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Lecture 20: Noncentral χ^{2}-, t-, and F-distributions

The chi-square (χ^{2}-), t-, and F-distributions previously defined are special cases of the non-central χ^{2}-, t-, and F-distributions, respectively, which are useful in statistics.

Definition (the noncentral chi-square distribution)

Let X_{1},...,X_{n} be independent random variables and X_{i} ∼ N(μ_{i},σ^{2}), i = 1,...,n. The distribution of the random variable Y = (X_{1}^{2} + ··· + X_{n}^{2})/σ^{2} is called the noncentral chi-square distribution with degrees of freedom n and the noncentrality parameter δ = (μ_{1}^{2} + ··· + μ_{n}^{2})/σ^{2}.

The chi-square distribution defined earlier is a special case of the noncentral chi-square distribution with δ = 0 and, therefore, is sometimes called a central chi-square distribution.

It follows from the definition of noncentral chi-square distributions that if Y_{1},...,Y_{k} are independent random variables and Y_{i} has the noncentral chi-square distribution with degrees of freedom n_{i} and the

UW-Madison (Statistics)  Stat 609 Lecture 20  2014  1 / 11

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Lecture 20: Noncentral χ^{2}-, t-, and F-distributions

The chi-square (χ^{2}-), t-, and F-distributions previously defined are special cases of the non-central χ^{2}-, t-, and F-distributions, respectively, which are useful in statistics.

Definition (the noncentral chi-square distribution)

Let X_{1},...,X_{n} be independent random variables and X_{i} ∼ N(μ_{i},σ^{2}), i = 1,...,n. The distribution of the random variable Y = (X_{1}^{2} + ··· + X_{n}^{2})/σ^{2} is called the noncentral chi-square distribution with degrees of freedom n and the noncentrality parameter δ = (μ_{1}^{2} + ··· + μ_{n}^{2})/σ^{2}.

The chi-square distribution defined earlier is a special case of the noncentral chi-square distribution with δ = 0 and, therefore, is sometimes called a central chi-square distribution.

It follows from the definition of noncentral chi-square distributions that if Y_{1},...,Y_{k} are independent random variables and Y_{i} has the noncentral chi-square distribution with degrees of freedom n_{i} and the

UW-Madison (Statistics)  Stat 609 Lecture 20  2014  1 / 11

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noncentrality parameter δ_{i}, i = 1,...,k, then Y = Y_{1} + ··· + Y_{k} has the noncentral chi-square distribution with degrees of freedom n_{1} + ··· + n_{k} and the noncentrality parameter δ_{1} + ··· + δ_{k}.

Theorem (properties of the noncentral chi-square distribution)

Let Y be a random variable having the noncentral chi-square distribution with degrees of freedom k and noncentrality parameter δ.

(i) The pdf of Y is

g_{δ,k}(x) = e^{−δ/2} ∑_{j=0}^{∞} \frac{(δ/2)^{j}}{j!} f_{2j+k}(x),

where f_{v}(x) is the pdf of the central chi-square distribution with degrees of freedom v, v = 1,2,...;

(ii) The mgf of Y is

\frac{e^{δt/(1−2t)}}{(1 − 2t)^{k/2}} , t < 1/2;

(iii) E(Y ) = k + δ; (iv) Var(Y ) = 2k + 4δ.

UW-Madison (Statistics)  Stat 609 Lecture 20  2014  2 / 11

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noncentrality parameter δ_{i}, i = 1,...,k, then Y = Y_{1} + ··· + Y_{k} has the noncentral chi-square distribution with degrees of freedom n_{1} + ··· + n_{k} and the noncentrality parameter δ_{1} + ··· + δ_{k}.

Theorem (properties of the noncentral chi-square distribution)

Let Y be a random variable having the noncentral chi-square distribution with degrees of freedom k and noncentrality parameter δ.

(i) The pdf of Y is

g_{δ,k}(x) = e^{−δ/2} ∑_{j=0}^{∞} \frac{(δ/2)^{j}}{j!} f_{2j+k}(x),

where f_{v}(x) is the pdf of the central chi-square distribution with degrees of freedom v, v = 1,2,...;

(ii) The mgf of Y is

\frac{e^{δt/(1−2t)}}{(1 − 2t)^{k/2}} , t < 1/2;

(iii) E(Y ) = k + δ; (iv) Var(Y ) = 2k + 4δ.

UW-Madison (Statistics)  Stat 609 Lecture 20  2014  2 / 11

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Proof.

We first prove result (ii). Let X_{k} be a random variable having the standard normal distribution and μ = \sqrt{δ} be a positive number. For t < 1/2, the mgf of (X_{k} + μ)^{2} is

ψ_{δ}(t) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−x^{2}/2} e^{t(x+μ)^{2}}dx
= \frac{e^{μ^{2}t/(1−2t)}}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−(1−2t)[x−2μt/(1−2t)]^{2}/2} dx
= \frac{e^{δt/(1−2t)}}{\sqrt{1 − 2t}} .

By definition, Y ∼ X_{1}^{2} + ··· + X_{k−1}^{2} + (X_{k} + \sqrt{δ})^{2}, where X_{i}’s are independent and have the standard normal distribution. From the obtained result, the mgf of Y is

E \{e^{t[X_{1}^{2}+··· +X_{k−1}^{2} +(X_{k}+\sqrt{δ})^{2}]}\} = [ψ_{0}(t)]^{k−1} ψ\sqrt{δ}(t) = \frac{e^{δt/(1−2t)}}{(1 − 2t)^{k/2}} .

UW-Madison (Statistics)  Stat 609 Lecture 20  2014  3 / 11

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We now use the proved result (ii) to show result (i). By the uniqueness of the mfg, it suffices to show that the pdf g_{δ,k}(x) in (i) has mgf exactly the same as the one in (ii). Note that the mgf of the central chi-square distribution with degrees of freedom v is,

\frac{1}{(1 − 2t)^{v/2}} , t < 1/2,

The mgf of the pdf g_{δ,k}(x) in (i) is, for t < 1/2,

∫_{−∞}^{∞} e^{tx}g_{δ,k}(x)dx = e^{−δ/2} ∫_{−∞}^{∞} ∑_{j=0}^{∞} \frac{(δ/2)^{j}}{j!} e^{tx}f_{2j+k}(x)dx
= e^{−δ/2} ∑_{j=0}^{∞} \frac{(δ/2)^{j}}{j!} ∫_{−∞}^{∞} e^{tx}f_{2j+k}(x)dx
= e^{−δ/2} ∑_{j=0}^{∞} \frac{(δ/2)^{j}}{j!(1 − 2t)^{(j+k/2)}}
= \frac{e^{−δ/2}}{(1 − 2t)^{k/2}} ∑_{j=0}^{∞} \frac{\{δ/[2(1 − 2t)]\}^{j}}{j!}

UW-Madison (Statistics)  Stat 609 Lecture 20  2014  4 / 11

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= \frac{e^{−δ/2+δ/[2(1−2t)]}}{(1 − 2t)^{k/2}} = \frac{e^{δt/(1−2t)}}{(1 − 2t)^{k/2}}

To establish (iii), let X_{i}’s be as defined in the proof of (i). Then,

E(Y ) = E(X_{1}^{2}) + ··· + E(X_{k−1}^{2} ) + E(X_{k} + \sqrt{δ})^{2}
= k − 1 + E(X_{k}^{2}) + δ + E(2\sqrt{δ}X_{k})
= k + δ

because E(X_{i}^{2}) = 1 and E(X_{i}) = 0. To show (iv), note that Var(X_{i}^{2}) = 2 and Cov(X_{k}^{2},X_{k}) = E(X_{k}^{3}) = 0. Then,

Var(Y ) = Var(X_{1}^{2}) + ··· + Var(X_{k−1}^{2} ) + Var((X_{k} + \sqrt{δ})^{2})
= 2(k − 1) + Var(X_{k}^{2} + 2\sqrt{δ}X_{k})
= 2(k − 1) + Var(X_{k}^{2}) + 4δVar(X_{k}) + 4\sqrt{δ}Cov(X_{k}^{2},X_{k})
= 2k + 4δ
UW-Madison (Statistics)  Stat 609 Lecture 20  2014  5 / 11

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Definition (the noncentral t-distribution)

Let X ∼ N(δ,1), δ ∈ R, U ∼ the central chi-square with degrees of freedom n, and X and U be independent. The distribution of T = X/\sqrt{U/n} is the noncentral t-distribution with degrees of freedom n and noncentrality parameter δ.

The t-distribution previously defined can be called a central t-distribution, since it is a special case of the noncentral t-distribution with δ = 0. Using the formula for the ratio of two independent random variables, we can show that T has the following pdf:

\frac{1}{2^{(n+1)/2}Γ(n/2)\sqrt{πn}} ∫_{0}^{∞} y^{(n−1)/2} e^{−[(x\sqrt{y/n}−δ)^{2}+y]/2} dy

Since U ∼ the central chi-square with degrees of freedom n, W = \sqrt{U/n} has pdf

\frac{2n^{n/2}}{Γ(n/2)2^{n/2}} w^{n−1} e^{−nw^{2}/2} , w > 0

Then, T has pdf

∫_{0}^{∞} w ( \frac{n^{n/2}}{Γ(n/2)2^{n/2−1}} w^{n−1} e^{−nw^{2}/2} ) ( \frac{1}{\sqrt{2π}} e^{−(xw−δ)^{2}/2} ) dw

= \frac{n^{n/2}}{Γ(n/2)2^{(n−1)/2} \sqrt{π}} ∫_{0}^{∞} w^{n}e^{[(xw−δ)^{2}+nw^{2}]/2} dw

Letting y = nw^{2}, we obtain the desired result. By the independence of X and U and the fact that X ∼ N(δ,1) and U ∼ the central chi-square,

E(T ) = E ( \frac{X}{\sqrt{U/n}}) = \sqrt{n}E(X)E ( \frac{1}{\sqrt{U}})
= \sqrt{n}δ ∫_{0}^{∞} \frac{1}{\sqrt{u}} ( \frac{1}{Γ(n/2)2^{n/2}} u^{n/2−1} e^{−u/2} ) du
= \frac{\sqrt{n}δ}{Γ(n/2)2^{n/2}} ∫_{0}^{∞} u^{(n−1)/2−1} e^{−u/2} du = \frac{\sqrt{n}δΓ((n − 1)/2)}{\sqrt{2}Γ(n/2)}

when n > 1 and is ∞ when n = 1.

beamer-tu-logo UW-Madison (Statistics)  Stat 609 Lecture 20  2014  7 / 11

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E(T^{2}) = E ( \frac{X^{2}}{U/n}) = nE(X^{2})E ( \frac{1}{U})
= n(1 + δ^{2})\frac{1}{Γ(n/2)2^{n/2}} ∫_{0}^{∞} u^{n/2−2} e^{−u/2} du
= \frac{n(1 + δ^{2})}{Γ(n/2)2^{n/2}} Γ(n/2 − 1)2^{n/2−1} = \frac{n(1 + δ^{2})}{n − 2}

when n > 2 and is ∞ when n ≤ 2. Hence, when n > 2,

Var(T ) = \frac{n(1 + δ^{2})}{n − 2} − \frac{nδ^{2}}{2} [ \frac{Γ((n − 1)/2)}{Γ(n/2)} ]^{2}

Definition (the noncentral F-distribution)

Let X_{1} ∼ the noncentral chi-square distribution with degrees of freedom n_{1} and noncertrality parameter δ ≥ 0, X_{2} ∼ the central chi-square distribution with degrees of freedom n_{2}, and X_{1} and X_{2} be independent. The distribution of F = (X_{1}/n_{1})/(X_{2}/n_{2}) is called the noncentral F-distribution with degrees of freedom n_{1} and n_{2} and noncentrality parameter δ.

UW-Madison (Statistics)  Stat 609 Lecture 20  2014  8 / 11

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E(T^{2}) = E ( \frac{X^{2}}{U/n}) = nE(X^{2})E ( \frac{1}{U})
= n(1 + δ^{2})\frac{1}{Γ(n/2)2^{n/2}} ∫_{0}^{∞} u^{n/2−2} e^{−u/2} du
= \frac{n(1 + δ^{2})}{Γ(n/2)2^{n/2}} Γ(n/2 − 1)2^{n/2−1} = \frac{n(1 + δ^{2})}{n − 2}

when n > 2 and is ∞ when n ≤ 2. Hence, when n > 2,

Var(T ) = \frac{n(1 + δ^{2})}{n − 2} − \frac{nδ^{2}}{2} [ \frac{Γ((n − 1)/2)}{Γ(n/2)} ]^{2}

Definition (the noncentral F-distribution)

Let X_{1} ∼ the noncentral chi-square distribution with degrees of freedom n_{1} and noncertrality parameter δ ≥ 0, X_{2} ∼ the central chi-square distribution with degrees of freedom n_{2}, and X_{1} and X_{2} be independent. The distribution of F = (X_{1}/n_{1})/(X_{2}/n_{2}) is called the noncentral F-distribution with degrees of freedom n_{1} and n_{2} and noncentrality parameter δ.

UW-Madison (Statistics)  Stat 609 Lecture 20  2014  8 / 11

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The F-distribution introduced previously can be called a central F-distribution, since it is a special case of the noncentral F-distribution with δ = 0.

Using the formula for the ratio of two independent random variables and the pdf of X_{1} we derived previously, we can obtain the pdf for F = (X_{1}/n_{1})/(X_{2}/n_{2}).

Let f_{v} denote the pdf of the central chi-square with degrees of freedom v and f_{k_{1},k_{2}}(x) be the pdf of the central F-distribution with degrees of freedom k_{1} and k_{2}.

Then the pdf of F is

∫_{0}^{∞} y [e^{−δ/2} ∑_{j=0}^{∞} \frac{(δ/2)^{j}}{j!n_{1}n_{2}}f_{2j+k}( \frac{xy}{n_{1}}) ] f_{n_{2}} ( \frac{y}{n_{2}}) dy

= e^{−δ/2} ∑_{j=0}^{∞} \frac{(δ/2)^{j}}{j!} ∫_{0}^{∞} \frac{y}{n_{1}n_{2}}f_{2j+k}( \frac{xy}{n_{1}}) f_{n_{2}} ( \frac{y}{n_{2}}) dy

= e^{−δ/2} ∑_{j=0}^{∞} \frac{(δ/2)^{j}}{j!} \frac{n_{1}}{(2j + n_{1})}f_{2j+n_{1},n_{2}} ( \frac{n_{1}x}{2j + n_{1}})

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Here, we used the following result: if g_{j}(x) ≥ 0 for all j = 0,1,2,... and x ∈ R, then

∫_{−∞}^{∞} ∑_{j=0}^{∞}g_{j}(x)dx = ∑_{j=0}^{∞}∫_{−∞}^{∞} g_{j}(x)dx

which holds even when one of side is ∞. To show this, note that G_{n}(x) = ∑ _{j=0}^{n}g_{j}(x) is increasing in n for each x. By the monotone convergence theorem,

∑_{j=0}^{∞}∫_{−∞}^{∞} g_{j}(x)dx = lim_{n→∞} ∑_{j=0}^{n}∫_{−∞}^{∞} g_{j}(x)dx = lim_{n→∞}∫_{−∞}^{∞} ∑_{j=0}^{n}g_{j}(x)dx
= lim_{n→∞}∫_{−∞}^{∞} G_{n}(x)dx = ∫_{−∞}^{∞} lim_{n→∞}G_{n}(x)dx
= ∫_{−∞}^{∞} lim_{n→∞} ∑_{j=0}^{n}g_{j}(x)dx = ∫_{−∞}^{∞} ∑_{j=0}^{∞}g_{j}(x)dx

Finally, let’s calculate the mean and variance of F.

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From the previous calculation,

E ( \frac{1}{X_{2}}) = \{ \begin{matrix}\frac{1}{n_{2}−2} & n_{2} > 2 \\ ∞ & n_{2} ≤ 2 \end{matrix}

Then, when n_{2} > 2,

E(F) = E ( \frac{X_{1}/n_{1}}{X_{2}/n_{2}}) = E ( \frac{X_{1}}{n_{1}}) E ( \frac{n_{2}}{X_{2}}) = \frac{n_{1} + δ}{n_{1}} \frac{n_{2}}{n_{2} − 2} = \frac{n_{2}(n_{1} + δ)}{n_{1}(n_{2} − 2)}

Also,

E ( \frac{1}{X_{2}^{2}}) = \frac{1}{Γ(n_{2}/2)2^{n_{2}/2}} ∫_{0}^{∞} x^{n_{2}/2−3} e^{−x/2} dx = \{ \begin{matrix}\frac{1}{(n_{2}−2)(n_{2}−4)} & n_{2} > 4 \\ ∞ & n_{2} ≤ 4 \end{matrix}

Thus, when n_{2} > 4,

Var(F) = E ( \frac{X_{1}^{2}/n_{1}^{2}}{X_{2}^{2}/n_{2}^{2}}) − [E(F)]^{2} = \frac{n_{2}^{2}}{n_{1}^{2}}E(X_{1}^{2})E ( \frac{1}{X_{2}}) − ( \frac{n_{2}(n_{1} + δ)}{n_{1}(n_{2} − 2)}) ^{2}
= \frac{n_{2}^{2}}{n_{1}^{2}} ( \frac{2n_{1} + 4δ + (n_{1} + δ)^{2}}{(n_{2} − 2)(n_{2} − 4)} − \frac{(n_{1} + δ)^{2}}{(n_{2} − 2)^{2}})
= \frac{2n_{2}^{2}[(n_{1} + δ)^{2} + (n_{2} − 2)(n_{1} + 2δ)]}{n_{1}^{2}(n_{2} − 2)^{2}(n_{2} − 4)}

UW-Madison (Statistics)  Stat 609 Lecture 20  2014  11 / 11