So my question is how do I prove that the use of 2y, 5y, 10y and 30y is justified for risk bucketing and not other alternate buckets?
Ok so just to pose a second viewpoint but why do you have to necessarily use PCA to do this?
You are basically trying to show that given any underlying swap portfolio $P$ you can find a set of trades / risk exposures in portfolio $Q$ such that the PnL of $P + Q$ is minimised over some metric, i.e. final PnL or norm of daily pnl changes.
The question you are then asking is if you restrict yourself to having only swaps in $Q$ from 2Y, 5Y, 10Y, 30Y buckets, is this effective, and is this more effective than some other combination of buckets.
It seems like a computationally intensive statistical analysis or machine learning task. Generate lots of random portfolios $P_i$, calculate the hedges $Q_i$ with bucket restrictions on $Q_i$ and see what you get. Note I suggest this since you might also be able to factor convexity/cross gamma in this scenario which is likely to be a concern in terms of accrued PnL.
Alternatively (for a linear analysis) you could use your covariance matrix and assume a multi-variate Guassian distribution of rates and possibly come up with the proof mathematically on paper. I might try that tomorrow and re post.
Modelling Under MultiVariate Normal
If you make the assumption that market movements are multivariate-normally distributed with mean zero and covariance estimator as above then,
$\mathbf{\Delta r} \sim \mathcal{N}(\mathbf{0}, \mathbf{\Sigma})$.
A risk metric associated with a portfolio of risk exposures, $P$, is the variance-covariance VaR standard deviation:
$$c = \mathbf{P^T \Sigma P} \;, \quad \text{where VaR-99pc is} \quad c_{99} = \Phi^{-1}(0.99)c$$
for $\Phi(x)$ the standard normal cumulative distribution function.
Suppose you want to hedge, or simulate, that portfolio with another portfolio of risk exposures, $\mathbf{Q}$, then the risk of the difference can be expressed as:
$$c^* = \mathbf{(P^T-Q^T) \Sigma (P-Q)} $$
Obviously if you can 'trade' all exposures then the minimal risk (i.e zero) is found when $\mathbf{P=Q}$. However, in your case you are restricting $\mathbf{Q}$ so that many of its elements are zero and only those corresponding to the 2y, 5y, 10y and 30y buckets are permitted, i.e. $\mathbf{Q} = [0,..,0,q_{2y},0,...,0,q_{5y},0,..]$
You seek the values $q_{2y}, q_{5y}, ..$ such that $c^*$ (the residual risk) is minimal. Since this is a quadratic form there is a unique solution. I will save the matrix calculus and simply state that:
$$\mathbf{\hat{Q}} = \mathbf{\hat{\hat{\Sigma}}^{-1}\hat{\Sigma}P}$$
where $\mathbf{\hat{Q}}$ is the non-zero elements of $\mathbf{Q}$, $\mathbf{\hat{\Sigma}}$ are the rows of $\mathbf{\Sigma}$ corresponding to the non-zero elements of $\mathbf{Q}$, and $\mathbf{\hat{\hat{\Sigma}}}$ are the rows and columns of $\mathbf{\Sigma}$ corresponding to the non-zero elements of $\mathbf{Q}$.
So now based on your covariance matrix you have the analytical hedge quantities, $q_{2y}, q_{5y},..$ which will minimise VaR for the given portfolio $\mathbf{P}$.
How is this useful
1) Firstly you now have a means of assessing the quality of your hedge. $c^*$ is the minimum residual VaR: are you happy about its size given your known portfolio, $\mathbf{P}$?
2) You can choose a different combinations of hedging instruments and calculate the residual VaR again. Perhaps it is better. Obviously the more instruments you include the better smaller the residual VaR will become.
3) If you are interested in the quality of a 2y,5y,10y,30y hedge in general then you would like to examine $E[c^*]$, where the expectation is measured with respect to the probability space of all possible portfolios. This is quite difficult to measure since you need to define a probability space of portfolios and that is non-trivial; a balanced portfolio (offsets) is much more likely than having every bucket with large risk exposure in the same direction. However, if you assumed a uniform distribution you could probably still come up with results, probably analytic ones.
4) You can extend 3) to ascertain for your initial assumptions what is the optimal combination of allowed hedging instruments for the general case of unknown portfolio given the initial assumptions.