment, for which a
necessary and sufficient condition when <mml:math id="E17" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝕏</mml:mi></mml:math> is finite is that at least
one generic holding time <mml:math id="E18" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math> in state <mml:math id="E19" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>j</mml:mi></mml:math>, with distribution function (DF)\
<mml:math id="E20" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math>, say, has infinite second moment (a simple example shows that this
condition is not necessary when <mml:math id="E21" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝕏</mml:mi></mml:math> is countably infinite).
Then, <mml:math id="E22" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mtext>Cox</mml:mtext></mml:mrow></mml:msub></mml:math> has the same Hurst index as the MRP <mml:math id="E23" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mtext>MRP</mml:mtext></mml:mrow></mml:msub></mml:math> that counts the jumps
of <mml:math id="E24" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>J</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>⋅</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:math>, while as <mml:math id="E25" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:math>,  for finite <mml:math id="E26" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝕏</mml:mi></mml:math>,
<mml:math id="E27" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>var</mml:mtext><mml:mtext> </mml:mtext><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mtext>MRP</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>∼</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>λ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mi>t</mml:mi></mml:msubsup><mml:mrow><mml:mi>𝒢</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mstyle><mml:mtext>d</mml:mtext><mml:mi>u</mml:mi></mml:math>,
<mml:math id="E28" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>var</mml:mtext><mml:mtext> </mml:mtext><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mtext>Cox</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>∼</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mi>t</mml:mi></mml:msubsup><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>𝕏</mml:mi></mml:mrow></mml:munder><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>ν</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo stretchy="true">¯</mml:mo></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mrow></mml:mstyle><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>ϖ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi>ℋ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>du</mml:mtext></mml:math>,
where
<mml:math id="E29" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo stretchy="true">¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:msub><mml:mo>∑</mml:mo><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:msub><mml:mi>ϖ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi>ν</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mtext>E</mml:mtext></mml:mrow></mml:mstyle><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>ξ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math>,
<mml:math id="E30" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ϖ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mtext>Pr</mml:mtext><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>J</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>λ</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:msub><mml:mo>∑</mml:mo><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>p</mml:mi><mml:mo stretchy="true">ˇ</mml:mo></mml:mover><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>μ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mstyle></mml:math>,
<mml:math id="E31" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>μ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mtext>E</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>,
<mml:math id="E32" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>p</mml:mi><mml:mo stretchy="true">ˇ</mml:mo></mml:mover><mml:mi>j</mml:mi></mml:msub><mml:mo>}</mml:mo></mml:mrow></mml:math>
is the stationary distribution for the embedded jump process
of the MRP, <mml:math id="E33" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℋ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mi>μ</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mi>∞</mml:mi></mml:msubsup><mml:mrow><mml:mo>min</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mtext>d</mml:mtext><mml:mi>u</mml:mi></mml:math>, and
<mml:math id="E34" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝒢</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mi>t</mml:mi></mml:msubsup><mml:mrow><mml:mo>min</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mtext>d</mml:mtext><mml:mi>u</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow><mml:mo>∼</mml:mo><mml:mstyle displaystyle="true"><mml:msub><mml:mo>∑</mml:mo><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:msub><mml:mi>ϖ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi>ℋ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mstyle><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>
where <mml:math id="E35" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>jj</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> is the
DF and <mml:math id="E36" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> the mean of the generic return time <mml:math id="E37" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> of the MRP
between successive
entries to the state <mml:math id="E38" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>j</mml:mi></mml:math>.  These two variances are of similar order
for <mml:math id="E39" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:math> only when each <mml:math id="E40" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>ℋ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mi>𝒢</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> converges to some
<mml:math id="E41" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>∞</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math>-valued constant, say, <mml:math id="E42" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>γ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math>, for <mml:math id="E43" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:math>.

D. J. Daley

T. Rolski

R. Vesilo

Long-Range Dependence in a Cox Process Directed by a Markov Renewal Process

We present an excursion with Jeff Hunter, visiting some of his research topics. Specifically, we 
    will present some facts about certain people whose work seems to have influenced Jeff in his scientific 
    career; we illustrate our presentation with postage stamps that have been issued in honour of these 
    people. Our main guide is Hunter’s two-volume book entitled <i>Mathematical Techniques 
    of Applied Probability</i> (Academic Press, 1983).

George P. H. Styan

Götz Trenkler

A Philatelic Excursion with Jeff Hunter in Probability and Matrix Theory

We will place certain parts of the theory of statistical efficiency into the author's 
operator trigonometry (1967), thereby providing new geometrical understanding of statistical efficiency. Important 
earlier results of Bloomfield and Watson, Durbin and Kendall, Rao and Rao, will be so interpreted. For
 example, worse case relative least squares efficiency corresponds to and is achieved by the maximal turning 
 antieigenvectors of the covariance matrix. Some little-known historical perspectives will also be exposed. 
 The overall view will be emphasized.

K. Gustafson

The Geometry of Statistical Efficiency and Matrix Statistics

Consider a network of parallel finite tandem queues with two stages, where each arrival attempts to
     minimize its own cost due to loss. It is known that the user optimal and asymptotic system optimal policies
      may differ—we give examples showing that they may differ for finite systems and that as the service
       rate is increased at the second stage the user optimal policy may change in such a way that the total expected
        cost due to loss increases.

Ilze Ziedins

A Paradox in a Queueing Network with State-Dependent Routing and Loss

In this paper, a procedure based on <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi></mml:math>-estimation to 
determine the number of regression
models for the problem of regression clustering is proposed. We have shown that the true
classification is attained when <mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> increases to infinity under certain mild conditions, for instance,
without assuming normality of the distribution of the random errors in each regression
model.

C. R. Rao

Y. Wu

Q. Shao

An <svg style="vertical-align:-0.0pt" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28&#xA;q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g></svg>-Estimation-Based Procedure for Determining the Number of Regression Models in Regression Clustering


Using a projection approach, we obtain an asymptotic information bound for estimates 
    of parameters in general regression models under choice-based and two-phase outcome-dependent 
    sampling. The asymptotic variances of the semiparametric estimates of Scott and Wild (1997, 2001) are 
    compared to these bounds and the estimates are found to be fully efficient.

Alan Lee

On the Semiparametric Efficiency of the Scott-Wild Estimator under Choice-Based and Two-Phase Sampling

Markov chain theory plays an important role in statistical inference both in the formulation 
of models for data and in the construction of efficient algorithms for inference. The use of Markov chains
in modeling data has a long history, however the use of Markov chain theory in developing algorithms for
statistical inference has only become popular recently. Using mark-recapture models as an illustration,
we show how Markov chains can be used for developing demographic models and also in developing
efficient algorithms for inference. We anticipate that a major area of future research involving mark-recapture
data will be the development of hierarchical models that lead to better demographic models that account
for all uncertainties in the analysis. A key issue is determining when the chains produced by Markov 
chain Monte Carlo sampling have converged.

Richard J. Barker

Matthew R. Schofield

Putting Markov Chains Back into Markov Chain Monte Carlo

Paul Cowpertwait

Graeme Wake

Robert D. Anderson

Howard Edwards

Shayle Searle

Statistics and Applied Probability: A Tribute to Jeffrey J. Hunter

Patch clamp recordings from ion channels often show <i>bursting</i> behaviour, that is, periods of repetitive activity, which are noticeably separated from each other by  periods of inactivity. A number of authors have obtained results for important properties of theoretical and empirical bursts when channel gating is modelled by
a continuous-time Markov chain with a finite-state space. We show how the use of marked continuous-time Markov chains can simplify the derivation of (i) the distributions of several burst properties, including the total open time, the total charge transfer, and the number of openings in a burst, and (ii) the form of these distributions when the underlying gating process is time reversible and in equilibrium.

Frank G. Ball

Robin K. Milne

Geoffrey F. Yeo

Marked Continuous-Time Markov Chain Modelling of Burst Behaviour for Single Ion Channels

We look at fitting regression models using data from stratified cluster samples when the
strata may depend in some way on the observed responses within clusters. One important
subclass of examples is that of family studies in genetic epidemiology, where the probability
of selecting a family into the study depends on the incidence of disease within the family.
We develop the survey-weighted estimating equation approach for this problem,
with particular emphasis on the estimation of superpopulation parameters. Full maximum
likelihood for this class of problems involves modelling the population distribution of the
covariates which is simply not feasible when there are a large number of potential covariates.
We discuss efficient semiparametric maximum likelihood methods in which the covariate
distribution is left completely unspecified. We further discuss the relative efficiencies of these
two approaches.

Alastair Scott

Chris Wild

Methods for Stratified Cluster Sampling with Informative Stratification

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Advances in Decision Sciences

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Published Special Issues

Advances in Decision Sciences

Statistics and Applied Probability: A Tribute to Jeffrey J. Hunter

Statistics and Applied Probability: A Tribute to Jeffrey J. Hunter

Articles

Long-Range Dependence in a Cox Process Directed by a Markov Renewal Process

A Philatelic Excursion with Jeff Hunter in Probability and Matrix Theory

The Geometry of Statistical Efficiency and Matrix Statistics

A Paradox in a Queueing Network with State-Dependent Routing and Loss

An -Estimation-Based Procedure for Determining the Number of Regression Models in Regression Clustering

On the Semiparametric Efficiency of the Scott-Wild Estimator under Choice-Based and Two-Phase Sampling

Putting Markov Chains Back into Markov Chain Monte Carlo

Statistics and Applied Probability: A Tribute to Jeffrey J. Hunter

Marked Continuous-Time Markov Chain Modelling of Burst Behaviour for Single Ion Channels

Methods for Stratified Cluster Sampling with Informative Stratification