Factoring the Joint Density Into Increments

Why Factor the Joint Density?

At successive interim looks, the cumulative $Z$ statistics are correlated because later looks reuse data from earlier ones. Their joint density can nevertheless be written using a sequence of independent normal quantities. This representation is useful for evaluating probabilities involving several looks and for checking calculations made from the multivariate normal density. The key is to transform the cumulative statistics to the appropriately scaled information increments; simply treating the cumulative $Z$ statistics themselves as independent would be incorrect.

The result here concerns the canonical joint normal model under $H_0$. It starts with the cumulative statistics and their canonical covariance structure, then derives the factorization by a change of variables. The information times are fixed design quantities, not random outcomes of the monitoring process.

Definition: Let $Z_1,\ldots,Z_K$ be jointly normal cumulative statistics at fixed information times $0<t_1<\cdots<t_K$. Under the canonical null model, $E(Z_j)=0$, $\operatorname{Var}(Z_j)=1$, and $\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}$ for $i<j$. Set $t_0=0$, $Z_0=0$, and $\Delta_j=t_j-t_{j-1}$. The standardized increment at look $j$ is $$ W_j=\frac{\sqrt{t_j}Z_j-\sqrt{t_{j-1}}Z_{j-1}}{\sqrt{\Delta_j}},\qquad j=1,\ldots,K. $$

The times need not be fractions that sum to any particular value; they only need to be strictly increasing and positive. Often the final information time is set to $t_K=1$, but the density factorization does not require that normalization. Joint normality is essential for the conclusion that uncorrelated increments are independent.

From the Canonical Covariance to the Density

Write $X_j=\sqrt{t_j}Z_j$, with $X_0=0$. The canonical covariance gives, for $i\leq j$, $$ \operatorname{Cov}(X_i,X_j) =\sqrt{t_i t_j}\operatorname{Cov}(Z_i,Z_j) =\sqrt{t_i t_j}\sqrt{\frac{t_i}{t_j}} =t_i. $$ Thus $\operatorname{Cov}(X_i,X_j)=\min(t_i,t_j)$. The increment $D_j=X_j-X_{j-1}$ has variance $$ \operatorname{Var}(D_j)=t_j-t_{j-1}=\Delta_j. $$ For $i<j$, the covariance of $D_i$ and $D_j$ is zero: the covariance of the two differences cancels because each of the four covariance terms is the minimum of its two time arguments. More explicitly, since $t_{i-1}\leq t_i\leq t_{j-1}\leq t_j$, $$ \operatorname{Cov}(D_i,D_j) =t_i-t_i-t_{i-1}+t_{i-1}=0. $$ The vector of increments is jointly normal, since it is a linear transformation of the jointly normal vector of cumulative statistics. Its components are therefore independent when their pairwise covariances are zero. Consequently $D_j\sim N(0,\Delta_j)$, and $W_j=D_j/\sqrt{\Delta_j}$ are independent standard normal variables.

This independence gives the product density in increment coordinates. To express the result as a density for $z_1,\ldots,z_K$, the change-of-variables factor must also be included. The map from $z$ to $w$ is lower triangular: its $j$th diagonal entry is $\sqrt{t_j/\Delta_j}$. Its determinant is the product of those entries. Hence the joint density of the cumulative statistics is:

f_Z(z_1,\ldots,z_K) = \prod_{j=1}^{K} \sqrt{\frac{t_j}{\Delta_j}}\, \phi\!\left( \frac{\sqrt{t_j}z_j-\sqrt{t_{j-1}}z_{j-1}}{\sqrt{\Delta_j}} \right),

where $\phi(u)=(2\pi)^{-1/2}\exp(-u^2/2)$ is the standard normal density. This is a density with respect to the cumulative-statistic coordinates $z_1,\ldots,z_K$. The multiplier is not optional: the product of the standard normal densities alone is the density in $w$-coordinates, not in $z$-coordinates.

Justification by covariance determinant: The covariance matrix $\Sigma$ of $Z=(Z_1,\ldots,Z_K)^\mathsf{T}$ has determinant $$ |\Sigma|=\frac{\prod_{j=1}^{K}\Delta_j}{\prod_{j=1}^{K}t_j}. $$ Indeed, the covariance matrix of $X=(X_1,\ldots,X_K)^\mathsf{T}$ is that of the successive sums of independent increments with variances $\Delta_j$, so its determinant is $\prod_j\Delta_j$. Since $X_j=\sqrt{t_j}Z_j$, the determinant for $Z$ is divided by $\prod_j t_j$. Thus $|\Sigma|^{-1/2}=\prod_j\sqrt{t_j/\Delta_j}$, exactly the Jacobian multiplier in the factorized density.

Worked Examples

Worked Example: Factoring a Two-Look Density

Suppose the information times are $t_1=0.25$ and $t_2=1$, and consider the hypothetical point $z_1=0.6,\ z_2=1.0$. Find the standardized increments and evaluate the joint density at this point.

Here $\Delta_1=0.25$ and $\Delta_2=1-0.25=0.75$. The first increment is $W_1=Z_1$, so its value is $0.6000$. For the second, $$ w_2=\frac{\sqrt{1}(1.0)-\sqrt{0.25}(0.6)}{\sqrt{0.75}} =\frac{1.0-0.3000}{0.8660} =0.8083. $$ The Jacobian multiplier is $\sqrt{t_1/\Delta_1}\sqrt{t_2/\Delta_2}=1\times1.1547$. Therefore $$ f_Z(0.6,1.0) =1.1547\,\phi(0.6000)\phi(0.8083) \approx 0.1107. $$ The calculation uses the density in the two cumulative-statistic coordinates. Omitting the multiplier $1.1547$ would give the density in increment coordinates instead.

Worked Example: Three Successive Increments

Take hypothetical information times $t_1=0.25,\ t_2=0.50,\ t_3=1.00$, and cumulative-statistic values $z_1=0.4,\ z_2=0.5,\ z_3=0.8$. Calculate the standardized increments and the density factorization.

The successive information increases are $0.25,\ 0.25,\ 0.50$. The scaled cumulative values are $x_1=\sqrt{0.25}(0.4)=0.2000$, $x_2=\sqrt{0.50}(0.5)=0.3536$, and $x_3=1(0.8)=0.8000$. Thus $$ w_1=\frac{0.2000}{\sqrt{0.25}}=0.4000,\qquad w_2=\frac{0.3536-0.2000}{\sqrt{0.25}}\approx0.3071, $$ and, using the unrounded value $x_2=0.5\sqrt{0.5}$, $$ w_3=\frac{0.8000-0.5\sqrt{0.5}}{\sqrt{0.50}} \approx0.6314. $$ The Jacobian multiplier is $$ \sqrt{\frac{0.25}{0.25}}\sqrt{\frac{0.50}{0.25}}\sqrt{\frac{1.00}{0.50}} =1\times\sqrt{2}\times\sqrt{2}=2. $$ Accordingly, the joint density at the stated point is $2\phi(0.4000)\phi(0.3071)\phi(0.6314)$, approximately $0.09160$. The increments are independent under the model, even though the three cumulative statistics are not.

Worked Example: Checking the Determinant and the Factorization

For another hypothetical three-look design, let $(t_1,t_2,t_3)=(0.2,0.6,1.0)$, and consider $(z_1,z_2,z_3)=(0,0.8165,0)$. Verify the increment values and the density multiplier in two ways.

The information increases are $(0.2,0.4,0.4)$. The scaled cumulative values are $x_1=0$, $x_2=\sqrt{0.6}(0.8165)\approx0.6325$, and $x_3=0$. Thus the standardized increments are approximately $w_1=0,\ w_2=0.6325/\sqrt{0.4}=1,\ w_3=-0.6325/\sqrt{0.4}=-1$. The change-of-variables multiplier is $$ \prod_{j=1}^3\sqrt{\frac{t_j}{\Delta_j}} =1\sqrt{1.5}\sqrt{2.5} =\sqrt{3.75}\approx1.9365. $$ As a determinant check, the canonical covariance determinant is $$ |\Sigma|=\frac{(0.2)(0.4)(0.4)}{(0.2)(0.6)(1.0)} =0.2667, $$ so $|\Sigma|^{-1/2}\approx1.9365$, the same multiplier. The density is $1.9365\,\phi(0)\phi(1)\phi(-1)\approx0.04524$. Agreement between the Jacobian and determinant calculations checks the scaling in the factorization.

Using and Checking the Factorization

The factorized form offers two equivalent routes to evaluate the canonical joint density. One can use the standard multivariate normal formula with the canonical covariance matrix, or transform to $W_1,\ldots,W_K$ and multiply the univariate densities together with the Jacobian. The first route is direct but may require manipulating a covariance matrix. The increment route exploits the structure of the canonical model, often making both algebra and computation simpler. They must give the same density when they use the same coordinate system and include the same determinant factor.

Check the model.
Confirm that the looks are jointly normal under the stated model and that their variances and correlations have the canonical form. The information times must be fixed and strictly increasing.

Form scaled cumulative values.
For each look, calculate $x_j=\sqrt{t_j}z_j$, and set $x_0=0$.

Standardize each increase.
Calculate $w_j=(x_j-x_{j-1})/\sqrt{\Delta_j}$. These are the independent standard normal coordinates under the canonical null model.

Evaluate in the right coordinates.
Multiply $\phi(w_j)$ across looks and include $\prod_j\sqrt{t_j/\Delta_j}$ for a density expressed in the original $z$-coordinates.

Several checks can reveal a faulty calculation. First, reconstruct each scaled cumulative value from the increments: $x_j=\sum_{r=1}^j\sqrt{\Delta_r}w_r$. Second, verify that the covariance determinant agrees with $\prod_j\Delta_j/\prod_jt_j$. Third, in a simulation check under the model, the computed $W_j$ values should have means near zero, variances near one, and pairwise correlations near zero, subject to simulation error. A substantial discrepancy may signal an incorrect information time, a scaling error, or a failure of the assumed canonical covariance structure.

Common Pitfalls and Limits

Calling the cumulative statistics independent. They are generally correlated because they share data. Independence applies to the standardized increments $W_j$, not to $Z_1,\ldots,Z_K$.
Dropping the Jacobian. The product $\prod_j\phi(w_j)$ is a density in increment coordinates. For a density in cumulative $Z$ coordinates, multiply by $\prod_j\sqrt{t_j/\Delta_j}$.
Using rounded intermediate values too early. Small rounding errors can noticeably alter a later increment when the time increase is small. Retain full precision during the calculation and round the reported result at the end.
Applying the result to a noncanonical covariance structure. The factorization depends on the specified covariance relationships and joint normality. Zero covariance alone does not imply independence for arbitrary, non-normal variables.
Allowing repeated information times. If $t_j=t_{j-1}$, then $\Delta_j=0$, so the stated standardization and density are undefined. A duplicated look adds no new information and must not be treated as a positive-variance increment.

The factorization is a statement about the joint density under the canonical model, not a guarantee that every monitoring design or statistic follows that model exactly. Departures from joint normality or from the assumed covariance pattern require an appropriate model check or alternative calculation. In particular, the determinant and increment formulas should not be transferred unchanged to arbitrary correlated statistics.

Key takeaway: In the canonical joint normal model, scale each cumulative statistic by the square root of its information time, take successive differences, and standardize by the square root of the information increase. Those increments have independent standard normal densities; the product becomes the density in cumulative-statistic coordinates only after including the Jacobian.

Check Your Understanding

Answer each question, then compare with the solutions below.

For $t_1=0.36,\ t_2=1$, write $W_1$ and $W_2$ in terms of $Z_1,Z_2$.
Why does joint normality matter when concluding that the standardized increments are independent?
For information times $0.25,1$, what is the Jacobian multiplier in the density factorization?
What fails in the stated factorization if two successive information times are equal?

Solutions. (1) $\Delta_1=0.36$ and $\Delta_2=0.64$, so $W_1=Z_1$ and $W_2=(Z_2-0.6Z_1)/0.8$. (2) The covariance calculation gives uncorrelated increments; joint normality makes uncorrelated components independent. (3) The multiplier is $\sqrt{0.36/0.36}\sqrt{1/0.64}=1.25$. (4) The corresponding $\Delta_j$ is zero, so division by $\sqrt{\Delta_j}$ is undefined; the repeated time does not provide a positive-variance increment.